Finite Group Databasebeta

← #29686948

Subgroups — #29686948

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455536C211yesmaximal, characteristic
2#21454720C211yesmaximal, characteristic, complemented
3C2×#16839877C211yesmaximal, characteristic, complemented
4#131×Dic61C211yesmaximal, characteristic, complemented
5#21446716C211yesmaximal, characteristic, complemented
6#21446606C211yesmaximal, characteristic, complemented
7#21446491C211yesmaximal, characteristic, complemented
8#21446487C211yesmaximal, characteristic, complemented
9#21446322C211yesmaximal, characteristic, complemented
10#21446318C211yesmaximal, characteristic, complemented
11#21445871C211yesmaximal, characteristic, complemented
12#21445699C211yesmaximal, characteristic, complemented
13#21445620C211yesmaximal, characteristic, complemented
14#21445619C211yesmaximal, characteristic, complemented
15C61×#2955C211yesmaximal, characteristic, complemented
16#16840560C2211yescharacteristic
17C2×#14519797C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19C2×#14519783C2211yescharacteristic
20#41×Dic61C2211yescharacteristic
21#16840035C412yescomplemented
22#16840030C412yescomplemented
23#16840027C412yescomplemented
24#16840023C412yescomplemented
25#16839882—22no—
26#16839877—21no—
27#16839877—21no—
28#16839877—21no—
29#16839877—21no—
30#16839877—22no—
31#16839810C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33C2×#14519712C2211yescharacteristic
34#16839798C2211yescharacteristic
35#16839779C2211yescharacteristic
36#16839778C2211yescharacteristic
37C2×#14519704C2211yescharacteristic
38C2×#14519703C2211yescharacteristic
39#16839746C2211yescharacteristic
40#16839744C2211yescharacteristic
41#16839736C2211yescharacteristic
42#16839735C2211yescharacteristic
43#16839720—21no—
44#16839720C2211yescharacteristic
45#16839717—21no—
46#16839717C2211yescharacteristic
47#16839717C2211yescharacteristic
48M16×Dic61—21no—
49M16×Dic61C2211yescharacteristic
50C2×C8×Dic61—21no—
51C2×C8×Dic61C2211yescharacteristic
52#16839534C2211yescharacteristic
53#16839533C2211yescharacteristic
54#16839533C2211yescharacteristic
55#16839532C2211yescharacteristic
56#16839523C2211yescharacteristic
57#16839521C2211yescharacteristic
58C61×#575C2211yescharacteristic
59C2×#131×C61C2211yescharacteristic
60C2×#131×C61C2211yescharacteristic
61C61×#444—22no—
62C61×#443—22no—
63C61×#430C2211yescharacteristic
64C61×#421C2211yescharacteristic
65C61×#421C2211yescharacteristic
66C61×#405C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519808—21no—
69#14519807C2×C412yes—
70#14519803—21no—
71#14519802C2×C412yes—
72#14519797—22no—
73#14519797—21no—
74#14519797—21no—
75#14519797—21no—
76#14519797—21no—
77C2×#11758202C2×C411yescharacteristic, complemented
78C2×#11758200C2×C411yescharacteristic, complemented
79C2×#11758201C2×C411yescharacteristic, complemented
80C2×#11758199C2×C411yescharacteristic, complemented
81#14519784—21no—
82#14519784—21no—
83#14519784C2×C412yes—
84#14519784C2×C412yes—
85#14519783—21no—
86#14519783—21no—
87#14519783—21no—
88#14519783—21no—
89C2×#11758194—21no—
90C2×#11758194C2311yescharacteristic
91C2×#11758194C2311yescharacteristic
92C2×#11758194C2311yescharacteristic
93C2×#11758189—21no—
94C22×#92418—21no—
95C22×#92418C2311yescharacteristic
96C2×C8×D61—21no—
97Q8×Dic61—21no—
98D4×Dic61—21no—
99D4×Dic61—22no—
100#14519736—21no—
101#14519736C2311yescharacteristic
102#14519736C2311yescharacteristic
103C2×C4×Dic61—21no—
104C2×C4×Dic61C2311yescharacteristic
105#14519712—22no—
106#14519712—21no—
107#14519712—21no—
108#14519712C2×C412yescomplemented
109#14519712C2×C412yescomplemented
110#14519710—21no—
111#14519708—21no—
112#14519704—22no—
113#14519704—21no—
114#14519704—21no—
115#14519703—22no—
116#14519703—21no—
117#14519703—21no—
118#14519696C2×C412yescomplemented
119#14519695C2×C412yescomplemented
120#14519694C2×C412yescomplemented
121#14519693C2×C412yescomplemented
122#14519692—22no—
123#14519691—22no—
124#14519687—21no—
125#14519687—21no—
126#14519687—21no—
127#14519687—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C8×Dic61—21no—
132C8×Dic61D411yescharacteristic
133C8×Dic61D411yescharacteristic
134C8×Dic61D411yescharacteristic
135C8×Dic61D411yescharacteristic
136C2×#41×C61C2311yescharacteristic
137C2×#41×C61C2311yescharacteristic
138#137×C61—21no—
139#137×C61C2×C412yes—
140AGL(1,ℤ/8)×C61—21no—
141AGL(1,ℤ/8)×C61C2×C412yes—
142#135×C61—22no—
143#131×C61—21no—
144#131×C61—21no—
145#131×C61—21no—
146#131×C61—21no—
147#131×C61—21no—
148#131×C61—21no—
149#131×C61D412yes—
150#131×C61D412yes—
151C2×M16×C61—21no—
152C2×M16×C61—21no—
153C2×M16×C61C2311yescharacteristic
154C2×M16×C61C2311yescharacteristic
155C2×M16×C61C2311yescharacteristic
156C22×C8×C61—21no—
157C22×C8×C61—21no—
158C22×C8×C61C2311yescharacteristic
159#117×C61C2311yescharacteristic
160#108×C61—22no—
161#104×C61—22no—
162#104×C61—22no—
163#97×C61—21no—
164C4×C8×C61—21no—
165#41×C61—21no—
166#41×C61—21no—
167#41×C61—21no—
168#41×C61—21no—
169#41×C61—21no—
170#41×C61—21no—
171#41×C61C22×C412yes—
172#41×C61C2×D412yes—
173C2×Q8×C61C22×C411yescharacteristic
174C2×Q8×C61#4111yescharacteristic
175C2×D4×C61—21no—
176C2×D4×C61—21no—
177C2×D4×C61C22×C411yescharacteristic
178C2×D4×C61#4111yescharacteristic
179C22×C4×C61—21no—
180C22×C4×C61—21no—
181C22×C4×C61C2411yescharacteristic
182C2×C22⋊D61—21no—
183C22×Dic61—22no—
184C22×Dic61—21no—
185#11758222—21no—
186#11758222—21no—
187#11758222—21no—
188#11758222—21no—
189C2×D244C22×C411yescharacteristic
190C2×C4×D61—21no—
191C2×Dic122C22×C411yescharacteristic
192Q16×C61—22no—
193SD16×C61—22no—
194SD16×C61—22no—
195D8×C61—22no—
196M16×C61—21no—
197M16×C61—21no—
198M16×C61—22no—
199M16×C61—22no—
200M16×C61—21no—
201M16×C61—21no—
202M16×C61—41no—
203M16×C61—41no—
204M16×C61C22×C412yes—
205M16×C61C2×D412yes—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C2×C8×C61—22no—
209C2×C8×C61—22no—
210C2×C8×C61—21no—
211C2×C8×C61—41no—
212C2×C8×C61—41no—
213C2×C8×C61C2×D411yescharacteristic
214C2×C8×C61C2×D411yescharacteristic
215C2×C8×C61C2×D411yescharacteristic
216C2×C8×C61C2×D411yescharacteristic
217C2×C8×C61#4111yescharacteristic
218C2×C8×C61#4111yescharacteristic
219C4⋊C4×C61—21no—
220C22⋊C4×C61—21no—
221C42×C61—21no—
222C22⋊Dic61—22no—
223C22⋊Dic61—21no—
224C22⋊Dic61#4112yes—
225#11758202—22no—
226#11758202—21no—
227#11758202—21no—
228#11758201—22no—
229#11758201—21no—
230#11758201—21no—
231#11758200—22no—
232#11758200—21no—
233#11758200—21no—
234#11758199—22no—
235#11758199—21no—
236#11758199—21no—
237C4⋊Dic61—22no—
238C4⋊Dic61—21no—
239C4⋊Dic61—21no—
240C4×Dic61—22no—
241C4×Dic61—21no—
242C4×Dic61—21no—
243C4×Dic61—21no—
244C4×Dic61C2×D411yescharacteristic
245C4×Dic61C2×D411yescharacteristic
246#11758194—21no—
247#11758194—21no—
248#11758194—22no—
249#11758194—22no—
250#11758194—21no—
251#11758194—21no—
252#11758194C22×C412yes—
253#11758194C22×C412yes—
254C2×#92418—21no—
255C2×#92418—21no—
256C2×#92418—22no—
257C2×#92418—22no—
258C2×#92418—21no—
259C2×#92418C22×C411yescharacteristic
260C2×#92418C22×C411yescharacteristic
261C2×#92418C2×D411yescharacteristic
262C2×#92418C2×D411yescharacteristic
263C2×#92418#4111yescharacteristic
264C2×#92418#4111yescharacteristic
265#11758189—41no—
266#11758189—41no—
267C8×D61—41no—
268C8×D61—41no—
269C23×C61—21no—
270C23×C61—21no—
271C23×C61C2×#4111yescharacteristic
272C22×D61—21no—
273Q8×C61—21no—
274Q8×C61—21no—
275Q8×C61—21no—
276Q8×C61C4×D412yes—
277D4×C61—22no—
278D4×C61—22no—
279D4×C61—21no—
280D4×C61—21no—
281D4×C61—21no—
282D4×C61—41no—
283D4×C61—41no—
284D4×C61C4×D412yes—
285C2×C4×C61—21no—
286C2×C4×C61—22no—
287C2×C4×C61—22no—
288C2×C4×C61—21no—
289C2×C4×C61—41no—
290C2×C4×C61—41no—
291C2×C4×C61—41no—
292C2×C4×C61C23×C411yescharacteristic
293C2×C4×C61C22×D411yescharacteristic
294C2×C4×C61C22×D411yescharacteristic
295C2×C4×C61C2×#4111yescharacteristic
296C2×C4×C61C2×#4111yescharacteristic
297C2×C4×C61C2×#4111yescharacteristic
298C22⋊D61—41no—
299C22⋊D61—41no—
300C2×Dic61—21no—
301C2×Dic61—21no—
302C2×Dic61—22no—
303C2×Dic61—21no—
304C2×Dic61—42no—
305C2×Dic61—44no—
306C2×Dic61—41no—
307D244—21no—
308D244—21no—
309C4×D61—41no—
310C4×D61—41no—
311Dic122—21no—
312Dic122—21no—
313C8×C61—22no—
314C8×C61—22no—
315C8×C61—41no—
316C8×C61#12111yescharacteristic
317C8×C61#12111yescharacteristic
318C8×C61#12111yescharacteristic
319C8×C61#12111yescharacteristic
320#92418—22no—
321#92418—22no—
322#92418—41no—
323#92418C4×D411yescharacteristic
324#92418C4×D411yescharacteristic
325#92418C4×D411yescharacteristic
326#92418C4×D411yescharacteristic
327#21558—611nomaximal, Sylow
328C22×C61—22no—
329C22×C61—41no—
330C22×C61—42no—
331C22×C61—41no—
332C22×C61—81no—
333C22×C61C4×#4111yescharacteristic
334C22×C61#52511yescharacteristic
335C22×C61#53211yescharacteristic
336C2×D61—41no—
337C2×D61—81no—
338C4×C61—22no—
339C4×C61—41no—
340C4×C61—81no—
341C4×C61C2×C4×D411yescharacteristic
342C4×C61C2×C4×D411yescharacteristic
343C4×C61C2×#12111yescharacteristic
344C4×C61#52911yescharacteristic
345Dic61—22no—
346Dic61—42no—
347Dic61—41no—
348Dic61—41no—
349Dic61—81no—
350#3067—611no—
351#2955—611no—
352#2955D6111yescharacteristic, complemented
353C2×#443—611no—
354C4×#131—611no—
355#1976—611no—
356#1953—611no—
357#1932—611no—
358#1932—611no—
359#1906—611no—
360#1906—611no—
361#1814—611no—
362#1772—611no—
363#1757—611no—
364#1757—611no—
365C2×C61—21no—
366C2×C61—21no—
367C2×C61—21no—
368C2×C61—42no—
369C2×C61—81no—
370C2×C61C4×#12111yescharacteristic
371D61—81no—
372#575—611no—
373#575C2×D6111yescharacteristic
374C2×#135—611no—
375C2×#131—611no—
376C2×#131—611no—
377C2×#131C2×D6111yescharacteristic
378C2×#131C2×D6111yescharacteristic
379C4×#41—611no—
380#473—612no—
381#472—612no—
382#471—612no—
383#471—612no—
384#444—22no—
385#444—1222no—
386#443—22no—
387#443—1221no—
388#443—1221no—
389#443—1221no—
390#443—1221no—
391#443—1222no—
392#430—611no—
393#430C2×D6111yescharacteristic
394C2×#108—611no—
395C2×#108—611no—
396#427—611no—
397#421—611no—
398#421—611no—
399#421C2×D6111yescharacteristic
400#421C2×D6111yescharacteristic
401C2×#104—611no—
402C2×#104—611no—
403#413—611no—
404#413—611no—
405#410—611no—
406#410—611no—
407#405—611no—
408#405—611no—
409#405—611no—
410#405—1221no—
411#405—1221no—
412#405C2×D6111yescharacteristic
413C4×M16—611no—
414C4×M16—1221no—
415C2×C4×C8—611no—
416C2×C4×C8—1221no—
417#344—611no—
418#344—611no—
419#344—611no—
420#344—611no—
421#339—611no—
422#339—611no—
423C61#2155811yescharacteristic, complemented, Sylow
424C2×#41—611no—
425C2×#41C22×D6111yescharacteristic
426C2×#41C22×D6111yescharacteristic
427#137—21no—
428#137—612no—
429#137—1221no—
430#137C4×D6112yes—
431AGL(1,ℤ/8)—21no—
432AGL(1,ℤ/8)—612no—
433AGL(1,ℤ/8)—1221no—
434AGL(1,ℤ/8)C4×D6112yes—
435#135—22no—
436#135—1221no—
437#135—1221no—
438#135—1221no—
439#135—1221no—
440#135—1222no—
441C2×Q16—611no—
442C2×SD16—611no—
443C2×SD16—611no—
444C2×D8—611no—
445#131—21no—
446#131—21no—
447#131—21no—
448#131—21no—
449#131—21no—
450#131—21no—
451#131—612no—
452#131—612no—
453#131—1221no—
454#131—1221no—
455#131—1221no—
456#131—1221no—
457#131—1221no—
458#131—1221no—
459#131C22⋊D6112yes—
460#131C22⋊D6112yes—
461C2×M16—21no—
462C2×M16—21no—
463C2×M16—611no—
464C2×M16—611no—
465C2×M16—611no—
466C2×M16—1221no—
467C2×M16—1221no—
468C2×M16C22×D6111yescharacteristic
469C2×M16C22×D6111yescharacteristic
470C2×M16C22×D6111yescharacteristic
471C22×C8—21no—
472C22×C8—21no—
473C22×C8—611no—
474C22×C8—1221no—
475C22×C8—1221no—
476C22×C8C22×D6111yescharacteristic
477C4×Q8—1221no—
478C4×D4—1222no—
479C4×D4—1221no—
480#117—611no—
481#117—611no—
482#117—1221no—
483#117C22×D6111yescharacteristic
484C2×C42—611no—
485C2×C42—1221no—
486#108—22no—
487#108—612no—
488#108—612no—
489#108—1221no—
490#108—1221no—
491#108—1222no—
492#107—1221no—
493#106—1221no—
494#104—22no—
495#104—22no—
496#104—1221no—
497#104—1221no—
498#104—1221no—
499#104—1221no—
500#104—1222no—
501#104—1222no—
502#101—612no—
503#101—612no—
504#100—612no—
505#100—612no—
506#99—1222no—
507#99—1222no—
508#97—21no—
509#97—1221no—
510#97—1221no—
511#97—1221no—
512#97—1221no—
513C4×C8—21no—
514C4×C8—611no—
515C4×C8—611no—
516C4×C8—611no—
517C4×C8—611no—
518C4×C8—1221no—
519C4×C8—1221no—
520C4×C8—1221no—
521C4×C8—1221no—
522#41—21no—
523#41—21no—
524#41—21no—
525#41—21no—
526#41—21no—
527#41—21no—
528#41—1221no—
529#41—1221no—
530#41—1221no—
531#41—1221no—
532#41C2×C4×D6112yes—
533#41C2×C22⋊D6112yes—
534C2×Q8—611no—
535C2×Q8C2×C4×D6111yescharacteristic
536C2×Q8#1175822211yescharacteristic
537C2×D4—21no—
538C2×D4—21no—
539C2×D4—611no—
540C2×D4—1221no—
541C2×D4C2×C4×D6111yescharacteristic
542C2×D4#1175822211yescharacteristic
543C22×C4—21no—
544C22×C4—21no—
545C22×C4—1222no—
546C22×C4—1221no—
547C22×C4—1221no—
548C22×C4C23×D6111yescharacteristic
549Q16—22no—
550Q16—1221no—
551Q16—1221no—
552Q16—1222no—
553SD16—22no—
554SD16—22no—
555SD16—1221no—
556SD16—1221no—
557SD16—1221no—
558SD16—1221no—
559SD16—1222no—
560SD16—1222no—
561D8—22no—
562D8—1221no—
563D8—1221no—
564D8—1222no—
565M16—21no—
566M16—21no—
567M16—22no—
568M16—22no—
569M16—21no—
570M16—21no—
571M16—41no—
572M16—41no—
573M16—612no—
574M16—612no—
575M16—1222no—
576M16—1222no—
577M16—1221no—
578M16—1221no—
579M16—1221no—
580M16—1221no—
581M16—2441no—
582M16—2441no—
583M16C2×C4×D6112yes—
584M16C2×C22⋊D6112yes—
585C2×C8—21no—
586C2×C8—21no—
587C2×C8—22no—
588C2×C8—22no—
589C2×C8—21no—
590C2×C8—41no—
591C2×C8—41no—
592C2×C8—611no—
593C2×C8—611no—
594C2×C8—611no—
595C2×C8—611no—
596C2×C8—611no—
597C2×C8—611no—
598C2×C8—1222no—
599C2×C8—1222no—
600C2×C8—1221no—
601C2×C8—1221no—
602C2×C8—1221no—
603C2×C8—2441no—
604C2×C8—2441no—
605C2×C8D4×D6111yescharacteristic
606C2×C8D4×D6111yescharacteristic
607C2×C8#1175822411yescharacteristic
608C2×C8#1175822411yescharacteristic
609C2×C8C2×C22⋊D6111yescharacteristic
610C2×C8C2×C22⋊D6111yescharacteristic
611C4⋊C4—21no—
612C4⋊C4—1222no—
613C4⋊C4—1221no—
614C4⋊C4—1221no—
615C22⋊C4—21no—
616C22⋊C4—612no—
617C22⋊C4—1222no—
618C22⋊C4—1221no—
619C42—21no—
620C42—611no—
621C42—611no—
622C42—1222no—
623C42—1221no—
624C42—1221no—
625C42—1221no—
626C23—21no—
627C23—21no—
628C23—1221no—
629C23#41×D6111yescharacteristic
630Q8—21no—
631Q8—21no—
632Q8—21no—
633Q8—1221no—
634Q8—1221no—
635Q8C4×C22⋊D6112yes—
636D4—22no—
637D4—22no—
638D4—21no—
639D4—21no—
640D4—21no—
641D4—41no—
642D4—41no—
643D4—1221no—
644D4—1221no—
645D4—2441no—
646D4—2441no—
647D4C4×C22⋊D6112yes—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—21no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—1221no—
656C2×C4—1221no—
657C2×C4—1221no—
658C2×C4—1222no—
659C2×C4—2442no—
660C2×C4—2444no—
661C2×C4—2441no—
662C2×C4—2441no—
663C2×C4—2441no—
664C2×C4C22×C4×D6111yescharacteristic
665C2×C4C2×D4×D6111yescharacteristic
666C2×C4C22×C22⋊D6111yescharacteristic
667C2×C4#41×D6111yescharacteristic
668C2×C4C2×#1175822411yescharacteristic
669C2×C4C2×#1175822211yescharacteristic
670C8—22no—
671C8—22no—
672C8—41no—
673C8—611no—
674C8—611no—
675C8—611no—
676C8—611no—
677C8—1222no—
678C8—1222no—
679C8—2441no—
680C8#1451974911yescharacteristic
681C8#1451974911yescharacteristic
682C8#1451974911yescharacteristic
683C8#1451974911yescharacteristic
684C22—22no—
685C22—41no—
686C22—42no—
687C22—41no—
688C22—81no—
689C22—4881no—
690C22—2441no—
691C22C4×#1175822411yescharacteristic
692C22#1684027111yescharacteristic
693C22#1684030811yescharacteristic
694C4—22no—
695C4—41no—
696C4—81no—
697C4—1222no—
698C4—4881no—
699C4—2441no—
700C4—2442no—
701C4—2441no—
702C4C4×D4×D6111yescharacteristic
703C4C2×C4×C22⋊D6111yescharacteristic
704C4C2×#1451974911yescharacteristic
705C4#1684029211yescharacteristic
706C2—21no—
707C2—21no—
708C2—21no—
709C2—42no—
710C2—81no—
711C2—4881no—
712C2C4×#1451974911yescharacteristic
subgroupG/H
centerC8#14519749
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2955C2
socleC2×C61C4×#121
prime radical#21455536C2
cosocleC22×C4×C61C24
hypercenter#2955D61
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2955

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454720C2semidirect
3. C2×#16839877C2semidirect
4. #131×Dic61C2semidirect
5. #21446716C2semidirect
6. #21446606C2semidirect
7. #21446491C2semidirect
8. #21446487C2semidirect
9. #21446322C2semidirect
10. #21446318C2semidirect
11. #21445871C2semidirect
12. #21445699C2semidirect
13. #21445620C2semidirect
14. #21445619C2semidirect
15. C61×#2955C2semidirect
21. #16840035C4semidirect
22. #16840030C4semidirect
23. #16840027C4semidirect
24. #16840023C4semidirect
77. C2×#11758202C2×C4semidirect
78. C2×#11758200C2×C4semidirect
79. C2×#11758201C2×C4semidirect
80. C2×#11758199C2×C4semidirect
108. #14519712C2×C4semidirect
109. #14519712C2×C4semidirect
118. #14519696C2×C4semidirect
119. #14519695C2×C4semidirect
120. #14519694C2×C4semidirect
121. #14519693C2×C4semidirect
352. #2955D61semidirect
423. C61#21558semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214555362
#214547202
C2×#168398772
#131×Dic612
#214467162
#214466062
#214464912
#214464872
#214463222
#214463182
#214458712
#214456992
#214456202
#214456192
C61×#29552
#2155861