Finite Group Databasebeta

← #30617052

Subgroups — #30617052

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×#6923C211yesmaximal, characteristic, complemented
2C2×#16840725C211yesmaximal, characteristic
3#21462306C212yesmaximal
4#21461965C211yesmaximal, characteristic, complemented
5#21461923C211yesmaximal, characteristic, complemented
6#21461910C212yesmaximal, complemented
7#21461834C212yesmaximal
8#21461696C211yesmaximal, characteristic
9#107×#6923C211yesmaximal, characteristic, complemented
10#21461592C212yesmaximal, complemented
11#467×D61C211yesmaximal, characteristic
12C2×D61⋊M16C2211yescharacteristic
13C2×Q8×#6923C2212yes—
14C2×#14519856C2211yescharacteristic
15Q16×#6923—24no—
16C2×#14519846C2211yescharacteristic
17C2×#14519846C2211yescharacteristic
18C2×#14519840C2212yes—
19C2×#14519840C2212yes—
20#16840725—24no—
21C2×#14519831C2211yescharacteristic
22C2×#14519831C2211yescharacteristic
23C2×C8×#6923C2211yescharacteristic
24C2×C8×#6923C2211yescharacteristic
25(C2×#6923)⋊Q8C2212yes—
26#16840713—22no—
27#16840713C2212yes—
28#16840712—22no—
29#16840712—22no—
30#16840711—22no—
31#16840704C2212yes—
32C4⋊C4×#6923C2212yes—
33#16840636C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
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×#11758213—21no—
48D61⋊M16—22no—
49D61⋊M16—22no—
50C2×#11758217—21no—
51C2×D61⋊C8C2311yescharacteristic
52C2×D61⋊C8C2311yescharacteristic
53Q8×#6923—24no—
54Q8×#6923—24no—
55#14519856—22no—
56#14519856—22no—
57C2×C4⋊#6923—22no—
58C2×C4⋊#6923—22no—
59C2×C4⋊#6923D411yescharacteristic
60C2×C4⋊#6923D411yescharacteristic
61C2×C4⋊#6923C2312yes—
62C2×C4⋊#6923C2312yes—
63C2×C22⋊#6923—22no—
64C2×C22⋊#6923D411yescharacteristic
65C2×C22⋊#6923D411yescharacteristic
66C2×C4×#6923—22no—
67C2×C4×#6923—22no—
68C2×C4×#6923C2311yescharacteristic
69C2×C4×#6923C2311yescharacteristic
70#14519846—22no—
71#14519846—22no—
72#14519840—24no—
73#14519840—24no—
74#14519831—22no—
75#14519831—22no—
76C8×#6923—22no—
77C8×#6923D414yes—
78(C2×#6923)⋊C4—22no—
79(C2×#6923)⋊C4—22no—
80(C2×#6923)⋊C4—22no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—22no—
83C2×Q8×D61C2312yes—
84C22×C4×D61C2311yescharacteristic
85C2×Dic244C2×C411yescharacteristic
86C2×#11758202C2×C412yes—
87Q16×D61—22no—
88Q16×D61—22no—
89C22×#92418C2×C411yescharacteristic
90C2×C8×D61—22no—
91C2×C8×D61—22no—
92C2×C8×D61C2×C411yescharacteristic
93C2×C8×D61C2×C411yescharacteristic
94C2×C8×D61C2311yescharacteristic
95C2×C8×D61C2311yescharacteristic
96#14519757C2×C412yes—
97#14519756C2×C412yes—
98#14519755—22no—
99#14519754—22no—
100#14519753C2×C412yes—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2312yes—
103C22⋊C4×D61—22no—
104#14519711C2×C411yescharacteristic
105#14519710C2×C412yes—
106#14519702C2×C412yes—
107#14519701C2×C412yes—
108#14519700C2×C412yes—
109C2×Q16×C61C2×C411yescharacteristic
110C22×C8×C61C2×C411yescharacteristic
111C22⋊Q8×C61C2×C412yes—
112#107×C61C2×C411yescharacteristic
113#103×C61C2×C412yes—
114C23×D61C2×D411yescharacteristic
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923C2×D411yescharacteristic
118C22×#6923C2×D411yescharacteristic
119C22×#6923C2×D411yescharacteristic
120C22×#6923C2×D411yescharacteristic
121C2×Q8×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×Dic61#4111yescharacteristic
124Q8×D61—22no—
125Q8×D61—22no—
126Q8×D61—22no—
127Q8×D61—22no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61C22×C411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61#4111yescharacteristic
137C2×C4×D61C2411yescharacteristic
138C2×Dic122—22no—
139C2×Dic122C22×C412yes—
140C22⋊#6923—22no—
141C22⋊#6923—22no—
142C22⋊#6923—82no—
143#11758217—24no—
144C2×#92420—21no—
145C2×#92420—21no—
146C4⋊#6923—22no—
147C4⋊#6923—22no—
148C4⋊#6923—22no—
149C4⋊#6923—22no—
150C4⋊#6923—22no—
151C4⋊#6923—22no—
152C4⋊#6923—44no—
153C4⋊#6923—82no—
154C4×#6923—21no—
155C4×#6923—21no—
156C4×#6923—44no—
157C4×#6923—82no—
158C4×#6923C2×D412yes—
159C4×#6923#4112yes—
160#11758213—24no—
161D61⋊C8—21no—
162D61⋊C8—21no—
163D61⋊C8C2×D412yes—
164D61⋊C8#4112yes—
165Q16×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61C22×C411yescharacteristic
168C2×C8×C61C22×C411yescharacteristic
169C4⋊C4×C61—22no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—22no—
172C22⋊Dic61—22no—
173#11758202—24no—
174#11758198—22no—
175#11758198—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C22×C412yes—
178#11758196—22no—
179#11758196—22no—
180#11758196—22no—
181C2×#92418—22no—
182C2×#92418C22×C411yescharacteristic
183C2×#92418C22×C411yescharacteristic
184Dic244—22no—
185C8×D61—21no—
186C8×D61—21no—
187C8×D61—22no—
188C8×D61—22no—
189C8×D61C2×D412yes—
190C8×D61#4112yes—
191C23×C61C4×D411yescharacteristic
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61C4×D411yescharacteristic
199C22×D61C22×D411yescharacteristic
200C2×#6923—21no—
201C2×#6923—21no—
202C2×#6923—21no—
203C2×#6923—21no—
204C2×#6923—21no—
205C2×#6923—21no—
206C2×#6923—21no—
207C2×#6923—21no—
208C2×#6923—21no—
209C2×#6923—21no—
210C2×#6923—42no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—42no—
216C2×#6923#12112yes—
217C2×#6923#12112yes—
218Q8×C61—22no—
219Q8×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C23×C411yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61C4×D411yescharacteristic
231C2×Dic61C2×#4111yescharacteristic
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61C22×D411yescharacteristic
243C4×D61C2×#4111yescharacteristic
244C4×D61C2×#4111yescharacteristic
245C4×D61C2×#4111yescharacteristic
246Dic122—22no—
247Dic122—22no—
248Dic122—42no—
249Dic122—42no—
250#92420—22no—
251#92420—22no—
252C8×C61—21no—
253C8×C61C4×D412yes—
254#92418—21no—
255#92418C4×D412yes—
256#21525—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61C2×C4×D411yescharacteristic
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—41no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61C2×#12111yescharacteristic
269C2×D61#53211yescharacteristic
270C2×D61#57611yescharacteristic
271C2×D61#57811yescharacteristic
272C2×D61#57811yescharacteristic
273C2×D61#57811yescharacteristic
274#6923—22no—
275#6923—22no—
276#6923—41no—
277#6923—41no—
278#6923—82no—
279#6923—82no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61C2×C4×D411yescharacteristic
284C4×C61C4×#4111yescharacteristic
285Dic61—21no—
286Dic61—42no—
287Dic61—42no—
288Dic61#52511yescharacteristic
289Dic61#52911yescharacteristic
290C2×#467—611no—
291C2×C4×Q16—611no—
292C2×#405—611no—
293#2323—612no—
294#1969—611no—
295#1942—611no—
296#1935—612no—
297#1894—612no—
298#1808—611no—
299C4×#107—611no—
300#1763—612no—
301C2×C61—41no—
302C2×C61C4×#12111yescharacteristic
303C2×C61#295011yescharacteristic
304C2×C61#295111yescharacteristic
305D61—41no—
306D61#305911yescharacteristic
307D61#306111yescharacteristic
308D61#317511yescharacteristic
309D61#317711yescharacteristic
310C22×Q16—611no—
311C22×M16—611no—
312C23×C8—611no—
313C2×C22⋊Q8—612no—
314C2×C4×Q8—612no—
315C2×#117—611no—
316#467—611no—
317#467—611no—
318#467—612no—
319#467—611no—
320#467—612no—
321#467#692311yescharacteristic, complemented
322C4×Q16—1224no—
323C2×#107—611no—
324C2×#107—611no—
325C2×#107—611no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—612no—
329#405—1224no—
330C2×#97—611no—
331C2×#97—611no—
332C2×C4×C8—611no—
333C2×C4×C8—611no—
334#391—612no—
335#389—612no—
336#389—1222no—
337#388—1222no—
338#388—1222no—
339#387—1222no—
340#380—612no—
341C4×C4⋊C4—612no—
342#340—612no—
343#340—612no—
344#340—612no—
345C61#2152511yescharacteristic, complemented, Sylow
346C22×Q8—612no—
347C23×C4—611no—
348C2×Q16—612no—
349C2×Q16—611no—
350C2×Q16—1222no—
351C2×Q16—1222no—
352C2×Q16C2×#692311yescharacteristic
353C2×M16—1222no—
354C2×M16—1222no—
355C2×M16—1221no—
356C2×M16—1221no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—1222no—
365C22×C8—1222no—
366C22×C8C2×#692311yescharacteristic
367C22⋊Q8—612no—
368C22⋊Q8—612no—
369C22⋊Q8—612no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8C2×#692312yes—
373C4×Q8—1224no—
374C4×Q8—1224no—
375#117—1222no—
376#117—1222no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—1222no—
392C2×C42—1222no—
393#107—611no—
394#107—612no—
395#107—1222no—
396#107—1222no—
397#107C2×#692311yescharacteristic
398#103—612no—
399#103—612no—
400#103—612no—
401#103—1224no—
402#103—1224no—
403#103C2×#692312yes—
404#97—1222no—
405#97—1222no—
406C4×C8—614no—
407C4×C8—1222no—
408#95—1222no—
409#95—1222no—
410#95—1222no—
411#95—1222no—
412#95—1222no—
413C24—611no—
414C2×Q8—612no—
415C2×Q8—1222no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8C22×#692312yes—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4C22×#692311yescharacteristic
439Q16—22no—
440Q16—1224no—
441Q16—1222no—
442M16—1224no—
443M16—1224no—
444C2×C8—22no—
445C2×C8—611no—
446C2×C8—612no—
447C2×C8—612no—
448C2×C8—612no—
449C2×C8—612no—
450C2×C8—611no—
451C2×C8—1222no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1222no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1222no—
460C2×C8C22×#692311yescharacteristic
461C2×C8C22×#692311yescharacteristic
462C4⋊C4—22no—
463C4⋊C4—612no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—4882no—
475C4⋊C4—2444no—
476C4⋊C4C22×#692312yes—
477C22⋊C4—22no—
478C22⋊C4—1222no—
479C22⋊C4—1222no—
480C22⋊C4—1222no—
481C22⋊C4—1222no—
482C22⋊C4—1222no—
483C22⋊C4—4882no—
484C42—612no—
485C42—612no—
486C42—1221no—
487C42—1221no—
488C42—4882no—
489C42—2444no—
490C23—611no—
491C23—611no—
492C23—1221no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23D4×#692311yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—1222no—
502Q8—1222no—
503Q8—2442no—
504Q8—2442no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—611no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—611no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4D4×#692311yescharacteristic
548C2×C4C23×#692311yescharacteristic
549C8—21no—
550C8—612no—
551C8—1222no—
552C8—1222no—
553C8—1221no—
554C8D4×#692312yes—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—611no—
559C22—611no—
560C22—611no—
561C22—611no—
562C22—611no—
563C22—611no—
564C22—1221no—
565C22—1221no—
566C22—1221no—
567C22—2441no—
568C22—2441no—
569C22—2441no—
570C22—2441no—
571C22C2×D4×#692311yescharacteristic
572C4—21no—
573C4—42no—
574C4—42no—
575C4—611no—
576C4—611no—
577C4—1222no—
578C4—1222no—
579C4—1221no—
580C4—4882no—
581C4—4882no—
582C4—2441no—
583C4—2441no—
584C4—2442no—
585C4—2442no—
586C4C2×D4×#692311yescharacteristic
587C4#41×#692311yescharacteristic
588C2—41no—
589C2—611no—
590C2—611no—
591C2—611no—
592C2—611no—
593C2—2441no—
594C2#121×#692311yescharacteristic
595C2#2146258011yescharacteristic
596C2#2146258111yescharacteristic
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#21525
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. #21461910C2semidirect
9. #107×#6923C2semidirect
10. #21461592C2semidirect
39. #16840004C4semidirect
40. #16840003C4semidirect
41. #16840002C4semidirect
42. #16840001C4semidirect
43. #16840000C4semidirect
46. C61×#467C4semidirect
321. #467#6923semidirect
345. C61#21525semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×#69232
C2×#168407252
#214623062
#214619652
#214619232
#214619102
#214618342
#214616962
#107×#69232
#214615922
#467×D612
#2152561