Finite Group Databasebeta

← #27150720

Subgroups — #27150720

Order 12416.

362 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261272C211yesmaximal, characteristic
2#20260866C211yesmaximal, characteristic, complemented
3#20260814C211yesmaximal, characteristic, complemented
4#20260812C211yesmaximal, characteristic, complemented
5#20260808C211yesmaximal, characteristic, complemented
6#20260808C211yesmaximal, characteristic, complemented
7#20260805C211yesmaximal, characteristic, complemented
8#20260805C211yesmaximal, characteristic, complemented
9#20260795C211yesmaximal, characteristic, complemented
10#20260795C211yesmaximal, characteristic, complemented
11#20260791C211yesmaximal, characteristic, complemented
12#20260692C211yesmaximal, characteristic, complemented
13#20260689C211yesmaximal, characteristic, complemented
14#20260687C211yesmaximal, characteristic, complemented
15#446×C97C211yesmaximal, characteristic, complemented
16C2×#13095494C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22#15690805C411yescharacteristic
23#15690805C411yescharacteristic
24#15690801C411yescharacteristic
25#15690799C411yescharacteristic
26#15690795C411yescharacteristic
27#15690795C411yescharacteristic
28#15690786C411yescharacteristic
29#15690785C411yescharacteristic
30C4×D388C2211yescharacteristic
31C4×D388C2211yescharacteristic
32C4×D388C2211yescharacteristic
33#15690767C2211yescharacteristic
34#15690767C2211yescharacteristic
35C42×D97C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37C4⋊#11715895C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39#15690722C2211yescharacteristic
40#15690722C2211yescharacteristic
41#15690722C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690722C2211yescharacteristic
44#15690722C2211yescharacteristic
45#15690721C2211yescharacteristic
46#15690721C2211yescharacteristic
47#15690720C2211yescharacteristic
48#15690720C2211yescharacteristic
49C4×#11715895C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C4×Q8×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic
54C4⋊C8×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic
56#97×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59C2×D388C2×C411yescharacteristic
60C2×D388C2×C411yescharacteristic
61C2×D388C2×C411yescharacteristic
62C2×C4×D97C2×C411yescharacteristic
63C2×C4×D97C2×C411yescharacteristic
64C2×C4×D97C2311yescharacteristic
65C2×C4×D97C2311yescharacteristic
66C2×C4×D97C2311yescharacteristic
67#13095494—22no—
68#13095494—22no—
69#13095494—22no—
70#13095494—22no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C8×D97—22no—
76C8×D97—22no—
77C4⋊Dic97C2×C411yescharacteristic
78C4⋊Dic97C2×C411yescharacteristic
79C4⋊Dic97C2×C411yescharacteristic
80#13095490C2×C411yescharacteristic
81#13095490C2×C411yescharacteristic
82#13095490C2×C411yescharacteristic
83#13095490C2×C411yescharacteristic
84#13095488C2×C411yescharacteristic
85#13095488C2×C411yescharacteristic
86#13095488C2×C411yescharacteristic
87#13095488C2×C411yescharacteristic
88#13095488C2×C411yescharacteristic
89#13095488C2×C411yescharacteristic
90#13095488C2×C411yescharacteristic
91#13095488C2×C411yescharacteristic
92#13095488C2×C411yescharacteristic
93#13095488C2×C411yescharacteristic
94C4×Dic97C2×C411yescharacteristic
95C4×Dic97C2×C411yescharacteristic
96C4×Dic97C2311yescharacteristic
97C2×Q8×C97C2×C411yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97C2×C411yescharacteristic
103C4⋊C4×C97C2×C411yescharacteristic
104C4⋊C4×C97C2×C411yescharacteristic
105C42×C97C2311yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C22×D97C22×C411yescharacteristic
109C22×D97C22×C411yescharacteristic
110C22×D97C22×C411yescharacteristic
111Q8×C97—41no—
112C2×C4×C97C22×C411yescharacteristic
113C2×C4×C97C22×C411yescharacteristic
114C2×C4×C97C22×C411yescharacteristic
115C2×C4×C97C22×C411yescharacteristic
116C2×C4×C97C22×C411yescharacteristic
117C2×C4×C97C22×C411yescharacteristic
118C2×C4×C97C2411yescharacteristic
119C2×Dic97C22×C411yescharacteristic
120C2×Dic97C22×C411yescharacteristic
121C2×Dic97C22×C411yescharacteristic
122C2×Dic97C22×C411yescharacteristic
123C2×Dic97C22×C411yescharacteristic
124D388—41no—
125D388—41no—
126D388—41no—
127C4×D97—21no—
128C4×D97—21no—
129C4×D97—21no—
130C4×D97—21no—
131C4×D97—21no—
132C4×D97—21no—
133C4×D97—41no—
134C4×D97—41no—
135C8×C97—21no—
136C8×C97—21no—
137C8×C97—21no—
138C8×C97#4112yes—
139#11715895—21no—
140#11715895—21no—
141#11715895—21no—
142#11715895#4112yes—
143C22×C97C23×C411yescharacteristic
144C2×D97—21no—
145C2×D97—21no—
146C2×D97—21no—
147C2×D97—21no—
148C2×D97—21no—
149C2×D97—21no—
150C2×D97—21no—
151C2×D97—21no—
152C2×D97—21no—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—21no—
159C4×C97C2×#4111yescharacteristic
160C4×C97C2×#4111yescharacteristic
161Dic97—21no—
162Dic97—21no—
163Dic97—21no—
164Dic97—21no—
165Dic97—21no—
166C2×C97C4×#4111yescharacteristic
167C2×C97#56811yescharacteristic
168C2×C97#56811yescharacteristic
169D97—41no—
170D97—41no—
171D97—41no—
172#2967—971nomaximal, Sylow
173C97#296711yescharacteristic, complemented, Sylow
174#529—971no—
175#446—971no—
176#446D9711yescharacteristic, complemented
177#436—971no—
178#436—971no—
179#435—971no—
180#435—971no—
181#435—971no—
182#435—971no—
183#433—971no—
184#433—971no—
185#432—971no—
186#405—971no—
187#405—971no—
188C4×M16—971no—
189C2×M16—971no—
190C2×M16—971no—
191C2×M16—971no—
192C2×M16—971no—
193C22×C8—971no—
194C22×C8—971no—
195#126—971no—
196#126—971no—
197#125—971no—
198#124—971no—
199#123—971no—
200#123—971no—
201C22⋊Q8—971no—
202#121—971no—
203C4×Q8C2×D9711yescharacteristic
204C4×D4—971no—
205C4×D4—971no—
206C4×D4—971no—
207#117—971no—
208#117—971no—
209C2×C42—971no—
210C4⋊C8—971no—
211C4⋊C8—971no—
212C4⋊C8—971no—
213C4⋊C8C2×D9711yescharacteristic
214C4⋊C8C2×D9711yescharacteristic
215C4⋊C8C2×D9711yescharacteristic
216C22⋊C8—971no—
217C22⋊C8—971no—
218C22⋊C8—971no—
219C22⋊C8—971no—
220C22⋊C8—971no—
221C22⋊C8—971no—
222#97—971no—
223#97—971no—
224#97—971no—
225#97—971no—
226#97C2×D9711yescharacteristic
227#97C2×D9711yescharacteristic
228C4×C8—971no—
229C4×C8—971no—
230C4×C8—971no—
231C4×C8C2×D9711yescharacteristic
232C2×Q8C4×D9711yescharacteristic
233C2×D4—971no—
234C2×D4—971no—
235C2×D4—971no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—971no—
241M16—1942no—
242M16—1942no—
243M16—1942no—
244M16—1942no—
245C2×C8—971no—
246C2×C8—971no—
247C2×C8—971no—
248C2×C8—971no—
249C2×C8—1942no—
250C2×C8—1942no—
251C2×C8C22×D9711yescharacteristic
252C2×C8C22×D9711yescharacteristic
253C2×C8C22×D9711yescharacteristic
254C2×C8C22×D9711yescharacteristic
255C4⋊C4—971no—
256C4⋊C4—971no—
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4C4×D9711yescharacteristic
263C4⋊C4C4×D9711yescharacteristic
264C4⋊C4C4×D9711yescharacteristic
265C22⋊C4—971no—
266C22⋊C4—971no—
267C22⋊C4—971no—
268C22⋊C4—971no—
269C22⋊C4—971no—
270C22⋊C4—971no—
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4—971no—
274C22⋊C4—971no—
275C42—971no—
276C42—971no—
277C42—971no—
278C42C22×D9711yescharacteristic
279C42C22×D9711yescharacteristic
280C42C22×D9711yescharacteristic
281C23—971no—
282C23—971no—
283C23—971no—
284Q8—41no—
285D4—3881no—
286D4—3881no—
287D4—3881no—
288C2×C4—971no—
289C2×C4—971no—
290C2×C4—971no—
291C2×C4—971no—
292C2×C4—971no—
293C2×C4—3881no—
294C2×C4—3881no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4C2×C4×D9711yescharacteristic
302C2×C4C2×C4×D9711yescharacteristic
303C2×C4C2×C4×D9711yescharacteristic
304C2×C4C2×C4×D9711yescharacteristic
305C2×C4C2×C4×D9711yescharacteristic
306C2×C4C2×C4×D9711yescharacteristic
307C2×C4C23×D9711yescharacteristic
308C8—21no—
309C8—21no—
310C8—21no—
311C8—972no—
312C8—1941no—
313C8—1941no—
314C8—1941no—
315C8#1309551212yes—
316C22—1941no—
317C22—1941no—
318C22—1941no—
319C22—1941no—
320C22—1941no—
321C22—1941no—
322C22—1941no—
323C22—1941no—
324C22—1941no—
325C22C22×C4×D9711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—1941no—
333C4—1941no—
334C4—1941no—
335C4—1941no—
336C4—1941no—
337C4#41×D9711yescharacteristic
338C4C2×#1309551211yescharacteristic
339C2—3881no—
340C2—3881no—
341C2—3881no—
342C2C4×#1309551211yescharacteristic
343C2#2026150011yescharacteristic
344C2#2026150111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#446×C97C2
prime radical#15690785C4
cosocleC2×C4×C97C24
hypercenter#446D97
hypocenterC97#2967
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #446

Complements

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

NH ≅ G/Ntype
2. #20260866C2semidirect
3. #20260814C2semidirect
4. #20260812C2semidirect
5. #20260808C2semidirect
6. #20260808C2semidirect
7. #20260805C2semidirect
8. #20260805C2semidirect
9. #20260795C2semidirect
10. #20260795C2semidirect
11. #20260791C2semidirect
12. #20260692C2semidirect
13. #20260689C2semidirect
14. #20260687C2semidirect
15. #446×C97C2semidirect
173. C97#2967semidirect
176. #446D97semidirect

Maximal subgroups

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

subgroupindex
#202612722
#202608662
#202608142
#202608122
#202608082
#202608052
#202607952
#202607912
#202606922
#202606892
#202606872
#446×C972
#296797