Finite Group Databasebeta

← #30611802

Subgroups — #30611802

Order 15616.

884 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462735C211yesmaximal, characteristic
2#21462516C211yesmaximal, characteristic
3#21461345C212yesmaximal, complemented
4#21461340C212yesmaximal, complemented
5#21461336C211yesmaximal, characteristic, complemented
6#21461329C211yesmaximal, characteristic, complemented
7#21461324C212yesmaximal, complemented
8#21461321C212yesmaximal, complemented
9#21461320C211yesmaximal, characteristic, complemented
10D61⋊#332C211yesmaximal, characteristic, complemented
11#21459699C211yesmaximal, characteristic
12#16840811—21no—
13#16840789—22no—
14D4×#92420—21no—
15#16840786—22no—
16#16840782C2212yes—
17#16840780C2211yescharacteristic
18#16840779C2212yes—
19#16840779C2212yes—
20#16840767C2211yescharacteristic
21C2×#14519844C2212yes—
22D61⋊(C4⋊C8)C2211yescharacteristic
23D61⋊(C4⋊C8)C2211yescharacteristic
24#16840736—21no—
25#16840727—21no—
26#16840622C2211yescharacteristic
27#16840622C2211yescharacteristic
28#16840622C2212yes—
29#16840621C2212yes—
30#16840621C2212yes—
31#16840621C2211yescharacteristic
32#16840621C2211yescharacteristic
33#16840621C2212yes—
34#16840621C2212yes—
35#16840621C2212yes—
36#16840491C411yescharacteristic
37#16840490—21no—
38#16840487C411yescharacteristic
39#16840482C411yescharacteristic
40#16840481—21no—
41#16840451C412yes—
42#16840448—22no—
43#16840446C412yes—
44#16840369—22no—
45#16840332—21no—
46#16840316C2211yescharacteristic
47#16840315C2212yes—
48#16840314C2212yes—
49C4⋊Q8×D61C2211yescharacteristic
50#16840238—21no—
51C4×#11758224C2211yescharacteristic
52C61×#556C411yescharacteristic
53C2×#11758213—21no—
54C2×D61⋊C8—21no—
55C2×D61⋊C8—21no—
56C22×#92420—22no—
57#14519844—22no—
58#14519844—21no—
59#14519844C2312yes—
60#14519844C2311yescharacteristic
61#14519844C2311yescharacteristic
62#14519844C2311yescharacteristic
63#14519844C2312yes—
64#14519844C2311yescharacteristic
65C4⋊#92420—21no—
66C4⋊#92420—21no—
67#14519834—21no—
68C22⋊#92420—22no—
69C22⋊#92420—22no—
70C22⋊#92420—22no—
71#14519832—22no—
72C4×#92420—22no—
73C4×#92420—22no—
74C4×#92420—21no—
75C2×#11758224D411yescharacteristic
76C2×Q8×D61—21no—
77#14519777—21no—
78#14519777C2×C411yescharacteristic
79#14519776—21no—
80#14519776C2312yes—
81#14519776C2311yescharacteristic
82#14519775—21no—
83#14519775—21no—
84#14519775C2311yescharacteristic
85#14519770—22no—
86#14519770—22no—
87#14519770—22no—
88#14519769—22no—
89#14519768—22no—
90#14519768C2×C412yes—
91#14519768C2×C412yes—
92#14519767—22no—
93#14519767—22no—
94#14519767—22no—
95#14519767—21no—
96#14519764—21no—
97#14519759—22no—
98#14519759—22no—
99#14519757C2×C411yescharacteristic
100#14519756C2×C412yes—
101#14519755—21no—
102#14519754—22no—
103#14519754—22no—
104#14519754—22no—
105#14519753C2×C411yescharacteristic
106Q8×Dic61—21no—
107Q8×Dic61C2×C412yes—
108C4×Dic122—21no—
109C4×Dic122—22no—
110C4×Dic122—21no—
111C4×Dic122—22no—
112C4×Dic122C2×C411yescharacteristic
113C4×Dic122C2×C411yescharacteristic
114#14519742—21no—
115#14519742—22no—
116#14519742C2×C412yes—
117#14519742D411yescharacteristic
118D4×Dic61D411yescharacteristic
119C4×C22⋊D61C2×C412yes—
120#14519737D412yes—
121#14519736C2×C412yes—
122#14519735—22no—
123#14519735D411yescharacteristic
124#14519734—21no—
125#14519734—22no—
126#14519733—22no—
127#14519733D412yes—
128C4⋊C4×D61—21no—
129C4⋊C4×D61C2×C412yes—
130C2×C4×Dic61C2312yes—
131C42×D61C2311yescharacteristic
132C4⋊Q8×C61C2×C411yescharacteristic
133#126×C61—22no—
134#124×C61—21no—
135C22⋊Q8×C61C2×C412yes—
136C4×Q8×C61—21no—
137C4×Q8×C61C2×C411yescharacteristic
138C4×D4×C61C2×C411yescharacteristic
139#117×C61C2×C412yes—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22×C412yes—
144C2×C22⋊D61C22⋊C412yes—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D412yes—
147Q8×D61—42no—
148Q8×D61—42no—
149#11758224—22no—
150#11758224—22no—
151C2×C4×D61—22no—
152C2×C4×D61C22×C411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×Dic122—21no—
156C2×Dic122—22no—
157C2×Dic122—21no—
158C2×Dic122C22⋊C411yescharacteristic
159C2×Dic122C22×C411yescharacteristic
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—22no—
163C2×#92420—22no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—82no—
167#11758213—42no—
168D61⋊C8—42no—
169D61⋊C8—81no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22⋊C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C412yes—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61C22⋊C412yes—
179C42×C61—22no—
180C42×C61—21no—
181C42×C61C22×C411yescharacteristic
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61C22⋊C412yes—
185#11758198—21no—
186#11758198—21no—
187#11758198C22⋊C412yes—
188C4⋊Dic61—21no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61C22⋊C411yescharacteristic
194C4⋊Dic61C22×C412yes—
195C4⋊Dic61C22×C411yescharacteristic
196#11758196—22no—
197#11758196—21no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—21no—
202#11758196—22no—
203#11758196—22no—
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—21no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2411yescharacteristic
219C23×C61C2×C22⋊C412yes—
220C22×D61C2×C22⋊C411yescharacteristic
221Q8×C61—41no—
222Q8×C61—42no—
223D4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—42no—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C22⋊D61—24no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—22no—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C22×D411yescharacteristic
244C2×Dic61C22×D411yescharacteristic
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—41no—
248C4×D61—44no—
249C4×D61#13512yes—
250C4×D61#13512yes—
251Dic122—22no—
252Dic122—42no—
253Dic122—42no—
254Dic122—42no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258#92420—41no—
259#92420—41no—
260#92420—42no—
261#92420—42no—
262#92420—41no—
263#92420—41no—
264#17725—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61C22×C22⋊C411yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61#41612yes—
272C4×C61—21no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61#41612yes—
279Dic61—21no—
280Dic61—22no—
281Dic61—22no—
282Dic61—41no—
283Dic61—42no—
284Dic61—41no—
285Dic61C2×#13511yescharacteristic
286Dic61C2×#13511yescharacteristic
287Dic61#57711yescharacteristic
288Dic61#57811yescharacteristic
289#3509—611no—
290#2988—611no—
291#2925—611no—
292#1554—612no—
293#1552—612no—
294#1550—611no—
295#1546—611no—
296#1544—612no—
297#1542—611no—
298#1542—612no—
299C2×#332—611no—
300C2×C61—42no—
301C2×C61#288911yescharacteristic
302C2×C61C2×#41611yescharacteristic
303C2×C61#289811yescharacteristic
304D61—22no—
305#557—611no—
306#557—611no—
307#557—1221no—
308#556—611no—
309#556—1221no—
310#556#692311yescharacteristic
311#552—612no—
312#552—612no—
313#552—1222no—
314#541—1222no—
315#536—1221no—
316#533—612no—
317#533—612no—
318#533—611no—
319C2×C4⋊Q8—611no—
320#519—1221no—
321C4×#41—611no—
322C8×Q8—1221no—
323C8×D4—1221no—
324C8×D4—1222no—
325#433—1222no—
326#432—612no—
327#431—612no—
328#431—611no—
329#431—612no—
330#423—611no—
331C2×C4⋊C8—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8—612no—
334#408—1221no—
335#405—1221no—
336#333—611no—
337#333—612no—
338#333—611no—
339#332—612no—
340#332—612no—
341#332—611no—
342#332—612no—
343#332—611no—
344#332—612no—
345#332—612no—
346C61#1772511yescharacteristic, complemented, Sylow
347C2×#41—611no—
348C22×Q8—1221no—
349C2×M16—1221no—
350C22×C8—1222no—
351C22×C8—1221no—
352C22×C8—1221no—
353C4⋊Q8—612no—
354C4⋊Q8—611no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8C2×#692311yescharacteristic
362#126—22no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#125—612no—
367#125—612no—
368#125—1222no—
369#125—1221no—
370#125—1222no—
371#125—1222no—
372#125—1222no—
373#125—1222no—
374#124—21no—
375#124—1221no—
376#123—1222no—
377#123—1222no—
378C22⋊Q8—611no—
379C22⋊Q8—612no—
380C22⋊Q8—611no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1221no—
385C22⋊Q8C2×#692312yes—
386C4×Q8—21no—
387C4×Q8—611no—
388C4×Q8—611no—
389C4×Q8—612no—
390C4×Q8—612no—
391C4×Q8—611no—
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×Q8—1222no—
397C4×Q8—1221no—
398C4×Q8—1221no—
399C4×Q8C2×#692311yescharacteristic
400C4×D4—611no—
401C4×D4—612no—
402C4×D4—612no—
403C4×D4C2×#692311yescharacteristic
404#117—611no—
405#117—612no—
406#117—612no—
407#117—1222no—
408#117—1221no—
409#117—1222no—
410#117—1222no—
411#117C2×#692312yes—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1221no—
414C2×C42—612no—
415C2×C42—611no—
416C4⋊C8—612no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—612no—
421C4⋊C8—611no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8—1222no—
425C4⋊C8—1221no—
426C22⋊C8—1222no—
427C22⋊C8—1222no—
428C22⋊C8—1222no—
429C22⋊C8—1221no—
430#97—1222no—
431C4×C8—1222no—
432C4×C8—1222no—
433C4×C8—1221no—
434#41—1222no—
435#41—1222no—
436C2×Q8—21no—
437C2×Q8—611no—
438C2×Q8—611no—
439C2×Q8—1221no—
440C2×Q8—1222no—
441C2×Q8—1221no—
442C2×Q8—2442no—
443C2×Q8—2442no—
444C2×Q8C22×#692311yescharacteristic
445C2×D4—612no—
446C2×D4C22⋊#692311yescharacteristic
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—1222no—
453C22×C4C22×#692312yes—
454M16—2442no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—4881no—
462C2×C8—4882no—
463C2×C8—2442no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—612no—
468C4⋊C4—612no—
469C4⋊C4—611no—
470C4⋊C4—612no—
471C4⋊C4—611no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4C22⋊#692311yescharacteristic
489C4⋊C4C22×#692312yes—
490C4⋊C4C22×#692311yescharacteristic
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—612no—
494C22⋊C4—612no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4C22⋊#692312yes—
500C42—22no—
501C42—21no—
502C42—612no—
503C42—612no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—612no—
508C42—1222no—
509C42—1222no—
510C42—1222no—
511C42—1221no—
512C42C22×#692311yescharacteristic
513C23—611no—
514C23C2×C22⋊#692312yes—
515Q8—41no—
516Q8—42no—
517Q8—1222no—
518Q8—2441no—
519Q8—2441no—
520Q8—2441no—
521Q8—2442no—
522Q8—2442no—
523Q8—2442no—
524D4—22no—
525D4—1224no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—2441no—
547C2×C4—2444no—
548C2×C4C2×C22⋊#692312yes—
549C2×C4C2×C22⋊#692311yescharacteristic
550C2×C4C2×C22⋊#692311yescharacteristic
551C2×C4C23×#692311yescharacteristic
552C8—2441no—
553C8—2442no—
554C8—2441no—
555C8—2441no—
556C8—2441no—
557C8—2442no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—612no—
562C22—1221no—
563C22—1221no—
564C22C22×C22⋊#692311yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—611no—
572C4—611no—
573C4—611no—
574C4—611no—
575C4—1221no—
576C4—1222no—
577C4—1222no—
578C4—2441no—
579C4—2442no—
580C4—2441no—
581C4#1684075212yes—
582C2—42no—
583C2—1222no—
584C2#2146243111yescharacteristic
585C2C2×#1684075211yescharacteristic
586C2#2146244111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#556C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758224D4
cosocleC4×Dic61C24
hypercenter#556#6923
hypocenterC61#17725
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
3. #21461345C2semidirect
4. #21461340C2semidirect
5. #21461336C2semidirect
6. #21461329C2semidirect
7. #21461324C2semidirect
8. #21461321C2semidirect
9. #21461320C2semidirect
10. D61⋊#332C2semidirect
346. C61#17725semidirect

Maximal subgroups

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

subgroupindex
#214627352
#214625162
#214613452
#214613402
#214613362
#214613292
#214613242
#214613212
#214613202
D61⋊#3322
#214596992
#1772561