Finite Group Databasebeta

← #30613733

Subgroups — #30613733

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462770C211yesmaximal, characteristic
2#21462728C211yesmaximal, characteristic, complemented
3#21461463C211yesmaximal, characteristic, complemented
4#21461457C211yesmaximal, characteristic, complemented
5#21461444C211yesmaximal, characteristic, complemented
6#21461435C211yesmaximal, characteristic, complemented
7#21461432C211yesmaximal, characteristic, complemented
8#21461427C211yesmaximal, characteristic, complemented
9#21461413C211yesmaximal, characteristic, complemented
10#21461403C211yesmaximal, characteristic, complemented
11#21461372C211yesmaximal, characteristic, complemented
12#21461132C211yesmaximal, characteristic
13#21461127C211yesmaximal, characteristic, complemented
14#21461038C211yesmaximal, characteristic, complemented
15#21456420C211yesmaximal, characteristic
16#16840814—21no—
17#16840813C2211yescharacteristic
18#16840813C2211yescharacteristic
19#16840812C2211yescharacteristic
20#16840811—21no—
21#16840811C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840791—21no—
26#16840790C2211yescharacteristic
27#16840789—21no—
28#16840787—21no—
29#16840786—21no—
30#16840786—21no—
31#16840779C2211yescharacteristic
32#16840779C2211yescharacteristic
33#16840778C2211yescharacteristic
34#16840769—21no—
35C2×C4⋊#92420C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840626C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840624C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46#16840610C2211yescharacteristic
47#16840610C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
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×D61⋊C8—21no—
69C22×#92420—21no—
70#14519844—21no—
71#14519844—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844D411yescharacteristic
75#14519844D411yescharacteristic
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88C22⋊#92420D411yescharacteristic
89C22⋊#92420D411yescharacteristic
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
154C2×#92420—21no—
155C2×#92420—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—81no—
161C2×#92420C2×D411yescharacteristic
162C2×#92420C2×D411yescharacteristic
163C2×#92420C2×D411yescharacteristic
164C2×#92420C2×D411yescharacteristic
165#11758213—81no—
166D61⋊C8—81no—
167Q16×C61—41no—
168M16×C61—41no—
169C2×C8×C61C22×C411yescharacteristic
170C2×C8×C61C22×C411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊Dic61—21no—
178#11758202—22no—
179#11758201—22no—
180#11758198—21no—
181#11758198—21no—
182#11758198—21no—
183#11758198—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61C22×C411yescharacteristic
188C4⋊Dic61C22×C411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61C22×C411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C4×Dic61C2411yescharacteristic
202#11758194—22no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205#11758190—41no—
206C23×C61C4×D411yescharacteristic
207C22×D61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C23×C411yescharacteristic
220C22⋊D61—42no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—41no—
227C2×Dic61C4×D411yescharacteristic
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61C22×D411yescharacteristic
230C2×Dic61C2×#4111yescharacteristic
231D244—21no—
232D244—21no—
233C4×D61—41no—
234C4×D61—41no—
235Dic122—21no—
236Dic122—21no—
237Dic122—42no—
238#92420—21no—
239#92420—21no—
240#92420—21no—
241#92420—21no—
242#92420—41no—
243#92420—41no—
244#92420—41no—
245#92420—41no—
246#92420—41no—
247#92420—41no—
248C8×C61—21no—
249C8×C61—21no—
250#92418—21no—
251#92418#13112yes—
252#18081—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61C2×C4×D411yescharacteristic
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—41no—
260C4×C61—21no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61C2×#13111yescharacteristic
266C4×C61#56811yescharacteristic
267Dic61—21no—
268Dic61—21no—
269Dic61—21no—
270Dic61—41no—
271Dic61—41no—
272Dic61—41no—
273Dic61#57711yescharacteristic
274Dic61#57811yescharacteristic
275#3172—611no—
276#2994—611no—
277#2986—611no—
278#1592—611no—
279#1591—611no—
280#1587—611no—
281#1585—611no—
282#1584—611no—
283#1583—611no—
284#1579—611no—
285C8×SD16—611no—
286#1565—611no—
287#1482—611no—
288#1481—611no—
289#1453—611no—
290C2×C61—41no—
291C2×C61#293511yescharacteristic
292C2×C61#294711yescharacteristic
293C2×C61#295511yescharacteristic
294D61—81no—
295#533—611no—
296#529—611no—
297#501—611no—
298#497—611no—
299#491—611no—
300#489—611no—
301#470—611no—
302#470#692311yescharacteristic
303#468—611no—
304#468—611no—
305#446—611no—
306#446—611no—
307#446—611no—
308#446—1221no—
309C8×Q8—611no—
310C8×Q8—611no—
311C8×Q8—611no—
312C8×Q8—1221no—
313C4×Q16—611no—
314C4×Q16—611no—
315C4×SD16—611no—
316C4×SD16—611no—
317#436—611no—
318#435—1221no—
319C8×D4—611no—
320C8×D4—1221no—
321#433—1221no—
322#433—1221no—
323#433—1221no—
324#431—611no—
325#431—611no—
326#431—611no—
327#424—1221no—
328C2×C4⋊C8—611no—
329C4×M16—611no—
330#335—611no—
331#335—611no—
332#334—611no—
333#334—611no—
334Q8⋊C8—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340D4⋊C8—611no—
341D4⋊C8—611no—
342#322—611no—
343#322—611no—
344#322—611no—
345C82—611no—
346C61#1808111yescharacteristic, complemented, Sylow
347C2×Q16—611no—
348C2×Q16C2×#692311yescharacteristic
349C2×SD16—611no—
350C2×SD16—611no—
351C2×M16—611no—
352C2×M16—1221no—
353C2×M16C2×#692311yescharacteristic
354C22×C8—1221no—
355C22×C8—1221no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358#126—1221no—
359#125—611no—
360#125—1221no—
361#124—611no—
362#123—1221no—
363C22⋊Q8—611no—
364C22⋊Q8—1221no—
365C22⋊Q8C2×#692311yescharacteristic
366C22⋊Q8C2×#692311yescharacteristic
367#121—611no—
368C4×Q8—611no—
369C4×Q8—611no—
370C4×Q8—611no—
371C4×Q8—1221no—
372C4×D4—611no—
373C4×D4—1221no—
374#117—1221no—
375#117—1221no—
376C2×C42—611no—
377#107—611no—
378#107—611no—
379#106—611no—
380#106C2×#692311yescharacteristic
381C4⋊C8—611no—
382C4⋊C8—611no—
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—1221no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395#103—611no—
396#103—611no—
397#103—611no—
398#103—611no—
399#103C2×#692311yescharacteristic
400#103C2×#692311yescharacteristic
401#102—611no—
402#102—611no—
403C22⋊C8—611no—
404C22⋊C8—611no—
405C22⋊C8—1221no—
406C22⋊C8—1221no—
407C22⋊C8—1221no—
408C22⋊C8—1221no—
409#97—611no—
410#97—611no—
411#97—1221no—
412#97—1221no—
413#97—1221no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8—1221no—
423C4×C8—1221no—
424C2×Q8—611no—
425C2×Q8—1221no—
426C2×Q8C22×#692311yescharacteristic
427C2×Q8C22×#692311yescharacteristic
428C2×D4—611no—
429C2×D4—1221no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—1221no—
433C22×C4C22×#692311yescharacteristic
434Q16—41no—
435Q16—1222no—
436SD16—1222no—
437SD16—2441no—
438M16—41no—
439M16—1222no—
440M16—4881no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—4881no—
454C2×C8—4881no—
455C2×C8C22×#692311yescharacteristic
456C2×C8C22×#692311yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—611no—
460C4⋊C4—611no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4C22×#692311yescharacteristic
471C4⋊C4C22×#692311yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C42—611no—
480C42—611no—
481C42—611no—
482C42—611no—
483C42—1221no—
484C42—1221no—
485C42—1221no—
486C23—1221no—
487C23D4×#692311yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—1221no—
493Q8—1221no—
494Q8—2442no—
495D4—1221no—
496D4—1221no—
497D4—2442no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4D4×#692311yescharacteristic
517C2×C4C23×#692311yescharacteristic
518C8—21no—
519C8—21no—
520C8—612no—
521C8—1221no—
522C8—1221no—
523C8—1221no—
524C8—1221no—
525C8—1221no—
526C8—2441no—
527C8—2441no—
528C8—2441no—
529C8—2441no—
530C8—2441no—
531C8—2441no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—2441no—
536C22—2441no—
537C22—2441no—
538C22C2×D4×#692311yescharacteristic
539C4—21no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—611no—
545C4—611no—
546C4—1221no—
547C4—1221no—
548C4—1221no—
549C4—2441no—
550C4—2441no—
551C4—2441no—
552C4D61⋊#13111yescharacteristic
553C4#1684084011yescharacteristic
554C2—41no—
555C2—4881no—
556C2#2146256311yescharacteristic
557C2#2146257711yescharacteristic
558C2#2146259211yescharacteristic
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#18081
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. #21462728C2semidirect
3. #21461463C2semidirect
4. #21461457C2semidirect
5. #21461444C2semidirect
6. #21461435C2semidirect
7. #21461432C2semidirect
8. #21461427C2semidirect
9. #21461413C2semidirect
10. #21461403C2semidirect
11. #21461372C2semidirect
13. #21461127C2semidirect
14. #21461038C2semidirect
346. C61#18081semidirect

Maximal subgroups

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

subgroupindex
#214627702
#214627282
#214614632
#214614572
#214614442
#214614352
#214614322
#214614272
#214614132
#214614032
#214613722
#214611322
#214611272
#214610382
#214564202
#1808161