Finite Group Databasebeta

← #30613090

Subgroups — #30613090

Order 15616.

566 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#21462762C211yesmaximal, characteristic
2#21462724C211yesmaximal, characteristic, complemented
3#21461448C211yesmaximal, characteristic, complemented
4#21461444C211yesmaximal, characteristic, complemented
5#21461444C211yesmaximal, characteristic, complemented
6#21461436C211yesmaximal, characteristic, complemented
7#21461432C211yesmaximal, characteristic, complemented
8#21461432C211yesmaximal, characteristic, complemented
9#21461428C211yesmaximal, characteristic, complemented
10#21461424C211yesmaximal, characteristic, complemented
11#21461385C211yesmaximal, characteristic, complemented
12#21461171C211yesmaximal, characteristic
13#21461168C211yesmaximal, characteristic, complemented
14#21461027C211yesmaximal, characteristic, complemented
15#21456422C211yesmaximal, characteristic
16#16840814—21no—
17#16840814—21no—
18#16840813C2211yescharacteristic
19#16840813C2211yescharacteristic
20#16840813C2211yescharacteristic
21Q8×#92420C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840797C2211yescharacteristic
25#16840795—21no—
26#16840791—21no—
27D4×#92420C2211yescharacteristic
28#16840787—21no—
29#16840787—21no—
30#16840786—21no—
31#16840784C2211yescharacteristic
32#16840780C2211yescharacteristic
33#16840768—21no—
34#16840766C2211yescharacteristic
35C4×D61⋊C8C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840626C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840624C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#92420⋊C8C2211yescharacteristic
51#92420⋊C8C2211yescharacteristic
52#16840317C2211yescharacteristic
53#16840291C2211yescharacteristic
54#16840171C411yescharacteristic
55#16840130C411yescharacteristic
56#16840121C411yescharacteristic
57#16840017C411yescharacteristic
58#16840017C411yescharacteristic
59#16840005C411yescharacteristic
60#16840005C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62Q16×Dic61C2211yescharacteristic
63SD16×Dic61C2211yescharacteristic
64SD16×Dic61C2211yescharacteristic
65#16839714C2211yescharacteristic
66C61×#497C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×D61⋊C8—21no—
70C22×#92420—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#14519834D411yescharacteristic
86#14519834D411yescharacteristic
87C22⋊#92420—21no—
88C22⋊#92420—21no—
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×#11758189C2×C411yescharacteristic
102C2×#11758189C2×C411yescharacteristic
103#14519777C2×C411yescharacteristic
104#14519770—21no—
105#14519769C2×C411yescharacteristic
106#14519768—21no—
107#14519759—21no—
108#14519757—21no—
109#14519753C2×C411yescharacteristic
110#14519753C2×C411yescharacteristic
111#14519753C2×C411yescharacteristic
112#14519748C2×C411yescharacteristic
113Q8×Dic61C2311yescharacteristic
114Q8×Dic61C2311yescharacteristic
115Q8×Dic61C2311yescharacteristic
116#14519742—21no—
117D4×Dic61C2311yescharacteristic
118#14519737—21no—
119#14519735—21no—
120#14519733—21no—
121C42×D61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519709C2×C411yescharacteristic
126#14519701C2×C411yescharacteristic
127#14519701C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134#14519688C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×Q16×C61C2×C411yescharacteristic
138C2×Q16×C61C2×C411yescharacteristic
139C2×SD16×C61C2×C411yescharacteristic
140C2×SD16×C61C2×C411yescharacteristic
141C4⋊Q8×C61C2×C411yescharacteristic
142#124×C61C2×C411yescharacteristic
143#97×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150C2×C4×D61C22×C411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61#4111yescharacteristic
153C2×Dic122—21no—
154#11758217—81no—
155C2×#92420—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—81no—
162C2×#92420C2×D411yescharacteristic
163C2×#92420C2×D411yescharacteristic
164C2×#92420C2×D411yescharacteristic
165C2×#92420C2×D411yescharacteristic
166#11758213—81no—
167D61⋊C8—42no—
168Q16×C61—41no—
169Q16×C61—41no—
170SD16×C61—41no—
171SD16×C61—41no—
172C2×C8×C61C22×C411yescharacteristic
173C2×C8×C61C22×C411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C42×C61C22×C411yescharacteristic
179C22⋊Dic61—21no—
180C22⋊Dic61—21no—
181#11758198—21no—
182#11758198—21no—
183#11758198—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22×C411yescharacteristic
189C4⋊Dic61C22×C411yescharacteristic
190C4⋊Dic61C22×C411yescharacteristic
191C4⋊Dic61C22×C411yescharacteristic
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196#11758196—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61#4111yescharacteristic
202C4×Dic61C2411yescharacteristic
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205#11758189—41no—
206#11758189—41no—
207C23×C61—21no—
208C22×D61C4×D411yescharacteristic
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215D4×C61—21no—
216D4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C23×C411yescharacteristic
223C22⋊D61—81no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61C4×D411yescharacteristic
231C2×Dic61C22×D411yescharacteristic
232C2×Dic61C2×#4111yescharacteristic
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—41no—
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—21no—
251#92418—21no—
252#92418—21no—
253#17988—611nomaximal, Sylow
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61C2×C4×D411yescharacteristic
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61#56811yescharacteristic
267C4×C61#56811yescharacteristic
268Dic61—21no—
269Dic61—21no—
270Dic61—41no—
271Dic61—41no—
272Dic61—41no—
273Dic61—41no—
274Dic61#57711yescharacteristic
275Dic61#57811yescharacteristic
276#3172—611no—
277#2993—611no—
278#2985—611no—
279#1588—611no—
280#1587—611no—
281#1587—611no—
282#1585—611no—
283#1584—611no—
284#1584—611no—
285#1583—611no—
286#1582—611no—
287#1569—611no—
288#1494—611no—
289#1493—611no—
290#1451—611no—
291C2×C61—81no—
292C2×C61#293211yescharacteristic
293C2×C61#294711yescharacteristic
294C2×C61#295611yescharacteristic
295D61—41no—
296#533—611no—
297#529—611no—
298#501—611no—
299#497#692311yescharacteristic
300#491—611no—
301#489—611no—
302#470—611no—
303#470—611no—
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—
345#322—611no—
346#322—611no—
347C61#1798811yescharacteristic, complemented, Sylow
348C2×Q16C2×#692311yescharacteristic
349C2×Q16C2×#692311yescharacteristic
350C2×SD16C2×#692311yescharacteristic
351C2×SD16C2×#692311yescharacteristic
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—1221no—
355C2×M16—1221no—
356C22×C8—1221no—
357C22×C8—1221no—
358C4⋊Q8—611no—
359C4⋊Q8C2×#692311yescharacteristic
360#126—1221no—
361#125—611no—
362#125—1221no—
363#124C2×#692311yescharacteristic
364#123—1221no—
365C22⋊Q8—611no—
366C22⋊Q8—611no—
367C22⋊Q8—611no—
368C22⋊Q8—1221no—
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#106—611no—
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#103—611no—
401#103—611no—
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—1221no—
411#97—1221no—
412#97—1221no—
413#97—1221no—
414#97C2×#692311yescharacteristic
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—1221no—
426C2×Q8C22×#692311yescharacteristic
427C2×Q8C22×#692311yescharacteristic
428C2×Q8C22×#692311yescharacteristic
429C2×D4—1221no—
430C2×D4C22×#692311yescharacteristic
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—1221no—
435Q16—41no—
436Q16—41no—
437SD16—41no—
438SD16—41no—
439M16—4881no—
440M16—4881no—
441M16—2441no—
442M16—2441no—
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—611no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
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—1221no—
485C42—1221no—
486C42—1221no—
487C42C22×#692311yescharacteristic
488C23—21no—
489C23—611no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—4881no—
497D4—21no—
498D4—21no—
499D4—4881no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4D4×#692311yescharacteristic
518C2×C4D4×#692311yescharacteristic
519C2×C4C23×#692311yescharacteristic
520C8—21no—
521C8—21no—
522C8—1221no—
523C8—1221no—
524C8—1221no—
525C8—1221no—
526C8—1222no—
527C8—1221no—
528C8—1221no—
529C8—2441no—
530C8—2441no—
531C8—2441no—
532C8—2441no—
533C8—2441no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—1221no—
538C22—1221no—
539C22—1221no—
540C22C2×D4×#692311yescharacteristic
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—611no—
547C4—611no—
548C4—1221no—
549C4—1221no—
550C4—2441no—
551C4—2441no—
552C4—2441no—
553C4—2441no—
554C4#1684084011yescharacteristic
555C4#1684084011yescharacteristic
556C2—81no—
557C2—2441no—
558C2#2146254811yescharacteristic
559C2#2146257711yescharacteristic
560C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#497C4
socleC22×C61C2×C4×D4
prime radical#14519748C2×C4
cosocleC4×Dic61C24
hypercenter#497#6923
hypocenterC61#17988
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #21462724C2semidirect
3. #21461448C2semidirect
4. #21461444C2semidirect
5. #21461444C2semidirect
6. #21461436C2semidirect
7. #21461432C2semidirect
8. #21461432C2semidirect
9. #21461428C2semidirect
10. #21461424C2semidirect
11. #21461385C2semidirect
13. #21461168C2semidirect
14. #21461027C2semidirect
347. C61#17988semidirect

Maximal subgroups

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

subgroupindex
#214627622
#214627242
#214614482
#214614442
#214614362
#214614322
#214614282
#214614242
#214613852
#214611712
#214611682
#214610272
#214564222
#1798861