Finite Group Databasebeta

← #29686930

Subgroups — #29686930

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
2#21454643C211yesmaximal, characteristic, complemented
3#21454632C211yesmaximal, characteristic
4C2×#16839861C211yesmaximal, characteristic, complemented
5#21454245C211yesmaximal, characteristic, complemented
6#21446720C211yesmaximal, characteristic, complemented
7#21446602C211yesmaximal, characteristic, complemented
8#21446479C211yesmaximal, characteristic, complemented
9#21446472C211yesmaximal, characteristic, complemented
10#21446332C211yesmaximal, characteristic, complemented
11#21446325C211yesmaximal, characteristic
12#21445727C211yesmaximal, characteristic, complemented
13#21445614C211yesmaximal, characteristic, complemented
14#21445610C211yesmaximal, characteristic, complemented
15C61×#1816C211yesmaximal, characteristic, complemented
16C2×#14519807C2211yescharacteristic
17C2×#14519802C2211yescharacteristic
18C22×#11758190C2211yescharacteristic
19C2×#14519784C2211yescharacteristic
20#41×Dic61C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16840025C418yescomplemented
24#16839871—21no—
25#16839866—21no—
26#16839865—21no—
27#16839861—21no—
28#16839861—21no—
29#16839861—21no—
30#16839861—21no—
31#16839859—21no—
32#16839854—21no—
33#16839853—21no—
34#16839847—21no—
35#16839842—21no—
36C2×#14519713C2211yescharacteristic
37#16839802C2211yescharacteristic
38#16839802C2211yescharacteristic
39#16839772C2211yescharacteristic
40#16839768C2211yescharacteristic
41C2×#14519700C2211yescharacteristic
42C2×#14519697C2211yescharacteristic
43C2×#14519695C2211yescharacteristic
44C2×#14519695C2211yescharacteristic
45C2×#14519693C2211yescharacteristic
46C2×#14519693C2211yescharacteristic
47#16839720—21no—
48#16839720C2211yescharacteristic
49#16839717—21no—
50#16839717C2211yescharacteristic
51M16×Dic61—21no—
52M16×Dic61C2211yescharacteristic
53M16×Dic61C2211yescharacteristic
54M16×Dic61C2211yescharacteristic
55C2×#14519687—21no—
56C2×#14519687C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839531C2211yescharacteristic
59#16839526C2211yescharacteristic
60#16839526C2211yescharacteristic
61C2×#131×C61C2211yescharacteristic
62C2×#108×C61C2211yescharacteristic
63C2×#107×C61C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#343C2211yescharacteristic
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—
83#14519784—22no—
84#14519784—22no—
85C2×#11758194—21no—
86C2×#11758194C2311yescharacteristic
87C2×#11758194C2311yescharacteristic
88C2×#11758194C2311yescharacteristic
89C22×#92418—21no—
90C22×#92418C2311yescharacteristic
91Q8×Dic61D411yescharacteristic
92Q8×Dic61D411yescharacteristic
93#14519742—21no—
94#14519742—21no—
95#14519742—21no—
96#14519742—21no—
97D4×Dic61—21no—
98D4×Dic61—21no—
99D4×Dic61D411yescharacteristic
100D4×Dic61D411yescharacteristic
101#14519739—21no—
102#14519739—21no—
103#14519739—21no—
104#14519739—21no—
105#14519736—21no—
106#14519736C2311yescharacteristic
107C2×#11758196—21no—
108C2×#11758198—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61C2311yescharacteristic
112#14519713C2×C414yescomplemented
113#14519710—21no—
114#14519710—21no—
115#14519708—21no—
116#14519708—21no—
117#14519702—21no—
118#14519702—21no—
119#14519700—21no—
120#14519700—21no—
121#14519699—21no—
122#14519699—21no—
123#14519697—21no—
124#14519697—21no—
125#14519695—22no—
126#14519695C2×C414yescomplemented
127#14519693—22no—
128#14519693C2×C414yescomplemented
129#14519687—21no—
130#14519687—21no—
131#14519687—21no—
132#14519687—21no—
133#14519687—21no—
134#14519687D411yescharacteristic
135#14519687D411yescharacteristic
136#14519687D411yescharacteristic
137#14519687D411yescharacteristic
138C8×Dic61—21no—
139C8×Dic61—21no—
140C8×Dic61—21no—
141C8×Dic61—21no—
142C8×Dic61—21no—
143C2×#41×C61C2311yescharacteristic
144#131×C61—22no—
145#131×C61C2×C414yescomplemented
146C2×M16×C61—21no—
147C2×M16×C61C2311yescharacteristic
148C2×M16×C61C2311yescharacteristic
149C2×M16×C61C2311yescharacteristic
150C22×C8×C61—21no—
151C22×C8×C61C2311yescharacteristic
152C2×C4⋊C4×C61C2311yescharacteristic
153C2×C4⋊C4×C61C2311yescharacteristic
154#108×C61—22no—
155#107×C61—21no—
156#107×C61—21no—
157#103×C61—21no—
158#103×C61—21no—
159#102×C61—21no—
160#102×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×C61C2411yescharacteristic
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61—21no—
176C22×Dic61#4111yescharacteristic
177C22×Dic61#4111yescharacteristic
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—21no—
194M16×C61—24no—
195M16×C61—21no—
196M16×C61—21no—
197M16×C61—21no—
198M16×C61—21no—
199M16×C61C22×C412yes—
200C2×C8×C61—21no—
201C2×C8×C61—24no—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C2×C8×C61C22×C412yes—
205C2×C8×C61C2×D411yescharacteristic
206C2×C8×C61C2×D411yescharacteristic
207C2×C8×C61#4112yes—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—21no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C22⋊Dic61—21no—
213C22⋊Dic61—21no—
214C22⋊Dic61—21no—
215#11758202—24no—
216#11758201—24no—
217#11758200—24no—
218#11758199—24no—
219#11758198—41no—
220#11758198—41no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61C2×D411yescharacteristic
224C4⋊Dic61C2×D411yescharacteristic
225#11758196—41no—
226#11758196—41no—
227C4×Dic61—21no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61—41no—
231C4×Dic61—41no—
232C4×Dic61C2×D411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236#11758194—24no—
237#11758194—21no—
238#11758194—21no—
239#11758194—21no—
240#11758194C22×C412yes—
241#11758194C22×C412yes—
242#11758194#4112yes—
243C2×#92418—24no—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—21no—
247C2×#92418—21no—
248C2×#92418—21no—
249C2×#92418C2×D411yescharacteristic
250C2×#92418C2×D411yescharacteristic
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—41no—
270C2×C4×C61—41no—
271C2×C4×C61—81no—
272C2×C4×C61—81no—
273C2×C4×C61C23×C411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×Dic61—22no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—81no—
287C2×Dic61—81no—
288C2×Dic61#12111yescharacteristic
289C2×Dic61#12111yescharacteristic
290C2×Dic61#12111yescharacteristic
291C2×Dic61#12111yescharacteristic
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—41no—
304C8×C61—41no—
305C8×C61C4×D412yes—
306C8×C61C4×D412yes—
307#92418—22no—
308#92418—22no—
309#92418—21no—
310#92418—22no—
311#92418—21no—
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—81no—
328C4×C61—81no—
329C4×C61C2×C4×D411yescharacteristic
330C4×C61C2×C4×D411yescharacteristic
331C4×C61#52511yescharacteristic
332C4×C61#53211yescharacteristic
333Dic61—21no—
334Dic61—21no—
335Dic61—21no—
336Dic61—21no—
337Dic61—42no—
338Dic61—41no—
339Dic61—41no—
340C2×#471—611no—
341C4×#137—611no—
342C4×AGL(1,ℤ/8)—611no—
343C2×#440—611no—
344#2875—611no—
345#1977—611no—
346#1952—611no—
347#1931—611no—
348#1930—611no—
349#1908—611no—
350#1907—611no—
351#1816D6111yescharacteristic, complemented
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×#131—611no—
366C2×#131C2×D6111yescharacteristic
367C4×#41—611no—
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×#108C2×D6111yescharacteristic
385#428—611no—
386#428—611no—
387C2×#107C2×D6111yescharacteristic
388#419—611no—
389#418—611no—
390#417C2×D6111yescharacteristic
391#417C2×D6111yescharacteristic
392C2×#103—611no—
393C2×#102—611no—
394C2×#101—611no—
395C2×#101—611no—
396C2×#100—611no—
397C2×#100—611no—
398#405—611no—
399#405—611no—
400#405—1221no—
401#405—1221no—
402C4×M16—611no—
403C4×M16—611no—
404C4×M16—611no—
405C4×M16—1221no—
406C2×#97—611no—
407C2×#97—1221no—
408#343—611no—
409#343—611no—
410#343C2×D6111yescharacteristic
411#343C2×D6111yescharacteristic
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—1221no—
428C2×SD16—1221no—
429C2×SD16—1222no—
430C2×D8—1221no—
431#131—22no—
432#131—1222no—
433#131—1222no—
434#131C4×D6114yescomplemented
435C2×M16—21no—
436C2×M16—611no—
437C2×M16—611no—
438C2×M16—611no—
439C2×M16—1221no—
440C2×M16C22×D6111yescharacteristic
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C22×C8—21no—
444C22×C8—611no—
445C22×C8—1221no—
446C22×C8C22×D6111yescharacteristic
447C4×Q8—611no—
448C4×Q8—611no—
449C4×Q8—1221no—
450C4×Q8—1221no—
451C4×Q8—1221no—
452C4×Q8—1221no—
453C4×D4—611no—
454C4×D4—611no—
455C4×D4—1221no—
456C4×D4—1221no—
457C4×D4—1221no—
458C4×D4—1221no—
459C4×D4—1221no—
460C4×D4—1221no—
461#117—611no—
462#117—1221no—
463C2×C4⋊C4—1221no—
464C2×C4⋊C4C22×D6111yescharacteristic
465C2×C4⋊C4C22×D6111yescharacteristic
466C2×C22⋊C4—1221no—
467C2×C42—611no—
468C2×C42—1221no—
469C2×C42—1221no—
470#108—22no—
471#108—614no—
472#107—21no—
473#107—21no—
474#107—1221no—
475#107—1221no—
476#106—1221no—
477#106—1221no—
478#103—21no—
479#103—21no—
480#103—1221no—
481#103—1221no—
482#103—1221no—
483#103—1221no—
484#102—21no—
485#102—21no—
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—611no—
495#97—611no—
496#97—611no—
497#97—611no—
498#97—1221no—
499#97—1221no—
500#97—1221no—
501#97—1221no—
502#97—1221no—
503C4×C8—1221no—
504C4×C8—1221no—
505C4×C8—1221no—
506C4×C8—1221no—
507C4×C8—1221no—
508C24—1221no—
509#41—21no—
510#41—21no—
511#41C2×C4×D6112yes—
512#41#1175822612yes—
513C2×Q8—612no—
514C2×Q8—1221no—
515C2×Q8—1221no—
516C2×Q8—1221no—
517C2×Q8—1221no—
518C2×Q8—1221no—
519C2×Q8D4×D6111yescharacteristic
520C2×D4—21no—
521C2×D4—612no—
522C2×D4—1221no—
523C2×D4—1221no—
524C2×D4—1221no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4D4×D6111yescharacteristic
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—611no—
532C22×C4—611no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—4882no—
537C22×C4—2441no—
538C22×C4—2441no—
539C22×C4C23×D6111yescharacteristic
540Q16—1224no—
541SD16—1224no—
542SD16—1224no—
543SD16—1224no—
544SD16—1224no—
545D8—1224no—
546M16—21no—
547M16—24no—
548M16—21no—
549M16—21no—
550M16—21no—
551M16—21no—
552M16—612no—
553M16—612no—
554M16—612no—
555M16—1224no—
556M16—1221no—
557M16—1221no—
558M16—1221no—
559M16C2×C4×D6112yes—
560C2×C8—21no—
561C2×C8—24no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—611no—
565C2×C8—611no—
566C2×C8—1224no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8—1221no—
571C2×C8—1221no—
572C2×C8C2×C4×D6112yes—
573C2×C8D4×D6111yescharacteristic
574C2×C8D4×D6111yescharacteristic
575C2×C8#1175822412yes—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—611no—
581C4⋊C4—611no—
582C4⋊C4—1221no—
583C4⋊C4—1221no—
584C4⋊C4—2441no—
585C4⋊C4—2441no—
586C22⋊C4—1221no—
587C22⋊C4—1221no—
588C22⋊C4—1221no—
589C22⋊C4—2441no—
590C22⋊C4—2441no—
591C42—611no—
592C42—611no—
593C42—611no—
594C42—611no—
595C42—1221no—
596C42—1221no—
597C42—1221no—
598C42—2441no—
599C42—2441no—
600C23—21no—
601C23—1222no—
602C23—4881no—
603C23—4881no—
604C23—2441no—
605C23—2441no—
606C23#41×D6111yescharacteristic
607Q8—21no—
608Q8—1222no—
609Q8—1222no—
610Q8—2441no—
611Q8—2441no—
612Q8#1451973912yes—
613D4—22no—
614D4—22no—
615D4—21no—
616D4—1222no—
617D4—1222no—
618D4—2441no—
619D4—2441no—
620D4#1451973912yes—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—611no—
629C2×C4—611no—
630C2×C4—611no—
631C2×C4—611no—
632C2×C4—1222no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—4882no—
636C2×C4—4882no—
637C2×C4—4881no—
638C2×C4—4881no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4C22×C4×D6111yescharacteristic
644C2×C4C2×D4×D6111yescharacteristic
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4#41×D6111yescharacteristic
647C2×C4C2×#1175822411yescharacteristic
648C2×C4C2×#1175822611yescharacteristic
649C8—22no—
650C8—41no—
651C8—41no—
652C8—1222no—
653C8—1222no—
654C8—1221no—
655C8—1221no—
656C8—1222no—
657C8#1451973712yes—
658C8#1451973712yes—
659C22—21no—
660C22—21no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—4882no—
665C22—4882no—
666C22—4881no—
667C22—4881no—
668C22—2442no—
669C22#1684023011yescharacteristic
670C22#121×D6111yescharacteristic
671C22#1684029811yescharacteristic
672C4—22no—
673C4—81no—
674C4—81no—
675C4—1221no—
676C4—1221no—
677C4—1221no—
678C4—1221no—
679C4—2442no—
680C4—2441no—
681C4—2441no—
682C4C2×#1451973711yescharacteristic
683C4C2×#1451973911yescharacteristic
684C4#1684027511yescharacteristic
685C4#1684030711yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—4882no—
690C2#2144832511yescharacteristic
691C2#2145554511yescharacteristic
692C2#2145555411yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1816C2
socleC22×C61C2×#121
prime radicalC2×#14519802C22
cosocleC22×C4×C61C24
hypercenter#1816D61
hypocenterC61#21557
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
2. #21454643C2semidirect
4. C2×#16839861C2semidirect
5. #21454245C2semidirect
6. #21446720C2semidirect
7. #21446602C2semidirect
8. #21446479C2semidirect
9. #21446472C2semidirect
10. #21446332C2semidirect
12. #21445727C2semidirect
13. #21445614C2semidirect
14. #21445610C2semidirect
15. C61×#1816C2semidirect
23. #16840025C4semidirect
72. #14519807C2×C4semidirect
79. C2×#11758190C2×C4semidirect
112. #14519713C2×C4semidirect
126. #14519695C2×C4semidirect
128. #14519693C2×C4semidirect
145. #131×C61C2×C4semidirect
351. #1816D61semidirect
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
#214546432
#214546322
C2×#168398612
#214542452
#214467202
#214466022
#214464792
#214464722
#214463322
#214463252
#214457272
#214456142
#214456102
C61×#18162
#2155761