Finite Group Databasebeta

← #30615958

Subgroups — #30615958

Order 15616.

1234 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
1C22×#14519839C211yesmaximal, characteristic
2C2×#16840728C211yesmaximal, characteristic, complemented
3#21462300C211yesmaximal, characteristic, complemented
4#21462292C211yesmaximal, characteristic
5#21461911C214yesmaximal, complemented
6#21461909C214yesmaximal, complemented
7#21461507C211yesmaximal, characteristic, complemented
8#21461504C211yesmaximal, characteristic, complemented
9#418×D61C211yesmaximal, characteristic
10C22×D61⋊C8C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13C22×C4⋊#6923C2211yescharacteristic
14C2×#14519839—24no—
15C2×#14519839C2212yes—
16C2×#14519839C2212yes—
17#16840728—24no—
18C2×#14519831C2212yescomplemented
19C2×C8×#6923C2212yescomplemented
20#16840714—21no—
21#16840714—21no—
22#16840714C2214yes—
23#16840713C2214yescomplemented
24#16840709—22no—
25#16840708—21no—
26#16840706—21no—
27#16840703—22no—
28#16840636C2214yescomplemented
29#16840635C2212yescomplemented
30#16840635C2212yescomplemented
31C22×D4×D61C2211yescharacteristic
32C2×M16×D61C2211yescharacteristic
33#117×D61C2211yescharacteristic
34#16839768C411yescharacteristic, complemented
35#16839767C411yescharacteristic, complemented
36#16839766C411yescharacteristic, complemented
37#16839765C414yescomplemented
38#102×D61C2214yes—
39C61×#418C411yescharacteristic, complemented
40C23×#6923—21no—
41C2×D61⋊C8—21no—
42C2×D61⋊C8—22no—
43C2×D61⋊C8D414yes—
44C2×D61⋊C8C2311yescharacteristic
45C2×D61⋊C8C2311yescharacteristic
46C22×#92420—21no—
47#14519856—22no—
48#14519856—22no—
49#14519856—22no—
50#14519856—22no—
51C2×C4⋊#6923—22no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—24no—
55C2×C4⋊#6923D411yescharacteristic
56C2×C4⋊#6923D411yescharacteristic
57C2×C4⋊#6923C2312yes—
58C2×C4⋊#6923C2312yes—
59C2×C22⋊#6923—21no—
60C2×C22⋊#6923—24no—
61C2×C22⋊#6923—24no—
62C2×C22⋊#6923D412yes—
63C2×C4×#6923—21no—
64C2×C4×#6923C2312yes—
65#14519839—24no—
66#14519839—24no—
67#14519831—24no—
68C8×#6923—24no—
69(C2×#6923)⋊C4—24no—
70C24×D61—21no—
71C22×D244C2×C411yescharacteristic
72C22×C22⋊D61—21no—
73C2×D4×D61—22no—
74C2×D4×D61—21no—
75C2×D4×D61—22no—
76C2×D4×D61—22no—
77C2×D4×D61—22no—
78C2×D4×D61—21no—
79C2×D4×D61C2×C412yes—
80C2×D4×D61C2312yes—
81C22×C4×D61C2311yescharacteristic
82C2×#11758194C2×C411yescharacteristic
83C2×#11758189C2×C412yes—
84M16×D61—22no—
85M16×D61—22no—
86C2×C8×D61C2312yes—
87#14519736C2×C411yescharacteristic
88#14519735C2×C412yes—
89#14519734—21no—
90#14519733—21no—
91C4⋊C4×D61C2312yes—
92C22⋊C4×D61—21no—
93C42×D61—21no—
94#14519699C2×C414yescomplemented
95#14519698C2×C414yescomplemented
96#14519697C2×C414yescomplemented
97C22×D4×C61C2×C411yescharacteristic
98C2×M16×C61C2×C411yescharacteristic
99#117×C61C2×C411yescharacteristic
100#102×C61C2×C414yescomplemented
101C24×C61—21no—
102C23×D61—21no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—82no—
108C23×D61—82no—
109C23×D61C2×D411yescharacteristic
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—42no—
113C22×#6923—82no—
114C22×#6923C2×D412yes—
115C22×#6923C2×D412yes—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—21no—
119C2×D4×C61C22×C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C22×Dic61#4111yescharacteristic
126D4×D61—24no—
127D4×D61—24no—
128D4×D61—22no—
129D4×D61—22no—
130D4×D61—22no—
131D4×D61—22no—
132D4×D61—42no—
133D4×D61—42no—
134D4×D61—42no—
135D4×D61—42no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—21no—
139C2×D244C22×C412yes—
140C2×C4×D61—22no—
141C2×C4×D61C22×C411yescharacteristic
142C2×C4×D61C2×D412yes—
143C2×C4×D61C2×D411yescharacteristic
144C2×C4×D61#4112yes—
145C2×C4×D61#4112yes—
146C2×C4×D61#4112yes—
147C2×C4×D61#4111yescharacteristic
148C2×C4×D61C2411yescharacteristic
149C22⋊#6923—24no—
150C22⋊#6923—42no—
151C22⋊#6923—84no—
152C22⋊#6923—84no—
153C2×#92420—24no—
154C2×#92420—21no—
155C2×#92420—21no—
156C4⋊#6923—22no—
157C4⋊#6923—22no—
158C4⋊#6923—22no—
159C4⋊#6923—22no—
160C4⋊#6923—22no—
161C4⋊#6923—22no—
162C4⋊#6923—42no—
163C4⋊#6923—84no—
164C4×#6923—22no—
165C4×#6923—22no—
166C4×#6923—42no—
167D61⋊C8—24no—
168D61⋊C8—21no—
169D61⋊C8—21no—
170D61⋊C8C2×D412yes—
171D61⋊C8C2×D412yes—
172M16×C61—22no—
173C2×C8×C61C22×C412yes—
174C4⋊C4×C61C22×C412yes—
175C22⋊C4×C61—21no—
176C42×C61—21no—
177C22⋊Dic61—21no—
178#11758198—22no—
179C4⋊Dic61C22×C412yes—
180#11758196—22no—
181C4×Dic61—21no—
182C4×Dic61—22no—
183#11758194—22no—
184C2×#92418C22×C412yes—
185#11758189—24no—
186C8×D61—22no—
187C8×D61—22no—
188C23×C61—22no—
189C23×C61—42no—
190C23×C61—82no—
191C23×C61C4×D411yescharacteristic
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61—44no—
201C22×D61—84no—
202C22×D61—88no—
203C22×D61—84no—
204C22×D61—84no—
205C22×D61—82no—
206C22×D61C4×D411yescharacteristic
207C22×D61#12112yes—
208C22×D61#12412yes—
209C22×D61C22×D411yescharacteristic
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—24no—
213C2×#6923—24no—
214C2×#6923—42no—
215C2×#6923—42no—
216C2×#6923—44no—
217C2×#6923—42no—
218C2×#6923—42no—
219C2×#6923—84no—
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—42no—
223C2×C4×C61—22no—
224C2×C4×C61#11712yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C23×C411yescharacteristic
228C22⋊D61—42no—
229C22⋊D61—42no—
230C22⋊D61—44no—
231C2×Dic61—22no—
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61#12312yes—
234C2×Dic61#12612yes—
235C2×Dic61C2×#4111yescharacteristic
236D244—22no—
237D244—22no—
238D244—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61#11712yes—
242C4×D61C4×D412yes—
243C4×D61#12112yes—
244C4×D61#12312yes—
245C4×D61#12412yes—
246C4×D61#12612yes—
247C4×D61C22×D411yescharacteristic
248C4×D61C2×#4111yescharacteristic
249C4×D61C2×#4111yescharacteristic
250C4×D61C2×#4111yescharacteristic
251#92420—22no—
252#92420—22no—
253C8×C61—22no—
254#92418—22no—
255#21091—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—44no—
258C22×C61—42no—
259C22×C61—84no—
260C22×C61#38812yes—
261C22×C61C2×C4×D411yescharacteristic
262C2×D61—22no—
263C2×D61—24no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—44no—
267C2×D61—42no—
268C2×D61—88no—
269C2×D61—84no—
270C2×D61—84no—
271C2×D61—82no—
272C2×D61—82no—
273C2×D61#38812yes—
274C2×D61C2×#12111yescharacteristic
275C2×D61C2×#12411yescharacteristic
276C2×D61#57612yes—
277C2×D61#57712yes—
278#6923—42no—
279#6923—42no—
280#6923—82no—
281#6923—82no—
282C4×C61—42no—
283C4×C61#38812yes—
284C4×C61C2×#11711yescharacteristic
285C4×C61C2×C4×D411yescharacteristic
286Dic61—42no—
287Dic61#38812yes—
288Dic61C2×#12311yescharacteristic
289Dic61C2×#12611yescharacteristic
290C2×#418—611no—
291C22×#102—611no—
292C2×#405—611no—
293#2317—611no—
294#2309—611no—
295#1936—614no—
296#1934—614no—
297#1736—611no—
298#1735—611no—
299C2×C61—22no—
300C2×C61—82no—
301C2×C61C2×#38811yescharacteristic
302C2×C61#294612yes—
303D61—22no—
304D61—82no—
305D61#305711yescharacteristic
306D61#305811yescharacteristic
307D61#313612yes—
308C23×D4—611no—
309C22×M16—611no—
310C23×C8—611no—
311C2×#117—611no—
312C2×#117—611no—
313C2×#117—611no—
314C22×C4⋊C4—611no—
315#418—611no—
316#418—611no—
317#418—611no—
318#418—614no—
319#418#692311yescharacteristic, complemented
320C2×#102—614no—
321C2×#102—612no—
322C2×#102—612no—
323C2×#102—1224no—
324#405—1224no—
325C2×#97—612no—
326C2×C4×C8—612no—
327#390—614no—
328#390—1221no—
329#390—1221no—
330#389—614no—
331#385—1222no—
332#384—1221no—
333#382—1221no—
334#379—1222no—
335#340—612no—
336#340—614no—
337#340—612no—
338C61#2109111yescharacteristic, complemented, Sylow
339C25—1221no—
340C22×D4—612no—
341C22×D4—612no—
342C22×D4—611no—
343C22×D4—1222no—
344C22×D4—1222no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—1221no—
348C22×D4—1221no—
349C22×D4—1221no—
350C22×D4C2×#692311yescharacteristic
351C23×C4—611no—
352C23×C4—1221no—
353C2×M16—611no—
354C2×M16—612no—
355C2×M16—1222no—
356C2×M16—1222no—
357C2×M16C2×#692311yescharacteristic
358C22×C8—614no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—612no—
362C22×C8—1221no—
363C22×C8—1222no—
364C22×C8—1221no—
365#117—611no—
366#117—612no—
367#117—1221no—
368#117—1221no—
369#117—1222no—
370#117—1222no—
371#117—1222no—
372#117—1222no—
373#117C2×#692311yescharacteristic
374C2×C4⋊C4—612no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1224no—
382C2×C4⋊C4—1221no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1224no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1221no—
388C2×C42—612no—
389C2×C42—1221no—
390C2×C42—1221no—
391#102—614no—
392#102—614no—
393#102—614no—
394#102—1224no—
395#102—1224no—
396#102C2×#692314yescomplemented
397#97—1224no—
398C4×C8—1224no—
399#95—1224no—
400C24—21no—
401C24—611no—
402C24—1222no—
403C24—1222no—
404C24—1221no—
405C24—4882no—
406C24—4882no—
407C24—2442no—
408C24—2442no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—612no—
413C2×D4—1222no—
414C2×D4—1224no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1224no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1221no—
424C2×D4—2442no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4—2442no—
428C2×D4—2442no—
429C2×D4—2442no—
430C2×D4C22×#692312yes—
431C22×C4—611no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—611no—
435C22×C4—612no—
436C22×C4—611no—
437C22×C4—612no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—4882no—
446C22×C4—2442no—
447C22×C4C22×#692311yescharacteristic
448M16—22no—
449M16—1224no—
450M16—1222no—
451C2×C8—612no—
452C2×C8—612no—
453C2×C8—612no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1224no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1224no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8C22×#692312yes—
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—4884no—
472C4⋊C4—2442no—
473C4⋊C4C22×#692312yes—
474C22⋊C4—21no—
475C22⋊C4—1222no—
476C22⋊C4—1224no—
477C22⋊C4—1221no—
478C22⋊C4—4884no—
479C22⋊C4—4884no—
480C22⋊C4—2442no—
481C42—21no—
482C42—1222no—
483C42—1222no—
484C42—1222no—
485C42—1221no—
486C42—2442no—
487C23—22no—
488C23—42no—
489C23—82no—
490C23—611no—
491C23—612no—
492C23—612no—
493C23—611no—
494C23—1222no—
495C23—1222no—
496C23—1222no—
497C23—4882no—
498C23—4884no—
499C23—4884no—
500C23—4888no—
501C23—4884no—
502C23—2444no—
503C23—2444no—
504C23—2442no—
505C23—2442no—
506C23—2444no—
507C23—2442no—
508C23D4×#692311yescharacteristic
509D4—22no—
510D4—22no—
511D4—42no—
512D4—1222no—
513D4—1222no—
514D4—2442no—
515D4—2442no—
516D4—2444no—
517D4—2442no—
518C2×C4—22no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—612no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1222no—
538C2×C4—4884no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2444no—
546C2×C4#1451985612yes—
547C2×C4D4×#692311yescharacteristic
548C2×C4D4×#692312yes—
549C2×C4C23×#692311yescharacteristic
550C8—22no—
551C8—1222no—
552C8—1222no—
553C8—1222no—
554C22—22no—
555C22—42no—
556C22—44no—
557C22—84no—
558C22—611no—
559C22—611no—
560C22—612no—
561C22—612no—
562C22—612no—
563C22—1222no—
564C22—1224no—
565C22—1222no—
566C22—1222no—
567C22—4884no—
568C22—4882no—
569C22—4884no—
570C22—4882no—
571C22—4888no—
572C22—2442no—
573C22—2444no—
574C22#1684071212yes—
575C22C2×D4×#692311yescharacteristic
576C4—42no—
577C4—611no—
578C4—611no—
579C4—612no—
580C4—4882no—
581C4—4882no—
582C4—2442no—
583C4—2442no—
584C4—2442no—
585C4#1684071212yes—
586C4C2×#1451985611yescharacteristic
587C4C2×D4×#692311yescharacteristic
588C2—22no—
589C2—82no—
590C2—611no—
591C2—612no—
592C2—611no—
593C2—1222no—
594C2—4882no—
595C2C2×#1684071211yescharacteristic
596C2#2146257612yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#418C4
socleC22×C61C2×C4×D4
prime radicalC22×D4×D61C22
cosocleC2×C4×D61C24
hypercenter#418#6923
hypocenterC61#21091
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#16840728C2semidirect
3. #21462300C2semidirect
5. #21461911C2semidirect
6. #21461909C2semidirect
7. #21461507C2semidirect
8. #21461504C2semidirect
18. C2×#14519831C22semidirect
19. C2×C8×#6923C22semidirect
23. #16840713C22semidirect
28. #16840636C22semidirect
29. #16840635C22semidirect
30. #16840635C22semidirect
34. #16839768C4semidirect
35. #16839767C4semidirect
36. #16839766C4semidirect
37. #16839765C4semidirect
39. C61×#418C4semidirect
94. #14519699C2×C4semidirect
95. #14519698C2×C4semidirect
96. #14519697C2×C4semidirect
100. #102×C61C2×C4semidirect
319. #418#6923semidirect
338. C61#21091semidirect
396. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C22×#145198392
C2×#168407282
#214623002
#214622922
#214619112
#214619092
#214615072
#214615042
#418×D612
#2109161