Finite Group Databasebeta

← #27153188

Subgroups — #27153188

Order 12416.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261346C211yesmaximal, characteristic
2#20261343C211yesmaximal, characteristic
3#20261044C211yesmaximal, characteristic, complemented
4#20261043C211yesmaximal, characteristic, complemented
5#20261042C211yesmaximal, characteristic, complemented
6#20261040C211yesmaximal, characteristic, complemented
7#20260992C211yesmaximal, characteristic
8#20260990C211yesmaximal, characteristic
9#20260975C211yesmaximal, characteristic
10#20260973C211yesmaximal, characteristic
11#20260943C211yesmaximal, characteristic, complemented
12#20260943C211yesmaximal, characteristic, complemented
13#20260942C211yesmaximal, characteristic, complemented
14#20260760C211yesmaximal, characteristic
15#462×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690808C2211yescharacteristic
21#15690805—21no—
22#15690805—21no—
23#15690805—21no—
24#15690804—21no—
25#15690802—21no—
26#15690802C2211yescharacteristic
27#15690801C2211yescharacteristic
28#15690800—21no—
29#15690800—21no—
30#15690800—21no—
31#15690790C2211yescharacteristic
32#15690786C2211yescharacteristic
33#15690786C2211yescharacteristic
34#15690786C2211yescharacteristic
35Q8×Dic97C2211yescharacteristic
36Q8×Dic97C2211yescharacteristic
37C4×Dic194C2211yescharacteristic
38#15690767C2211yescharacteristic
39#15690744C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690741C2211yescharacteristic
43C4⋊#11715895C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690734C2211yescharacteristic
49#15690734C2211yescharacteristic
50#15690734C2211yescharacteristic
51#15690734C2211yescharacteristic
52C2×Q16×C97C2211yescharacteristic
53C2×Q16×C97C2211yescharacteristic
54C4⋊Q8×C97C2211yescharacteristic
55C4×Q8×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59C2×Dic194C2311yescharacteristic
60C2×C4×D97C2311yescharacteristic
61#13095501—41no—
62#13095501—41no—
63#13095494—41no—
64C2×#11715895C2311yescharacteristic
65C2×#11715895C2311yescharacteristic
66C8×D97—41no—
67C4⋊Dic97—21no—
68C4⋊Dic97—21no—
69C4⋊Dic97—21no—
70C4⋊Dic97—21no—
71C4⋊Dic97C2311yescharacteristic
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74#13095490—21no—
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490D411yescharacteristic
84#13095490D411yescharacteristic
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488D411yescharacteristic
90#13095488D411yescharacteristic
91C4×Dic97—21no—
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97C2311yescharacteristic
95C2×Q8×C97C2311yescharacteristic
96C2×Q8×C97C2311yescharacteristic
97C2×Q8×C97C2311yescharacteristic
98Q16×C97—41no—
99Q16×C97—41no—
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97—21no—
103C4⋊C4×C97—21no—
104C4⋊C4×C97—21no—
105C4⋊C4×C97C2311yescharacteristic
106C42×C97—21no—
107C42×C97C2311yescharacteristic
108C22×D97C2×D411yescharacteristic
109Q8×C97—21no—
110Q8×C97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97—21no—
114Q8×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97C2×D411yescharacteristic
120C2×C4×C97C2×D411yescharacteristic
121C2×C4×C97C2411yescharacteristic
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97C2×D411yescharacteristic
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129C4×D97—21no—
130C4×D97—21no—
131Dic194—21no—
132Dic194—21no—
133C8×C97—21no—
134C8×C97—21no—
135#11715895—21no—
136#11715895—21no—
137C22×C97C22×D411yescharacteristic
138C2×D97—21no—
139C2×D97—21no—
140C2×D97—21no—
141C4×C97—21no—
142C4×C97—21no—
143C4×C97—41no—
144C4×C97—41no—
145C4×C97—41no—
146C4×C97—41no—
147C4×C97H(5,𝔽2)11yescharacteristic
148C4×C972−1+411yescharacteristic
149Dic97—21no—
150Dic97—21no—
151Dic97—21no—
152Dic97—41no—
153Dic97—41no—
154Dic97—41no—
155Dic97—41no—
156C2×C97#53711yescharacteristic
157C2×C97#57811yescharacteristic
158C2×C97#57811yescharacteristic
159D97—41no—
160#3261—971nomaximal, Sylow
161C97#326111yescharacteristic, complemented, Sylow
162#541—971no—
163#541—971no—
164#479—971no—
165#479—971no—
166#479—971no—
167#479—971no—
168#470—971no—
169#470—971no—
170#467—971no—
171#467—971no—
172#462—971no—
173#462—971no—
174#462—971no—
175#462D9711yescharacteristic, complemented
176#424—971no—
177C2×Q16—971no—
178C2×Q16—971no—
179C2×Q16C2×D9711yescharacteristic
180C2×Q16C2×D9711yescharacteristic
181C2×M16—971no—
182C22×C8—971no—
183C4⋊Q8—971no—
184C4⋊Q8C2×D9711yescharacteristic
185#126—1941no—
186#126—1941no—
187#126—1941no—
188#126—1941no—
189#125—971no—
190#125—971no—
191#125—1941no—
192#125—1941no—
193#125—1941no—
194#125—1941no—
195C22⋊Q8—971no—
196C22⋊Q8—971no—
197C22⋊Q8—971no—
198C22⋊Q8—971no—
199C4×Q8—971no—
200C4×Q8—971no—
201C4×Q8—971no—
202C4×Q8C2×D9711yescharacteristic
203#117—971no—
204#107—971no—
205#107—971no—
206#106—971no—
207#106—971no—
208C4⋊C8—971no—
209C4⋊C8—971no—
210C4⋊C8—971no—
211C4⋊C8C2×D9711yescharacteristic
212#103—971no—
213#103—971no—
214#103—971no—
215#103—971no—
216#103—971no—
217#103—971no—
218#103C2×D9711yescharacteristic
219#103C2×D9711yescharacteristic
220C2×Q8—971no—
221C2×Q8C22×D9711yescharacteristic
222C2×Q8C22×D9711yescharacteristic
223C2×Q8C22×D9711yescharacteristic
224C22×C4—971no—
225Q16—41no—
226Q16—41no—
227Q16—3881no—
228Q16—3881no—
229M16—3881no—
230C2×C8—971no—
231C2×C8—971no—
232C2×C8—3881no—
233C2×C8C22×D9711yescharacteristic
234C2×C8C22×D9711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—971no—
239C4⋊C4—971no—
240C4⋊C4—971no—
241C4⋊C4—971no—
242C4⋊C4—971no—
243C4⋊C4—1941no—
244C4⋊C4—1941no—
245C4⋊C4—1941no—
246C4⋊C4—1941no—
247C4⋊C4—1941no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4C22×D9711yescharacteristic
257C22⋊C4—971no—
258C22⋊C4—971no—
259C22⋊C4—1941no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C42—21no—
264C42—971no—
265C42—1941no—
266C42—1941no—
267C42—1941no—
268C42C22×D9711yescharacteristic
269C23—971no—
270Q8—21no—
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—1941no—
277Q8—1941no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—971no—
283C2×C4—971no—
284C2×C4—971no—
285C2×C4—1941no—
286C2×C4—1941no—
287C2×C4—1941no—
288C2×C4—1941no—
289C2×C4—1941no—
290C2×C4—1941no—
291C2×C4D4×D9711yescharacteristic
292C2×C4D4×D9711yescharacteristic
293C2×C4C23×D9711yescharacteristic
294C8—21no—
295C8—21no—
296C8—1941no—
297C8—1941no—
298C22—1941no—
299C22—1941no—
300C22—1941no—
301C22C2×D4×D9711yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—3881no—
309C4—3881no—
310C4—3881no—
311C4—3881no—
312C4—1941no—
313C4—1941no—
314C4—1941no—
315C4#1569085511yescharacteristic
316C4#1569085711yescharacteristic
317C2—3881no—
318C2#2026131411yescharacteristic
319C2#2026154811yescharacteristic
320C2#2026155111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#462×C97C2
socleC22×C97C22×D4
prime radicalC22×D97C2×D4
hypercenter#462D97
hypocenterC97#3261
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #20261044C2semidirect
4. #20261043C2semidirect
5. #20261042C2semidirect
6. #20261040C2semidirect
11. #20260943C2semidirect
12. #20260943C2semidirect
13. #20260942C2semidirect
15. #462×C97C2semidirect
161. C97#3261semidirect
175. #462D97semidirect

Maximal subgroups

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

subgroupindex
#202613462
#202613432
#202610442
#202610432
#202610422
#202610402
#202609922
#202609902
#202609752
#202609732
#202609432
#202609422
#202607602
#462×C972
#326197