Finite Group Databasebeta

← #29683570

Subgroups — #29683570

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839809C211yesmaximal, characteristic
2#21454406C211yesmaximal, characteristic
3C2×#16839803C211yesmaximal, characteristic
4#428×D61C211yesmaximal, characteristic
5#21454235C211yesmaximal, characteristic
6#21446603C211yesmaximal, characteristic, complemented
7#21446602C211yesmaximal, characteristic, complemented
8#21446601C211yesmaximal, characteristic, complemented
9#21446600C211yesmaximal, characteristic, complemented
10#21446599C211yesmaximal, characteristic, complemented
11#21446598C211yesmaximal, characteristic, complemented
12#21446596C211yesmaximal, characteristic, complemented
13#21445556C211yesmaximal, characteristic
14#343×D61C211yesmaximal, characteristic
15C61×#1952C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×M16×D61C2211yescharacteristic
18C2×M16×D61C2211yescharacteristic
19C2×M16×D61C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21#117×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839809C418yescomplemented
24C2×#14519713C2211yescharacteristic
25C2×#14519712C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27#16839805C2211yescharacteristic
28#16839805C2211yescharacteristic
29#16839804C2211yescharacteristic
30#16839804C2211yescharacteristic
31#16839803—21no—
32#16839803—21no—
33#16839803—21no—
34#16839803—21no—
35#16839803—21no—
36#16839803—21no—
37#16839802C2211yescharacteristic
38#16839802C2211yescharacteristic
39#16839801—21no—
40#16839801—21no—
41#16839799—21no—
42#16839797—21no—
43C2×#14519710C2211yescharacteristic
44#107×D61—21no—
45C2×#14519709C2211yescharacteristic
46C2×#14519708C2211yescharacteristic
47#106×D61—21no—
48#16839720C2211yescharacteristic
49#16839719—21no—
50#16839718C2211yescharacteristic
51#16839717C2211yescharacteristic
52#16839716—21no—
53#16839714—21no—
54C4×#11758189—21no—
55#16839531C2211yescharacteristic
56#16839531C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839530C2211yescharacteristic
61C2×#108×C61C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C61×#428C2211yescharacteristic
64C2×#107×C61C2211yescharacteristic
65C61×#405C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#343C2211yescharacteristic
68C22×C4×D61—21no—
69C22×C4×D61C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C2×#11758189—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76C2×#11758189C2×C414yes—
77C2×#11758189D411yescharacteristic
78C2×#11758189D411yescharacteristic
79C2×#11758189D411yescharacteristic
80C2×#11758189D411yescharacteristic
81C2×#11758189Q814yes—
82M16×D61—22no—
83M16×D61—22no—
84M16×D61—22no—
85M16×D61C2×C414yes—
86M16×D61C2×C414yes—
87M16×D61Q814yes—
88C22×#92418C2311yescharacteristic
89C2×C8×D61—21no—
90C2×C8×D61—21no—
91C2×C8×D61—21no—
92#14519736C2311yescharacteristic
93#14519735—21no—
94#14519735—21no—
95#14519735—21no—
96#14519735—21no—
97#14519735D411yescharacteristic
98#14519735D411yescharacteristic
99#14519734—21no—
100#14519733—21no—
101C2×C4⋊Dic61C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C2×#11758196—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61—21no—
108C4⋊C4×D61D411yescharacteristic
109C4⋊C4×D61D411yescharacteristic
110C2×#11758198—21no—
111C22⋊C4×D61—21no—
112C2×C4×Dic61—21no—
113C42×D61—21no—
114#14519713C2×C414yescomplemented
115#14519712C2×C414yescomplemented
116#14519712C2×C414yescomplemented
117#14519711—21no—
118#14519711—21no—
119#14519710—21no—
120#14519710—21no—
121#14519710—21no—
122#14519710—21no—
123#14519710—21no—
124#14519710—21no—
125#14519709—21no—
126#14519709—21no—
127#14519709—21no—
128#14519709—21no—
129#14519708—21no—
130#14519708—21no—
131#14519708—21no—
132#14519708—21no—
133#14519708—21no—
134#14519708—21no—
135#14519688—21no—
136#14519687—21no—
137#14519687—21no—
138C4×#92418—21no—
139C8×Dic61—21no—
140C8×Dic61—21no—
141C2×M16×C61C2311yescharacteristic
142C2×M16×C61C2311yescharacteristic
143C2×M16×C61C2311yescharacteristic
144C22×C8×C61C2311yescharacteristic
145#117×C61C2311yescharacteristic
146C2×C4⋊C4×C61C2311yescharacteristic
147C2×C4⋊C4×C61C2311yescharacteristic
148#108×C61C2×C414yescomplemented
149#107×C61—21no—
150#107×C61—21no—
151#107×C61—21no—
152#107×C61—21no—
153#106×C61—21no—
154#106×C61—21no—
155#97×C61—21no—
156C4×C8×C61—21no—
157C23×D61#4111yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61#4111yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—41no—
167C2×C4×D61—41no—
168C2×C4×D61—82no—
169C2×C4×D61C22×C412yes—
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×Q812yes—
175C2×C4×D61C2×Q812yes—
176C2×C4×D61#4112yes—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61C22×C412yes—
181M16×C61C22×C412yes—
182M16×C61C2×Q812yes—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61C22×C412yes—
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×Q812yes—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C22⋊C4×C61—21no—
197C42×C61—21no—
198C22⋊Dic61—21no—
199#11758198—21no—
200#11758198—21no—
201#11758198—41no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D411yescharacteristic
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—21no—
210#11758196—21no—
211#11758196—41no—
212#11758196—41no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—41no—
217C4×Dic61—41no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194C22×C412yes—
222#11758194C22×C412yes—
223#11758194C2×Q812yes—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418C22×C412yes—
228C2×#92418C2×D411yescharacteristic
229C2×#92418C2×D411yescharacteristic
230C2×#92418C2×Q812yes—
231#11758189—24no—
232#11758189—24no—
233#11758189—24no—
234#11758189C4⋊C414yes—
235#11758189C4⋊C414yes—
236#11758189C4⋊C414yes—
237#11758189C4⋊C414yes—
238C8×D61—24no—
239C8×D61—24no—
240C8×D61—24no—
241C23×C61C2×#4111yescharacteristic
242C22×D61—41no—
243C22×D61—41no—
244C22×D61C4×Q812yes—
245C22×D61#12111yescharacteristic
246C22×D61#12111yescharacteristic
247C22×D61#12512yes—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—81no—
253C2×C4×C61—81no—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×Q811yescharacteristic
258C2×C4×C61C22×Q811yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—81no—
265C2×Dic61—81no—
266C2×Dic61C4×Q812yes—
267C2×Dic61#12111yescharacteristic
268C2×Dic61#12111yescharacteristic
269C2×Dic61#12512yes—
270C4×D61—81no—
271C4×D61—81no—
272C4×D61—82no—
273C4×D61—82no—
274C4×D61C2×C4⋊C412yes—
275C4×D61C2×C4⋊C412yes—
276C4×D61C4×D412yes—
277C4×D61C4×D412yes—
278C4×D61C22⋊Q812yes—
279C4×D61C22⋊Q812yes—
280C4×D61C4⋊Q812yes—
281C4×D61C4⋊Q812yes—
282C8×C61—22no—
283C8×C61—22no—
284C8×C61—22no—
285C8×C61C2×C4⋊C412yes—
286C8×C61C2×C4⋊C412yes—
287#92418—22no—
288#92418—22no—
289#92418—22no—
290#92418C2×C4⋊C412yes—
291#92418C2×C4⋊C412yes—
292#21182—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61C2×C4×Q811yescharacteristic
296C22×C61C2×#12111yescharacteristic
297C22×C61C2×#12511yescharacteristic
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61#38912yes—
305C2×D61#38912yes—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61C22×C4⋊C411yescharacteristic
311C4×C61C2×C4×D411yescharacteristic
312C4×C61C2×C22⋊Q811yescharacteristic
313C4×C61C2×C4⋊Q811yescharacteristic
314Dic61—41no—
315Dic61—41no—
316Dic61—81no—
317Dic61—81no—
318Dic61#38912yes—
319Dic61#38912yes—
320C2×#430—611no—
321#2911—611no—
322C2×#428—611no—
323C2×#428—611no—
324#2873—611no—
325#1952—611no—
326#1952—611no—
327#1952—611no—
328#1952D6111yescharacteristic, complemented
329#1951—611no—
330#1951—611no—
331#1951—611no—
332#1951—611no—
333#1742—611no—
334C2×#343—611no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61C2×#38911yescharacteristic
338C2×C61#306811yescharacteristic
339C2×C61#306911yescharacteristic
340D61—22no—
341D61—22no—
342C22×M16—611no—
343C22×M16—611no—
344C22×M16—611no—
345C22×M16—611no—
346C2×#117—611no—
347C2×#117—611no—
348C22×C4⋊C4—611no—
349#430—618no—
350C2×#108—611no—
351C2×#108—611no—
352C2×#108—611no—
353C2×#108C2×D6111yescharacteristic
354#428—611no—
355#428—611no—
356#428—611no—
357#428—611no—
358#428—611no—
359#428—611no—
360#428—1221no—
361#428—1221no—
362#428—1221no—
363#428—1221no—
364#428—1221no—
365#428—1221no—
366#428—1221no—
367#428—1221no—
368#428C2×D6111yescharacteristic
369#428C2×D6111yescharacteristic
370#427—1221no—
371#427—1221no—
372C2×#107—611no—
373C2×#107—1221no—
374C2×#107C2×D6111yescharacteristic
375C2×#106—611no—
376C2×#106—611no—
377C2×#106—1221no—
378#405—611no—
379#405—611no—
380#405—611no—
381#405—1221no—
382#405—1221no—
383#405C2×D6111yescharacteristic
384C4×M16—1221no—
385C4×M16—1221no—
386#343—611no—
387#343—611no—
388#343—611no—
389#343—611no—
390#343—611no—
391#343—611no—
392#343C2×D6111yescharacteristic
393#343C2×D6111yescharacteristic
394C61#2118211yescharacteristic, complemented, Sylow
395C23×C4—611no—
396C23×C4—1221no—
397C2×M16—614no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—611no—
401C2×M16—611no—
402C2×M16—614no—
403C2×M16—611no—
404C2×M16—611no—
405C2×M16—611no—
406C2×M16—614no—
407C2×M16—614no—
408C2×M16—614no—
409C2×M16—1221no—
410C2×M16—1221no—
411C2×M16—1221no—
412C2×M16—1222no—
413C2×M16—1222no—
414C2×M16—1222no—
415C2×M16C22×D6111yescharacteristic
416C2×M16C22×D6111yescharacteristic
417C2×M16C22×D6111yescharacteristic
418C22×C8—611no—
419C22×C8—1221no—
420C22×C8—1221no—
421C22×C8—1221no—
422C22×C8C22×D6111yescharacteristic
423#117—611no—
424#117—611no—
425#117—611no—
426#117—1221no—
427#117—1221no—
428#117—1221no—
429#117—1221no—
430#117—1221no—
431#117—1221no—
432#117C22×D6111yescharacteristic
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448#108—614no—
449#108—614no—
450#108—614no—
451#108C4×D6114yescomplemented
452#107—21no—
453#107—21no—
454#107—21no—
455#107—21no—
456#107—1221no—
457#107—1221no—
458#107—1221no—
459#107—1221no—
460#107—1221no—
461#107—1221no—
462#107—1221no—
463#107—1221no—
464#106—21no—
465#106—21no—
466#106—1221no—
467#106—1221no—
468#106—1221no—
469#106—1221no—
470#106—1221no—
471#106—1221no—
472#106—1221no—
473#106—1221no—
474#106—1221no—
475#106—1221no—
476#97—21no—
477#97—1221no—
478#97—1221no—
479#97—1221no—
480C4×C8—21no—
481C4×C8—1221no—
482C4×C8—1221no—
483C4×C8—1221no—
484C24—611no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—611no—
488C22×C4—612no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—612no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—4882no—
501C22×C4—2441no—
502C22×C4—2441no—
503C22×C4C23×D6111yescharacteristic
504M16—21no—
505M16—21no—
506M16—21no—
507M16—614no—
508M16—614no—
509M16—614no—
510M16—614no—
511M16—612no—
512M16—612no—
513M16—612no—
514M16—1224no—
515M16—1224no—
516M16—1224no—
517M16—1221no—
518M16—1221no—
519M16—1221no—
520M16C2×C4×D6112yes—
521M16C2×C4×D6112yes—
522M16Q8×D6112yes—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—612no—
527C2×C8—611no—
528C2×C8—611no—
529C2×C8—612no—
530C2×C8—1224no—
531C2×C8—1224no—
532C2×C8—1224no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8C2×C4×D6112yes—
537C2×C8D4×D6111yescharacteristic
538C2×C8D4×D6111yescharacteristic
539C2×C8Q8×D6112yes—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—611no—
545C4⋊C4—611no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—2441no—
553C4⋊C4—2441no—
554C4⋊C4D4×D6111yescharacteristic
555C4⋊C4D4×D6111yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—1221no—
558C22⋊C4—1221no—
559C22⋊C4—1221no—
560C22⋊C4—2441no—
561C22⋊C4—2441no—
562C42—21no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—2441no—
567C42—2441no—
568C23—611no—
569C23—611no—
570C23—612no—
571C23—612no—
572C23—2441no—
573C23—2441no—
574C23#41×D6111yescharacteristic
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—612no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—4881no—
596C2×C4—4881no—
597C2×C4—4882no—
598C2×C4—4882no—
599C2×C4—4881no—
600C2×C4—4881no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4C22×C4×D6111yescharacteristic
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×Q8×D6111yescharacteristic
607C2×C4C2×Q8×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C8—22no—
610C8—22no—
611C8—22no—
612C8—612no—
613C8—612no—
614C8—1222no—
615C8—1222no—
616C8—1222no—
617C8C4⋊C4×D6112yes—
618C8C4⋊C4×D6112yes—
619C22—21no—
620C22—21no—
621C22—612no—
622C22—612no—
623C22—1221no—
624C22—1221no—
625C22—1221no—
626C22—1221no—
627C22—2442no—
628C22—2442no—
629C22C4×Q8×D6111yescharacteristic
630C22#121×D6111yescharacteristic
631C22#125×D6111yescharacteristic
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—612no—
637C4—612no—
638C4—4881no—
639C4—4881no—
640C4—2441no—
641C4—2441no—
642C4C2×C4⋊C4×D6111yescharacteristic
643C4C4×D4×D6111yescharacteristic
644C4C22⋊Q8×D6111yescharacteristic
645C4C4⋊Q8×D6111yescharacteristic
646C2—21no—
647C2—21no—
648C2—1222no—
649C2—1222no—
650C2#389×D6111yescharacteristic
651C2#2145554511yescharacteristic
652C2#2145555411yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1952C2
socleC22×C61C2×#121
prime radicalC23×D61#41
cosocleC22×C4×C61C24
hypercenter#1952D61
hypocenterC61#21182
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
6. #21446603C2semidirect
7. #21446602C2semidirect
8. #21446601C2semidirect
9. #21446600C2semidirect
10. #21446599C2semidirect
11. #21446598C2semidirect
12. #21446596C2semidirect
15. C61×#1952C2semidirect
23. #16839809C4semidirect
114. #14519713C2×C4semidirect
115. #14519712C2×C4semidirect
116. #14519712C2×C4semidirect
148. #108×C61C2×C4semidirect
328. #1952D61semidirect
394. C61#21182semidirect
451. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398092
#214544062
C2×#168398032
#428×D612
#214542352
#214466032
#214466022
#214466012
#214466002
#214465992
#214465982
#214465962
#214455562
#343×D612
C61×#19522
#2118261