Finite Group Databasebeta

← #30622249

Subgroups — #30622249

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×#6923C211yesmaximal, characteristic, complemented
2C2×#16840772C211yesmaximal, characteristic
3#21462345C212yesmaximal
4#21461965C211yesmaximal, characteristic, complemented
5#21461923C211yesmaximal, characteristic, complemented
6#21461898C212yesmaximal, complemented
7#21461805C212yesmaximal
8#21461785C212yesmaximal, complemented
9#107×#6923C211yesmaximal, characteristic, complemented
10C8×C22⋊#6923C211yesmaximal, characteristic
11#467×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×#14519856C2212yes—
14Q16×#6923—24no—
15#16840772—24no—
16C2×#14519846C2211yescharacteristic
17C2×#14519846C2211yescharacteristic
18C2×#14519845C2212yes—
19C2×#14519840C2212yes—
20C2×#14519840C2212yes—
21D61⋊(C22⋊C8)C2211yescharacteristic
22C2×C8×#6923C2211yescharacteristic
23C2×C8×#6923C2211yescharacteristic
24(C2×#6923)⋊Q8C2212yes—
25#16840713C2212yes—
26#16840712—22no—
27#16840712—22no—
28#16840711—22no—
29C4⋊C4×#6923—22no—
30C4⋊C4×#6923C2212yes—
31C4×C22⋊#6923C2211yescharacteristic
32#16840635C2212yes—
33#16840635C2212yes—
34(C2×#6923)⋊C8C2211yescharacteristic
35(C2×#6923)⋊C8C2211yescharacteristic
36C2×Q16×D61C2211yescharacteristic
37C22×C8×D61C2211yescharacteristic
38C22⋊Q8×D61C2212yes—
39#16840004C411yescharacteristic, complemented
40#16840003C411yescharacteristic, complemented
41#16840002C412yescomplemented
42#16840001C412yescomplemented
43#16840000C411yescharacteristic, complemented
44#107×D61C2211yescharacteristic
45#103×D61C2212yes—
46C61×#467C411yescharacteristic, complemented
47C2×D61⋊C8C2311yescharacteristic
48C2×D61⋊C8C2311yescharacteristic
49Q8×#6923—24no—
50Q8×#6923—42no—
51#14519856—24no—
52#14519856—42no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923—22no—
55C2×C4⋊#6923C2312yes—
56C2×C4⋊#6923C2312yes—
57C2×C22⋊#6923—22no—
58C2×C22⋊#6923D411yescharacteristic
59C2×C22⋊#6923D411yescharacteristic
60C2×C4×#6923—22no—
61C2×C4×#6923—22no—
62C2×C4×#6923—22no—
63C2×C4×#6923C2311yescharacteristic
64C2×C4×#6923C2311yescharacteristic
65#14519846—22no—
66#14519846D414yes—
67#14519845—24no—
68#14519840—24no—
69#14519840—42no—
70#14519834—21no—
71C22⋊#92420—21no—
72C8×#6923—22no—
73C8×#6923—22no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77(C2×#6923)⋊C4—22no—
78(C2×#6923)⋊C4D411yescharacteristic
79(C2×#6923)⋊C4D411yescharacteristic
80C2×Q8×D61C2312yes—
81C22×C4×D61C2311yescharacteristic
82C2×Dic244C2×C411yescharacteristic
83C2×#11758202C2×C412yes—
84Q16×D61—22no—
85Q16×D61—22no—
86C22×#92418C2×C411yescharacteristic
87C2×C8×D61—22no—
88C2×C8×D61—22no—
89C2×C8×D61C2×C411yescharacteristic
90C2×C8×D61C2×C411yescharacteristic
91C2×C8×D61C2311yescharacteristic
92C2×C8×D61C2311yescharacteristic
93#14519757C2×C412yes—
94#14519756C2×C412yes—
95#14519755—22no—
96#14519754—22no—
97#14519753C2×C412yes—
98C4⋊C4×D61—22no—
99C4⋊C4×D61C2312yes—
100C22⋊C4×D61—22no—
101#14519711C2×C411yescharacteristic
102#14519710C2×C412yes—
103#14519702C2×C412yes—
104#14519701C2×C412yes—
105#14519700C2×C412yes—
106C2×Q16×C61C2×C411yescharacteristic
107C22×C8×C61C2×C411yescharacteristic
108C22⋊Q8×C61C2×C412yes—
109#107×C61C2×C411yescharacteristic
110#103×C61C2×C412yes—
111C23×D61C2×D411yescharacteristic
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923C2×D411yescharacteristic
115C22×#6923C2×D411yescharacteristic
116C22×#6923C2×D411yescharacteristic
117C22×#6923C2×D411yescharacteristic
118C2×Q8×C61C22×C412yes—
119C22×C4×C61C22×C411yescharacteristic
120C22×Dic61#4111yescharacteristic
121Q8×D61—22no—
122Q8×D61—22no—
123Q8×D61—22no—
124Q8×D61—22no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61C22×C411yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×C4×D61#4111yescharacteristic
134C2×C4×D61C2411yescharacteristic
135C2×Dic122—22no—
136C2×Dic122C22×C412yes—
137C22⋊#6923—22no—
138C22⋊#6923—22no—
139C22⋊#6923—44no—
140C2×#92420—21no—
141C2×#92420—21no—
142C4⋊#6923—22no—
143C4⋊#6923—22no—
144C4⋊#6923—22no—
145C4⋊#6923—44no—
146C4⋊#6923—44no—
147C4⋊#6923C2×D414yes—
148C4×#6923—21no—
149C4×#6923—21no—
150C4×#6923—21no—
151C4×#6923—21no—
152C4×#6923—44no—
153C4×#6923—44no—
154C4×#6923—44no—
155D61⋊C8—21no—
156D61⋊C8—21no—
157D61⋊C8—21no—
158D61⋊C8—21no—
159Q16×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61C22×C411yescharacteristic
162C2×C8×C61C22×C411yescharacteristic
163C4⋊C4×C61—22no—
164C4⋊C4×C61C22×C412yes—
165C22⋊C4×C61—22no—
166C22⋊Dic61—22no—
167#11758202—24no—
168#11758198—22no—
169#11758198—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61C22×C412yes—
172#11758196—22no—
173#11758196—22no—
174#11758196—22no—
175C2×#92418—22no—
176C2×#92418C22×C411yescharacteristic
177C2×#92418C22×C411yescharacteristic
178Dic244—22no—
179C8×D61—21no—
180C8×D61—21no—
181C8×D61—22no—
182C8×D61—22no—
183C8×D61C2×D412yes—
184C8×D61#4112yes—
185C23×C61C4×D411yescharacteristic
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61C4×D411yescharacteristic
193C22×D61C22×D411yescharacteristic
194C2×#6923—21no—
195C2×#6923—21no—
196C2×#6923—21no—
197C2×#6923—21no—
198C2×#6923—21no—
199C2×#6923—24no—
200C2×#6923—21no—
201C2×#6923—21no—
202C2×#6923—21no—
203C2×#6923—21no—
204C2×#6923—21no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923#13512yes—
211C2×#6923#13512yes—
212Q8×C61—22no—
213Q8×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C23×C411yescharacteristic
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61C4×D411yescharacteristic
225C2×Dic61C2×#4111yescharacteristic
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61C22×D411yescharacteristic
237C4×D61C2×#4111yescharacteristic
238C4×D61H(5,𝔽2)11yescharacteristic
239C4×D612−1+411yescharacteristic
240Dic122—22no—
241Dic122—22no—
242Dic122—42no—
243Dic122—42no—
244#92420—41no—
245#92420—41no—
246C8×C61—21no—
247C8×C61C4×D412yes—
248#92418—21no—
249#92418C4×D412yes—
250#23026—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61C2×C4×D411yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—41no—
262C2×D61D4211yescharacteristic
263C2×D61#54711yescharacteristic
264C2×D61C2×#13511yescharacteristic
265C2×D61C2×#13511yescharacteristic
266C2×D61#57811yescharacteristic
267C2×D61#57811yescharacteristic
268#6923—22no—
269#6923—22no—
270#6923—41no—
271#6923—41no—
272#6923—44no—
273#6923—82no—
274C4×C61—21no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61C2×C4×D411yescharacteristic
278C4×C61#52011yescharacteristic
279Dic61—21no—
280Dic61—42no—
281Dic61—42no—
282Dic61#54811yescharacteristic
283Dic61#55311yescharacteristic
284C2×#467—611no—
285C2×C4×Q16—611no—
286C2×#427—611no—
287#2362—612no—
288#1969—611no—
289#1942—611no—
290#1927—612no—
291#1873—612no—
292#1865—612no—
293C4×#107—611no—
294C8×C22⋊C4—611no—
295C2×C61—41no—
296C2×C61#236111yescharacteristic
297C2×C61C4×#13511yescharacteristic
298C2×C61#295011yescharacteristic
299D61—41no—
300D61#328311yescharacteristic
301D61#328511yescharacteristic
302D61#333611yescharacteristic
303D61#334211yescharacteristic
304C22×Q16—611no—
305C23×C8—611no—
306C2×C22⋊Q8—612no—
307C2×C4×Q8—612no—
308C2×#117—612no—
309#467—611no—
310#467—611no—
311#467—612no—
312#467—611no—
313#467—612no—
314#467#692311yescharacteristic, complemented
315C4×Q16—1224no—
316#427—1224no—
317C2×#107—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×#106—612no—
321C2×#103—612no—
322C2×#103—612no—
323C2×#103—612no—
324C2×C22⋊C8—611no—
325C2×C4×C8—611no—
326C2×C4×C8—611no—
327#391—612no—
328#389—612no—
329#388—1222no—
330#388—1222no—
331#387—1222no—
332C4×C4⋊C4—612no—
333C4×C4⋊C4—1222no—
334C4×C22⋊C4—611no—
335#340—612no—
336#340—612no—
337#336—611no—
338#336—611no—
339C61#2302611yescharacteristic, complemented, Sylow
340C22×Q8—612no—
341C23×C4—611no—
342C2×Q16—612no—
343C2×Q16—611no—
344C2×Q16—1222no—
345C2×Q16—1222no—
346C2×Q16C2×#692311yescharacteristic
347C22×C8—611no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8—611no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8—611no—
354C22×C8—1222no—
355C22×C8—1222no—
356C22×C8C2×#692311yescharacteristic
357C22⋊Q8—612no—
358C22⋊Q8—612no—
359C22⋊Q8—612no—
360C22⋊Q8—1222no—
361C22⋊Q8—1222no—
362C22⋊Q8C2×#692312yes—
363C4×Q8—1224no—
364C4×Q8—2442no—
365#117—1224no—
366#117—2442no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C22⋊C4—611no—
374C2×C22⋊C4—611no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—1222no—
380C2×C42—1222no—
381C2×C42—1222no—
382#107—614no—
383#107—611no—
384#107—612no—
385#107—1222no—
386#107C2×#692311yescharacteristic
387#106—1224no—
388#103—612no—
389#103—612no—
390#103—612no—
391#103—1224no—
392#103—2442no—
393#103C2×#692312yes—
394C22⋊C8—1221no—
395C22⋊C8—1221no—
396C4×C8—1222no—
397C4×C8—1222no—
398#95—611no—
399#95—611no—
400#95—1222no—
401#95—1222no—
402#95—1222no—
403#95—1222no—
404C24—611no—
405C2×Q8—612no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8C22×#692312yes—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4C22×#692311yescharacteristic
430Q16—22no—
431Q16—1224no—
432Q16—1222no—
433C2×C8—22no—
434C2×C8—611no—
435C2×C8—612no—
436C2×C8—612no—
437C2×C8—611no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8C22×#692311yescharacteristic
450C2×C8C22×#692311yescharacteristic
451C4⋊C4—22no—
452C4⋊C4—614no—
453C4⋊C4—612no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—2444no—
462C4⋊C4—2444no—
463C4⋊C4C22×#692312yes—
464C22⋊C4—22no—
465C22⋊C4—1222no—
466C22⋊C4—1222no—
467C22⋊C4—1222no—
468C22⋊C4—1222no—
469C22⋊C4—1222no—
470C22⋊C4—2444no—
471C42—1221no—
472C42—1221no—
473C42—1221no—
474C42—1221no—
475C42—2444no—
476C42—2444no—
477C42—2444no—
478C23—611no—
479C23—611no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—1221no—
485C23—1221no—
486C23D4×#692311yescharacteristic
487Q8—22no—
488Q8—22no—
489Q8—1222no—
490Q8—1222no—
491Q8—2442no—
492Q8—2442no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—611no—
498C2×C4—611no—
499C2×C4—611no—
500C2×C4—611no—
501C2×C4—611no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—611no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1224no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4D4×#692311yescharacteristic
536C2×C4C23×#692311yescharacteristic
537C8—21no—
538C8—612no—
539C8—1221no—
540C8—2441no—
541C8—2441no—
542C8D4×#692312yes—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—611no—
547C22—611no—
548C22—611no—
549C22—611no—
550C22—611no—
551C22—611no—
552C22—1221no—
553C22—1221no—
554C22—1221no—
555C22—2441no—
556C22—2441no—
557C22—2441no—
558C22—2441no—
559C22C2×D4×#692311yescharacteristic
560C4—21no—
561C4—42no—
562C4—42no—
563C4—611no—
564C4—611no—
565C4—1222no—
566C4—1222no—
567C4—1221no—
568C4—4882no—
569C4—2441no—
570C4—2441no—
571C4—2444no—
572C4—2442no—
573C4—2442no—
574C4C2×D4×#692311yescharacteristic
575C4#1684083111yescharacteristic
576C2—41no—
577C2—611no—
578C2—611no—
579C2—611no—
580C2—611no—
581C2—2441no—
582C2#2146234411yescharacteristic
583C2#135×#692311yescharacteristic
584C2#2146258011yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#467C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#467#6923
hypocenterC61#23026
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×Q16×#6923C2semidirect
4. #21461965C2semidirect
5. #21461923C2semidirect
6. #21461898C2semidirect
8. #21461785C2semidirect
9. #107×#6923C2semidirect
39. #16840004C4semidirect
40. #16840003C4semidirect
41. #16840002C4semidirect
42. #16840001C4semidirect
43. #16840000C4semidirect
46. C61×#467C4semidirect
314. #467#6923semidirect
339. C61#23026semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×#69232
C2×#168407722
#214623452
#214619652
#214619232
#214618982
#214618052
#214617852
#107×#69232
C8×C22⋊#69232
#467×D612
#2302661