Finite Group Databasebeta

← #27151653

Subgroups — #27151653

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261247C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic, complemented
3#20261079C211yesmaximal, characteristic, complemented
4#20261078C211yesmaximal, characteristic, complemented
5#20261072C211yesmaximal, characteristic, complemented
6#20261067C211yesmaximal, characteristic, complemented
7#20261062C211yesmaximal, characteristic, complemented
8#20261061C211yesmaximal, characteristic, complemented
9#20261055C211yesmaximal, characteristic, complemented
10#20261050C211yesmaximal, characteristic, complemented
11#20260771C211yesmaximal, characteristic, complemented
12#20260732C211yesmaximal, characteristic, complemented
13#20260729C211yesmaximal, characteristic, complemented
14C2×#15690722C211yesmaximal, characteristic, complemented
15#427×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×#11715895C2211yescharacteristic
19#15690794—21no—
20#15690790C2211yescharacteristic
21#15690790C2211yescharacteristic
22#15690785C2211yescharacteristic
23#15690785C2211yescharacteristic
24#15690780—21no—
25C4×D388—21no—
26C4×D388—21no—
27#15690769C2211yescharacteristic
28#15690768C2211yescharacteristic
29#15690768C2211yescharacteristic
30#15690767—21no—
31#15690766—21no—
32C4⋊C4×D97C2211yescharacteristic
33C4⋊C4×D97C2211yescharacteristic
34C22⋊C4×D97—21no—
35C42×D97—21no—
36#15690743C2211yescharacteristic, complemented
37#15690743C2211yescharacteristic, complemented
38#15690741C2211yescharacteristic, complemented
39#15690741C2211yescharacteristic, complemented
40#15690735C2211yescharacteristic, complemented
41#15690735C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690733C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic, complemented
46#15690730C2211yescharacteristic, complemented
47#15690730C2211yescharacteristic, complemented
48#15690722C2211yescharacteristic
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690722C2211yescharacteristic
52C22×C8×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic
54#117×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic, complemented
56#107×C97C2211yescharacteristic, complemented
57#106×C97C2211yescharacteristic, complemented
58#106×C97C2211yescharacteristic, complemented
59C23×D97D411yescharacteristic
60#13095511—22no—
61#13095511—22no—
62C2×Dic194C2311yescharacteristic
63C2×D388—21no—
64C2×D388—21no—
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C22×Dic97D411yescharacteristic
68C2×C4×D97—22no—
69C2×C4×D97—22no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97D411yescharacteristic
74C2×C4×D97D411yescharacteristic
75C2×C4×D97C2311yescharacteristic
76C2×C4×D97C2311yescharacteristic
77C2×#11715895—22no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C4⋊Dic97—21no—
81C4⋊Dic97—21no—
82C4⋊Dic97C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84#13095490—21no—
85#13095490—21no—
86C22⋊Dic97—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C22×C4×C97C2311yescharacteristic
97C2×C8×C97—22no—
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C42×C97—21no—
107C42×C97—21no—
108C23×C97C2×D411yescharacteristic
109C22×D97—22no—
110C22×D97—22no—
111C22×D97—21no—
112C22×D97—41no—
113C22×D97C2×D411yescharacteristic
114C22×D97C2×D411yescharacteristic
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97C2×D411yescharacteristic
120C2×C4×C97#4112yes—
121C2×C4×C97#4112yes—
122C2×C4×C97C2411yescharacteristic
123C22⋊D97—42no—
124C2×Dic97—22no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97C2×D411yescharacteristic
129C2×Dic97C2×D411yescharacteristic
130D388—21no—
131D388—21no—
132D388—42no—
133D388—42no—
134C4×D97—22no—
135C4×D97—22no—
136C4×D97—21no—
137C4×D97—21no—
138C4×D97—21no—
139C4×D97—21no—
140C4×D97—42no—
141C4×D97—42no—
142C4×D97—42no—
143Dic194—21no—
144Dic194—21no—
145C8×C97—21no—
146C8×C97—21no—
147#11715895—21no—
148#11715895—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97#12312yes—
152C22×C97C22×D411yescharacteristic
153C2×D97—21no—
154C2×D97—21no—
155C2×D97—21no—
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—44no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97#13512yes—
163C4×C97—41no—
164C4×C97—41no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97#12312yes—
168C4×C97C2×#4111yescharacteristic
169C4×C97C2×#4111yescharacteristic
170Dic97—21no—
171Dic97—21no—
172Dic97—41no—
173Dic97—41no—
174Dic97—41no—
175C2×C97—22no—
176C2×C97C2×#12311yescharacteristic
177C2×C97C2×#13511yescharacteristic
178C2×C97#57511yescharacteristic
179D97—22no—
180D97—41no—
181D97—81no—
182#3091—971nomaximal, Sylow
183C97#309111yescharacteristic, complemented, Sylow
184#525—971no—
185C2×#117—971no—
186#485—971no—
187#485—971no—
188#484—971no—
189#483—971no—
190#482—971no—
191#482—971no—
192#481—971no—
193#480—971no—
194#427—971no—
195#427D9711yescharacteristic, complemented
196#416—971no—
197#416—971no—
198C2×C22⋊C8—971no—
199C2×#41—971no—
200C23×C4—971no—
201C22×C8—971no—
202C22×C8C2×D9711yescharacteristic
203#123—1941no—
204C22⋊Q8—971no—
205C22⋊Q8—971no—
206#121—971no—
207#121—971no—
208#120—1941no—
209C4×D4—1941no—
210C4×D4—1941no—
211#117—971no—
212#117—971no—
213#117—971no—
214#117—1941no—
215#117—1941no—
216#117C2×D9711yescharacteristic
217#117C2×D9711yescharacteristic
218C2×C4⋊C4—971no—
219C2×C4⋊C4—971no—
220C2×C22⋊C4—1941no—
221C2×C42—1941no—
222#107—971no—
223#107—971no—
224#107C2×D9711yescharacteristic, complemented
225#107C2×D9711yescharacteristic, complemented
226#106—971no—
227#106—971no—
228#106C2×D9711yescharacteristic, complemented
229#106C2×D9711yescharacteristic, complemented
230#103—971no—
231#103—971no—
232#103—971no—
233#103—971no—
234#102—971no—
235#102—971no—
236#102—971no—
237#102—971no—
238C22⋊C8—971no—
239C22⋊C8—971no—
240C22⋊C8—971no—
241C22⋊C8—971no—
242C24—971no—
243#41—1942no—
244#41—1942no—
245C2×Q8—971no—
246C2×D4—971no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4—1941no—
250C22×C4—971no—
251C22×C4—971no—
252C22×C4—971no—
253C22×C4—971no—
254C22×C4—971no—
255C22×C4—1941no—
256C22×C4—1941no—
257C22×C4—1941no—
258C22×C4—1942no—
259C22×C4—1942no—
260C22×C4C22×D9711yescharacteristic
261C2×C8—22no—
262C2×C8—971no—
263C2×C8—971no—
264C2×C8—1942no—
265C2×C8C22×D9711yescharacteristic
266C2×C8C22×D9711yescharacteristic
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4C22×D9711yescharacteristic
274C4⋊C4C22×D9711yescharacteristic
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C42—21no—
287C42—21no—
288C42—1941no—
289C42—1941no—
290C42—1941no—
291C23—971no—
292C23—971no—
293C23—3881no—
294C23—1942no—
295C23—1941no—
296C23—1942no—
297C23D4×D9711yescharacteristic
298Q8—1941no—
299Q8—1941no—
300D4—3882no—
301D4—3882no—
302D4—3882no—
303D4—1941no—
304D4—1941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—971no—
310C2×C4—971no—
311C2×C4—3882no—
312C2×C4—3882no—
313C2×C4—3882no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1942no—
318C2×C4—1942no—
319C2×C4—1942no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4D4×D9711yescharacteristic
325C2×C4#1309551112yes—
326C2×C4#1309551212yes—
327C2×C4C23×D9711yescharacteristic
328C8—21no—
329C8—21no—
330C8—1941no—
331C8—1941no—
332C22—21no—
333C22—21no—
334C22—972no—
335C22—3884no—
336C22—3881no—
337C22—3881no—
338C22—3881no—
339C22—1941no—
340C22—1941no—
341C22—1941no—
342C22—1941no—
343C22—1941no—
344C22#1569079512yes—
345C22C2×D4×D9711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3881no—
351C4—3881no—
352C4—3881no—
353C4—1941no—
354C4—1941no—
355C4#1569079512yes—
356C4C2×#1309551111yescharacteristic
357C4C2×#1309551211yescharacteristic
358C2—22no—
359C2—7761no—
360C2—3881no—
361C2—1942no—
362C2C2×#1569079511yescharacteristic
363C2#135×D9711yescharacteristic
364C2#2026153211yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#427×C97C2
socleC22×C97C22×D4
prime radical#20261247C2
hypercenter#427D97
hypocenterC97#3091
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #117×D97C2semidirect
3. #20261079C2semidirect
4. #20261078C2semidirect
5. #20261072C2semidirect
6. #20261067C2semidirect
7. #20261062C2semidirect
8. #20261061C2semidirect
9. #20261055C2semidirect
10. #20261050C2semidirect
11. #20260771C2semidirect
12. #20260732C2semidirect
13. #20260729C2semidirect
14. C2×#15690722C2semidirect
15. #427×C97C2semidirect
36. #15690743C22semidirect
37. #15690743C22semidirect
38. #15690741C22semidirect
39. #15690741C22semidirect
40. #15690735C22semidirect
41. #15690735C22semidirect
42. #15690733C22semidirect
43. #15690733C22semidirect
44. #15690732C22semidirect
45. #15690732C22semidirect
46. #15690730C22semidirect
47. #15690730C22semidirect
55. #107×C97C22semidirect
56. #107×C97C22semidirect
57. #106×C97C22semidirect
58. #106×C97C22semidirect
183. C97#3091semidirect
195. #427D97semidirect
224. #107C2×D97semidirect
225. #107C2×D97semidirect
228. #106C2×D97semidirect
229. #106C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612472
#117×D972
#202610792
#202610782
#202610722
#202610672
#202610622
#202610612
#202610552
#202610502
#202607712
#202607322
#202607292
C2×#156907222
#427×C972
#309197