Finite Group Databasebeta

← #30612985

Subgroups — #30612985

Order 15616.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462764C211yesmaximal, characteristic
2#21462723C211yesmaximal, characteristic, complemented
3#21461466C211yesmaximal, characteristic, complemented
4#21461461C211yesmaximal, characteristic, complemented
5#21461457C211yesmaximal, characteristic, complemented
6#21461453C211yesmaximal, characteristic, complemented
7#21461413C211yesmaximal, characteristic, complemented
8#21461409C211yesmaximal, characteristic, complemented
9Q16×#92420C211yesmaximal, characteristic, complemented
10#21461404C211yesmaximal, characteristic, complemented
11#21461383C211yesmaximal, characteristic, complemented
12#21461172C211yesmaximal, characteristic
13#21461167C211yesmaximal, characteristic, complemented
14M16×#92420C211yesmaximal, characteristic, complemented
15#21456420C211yesmaximal, characteristic
16#16840814—21no—
17#16840813C2211yescharacteristic
18#16840813C2211yescharacteristic
19#16840812—21no—
20#16840812C2211yescharacteristic
21#16840811C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840795—21no—
26#16840794—21no—
27#16840790C2211yescharacteristic
28#16840787—21no—
29#16840786—21no—
30#16840786—21no—
31#16840783C2211yescharacteristic
32#16840778C2211yescharacteristic
33#16840769—21no—
34#16840765C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46#16840610C2211yescharacteristic
47#16840610C2211yescharacteristic
48#16840603C2211yescharacteristic
49#16840603C2211yescharacteristic
50C8×#92420C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840314C2211yescharacteristic
53#16840298C2211yescharacteristic
54#16840169C411yescharacteristic
55#16840155C411yescharacteristic
56#16840129C411yescharacteristic
57#16840123C411yescharacteristic
58#16840018C411yescharacteristic
59#16840010C411yescharacteristic
60#16840008C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62#16839854C2211yescharacteristic
63#16839849C2211yescharacteristic
64#16839847C2211yescharacteristic
65M16×Dic61C2211yescharacteristic
66C61×#470C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×D61⋊C8—21no—
70C22×#92420—21no—
71#14519844—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844—21no—
75C4⋊#92420—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420D411yescharacteristic
79C4⋊#92420D411yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86C22⋊#92420—21no—
87C22⋊#92420D411yescharacteristic
88C22⋊#92420D411yescharacteristic
89#14519832—21no—
90#14519832—21no—
91#14519832—21no—
92#14519832—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758202C2×C411yescharacteristic
102C2×#11758190C2×C411yescharacteristic
103C2×#11758201C2×C411yescharacteristic
104C2×#11758194C2×C411yescharacteristic
105#14519776C2×C411yescharacteristic
106#14519775C2×C411yescharacteristic
107#14519772—21no—
108#14519768C2×C411yescharacteristic
109#14519767—21no—
110#14519766C2×C411yescharacteristic
111#14519761—21no—
112#14519756C2×C411yescharacteristic
113#14519754—21no—
114#14519751C2×C411yescharacteristic
115Q8×Dic61C2311yescharacteristic
116Q8×Dic61C2311yescharacteristic
117#14519742—21no—
118#14519742C2311yescharacteristic
119#14519739C2311yescharacteristic
120#14519737—21no—
121#14519735—21no—
122#14519733—21no—
123C2×C4×Dic61C2311yescharacteristic
124#14519711C2×C411yescharacteristic
125#14519710C2×C411yescharacteristic
126#14519708C2×C411yescharacteristic
127#14519702C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519700C2×C411yescharacteristic
131#14519699C2×C411yescharacteristic
132#14519697C2×C411yescharacteristic
133#14519687C2×C411yescharacteristic
134#14519687C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×Q16×C61C2×C411yescharacteristic
138C2×M16×C61C2×C411yescharacteristic
139C22⋊Q8×C61C2×C411yescharacteristic
140C22⋊Q8×C61C2×C411yescharacteristic
141#106×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#103×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C2×C22⋊D61—21no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61#4111yescharacteristic
150C2×D244C22×C411yescharacteristic
151C2×C4×D61—21no—
152C2×Dic122—21no—
153C2×Dic122C22×C411yescharacteristic
154#11758217—81no—
155C2×#92420—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—42no—
162C2×#92420C2×D411yescharacteristic
163C2×#92420C2×D411yescharacteristic
164C2×#92420C2×D411yescharacteristic
165C2×#92420C2×D411yescharacteristic
166#11758213—81no—
167D61⋊C8—81no—
168Q16×C61—22no—
169M16×C61—22no—
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C22×C411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊Dic61—21no—
179#11758202—22no—
180#11758201—22no—
181#11758198—21no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22×C411yescharacteristic
189C4⋊Dic61C22×C411yescharacteristic
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61C22×C411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61#4111yescharacteristic
202C4×Dic61C2411yescharacteristic
203#11758194—22no—
204C2×#92418C22×C411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206#11758190—22no—
207C23×C61C4×D411yescharacteristic
208C22×D61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C23×C411yescharacteristic
221C22⋊D61—81no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C22×D411yescharacteristic
231C2×Dic61C2×#4111yescharacteristic
232D244—21no—
233D244—21no—
234C4×D61—41no—
235C4×D61—41no—
236Dic122—21no—
237Dic122—21no—
238Dic122—81no—
239#92420—22no—
240#92420—21no—
241#92420—21no—
242#92420—21no—
243#92420—21no—
244#92420—41no—
245#92420—41no—
246#92420—41no—
247#92420—41no—
248#92420—41no—
249C8×C61—21no—
250C8×C61#13112yes—
251#92418—21no—
252#92418#13112yes—
253#17969—611nomaximal, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61C2×C4×D411yescharacteristic
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—41no—
261C4×C61—21no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61C2×#13111yescharacteristic
267C4×C61C2×#13111yescharacteristic
268Dic61—21no—
269Dic61—21no—
270Dic61—21no—
271Dic61—41no—
272Dic61—41no—
273Dic61—41no—
274Dic61#57711yescharacteristic
275Dic61#57811yescharacteristic
276#3172—611no—
277#2993—611no—
278#2985—611no—
279#1593—611no—
280#1592—611no—
281#1591—611no—
282#1590—611no—
283#1579—611no—
284SD16⋊C8—611no—
285C8×Q16—611no—
286C8×SD16—611no—
287#1569—611no—
288#1494—611no—
289#1493—611no—
290C8×M16—611no—
291C2×C61—41no—
292C2×C61#293111yescharacteristic
293C2×C61#294711yescharacteristic
294C2×C61#295611yescharacteristic
295D61—81no—
296#533—611no—
297#529—611no—
298#501—611no—
299#497—611no—
300#491—611no—
301#489—611no—
302#470—611no—
303#470#692311yescharacteristic
304#468—611no—
305#468—611no—
306#446—611no—
307#446—611no—
308#446—611no—
309#446—1221no—
310#446—1221no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—611no—
314C4×Q16—611no—
315C4×Q16—611no—
316C4×SD16—611no—
317C4×SD16—611no—
318#436—611no—
319#436—1221no—
320#435—1221no—
321C8×D4—611no—
322#433—1221no—
323#433—1221no—
324#433—1221no—
325#432—611no—
326#431—611no—
327#424—1221no—
328#423—611no—
329C4×M16—611no—
330C2×C4×C8—611no—
331#335—611no—
332#335—611no—
333#334—611no—
334#334—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341D4⋊C8—611no—
342D4⋊C8—611no—
343#322—611no—
344#322—611no—
345C82—611no—
346C82—611no—
347C61#1796911yescharacteristic, complemented, Sylow
348C2×Q16—611no—
349C2×Q16C2×#692311yescharacteristic
350C2×SD16—611no—
351C2×SD16—611no—
352C2×M16—611no—
353C2×M16—1221no—
354C2×M16—1221no—
355C2×M16C2×#692311yescharacteristic
356C22×C8—1221no—
357C22×C8—1221no—
358C4⋊Q8—611no—
359C4⋊Q8—611no—
360#126—1221no—
361#125—611no—
362#125—1221no—
363#124—611no—
364#123—1221no—
365C22⋊Q8—611no—
366C22⋊Q8—1221no—
367C22⋊Q8C2×#692311yescharacteristic
368C22⋊Q8C2×#692311yescharacteristic
369#121—611no—
370C4×Q8—611no—
371C4×Q8—611no—
372C4×Q8—611no—
373C4×Q8—1221no—
374C4×D4—611no—
375C4×D4—1221no—
376#117—1221no—
377#117—1221no—
378C2×C42—611no—
379#107—611no—
380#107—611no—
381#106—611no—
382#106C2×#692311yescharacteristic
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396#103—611no—
397#103—611no—
398#103—611no—
399#103—611no—
400#103C2×#692311yescharacteristic
401#103C2×#692311yescharacteristic
402#102—611no—
403#102—611no—
404C22⋊C8—611no—
405C22⋊C8—611no—
406C22⋊C8—1221no—
407C22⋊C8—1221no—
408C22⋊C8—1221no—
409#97—611no—
410#97—611no—
411#97—1221no—
412#97—1221no—
413#97—1221no—
414#97—1221no—
415C4×C8—611no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—1221no—
422C4×C8—1221no—
423C4×C8—1221no—
424C4×C8—1221no—
425C2×Q8—611no—
426C2×Q8—1221no—
427C2×Q8C22×#692311yescharacteristic
428C2×Q8C22×#692311yescharacteristic
429C2×D4—611no—
430C2×D4—1221no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—1221no—
434C22×C4C22×#692311yescharacteristic
435Q16—22no—
436Q16—1222no—
437SD16—1222no—
438SD16—1222no—
439M16—22no—
440M16—1222no—
441M16—4881no—
442M16—4881no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—4881no—
456C2×C8—2442no—
457C2×C8C22×#692311yescharacteristic
458C2×C8C22×#692311yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—611no—
462C4⋊C4—611no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4C22×#692311yescharacteristic
473C4⋊C4C22×#692311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C22⋊C4—1221no—
480C22⋊C4—1221no—
481C42—611no—
482C42—611no—
483C42—611no—
484C42—611no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C23—1221no—
489C23D4×#692311yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—1221no—
495Q8—1221no—
496Q8—4881no—
497D4—1221no—
498D4—1221no—
499D4—4881no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4D4×#692311yescharacteristic
519C2×C4C23×#692311yescharacteristic
520C8—21no—
521C8—612no—
522C8—1221no—
523C8—1221no—
524C8—1221no—
525C8—1221no—
526C8—1222no—
527C8—1221no—
528C8—2441no—
529C8—2441no—
530C8—2441no—
531C8—2441no—
532C8—2441no—
533C8#1451986412yes—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2441no—
538C22—2441no—
539C22—2441no—
540C22C2×D4×#692311yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—611no—
547C4—611no—
548C4—1221no—
549C4—1221no—
550C4—1221no—
551C4—2441no—
552C4—2441no—
553C4—2441no—
554C4C2×#1451986411yescharacteristic
555C4D61⋊#13111yescharacteristic
556C2—41no—
557C2—4881no—
558C2#2146254211yescharacteristic
559C2#2146257711yescharacteristic
560C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#470C4
socleC22×C61C2×C4×D4
prime radical#14519751C2×C4
cosocleC4×Dic61C24
hypercenter#470#6923
hypocenterC61#17969
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #21462723C2semidirect
3. #21461466C2semidirect
4. #21461461C2semidirect
5. #21461457C2semidirect
6. #21461453C2semidirect
7. #21461413C2semidirect
8. #21461409C2semidirect
9. Q16×#92420C2semidirect
10. #21461404C2semidirect
11. #21461383C2semidirect
13. #21461167C2semidirect
14. M16×#92420C2semidirect
347. C61#17969semidirect

Maximal subgroups

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

subgroupindex
#214627642
#214627232
#214614662
#214614612
#214614572
#214614532
#214614132
#214614092
Q16×#924202
#214614042
#214613832
#214611722
#214611672
M16×#924202
#214564202
#1796961