Finite Group Databasebeta

← #30612716

Subgroups — #30612716

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462737C211yesmaximal, characteristic
2#21462727C211yesmaximal, characteristic, complemented
3#21461442C211yesmaximal, characteristic, complemented
4#21461435C211yesmaximal, characteristic, complemented
5#21461430C211yesmaximal, characteristic, complemented
6#21461423C211yesmaximal, characteristic, complemented
7#21461413C211yesmaximal, characteristic, complemented
8#21461411C211yesmaximal, characteristic, complemented
9#21461406C211yesmaximal, characteristic, complemented
10#21461404C211yesmaximal, characteristic, complemented
11#21461393C211yesmaximal, characteristic, complemented
12#21461153C211yesmaximal, characteristic
13#21461129C211yesmaximal, characteristic, complemented
14#21461041C211yesmaximal, characteristic, complemented
15#21455893C211yesmaximal, characteristic
16#16840813—21no—
17#16840813—21no—
18#16840812C2211yescharacteristic
19#16840812C2211yescharacteristic
20#16840811C2211yescharacteristic
21#16840811C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840790—21no—
25#16840790—21no—
26#16840790C2211yescharacteristic
27#16840790C2211yescharacteristic
28#16840787—21no—
29#16840787—21no—
30#16840784C2211yescharacteristic
31#16840781C2211yescharacteristic
32#16840780—21no—
33#16840780C2211yescharacteristic
34D61⋊(C4⋊C8)—21no—
35D61⋊(C4⋊C8)C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42#16840612C2211yescharacteristic
43#16840612C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46#16840610C2211yescharacteristic
47#16840610C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840317C2211yescharacteristic
53C4×#11758226C2211yescharacteristic
54#16840106C411yescharacteristic
55#16840106C411yescharacteristic
56#16840039C411yescharacteristic
57#16839972C411yescharacteristic
58#16839968C411yescharacteristic
59#16839927C411yescharacteristic
60#16839927C411yescharacteristic
61#16839854C2211yescharacteristic
62#16839854C2211yescharacteristic
63#16839849C2211yescharacteristic
64#16839849C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#475C411yescharacteristic
67C2×D61⋊C8—21no—
68C2×D61⋊C8—21no—
69C2×D61⋊C8—21no—
70#14519844—21no—
71#14519844—21no—
72#14519844—21no—
73#14519844D411yescharacteristic
74#14519844D411yescharacteristic
75C4⋊#92420—21no—
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—
86#14519834—21no—
87#14519834—21no—
88#14519834D411yescharacteristic
89#14519834D411yescharacteristic
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×#11758226C2×C411yescharacteristic
102C2×#11758202C2×C411yescharacteristic
103C2×#11758202C2×C411yescharacteristic
104C2×#11758190C2×C411yescharacteristic
105C2×#11758190C2×C411yescharacteristic
106#14519777C2×C411yescharacteristic
107#14519768—21no—
108#14519757C2×C411yescharacteristic
109#14519757C2×C411yescharacteristic
110#14519753—21no—
111Q8×Dic61—21no—
112Q8×Dic61C2311yescharacteristic
113#14519742C2311yescharacteristic
114#14519742C2311yescharacteristic
115C4×D244—21no—
116C4×D244C2×C411yescharacteristic
117#14519739—21no—
118#14519739C2311yescharacteristic
119#14519735—21no—
120#14519735—21no—
121#14519735C2×C411yescharacteristic
122C42×D61—21no—
123C42×D61C2311yescharacteristic
124#14519710C2×C411yescharacteristic
125#14519710C2×C411yescharacteristic
126C4⋊#92418C2×C411yescharacteristic
127#14519702C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519700C2×C411yescharacteristic
130#14519700C2×C411yescharacteristic
131#14519697C2×C411yescharacteristic
132#14519697C2×C411yescharacteristic
133#14519689C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C4⋊Q8×C61C2×C411yescharacteristic
138C4×Q8×C61C2×C411yescharacteristic
139#106×C61C2×C411yescharacteristic
140#106×C61C2×C411yescharacteristic
141C4⋊C8×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#103×C61C2×C411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61C22×C411yescharacteristic
146#11758226—22no—
147#11758226—22no—
148C2×D244—21no—
149C2×D244C22×C411yescharacteristic
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×Dic122C22×C411yescharacteristic
157C2×Dic122C22×C411yescharacteristic
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420C2×D411yescharacteristic
165C2×#92420C2×D411yescharacteristic
166C2×#92420C2×D411yescharacteristic
167C2×#92420C2×D411yescharacteristic
168D61⋊C8—81no—
169D61⋊C8—81no—
170D61⋊C8—81no—
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C22×C411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C411yescharacteristic
176C4⋊C4×C61C22×C411yescharacteristic
177C4⋊C4×C61C22×C411yescharacteristic
178C42×C61—21no—
179C42×C61C22×C411yescharacteristic
180#11758202—22no—
181#11758202—22no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185#11758198—21no—
186#11758198—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C22×C411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C4×Dic61C2411yescharacteristic
202C2×#92418C22×C411yescharacteristic
203C2×#92418C22×C411yescharacteristic
204#11758190—22no—
205#11758190—22no—
206C22×D61—21no—
207C22×D61C4×D411yescharacteristic
208Q8×C61—21no—
209Q8×C61—81no—
210Q8×C61#13112yes—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61C4×D411yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C23×C411yescharacteristic
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61C4×D411yescharacteristic
222C2×Dic61C22×D411yescharacteristic
223C2×Dic61C2×#4111yescharacteristic
224D244—24no—
225D244—21no—
226D244#13112yes—
227C4×D61—24no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—41no—
235C4×D61—81no—
236Dic122—21no—
237Dic122—21no—
238Dic122—21no—
239Dic122—21no—
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—
249#92420—41no—
250C8×C61—21no—
251C8×C61—21no—
252#92418—21no—
253#92418—21no—
254#17921—611nomaximal, Sylow
255C22×C61C2×C4×D411yescharacteristic
256C2×D61—21no—
257C2×D61—22no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—41no—
261C2×D61—41no—
262C4×C61—22no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61C2×#13111yescharacteristic
269C4×C61C2×#13111yescharacteristic
270Dic61—21no—
271Dic61—21no—
272Dic61—41no—
273Dic61—41no—
274Dic61—41no—
275Dic61#57711yescharacteristic
276Dic61#57811yescharacteristic
277#3107—611no—
278#2988—611no—
279#2985—611no—
280#1587—611no—
281#1585—611no—
282#1584—611no—
283#1582—611no—
284#1579—611no—
285SD16⋊C8—611no—
286C8×Q16—611no—
287C8×SD16—611no—
288#1573—611no—
289#1488—611no—
290#1481—611no—
291#1454—611no—
292C2×C61#292911yescharacteristic
293C2×C61#294711yescharacteristic
294C2×C61#295711yescharacteristic
295D61—41no—
296D61—42no—
297#533—611no—
298C4×#41—611no—
299#485—611no—
300#485—611no—
301#475#692311yescharacteristic
302#474—611no—
303#462—611no—
304#461—611no—
305#452—611no—
306#452—611no—
307#446—611no—
308#446—611no—
309#446—1221no—
310#446—1221no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—611no—
314C8×Q8—611no—
315C4×Q16—611no—
316C4×Q16—611no—
317C4×SD16—611no—
318C4×SD16—611no—
319C8×D4—611no—
320C8×D4—611no—
321C8×D4—1221no—
322C8×D4—1221no—
323#433—1221no—
324#433—1221no—
325#432—611no—
326#431—611no—
327#431—611no—
328#431—611no—
329#431—1221no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8—1221no—
332#335—611no—
333#335—611no—
334#334—611no—
335#334—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342D4⋊C8—611no—
343D4⋊C8—611no—
344#322—611no—
345#322—611no—
346#322—611no—
347C82—611no—
348C61#1792111yescharacteristic, complemented, Sylow
349C2×#41—611no—
350C2×Q16—611no—
351C2×Q16—611no—
352C2×SD16—611no—
353C2×SD16—611no—
354C22×C8—1221no—
355C22×C8—1221no—
356C22×C8—1221no—
357C4⋊Q8—611no—
358C4⋊Q8C2×#692311yescharacteristic
359#125—1221no—
360C22⋊Q8—611no—
361C22⋊Q8—611no—
362C22⋊Q8—1221no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—611no—
366C4×Q8—1221no—
367C4×Q8C2×#692311yescharacteristic
368C4×D4—611no—
369C4×D4—611no—
370C4×D4—1221no—
371C4×D4—1221no—
372#117—611no—
373#117—1221no—
374#117—1221no—
375C2×C42—611no—
376C2×C42—1221no—
377#107—611no—
378#107—611no—
379#106C2×#692311yescharacteristic
380#106C2×#692311yescharacteristic
381C4⋊C8—611no—
382C4⋊C8—611no—
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8C2×#692311yescharacteristic
397#103—611no—
398#103—611no—
399#103—611no—
400#103—611no—
401#103C2×#692311yescharacteristic
402#103C2×#692311yescharacteristic
403#102—611no—
404#102—611no—
405C22⋊C8—611no—
406C22⋊C8—611no—
407C22⋊C8—611no—
408C22⋊C8—611no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413#97—1221no—
414#97—1221no—
415#97—1221no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—1221no—
423C4×C8—1221no—
424C4×C8—1221no—
425C4×C8—1221no—
426#41—1222no—
427#41—1222no—
428C2×Q8—21no—
429C2×Q8—611no—
430C2×Q8—611no—
431C2×Q8C22×#692311yescharacteristic
432C2×D4—611no—
433C2×D4—1221no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440Q16—1222no—
441Q16—1222no—
442SD16—1222no—
443SD16—1222no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—4881no—
457C2×C8—4881no—
458C2×C8—4881no—
459C2×C8C22×#692311yescharacteristic
460C2×C8C22×#692311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—611no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4C22×#692311yescharacteristic
472C4⋊C4C22×#692311yescharacteristic
473C4⋊C4C22×#692311yescharacteristic
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C42—21no—
480C42—611no—
481C42—611no—
482C42—611no—
483C42—1221no—
484C42—1221no—
485C42—1221no—
486C42—1221no—
487C42C22×#692311yescharacteristic
488C23—611no—
489C23—1221no—
490Q8—21no—
491Q8—81no—
492Q8—1221no—
493Q8—1221no—
494Q8—1221no—
495Q8—1221no—
496Q8#1451986512yes—
497D4—612no—
498D4—1224no—
499D4—1221no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1224no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—4881no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4D4×#692311yescharacteristic
520C2×C4D4×#692311yescharacteristic
521C2×C4C23×#692311yescharacteristic
522C8—21no—
523C8—21no—
524C8—1221no—
525C8—1221no—
526C8—1221no—
527C8—1221no—
528C8—1221no—
529C8—1221no—
530C8—2441no—
531C8—2441no—
532C8—2441no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C22—1222no—
537C22—1221no—
538C22—1221no—
539C22—1221no—
540C22—2441no—
541C22—2441no—
542C22C2×D4×#692311yescharacteristic
543C4—22no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—611no—
550C4—611no—
551C4—1221no—
552C4—1221no—
553C4—2441no—
554C4—2441no—
555C4—2441no—
556C4D61⋊#13111yescharacteristic
557C4C2×#1451986511yescharacteristic
558C2—2441no—
559C2—2442no—
560C2#2146253511yescharacteristic
561C2#2146257711yescharacteristic
562C2#2146259711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#475C4
socleC22×C61C2×C4×D4
prime radicalC2×#11758226C2×C4
cosocleC4×Dic61C24
hypercenter#475#6923
hypocenterC61#17921
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #21462727C2semidirect
3. #21461442C2semidirect
4. #21461435C2semidirect
5. #21461430C2semidirect
6. #21461423C2semidirect
7. #21461413C2semidirect
8. #21461411C2semidirect
9. #21461406C2semidirect
10. #21461404C2semidirect
11. #21461393C2semidirect
13. #21461129C2semidirect
14. #21461041C2semidirect
348. C61#17921semidirect

Maximal subgroups

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

subgroupindex
#214627372
#214627272
#214614422
#214614352
#214614302
#214614232
#214614132
#214614112
#214614062
#214614042
#214613932
#214611532
#214611292
#214610412
#214558932
#1792161