Finite Group Databasebeta

← #27150839

Subgroups — #27150839

Order 12416.

314 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261289C211yesmaximal, characteristic
2#20260866C211yesmaximal, characteristic, complemented
3#20260865C211yesmaximal, characteristic, complemented
4#20260864C211yesmaximal, characteristic, complemented
5#20260864C211yesmaximal, characteristic, complemented
6#20260863C211yesmaximal, characteristic, complemented
7#20260863C211yesmaximal, characteristic, complemented
8#20260862C211yesmaximal, characteristic, complemented
9#20260797C211yesmaximal, characteristic
10#20260795C211yesmaximal, characteristic
11#20260795C211yesmaximal, characteristic
12#20260794C211yesmaximal, characteristic
13#20260791C211yesmaximal, characteristic
14#20260790C211yesmaximal, characteristic
15#446×C97C211yesmaximal, characteristic, complemented
16#15690809C411yescharacteristic
17#15690808C411yescharacteristic
18#15690800C411yescharacteristic
19#15690800C411yescharacteristic
20#15690790C411yescharacteristic
21#15690788C411yescharacteristic
22#15690788C411yescharacteristic
23#15690786C411yescharacteristic
24C4×Dic194C2211yescharacteristic
25C4×Dic194C2211yescharacteristic
26C4×Dic194C2211yescharacteristic
27#15690767C2211yescharacteristic
28#15690767C2211yescharacteristic
29C42×D97C2211yescharacteristic
30C4⋊#11715895C2211yescharacteristic
31C4⋊#11715895C2211yescharacteristic
32C4⋊#11715895C2211yescharacteristic
33#15690739C2211yescharacteristic
34#15690739C2211yescharacteristic
35#15690739C2211yescharacteristic
36#15690739C2211yescharacteristic
37#15690739C2211yescharacteristic
38#15690739C2211yescharacteristic
39#15690722C2211yescharacteristic
40#15690722C2211yescharacteristic
41#15690722C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690721C2211yescharacteristic
44#15690721C2211yescharacteristic
45#15690720C2211yescharacteristic
46#15690720C2211yescharacteristic
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×Dic194C2×C411yescharacteristic
60C2×Dic194C2×C411yescharacteristic
61C2×Dic194C2×C411yescharacteristic
62C2×C4×D97C2×C411yescharacteristic
63C2×C4×D97C2×C411yescharacteristic
64C2×C4×D97C2311yescharacteristic
65C2×#11715895C2311yescharacteristic
66C2×#11715895C2311yescharacteristic
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C4⋊Dic97C2×C411yescharacteristic
70C4⋊Dic97C2×C411yescharacteristic
71C4⋊Dic97C2×C411yescharacteristic
72#13095490C2×C411yescharacteristic
73#13095490C2×C411yescharacteristic
74#13095490C2×C411yescharacteristic
75#13095490C2×C411yescharacteristic
76#13095490C2×C411yescharacteristic
77#13095490C2×C411yescharacteristic
78#13095490C2×C411yescharacteristic
79#13095490C2×C411yescharacteristic
80#13095490C2×C411yescharacteristic
81#13095490C2×C411yescharacteristic
82#13095488C2×C411yescharacteristic
83#13095488C2×C411yescharacteristic
84#13095488C2×C411yescharacteristic
85#13095488C2×C411yescharacteristic
86C4×Dic97C2×C411yescharacteristic
87C4×Dic97C2×C411yescharacteristic
88C4×Dic97C2311yescharacteristic
89C4×Dic97C2311yescharacteristic
90C4×Dic97C2311yescharacteristic
91C2×Q8×C97C2×C411yescharacteristic
92C2×C8×C97C2311yescharacteristic
93C2×C8×C97C2311yescharacteristic
94C2×C8×C97C2311yescharacteristic
95C2×C8×C97C2311yescharacteristic
96C4⋊C4×C97C2×C411yescharacteristic
97C4⋊C4×C97C2×C411yescharacteristic
98C4⋊C4×C97C2×C411yescharacteristic
99C42×C97C2311yescharacteristic
100C42×C97C2311yescharacteristic
101C42×C97C2311yescharacteristic
102C22×D97C22×C411yescharacteristic
103Q8×C97—41no—
104C2×C4×C97C22×C411yescharacteristic
105C2×C4×C97C22×C411yescharacteristic
106C2×C4×C97C22×C411yescharacteristic
107C2×C4×C97C22×C411yescharacteristic
108C2×C4×C97C22×C411yescharacteristic
109C2×C4×C97C22×C411yescharacteristic
110C2×C4×C97C2411yescharacteristic
111C2×Dic97C22×C411yescharacteristic
112C2×Dic97C22×C411yescharacteristic
113C2×Dic97C22×C411yescharacteristic
114C2×Dic97C22×C411yescharacteristic
115C2×Dic97C22×C411yescharacteristic
116C2×Dic97C22×C411yescharacteristic
117C2×Dic97C22×C411yescharacteristic
118C4×D97—21no—
119C4×D97—21no—
120C4×D97—41no—
121C4×D97—41no—
122Dic194—41no—
123Dic194—41no—
124Dic194—41no—
125C8×C97—21no—
126C8×C97—21no—
127C8×C97—21no—
128C8×C97—21no—
129#11715895—21no—
130#11715895—21no—
131#11715895—21no—
132#11715895—21no—
133C22×C97C23×C411yescharacteristic
134C2×D97—21no—
135C2×D97—21no—
136C2×D97—21no—
137C4×C97—21no—
138C4×C97—21no—
139C4×C97—21no—
140C4×C97—21no—
141C4×C97—21no—
142C4×C97—21no—
143C4×C972−1+411yescharacteristic
144C4×C972−1+411yescharacteristic
145Dic97—21no—
146Dic97—21no—
147Dic97—21no—
148Dic97—21no—
149Dic97—21no—
150Dic97—21no—
151Dic97—21no—
152C2×C97#51911yescharacteristic
153C2×C97#56811yescharacteristic
154C2×C97#56811yescharacteristic
155D97—41no—
156#2987—971nomaximal, Sylow
157C97#298711yescharacteristic, complemented, Sylow
158#533—971no—
159#446—971no—
160#446—971no—
161#446—971no—
162#446—971no—
163#446—971no—
164#446—971no—
165#446—971no—
166#446D9711yescharacteristic, complemented
167#433—971no—
168#433—971no—
169#433—971no—
170#433—971no—
171#432—971no—
172#432—971no—
173C4⋊Q8—971no—
174C4⋊Q8—971no—
175#125—971no—
176#125—971no—
177C22⋊Q8—971no—
178C22⋊Q8—971no—
179C22⋊Q8—971no—
180C22⋊Q8—971no—
181C4×Q8—971no—
182C4×Q8—971no—
183C4×Q8—971no—
184C4×Q8C2×D9711yescharacteristic
185#117—971no—
186#117—971no—
187C2×C42—971no—
188C4⋊C8—971no—
189C4⋊C8—971no—
190C4⋊C8—971no—
191C4⋊C8—971no—
192C4⋊C8—971no—
193C4⋊C8—971no—
194C4⋊C8—971no—
195C4⋊C8—971no—
196C4⋊C8—971no—
197C4⋊C8C2×D9711yescharacteristic
198C4⋊C8C2×D9711yescharacteristic
199C4⋊C8C2×D9711yescharacteristic
200C22⋊C8—971no—
201C22⋊C8—971no—
202C22⋊C8—971no—
203C22⋊C8—971no—
204#97—971no—
205#97—971no—
206#97—971no—
207#97—971no—
208#97—971no—
209#97—971no—
210#97C2×D9711yescharacteristic
211#97C2×D9711yescharacteristic
212C4×C8—971no—
213C4×C8—971no—
214C4×C8—971no—
215C4×C8C2×D9711yescharacteristic
216C2×Q8—971no—
217C2×Q8—971no—
218C2×Q8—971no—
219C2×Q8C4×D9711yescharacteristic
220C22×C4—971no—
221C22×C4—971no—
222C22×C4—971no—
223C2×C8—971no—
224C2×C8—971no—
225C2×C8—971no—
226C2×C8—971no—
227C2×C8C22×D9711yescharacteristic
228C2×C8C22×D9711yescharacteristic
229C2×C8C22×D9711yescharacteristic
230C2×C8C22×D9711yescharacteristic
231C4⋊C4—971no—
232C4⋊C4—971no—
233C4⋊C4—971no—
234C4⋊C4—971no—
235C4⋊C4—971no—
236C4⋊C4—971no—
237C4⋊C4—971no—
238C4⋊C4—971no—
239C4⋊C4—971no—
240C4⋊C4—971no—
241C4⋊C4—971no—
242C4⋊C4—971no—
243C4⋊C4—971no—
244C4⋊C4C4×D9711yescharacteristic
245C4⋊C4C4×D9711yescharacteristic
246C4⋊C4C4×D9711yescharacteristic
247C22⋊C4—971no—
248C22⋊C4—971no—
249C22⋊C4—971no—
250C22⋊C4—971no—
251C42—971no—
252C42—971no—
253C42—971no—
254C42—971no—
255C42—971no—
256C42C22×D9711yescharacteristic
257C42C22×D9711yescharacteristic
258C42C22×D9711yescharacteristic
259C23—971no—
260Q8—41no—
261Q8—3881no—
262Q8—3881no—
263Q8—3881no—
264C2×C4—971no—
265C2×C4—971no—
266C2×C4—971no—
267C2×C4—971no—
268C2×C4—971no—
269C2×C4—971no—
270C2×C4—971no—
271C2×C4—3881no—
272C2×C4—3881no—
273C2×C4—1941no—
274C2×C4—1941no—
275C2×C4C2×C4×D9711yescharacteristic
276C2×C4C2×C4×D9711yescharacteristic
277C2×C4C2×C4×D9711yescharacteristic
278C2×C4C2×C4×D9711yescharacteristic
279C2×C4C2×C4×D9711yescharacteristic
280C2×C4C2×C4×D9711yescharacteristic
281C2×C4C23×D9711yescharacteristic
282C8—21no—
283C8—21no—
284C8—21no—
285C8—21no—
286C8—1941no—
287C8—1941no—
288C8—1941no—
289C8—1941no—
290C22—1941no—
291C22—1941no—
292C22—1941no—
293C22C22×C4×D9711yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—1941no—
301C4—1941no—
302C4—1941no—
303C4—1941no—
304C4—1941no—
305C4—1941no—
306C4—1941no—
307C4#1569085711yescharacteristic
308C4#1569085811yescharacteristic
309C2—3881no—
310C2#2026121111yescharacteristic
311C2#2026150011yescharacteristic
312C2#2026150111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#446×C97C2
prime radicalC22×D97C22×C4
cosocleC2×C4×C97C24
hypercenter#446D97
hypocenterC97#2987
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. #20260865C2semidirect
4. #20260864C2semidirect
5. #20260864C2semidirect
6. #20260863C2semidirect
7. #20260863C2semidirect
8. #20260862C2semidirect
15. #446×C97C2semidirect
157. C97#2987semidirect
166. #446D97semidirect

Maximal subgroups

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

subgroupindex
#202612892
#202608662
#202608652
#202608642
#202608632
#202608622
#202607972
#202607952
#202607942
#202607912
#202607902
#446×C972
#298797