Finite Group Databasebeta

← #30616707

Subgroups — #30616707

Order 15616.

1086 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#433C211yesmaximal, characteristic
2#21462256C211yesmaximal, characteristic
3#21461999C212yesmaximal, complemented
4#21461985C212yesmaximal, complemented
5#21461805C212yesmaximal
6#21461804C212yesmaximal
7#21461591C212yesmaximal, complemented
8C4×#14519840C212yesmaximal, complemented
9#541×D61C211yesmaximal, characteristic
10C2×#14519856C2211yescharacteristic
11C22×C4⋊#6923C2211yescharacteristic
12#16840787—22no—
13#16840786—22no—
14D61⋊(C4⋊C8)C2212yes—
15C2×#14519840C2212yes—
16C2×#14519840C2212yes—
17C2×#14519840C2212yes—
18C2×#14519840C2212yes—
19D61⋊(C22⋊C8)C2212yes—
20#16840721C2211yescharacteristic
21C4×D61⋊C8C2211yescharacteristic
22C4⋊C4×#6923—22no—
23C4×C4⋊#6923C2212yes—
24C4×C4⋊#6923C2212yes—
25C22⋊C4×#6923—22no—
26#16840635C2212yes—
27#16840635C2212yes—
28#16840635C2212yes—
29#16840635C2212yes—
30#16840375C411yescharacteristic, complemented
31#16840374C411yescharacteristic, complemented
32#16840373C411yescharacteristic, complemented
33#16840372C412yescomplemented
34#16840371—22no—
35#16840370C412yescomplemented
36#16840369—22no—
37C4⋊Q8×D61C2211yescharacteristic
38#126×D61—22no—
39#125×D61—22no—
40#125×D61C2211yescharacteristic
41C22⋊Q8×D61C2212yes—
42C4×Q8×D61C2212yes—
43#117×D61C2211yescharacteristic
44C61×#541C411yescharacteristic, complemented
45C23×#6923—21no—
46C2×D61⋊C8C2312yes—
47C2×D61⋊C8C2312yes—
48#14519856—24no—
49C2×C4⋊#6923—21no—
50C2×C4⋊#6923—24no—
51C2×C4⋊#6923C2312yes—
52C2×C4⋊#6923C2312yes—
53C2×C22⋊#6923—21no—
54C2×C4×#6923—21no—
55C2×C4×#6923—22no—
56C2×C4×#6923—22no—
57#14519844—22no—
58C4⋊#92420—22no—
59#14519840—24no—
60#14519840—24no—
61#14519840—24no—
62#14519840—24no—
63#14519834—22no—
64C22⋊#92420—22no—
65#14519832—22no—
66C4×#92420—22no—
67(C2×#6923)⋊C4—22no—
68(C2×#6923)⋊C4—22no—
69C2×Q8×D61C2312yes—
70C22×C4×D61C2311yescharacteristic
71#14519777C2×C411yescharacteristic, complemented
72#14519776—22no—
73#14519775C2×C412yescomplemented
74#14519772—22no—
75#14519772—22no—
76#14519772—22no—
77#14519771—22no—
78#14519770—22no—
79#14519770—22no—
80#14519770—22no—
81#14519769—22no—
82#14519769C2×C411yescharacteristic, complemented
83#14519768—22no—
84#14519768—22no—
85#14519768C2×C412yescomplemented
86#14519767—22no—
87#14519767—22no—
88#14519767—22no—
89#14519767—22no—
90#14519767—22no—
91#14519757C2×C412yes—
92#14519756C2×C412yes—
93#14519755—22no—
94#14519754—22no—
95#14519753C2×C412yes—
96Q8×Dic61C2×C412yescomplemented
97C4×Dic122—22no—
98C4×Dic122C2×C412yescomplemented
99#14519742—22no—
100#14519742C2×C412yescomplemented
101#14519736C2×C411yescharacteristic
102#14519735D412yes—
103#14519734C2×C411yescharacteristic
104#14519734C2×C411yescharacteristic
105#14519733D412yes—
106C4⋊C4×D61—22no—
107C4⋊C4×D61—22no—
108C4⋊C4×D61—22no—
109C4⋊C4×D61—22no—
110C4⋊C4×D61D412yes—
111C4⋊C4×D61C2312yes—
112C22⋊C4×D61—22no—
113C22⋊C4×D61D412yes—
114C42×D61—22no—
115C42×D61C2311yescharacteristic
116C42×D61C2311yescharacteristic
117C4⋊Q8×C61C2×C411yescharacteristic, complemented
118#126×C61—22no—
119#125×C61—22no—
120#125×C61C2×C411yescharacteristic, complemented
121C22⋊Q8×C61C2×C412yes—
122C4×Q8×C61C2×C412yescomplemented
123#117×C61C2×C411yescharacteristic
124C23×D61C2×D411yescharacteristic
125C22×#6923—22no—
126C22×#6923—22no—
127C22×#6923—24no—
128C22×#6923—82no—
129C2×Q8×C61C22×C412yes—
130C22×C4×C61C22×C411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132Q8×D61—22no—
133Q8×D61—22no—
134Q8×D61—22no—
135Q8×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61C22×C411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×D412yes—
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D411yescharacteristic
145C2×C4×D61C2411yescharacteristic
146C2×Dic122—22no—
147C2×Dic122C22×C412yes—
148C22⋊#6923—24no—
149C2×#92420—22no—
150C2×#92420—22no—
151C4⋊#6923—24no—
152C4⋊#6923—22no—
153C4⋊#6923—22no—
154C4⋊#6923#4114yes—
155C4⋊#6923#4114yes—
156C4×#6923—24no—
157C4×#6923—82no—
158C4×#6923—82no—
159D61⋊C8—22no—
160D61⋊C8—22no—
161D61⋊C8—22no—
162D61⋊C8—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61C22⋊C412yes—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—22no—
170C22⋊C4×C61C22⋊C412yes—
171C42×C61—22no—
172C42×C61C22×C411yescharacteristic
173C42×C61C22×C411yescharacteristic
174C22⋊Dic61—22no—
175C22⋊Dic61C22⋊C412yes—
176#11758198—22no—
177#11758198—22no—
178#11758198C22⋊C412yes—
179#11758198C22⋊C412yes—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61C22⋊C412yes—
185C4⋊Dic61C22×C412yes—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196C22⋊C412yes—
196#11758196C22⋊C412yes—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61C22×C411yescharacteristic
201C4×Dic61C22×C411yescharacteristic
202C4×Dic61C2×D412yes—
203C4×Dic61C2×D412yes—
204C23×C61C2×C22⋊C411yescharacteristic
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61C2×C22⋊C411yescharacteristic
210C22×D61C22×D411yescharacteristic
211C2×#6923—24no—
212C2×#6923—24no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—42no—
216C2×#6923—42no—
217C2×#6923—84no—
218Q8×C61—22no—
219Q8×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C23×C411yescharacteristic
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61C2×C22⋊C412yes—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C411yescharacteristic
233C2×Dic61C22×D411yescharacteristic
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61C2×#4112yes—
247C4×D61C2×#4112yes—
248Dic122—22no—
249Dic122—22no—
250Dic122—42no—
251Dic122—42no—
252#92420—42no—
253#92420—42no—
254#21398—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—21no—
257C22×C61C22×C22⋊C411yescharacteristic
258C2×D61—22no—
259C2×D61—21no—
260C2×D61—42no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61#52511yescharacteristic
264C2×D61#52511yescharacteristic
265C2×D61#57712yes—
266C2×D61#57812yes—
267#6923—44no—
268#6923—44no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61C4×#4112yes—
275Dic61—22no—
276Dic61—22no—
277Dic61—21no—
278Dic61—42no—
279Dic61—42no—
280Dic61#53212yes—
281C2×#541—611no—
282C2×#433—611no—
283#2274—611no—
284#1989—612no—
285#1981—612no—
286#1873—612no—
287#1872—612no—
288#1763—612no—
289C4×#103—612no—
290C2×C61—41no—
291C2×C61#229711yescharacteristic
292C2×C61#289812yes—
293D61—41no—
294D61#312012yes—
295D61#312311yescharacteristic
296D61#312711yescharacteristic
297#541—611no—
298#541—611no—
299#541—612no—
300#541—612no—
301#541—611no—
302#541—1222no—
303#541—1222no—
304#541#692311yescharacteristic, complemented
305C2×C4⋊Q8—611no—
306C2×#126—1222no—
307C2×#125—611no—
308C2×#125—1222no—
309C2×C22⋊Q8—612no—
310C2×C4×Q8—612no—
311C2×#117—611no—
312C2×#117—611no—
313C22×C4⋊C4—611no—
314#433—1222no—
315#433—1222no—
316C2×C4⋊C8—612no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103—612no—
320C2×#103—612no—
321C2×C22⋊C8—612no—
322C2×#97—611no—
323C2×C4×C8—611no—
324C4×C4⋊C4—612no—
325C4×C4⋊C4—612no—
326C4×C4⋊C4—1222no—
327C4×C22⋊C4—1222no—
328#340—612no—
329#340—612no—
330#340—612no—
331#340—612no—
332C61#2139811yescharacteristic, complemented, Sylow
333C22×Q8—612no—
334C23×C4—611no—
335C23×C4—1221no—
336C22×C8—612no—
337C22×C8—612no—
338C4⋊Q8—612no—
339C4⋊Q8—611no—
340C4⋊Q8—1222no—
341C4⋊Q8C2×#692311yescharacteristic, complemented
342#126—22no—
343#126—1222no—
344#126—1222no—
345#126—1222no—
346#126—1222no—
347#126—1222no—
348#126—1222no—
349#126—1222no—
350#125—22no—
351#125—612no—
352#125—611no—
353#125—1222no—
354#125—1222no—
355#125—1222no—
356#125—1222no—
357#125—1222no—
358#125—1222no—
359#125—1222no—
360#125—1222no—
361#125C2×#692311yescharacteristic, complemented
362C22⋊Q8—612no—
363C22⋊Q8—612no—
364C22⋊Q8—612no—
365C22⋊Q8—1222no—
366C22⋊Q8—1222no—
367C22⋊Q8C2×#692312yes—
368C4×Q8—612no—
369C4×Q8—612no—
370C4×Q8—612no—
371C4×Q8—1222no—
372C4×Q8—1222no—
373C4×Q8C2×#692312yescomplemented
374#117—611no—
375#117—612no—
376#117—612no—
377#117—611no—
378#117—611no—
379#117—1224no—
380#117C2×#692311yescharacteristic
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—1221no—
391C2×C22⋊C4—612no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1221no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—1222no—
397C2×C42—1221no—
398C2×C42—1222no—
399C2×C42—1222no—
400C4⋊C8—1222no—
401C4⋊C8—1222no—
402#103—1224no—
403#103—1224no—
404#103—1224no—
405#103—1224no—
406C22⋊C8—1222no—
407C22⋊C8—1222no—
408#97—1222no—
409C4×C8—1222no—
410#95—1222no—
411#95—1222no—
412C24—611no—
413C2×Q8—612no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×Q8C22×#692312yes—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1224no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—4882no—
435C22×C4C22×#692311yescharacteristic
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1222no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—612no—
447C4⋊C4—612no—
448C4⋊C4—612no—
449C4⋊C4—614no—
450C4⋊C4—614no—
451C4⋊C4—612no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1224no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4C22⋊#692312yes—
469C4⋊C4C22×#692312yes—
470C22⋊C4—22no—
471C22⋊C4—612no—
472C22⋊C4—612no—
473C22⋊C4—612no—
474C22⋊C4—1222no—
475C22⋊C4—1222no—
476C22⋊C4—1224no—
477C22⋊C4—1222no—
478C22⋊C4C22⋊#692312yes—
479C42—22no—
480C42—612no—
481C42—612no—
482C42—611no—
483C42—611no—
484C42—1222no—
485C42—1222no—
486C42—1224no—
487C42—1222no—
488C42—4882no—
489C42—4882no—
490C42C22×#692311yescharacteristic
491C42C22×#692311yescharacteristic
492C23—611no—
493C23—611no—
494C23—1221no—
495C23—1221no—
496C23—1222no—
497C23—1222no—
498C23C2×C22⋊#692311yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—1222no—
502Q8—1222no—
503Q8—2442no—
504Q8—2442no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—611no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—611no—
514C2×C4—1222no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1221no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—4884no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4C2×C22⋊#692312yes—
537C2×C4C2×C22⋊#692312yes—
538C2×C4C2×C22⋊#692311yescharacteristic
539C2×C4C23×#692311yescharacteristic
540C8—2442no—
541C8—2442no—
542C22—21no—
543C22—22no—
544C22—611no—
545C22—611no—
546C22—612no—
547C22—612no—
548C22—1221no—
549C22—1222no—
550C22—2441no—
551C22—2442no—
552C22—2441no—
553C22C22×C22⋊#692311yescharacteristic
554C4—21no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—42no—
559C4—612no—
560C4—1221no—
561C4—1222no—
562C4—1222no—
563C4—2444no—
564C4—2444no—
565C4—2442no—
566C4—2442no—
567C4#41×#692312yes—
568C2—41no—
569C2—611no—
570C2—612no—
571C2—611no—
572C2—2441no—
573C2#2146228011yescharacteristic
574C2#2146244112yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#541C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#541#6923
hypocenterC61#21398
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
3. #21461999C2semidirect
4. #21461985C2semidirect
7. #21461591C2semidirect
8. C4×#14519840C2semidirect
30. #16840375C4semidirect
31. #16840374C4semidirect
32. #16840373C4semidirect
33. #16840372C4semidirect
35. #16840370C4semidirect
44. C61×#541C4semidirect
71. #14519777C2×C4semidirect
73. #14519775C2×C4semidirect
82. #14519769C2×C4semidirect
85. #14519768C2×C4semidirect
96. Q8×Dic61C2×C4semidirect
98. C4×Dic122C2×C4semidirect
100. #14519742C2×C4semidirect
117. C4⋊Q8×C61C2×C4semidirect
120. #125×C61C2×C4semidirect
122. C4×Q8×C61C2×C4semidirect
304. #541#6923semidirect
332. C61#21398semidirect
341. C4⋊Q8C2×#6923semidirect
361. #125C2×#6923semidirect
373. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4332
#214622562
#214619992
#214619852
#214618052
#214618042
#214615912
C4×#145198402
#541×D612
#2139861