Finite Group Databasebeta

← #29686937

Subgroups — #29686937

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840025C211yesmaximal, characteristic
2C4×#14519807C211yesmaximal, characteristic, complemented
3C4×#14519802C211yesmaximal, characteristic
4C2×#16839864C211yesmaximal, characteristic, complemented
5#21446722C211yesmaximal, characteristic, complemented
6#21446604C211yesmaximal, characteristic, complemented
7#21446481C211yesmaximal, characteristic, complemented
8#21446474C211yesmaximal, characteristic, complemented
9#21446334C211yesmaximal, characteristic, complemented
10#21446327C211yesmaximal, characteristic
11#21445883C211yesmaximal, characteristic, complemented
12#21445725C211yesmaximal, characteristic, complemented
13C4×#14519695C211yesmaximal, characteristic, complemented
14C4×#14519693C211yesmaximal, characteristic, complemented
15C61×#2875C211yesmaximal, characteristic, complemented
16C2×#14519807C2211yescharacteristic
17C2×#14519802C2211yescharacteristic
18C22×#11758190C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16840025C418yescomplemented
22#16839872—21no—
23#16839867—21no—
24#16839864—21no—
25#16839864—21no—
26#16839864—21no—
27#16839864—21no—
28#16839862—21no—
29#16839860—21no—
30C4×#11758202—21no—
31C4×#11758201—21no—
32C4×#11758200—21no—
33C4×#11758199—21no—
34C2×#14519713C2211yescharacteristic
35C2×#14519713C2211yescharacteristic
36#16839805C2211yescharacteristic
37#16839805C2211yescharacteristic
38C2×#14519711C2211yescharacteristic
39#16839770C2211yescharacteristic
40#16839766C2211yescharacteristic
41#16839763C2211yescharacteristic
42#16839763C2211yescharacteristic
43C2×#14519702C2211yescharacteristic
44C2×#14519699C2211yescharacteristic
45C2×#14519695C2211yescharacteristic
46C2×#14519695C2211yescharacteristic
47C2×#14519693C2211yescharacteristic
48C2×#14519693C2211yescharacteristic
49C4×#11758194C2211yescharacteristic
50C4×#11758194C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839530C2211yescharacteristic
53#16839530C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839527C2211yescharacteristic
56#16839527C2211yescharacteristic
57C2×#131×C61C2211yescharacteristic
58C2×#131×C61C2211yescharacteristic
59C4×#41×C61C2211yescharacteristic
60C61×#405—21no—
61C61×#405—21no—
62C61×#405C2211yescharacteristic
63C61×#405C2211yescharacteristic
64C4×M16×C61—21no—
65C4×M16×C61C2211yescharacteristic
66C2×#97×C61—21no—
67C2×#97×C61C2211yescharacteristic
68C22×Dic122C2311yescharacteristic
69C22×D244C2311yescharacteristic
70C22×C4×D61—21no—
71#14519807—22no—
72#14519807C2×C414yescomplemented
73#14519802—22no—
74#14519802C2×C414yes—
75C2×#11758202—21no—
76C2×#11758190—21no—
77C2×#11758190—21no—
78C2×#11758190—22no—
79C2×#11758190C2×C414yescomplemented
80C2×#11758200—21no—
81C2×#11758201—21no—
82C2×#11758199—21no—
83C2×#11758194C2311yescharacteristic
84C2×#11758194C2311yescharacteristic
85C2×#11758194C2311yescharacteristic
86C2×#11758194C2311yescharacteristic
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×Dic122—21no—
91C4×D244—21no—
92C4×D244—21no—
93C4×D244—21no—
94C4×D244—21no—
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100#14519713—22no—
101#14519713C2×C414yescomplemented
102#14519711—21no—
103#14519711—21no—
104#14519710—21no—
105#14519710—21no—
106#14519708—21no—
107#14519708—21no—
108#14519702—21no—
109#14519702—21no—
110#14519701—21no—
111#14519701—21no—
112#14519700—21no—
113#14519700—21no—
114#14519699—21no—
115#14519699—21no—
116#14519698—21no—
117#14519698—21no—
118#14519697—21no—
119#14519697—21no—
120#14519695—22no—
121#14519695C2×C414yescomplemented
122#14519693—22no—
123#14519693C2×C414yescomplemented
124#14519688—21no—
125#14519688—21no—
126C4×#92418—21no—
127C4×#92418—21no—
128C2×#41×C61C2311yescharacteristic
129#131×C61—22no—
130#131×C61—22no—
131#131×C61—22no—
132#131×C61C2×C414yescomplemented
133C2×M16×C61—21no—
134C2×M16×C61—21no—
135C2×M16×C61C2311yescharacteristic
136C2×M16×C61C2311yescharacteristic
137C22×C8×C61—21no—
138C22×C8×C61—21no—
139C22×C8×C61C2311yescharacteristic
140C22×C8×C61C2311yescharacteristic
141C4×Q8×C61D411yescharacteristic
142C4×Q8×C61D411yescharacteristic
143C4×D4×C61—21no—
144C4×D4×C61—21no—
145C4×D4×C61D411yescharacteristic
146C4×D4×C61D411yescharacteristic
147#117×C61—21no—
148#117×C61C2311yescharacteristic
149C2×C42×C61—21no—
150C2×C42×C61C2311yescharacteristic
151#97×C61—21no—
152#97×C61—21no—
153#97×C61—21no—
154#97×C61D411yescharacteristic
155#97×C61D411yescharacteristic
156#97×C61D411yescharacteristic
157#97×C61D411yescharacteristic
158C4×C8×C61—21no—
159C4×C8×C61—21no—
160C4×C8×C61—21no—
161C23×D61—21no—
162#41×C61—21no—
163#41×C61—21no—
164#41×C61C22×C412yes—
165#41×C61#4112yes—
166C2×Q8×C61C2×D411yescharacteristic
167C2×D4×C61—21no—
168C2×D4×C61C2×D411yescharacteristic
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61#4111yescharacteristic
173C22×C4×C61#4111yescharacteristic
174C22×C4×C61C2411yescharacteristic
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61—21no—
178C2×D244—21no—
179C2×D244—21no—
180C2×D244—21no—
181C2×D244—21no—
182C2×D244—21no—
183C2×D244C22×C412yes—
184C2×C4×D61—41no—
185C2×C4×D61—41no—
186C2×C4×D61—82no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—21no—
192C2×Dic122C22×C412yes—
193M16×C61—24no—
194M16×C61—21no—
195M16×C61—21no—
196M16×C61—24no—
197M16×C61C22×C412yes—
198M16×C61#4112yes—
199C2×C8×C61—24no—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C2×C8×C61—24no—
205C2×C8×C61C22×C412yes—
206C2×C8×C61C2×D411yescharacteristic
207C2×C8×C61C2×D411yescharacteristic
208C2×C8×C61C2×D411yescharacteristic
209C2×C8×C61C2×D411yescharacteristic
210C2×C8×C61#4112yes—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61C2×D411yescharacteristic
214C4⋊C4×C61C2×D411yescharacteristic
215C22⋊C4×C61—21no—
216C22⋊C4×C61—21no—
217C22⋊C4×C61—21no—
218C42×C61—21no—
219C42×C61—21no—
220C42×C61—21no—
221C42×C61C2×D411yescharacteristic
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2×D411yescharacteristic
225#11758202—24no—
226#11758201—24no—
227#11758200—24no—
228#11758199—24no—
229#11758198—41no—
230#11758198—41no—
231C4⋊Dic61—21no—
232C4⋊Dic61—21no—
233C4⋊Dic61—21no—
234C4⋊Dic61—21no—
235#11758196—41no—
236#11758196—41no—
237C4×Dic61—41no—
238C4×Dic61—41no—
239#11758194—21no—
240#11758194—21no—
241#11758194—21no—
242#11758194—21no—
243#11758194—21no—
244#11758194—21no—
245#11758194C22×C412yes—
246#11758194C22×C412yes—
247C2×#92418—21no—
248C2×#92418—21no—
249C2×#92418—21no—
250C2×#92418—21no—
251#11758190—24no—
252#11758190—24no—
253C23×C61—21no—
254C23×C61C2×#4111yescharacteristic
255C22×D61—22no—
256C22×D61—41no—
257C22×D61—41no—
258C22×D61—81no—
259C22×D61—81no—
260Q8×C61—21no—
261Q8×C61C4×D412yes—
262D4×C61—22no—
263D4×C61—22no—
264D4×C61—21no—
265D4×C61C4×D412yes—
266C2×C4×C61—21no—
267C2×C4×C61—22no—
268C2×C4×C61—22no—
269C2×C4×C61—21no—
270C2×C4×C61—21no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—81no—
274C2×C4×C61—81no—
275C2×C4×C61#12111yescharacteristic
276C2×C4×C61#12111yescharacteristic
277C2×C4×C61#12111yescharacteristic
278C2×C4×C61#12111yescharacteristic
279C2×C4×C61C23×C411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C2×#4111yescharacteristic
283C2×C4×C61C2×#4111yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×Dic61—22no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—81no—
291C2×Dic61—81no—
292D244—22no—
293D244—22no—
294D244—41no—
295D244—41no—
296C4×D61—82no—
297C4×D61—82no—
298Dic122—22no—
299Dic122—22no—
300Dic122—41no—
301Dic122—41no—
302C8×C61—22no—
303C8×C61—21no—
304C8×C61—22no—
305C8×C61—21no—
306C8×C61C4×D412yes—
307C8×C61C4×D412yes—
308#92418—22no—
309#92418—22no—
310#92418—41no—
311#92418—41no—
312#21557—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—22no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61C4×#4111yescharacteristic
319C22×C61C2×#12111yescharacteristic
320C22×C61#52911yescharacteristic
321C2×D61—42no—
322C2×D61—82no—
323C2×D61—81no—
324C2×D61—81no—
325C2×D61—82no—
326C4×C61—22no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—41no—
332C4×C61—41no—
333C4×C61C2×C4×D411yescharacteristic
334C4×C61C2×C4×D411yescharacteristic
335C4×C61#52511yescharacteristic
336C4×C61#53211yescharacteristic
337Dic61—42no—
338Dic61—81no—
339Dic61—81no—
340C2×#471—611no—
341C4×#137—611no—
342C4×AGL(1,ℤ/8)—611no—
343C2×#440—611no—
344#2875D6111yescharacteristic, complemented
345#1977—611no—
346#1952—611no—
347#1931—611no—
348#1930—611no—
349#1908—611no—
350#1907—611no—
351#1816—611no—
352#1778—611no—
353C4×#101—611no—
354C4×#100—611no—
355C2×C61—21no—
356C2×C61—21no—
357C2×C61—42no—
358C2×C61C4×#12111yescharacteristic
359C2×C61#306811yescharacteristic
360C2×C61#306911yescharacteristic
361D61—82no—
362C2×#137—611no—
363C2×AGL(1,ℤ/8)—611no—
364C22×SD16—611no—
365C2×#131C2×D6111yescharacteristic
366C2×#131C2×D6111yescharacteristic
367C4×#41C2×D6111yescharacteristic
368C2×C4×Q8—611no—
369C2×C4×D4—611no—
370#471—618no—
371#442—1221no—
372#441—1221no—
373#440—1221no—
374#440—1221no—
375#440—1221no—
376#440—1221no—
377#440—1221no—
378#440—1221no—
379C4×Q16—1221no—
380C4×SD16—1221no—
381C4×SD16—1221no—
382C4×D8—1221no—
383C2×#108—611no—
384C2×#108—611no—
385#428—611no—
386#428—611no—
387C2×#107—611no—
388#419—611no—
389#418—611no—
390#417—611no—
391#417—611no—
392C2×#103—611no—
393C2×#102—611no—
394C2×#101—611no—
395C2×#101—611no—
396C2×#100—611no—
397C2×#100—611no—
398#405—21no—
399#405—21no—
400#405C2×D6111yescharacteristic
401#405C2×D6111yescharacteristic
402C4×M16—21no—
403C4×M16—611no—
404C4×M16—611no—
405C4×M16C2×D6111yescharacteristic
406C2×#97—21no—
407C2×#97C2×D6111yescharacteristic
408#343—611no—
409#343—611no—
410#343—611no—
411#343—611no—
412#341—611no—
413#341—611no—
414C61#2155711yescharacteristic, complemented, Sylow
415C2×#41C22×D6111yescharacteristic
416C22×Q8—611no—
417C22×D4—611no—
418C23×C4—1221no—
419#137—614no—
420#137—1222no—
421AGL(1,ℤ/8)—614no—
422AGL(1,ℤ/8)—1222no—
423C2×Q16—1221no—
424C2×SD16—614no—
425C2×SD16—1221no—
426C2×SD16—1221no—
427C2×SD16—1222no—
428C2×SD16—1221no—
429C2×SD16—1221no—
430C2×D8—1221no—
431#131—22no—
432#131—22no—
433#131—22no—
434#131C4×D6114yescomplemented
435C2×M16—21no—
436C2×M16—21no—
437C2×M16—611no—
438C2×M16—611no—
439C2×M16—611no—
440C2×M16—611no—
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C22×C8—21no—
444C22×C8—21no—
445C22×C8C22×D6111yescharacteristic
446C22×C8C22×D6111yescharacteristic
447C4×Q8—1221no—
448C4×Q8—1221no—
449C4×Q8—1221no—
450C4×Q8—1221no—
451C4×Q8D24411yescharacteristic
452C4×Q8D24411yescharacteristic
453C4×D4—21no—
454C4×D4—21no—
455C4×D4—1221no—
456C4×D4—1221no—
457C4×D4—1221no—
458C4×D4—1221no—
459C4×D4D24411yescharacteristic
460C4×D4D24411yescharacteristic
461#117—21no—
462#117C22×D6111yescharacteristic
463C2×C4⋊C4—611no—
464C2×C4⋊C4—611no—
465C2×C4⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C42—21no—
468C2×C42—1221no—
469C2×C42C22×D6111yescharacteristic
470#108—614no—
471#108—1222no—
472#107—1221no—
473#107—1221no—
474#107—1221no—
475#107—1221no—
476#106—1221no—
477#106—1221no—
478#103—1221no—
479#103—1221no—
480#103—1221no—
481#103—1221no—
482#103—1221no—
483#103—1221no—
484#102—1221no—
485#102—1221no—
486#102—1221no—
487#102—1221no—
488#102—1221no—
489#102—1221no—
490#101—614no—
491#101—1222no—
492#100—614no—
493#100—1222no—
494#97—21no—
495#97—21no—
496#97—21no—
497#97—1221no—
498#97—1221no—
499#97C22⋊D6111yescharacteristic
500#97C22⋊D6111yescharacteristic
501#97C22⋊D6111yescharacteristic
502#97C22⋊D6111yescharacteristic
503C4×C8—21no—
504C4×C8—21no—
505C4×C8—21no—
506C4×C8—1221no—
507C4×C8—1221no—
508C24—1221no—
509#41—21no—
510#41—21no—
511#41C2×C4×D6112yes—
512#41#1175822212yes—
513C2×Q8—612no—
514C2×Q8—1221no—
515C2×Q8—1221no—
516C2×Q8—1221no—
517C2×Q8—1221no—
518C2×Q8—1221no—
519C2×Q8C2×D24411yescharacteristic
520C2×D4—21no—
521C2×D4—612no—
522C2×D4—1221no—
523C2×D4—1221no—
524C2×D4—1221no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4C2×D24411yescharacteristic
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—4882no—
535C22×C4—2441no—
536C22×C4—2441no—
537C22×C4#1175822211yescharacteristic
538C22×C4#1175822211yescharacteristic
539C22×C4C23×D6111yescharacteristic
540Q16—1224no—
541SD16—1224no—
542SD16—1224no—
543SD16—1224no—
544SD16—1224no—
545D8—1224no—
546M16—24no—
547M16—21no—
548M16—21no—
549M16—24no—
550M16—612no—
551M16—612no—
552M16—1221no—
553M16—1221no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16C2×C4×D6112yes—
559M16#1175822212yes—
560C2×C8—24no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—21no—
565C2×C8—24no—
566C2×C8—1221no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8C2×C4×D6112yes—
571C2×C8#1175822212yes—
572C2×C8C2×C22⋊D6111yescharacteristic
573C2×C8C2×C22⋊D6111yescharacteristic
574C2×C8C2×C22⋊D6111yescharacteristic
575C2×C8C2×C22⋊D6111yescharacteristic
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—1221no—
579C4⋊C4—1221no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—2441no—
583C4⋊C4—2441no—
584C4⋊C4C2×D24411yescharacteristic
585C4⋊C4C2×D24411yescharacteristic
586C22⋊C4—21no—
587C22⋊C4—21no—
588C22⋊C4—21no—
589C22⋊C4—2441no—
590C22⋊C4—2441no—
591C42—21no—
592C42—21no—
593C42—21no—
594C42—2441no—
595C42—2441no—
596C42C2×D24411yescharacteristic
597C42C2×D24411yescharacteristic
598C42C2×C22⋊D6111yescharacteristic
599C42C2×C22⋊D6111yescharacteristic
600C23—21no—
601C23—1222no—
602C23—4881no—
603C23—4881no—
604C23—2441no—
605C23—2441no—
606C23C2×#1175822211yescharacteristic
607Q8—21no—
608Q8—1222no—
609Q8—1222no—
610Q8—2441no—
611Q8—2441no—
612Q8C4×D24412yes—
613D4—22no—
614D4—22no—
615D4—21no—
616D4—1222no—
617D4—1222no—
618D4—2441no—
619D4—2441no—
620D4C4×D24412yes—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—1222no—
631C2×C4—4882no—
632C2×C4—4882no—
633C2×C4—4881no—
634C2×C4—4881no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4#1451975111yescharacteristic
640C2×C4#1451975111yescharacteristic
641C2×C4#1451975111yescharacteristic
642C2×C4#1451975111yescharacteristic
643C2×C4C22×C4×D6111yescharacteristic
644C2×C4C22×C22⋊D6111yescharacteristic
645C2×C4C22×D24411yescharacteristic
646C2×C4C2×#1175822211yescharacteristic
647C2×C4C2×#1175822211yescharacteristic
648C2×C4C2×#1175822211yescharacteristic
649C8—22no—
650C8—21no—
651C8—22no—
652C8—21no—
653C8—1222no—
654C8—1222no—
655C8—2441no—
656C8—2441no—
657C8C4×C22⋊D6112yes—
658C8C4×C22⋊D6112yes—
659C22—21no—
660C22—21no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—4882no—
665C22—4882no—
666C22—4881no—
667C22—4881no—
668C22—2442no—
669C22C4×#1175822211yescharacteristic
670C22C2×#1451975111yescharacteristic
671C22#1684029611yescharacteristic
672C4—22no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—41no—
678C4—41no—
679C4—4881no—
680C4—4881no—
681C4—2442no—
682C4C2×C4×C22⋊D6111yescharacteristic
683C4C2×C4×D24411yescharacteristic
684C4#1684027311yescharacteristic
685C4#1684031211yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—4882no—
690C2C4×#1451975111yescharacteristic
691C2#2145554711yescharacteristic
692C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2875C2
socleC22×C61C2×#121
prime radicalC2×#14519802C22
cosocleC22×C4×C61C24
hypercenter#2875D61
hypocenterC61#21557
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. C4×#14519807C2semidirect
4. C2×#16839864C2semidirect
5. #21446722C2semidirect
6. #21446604C2semidirect
7. #21446481C2semidirect
8. #21446474C2semidirect
9. #21446334C2semidirect
11. #21445883C2semidirect
12. #21445725C2semidirect
13. C4×#14519695C2semidirect
14. C4×#14519693C2semidirect
15. C61×#2875C2semidirect
21. #16840025C4semidirect
72. #14519807C2×C4semidirect
79. C2×#11758190C2×C4semidirect
101. #14519713C2×C4semidirect
121. #14519695C2×C4semidirect
123. #14519693C2×C4semidirect
132. #131×C61C2×C4semidirect
344. #2875D61semidirect
414. C61#21557semidirect
434. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400252
C4×#145198072
C4×#145198022
C2×#168398642
#214467222
#214466042
#214464812
#214464742
#214463342
#214463272
#214458832
#214457252
C4×#145196952
C4×#145196932
C61×#28752
#2155761