Finite Group Databasebeta

← #27149997

Subgroups — #27149997

Order 12416.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20260676C211yesmaximal, characteristic, complemented
2#20260673C211yesmaximal, characteristic, complemented
3#20260671C211yesmaximal, characteristic
4#20260668C211yesmaximal, characteristic, complemented
5#20260666C211yesmaximal, characteristic, complemented
6#20260666C211yesmaximal, characteristic, complemented
7#20260665C211yesmaximal, characteristic, complemented
8#20260658C211yesmaximal, characteristic
9#20260656C211yesmaximal, characteristic, complemented
10#20260655C211yesmaximal, characteristic, complemented
11#20260647C211yesmaximal, characteristic
12#20260596C211yesmaximal, characteristic
13#20260593C211yesmaximal, characteristic
14#20260593C211yesmaximal, characteristic
15#396×C97C211yesmaximal, characteristic
16C22×Dic194C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C2×C4⋊Dic97C2211yescharacteristic
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31#15690717C2211yescharacteristic
32#15690717C2211yescharacteristic
33#15690717C2211yescharacteristic
34#15690717C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44C2×C4⋊C4×C97C2211yescharacteristic
45C2×C4⋊C4×C97C2211yescharacteristic
46C2×C4⋊C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51C2×Dic194—41no—
52C2×Dic194—41no—
53C2×Dic194—41no—
54C22×Dic97—41no—
55C22×Dic97—41no—
56C22×Dic97Q814yescomplemented
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C4⋊Dic97—41no—
66C4⋊Dic97—41no—
67#13095490—41no—
68#13095490—41no—
69#13095490—41no—
70#13095490—41no—
71#13095490—41no—
72#13095490—41no—
73#13095490—41no—
74#13095490—41no—
75C22⋊Dic97—41no—
76C22⋊Dic97—41no—
77C22⋊Dic97—41no—
78C24×C97C2311yescharacteristic
79C22×C4×C97C2311yescharacteristic
80C22×C4×C97C2311yescharacteristic
81C22×C4×C97C2311yescharacteristic
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C22×C4×C97C2311yescharacteristic
85C4⋊C4×C97—41no—
86C4⋊C4×C97—41no—
87C4⋊C4×C97—41no—
88C22⋊C4×C97—41no—
89C22⋊C4×C97—41no—
90C22⋊C4×C97—41no—
91C23×C97—21no—
92C23×C97—21no—
93C23×C97—21no—
94C23×C97—21no—
95C23×C97—21no—
96C23×C97—21no—
97C23×C97C2×Q812yes—
98C23×C97C2411yescharacteristic
99C2×C4×C97—21no—
100C2×C4×C97—21no—
101C2×C4×C97—21no—
102C2×C4×C97—21no—
103C2×C4×C97—21no—
104C2×C4×C97—21no—
105C2×C4×C97—21no—
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—44no—
140C2×Dic97C2×Q812yes—
141C2×Dic97C2×Q812yes—
142Dic194—44no—
143C22×C97—22no—
144C22×C97—22no—
145C22×C97—22no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97—41no—
149C22×C97—41no—
150C22×C97C22×Q811yescharacteristic
151C22×C97H(5,𝔽2)11yescharacteristic
152C22×C97H(5,𝔽2)11yescharacteristic
153C22×C97H(5,𝔽2)11yescharacteristic
154C22×C97H(5,𝔽2)11yescharacteristic
155C22×C972−1+411yescharacteristic
156C22×C972−1+411yescharacteristic
157C4×C97—41no—
158C4×C97—41no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97—41no—
163Dic97—22no—
164Dic97—22no—
165Dic97—41no—
166Dic97—41no—
167Dic97—41no—
168Dic97—41no—
169Dic97—41no—
170Dic97—41no—
171C2×C97—42no—
172C2×C97#54311yescharacteristic
173C2×C97#54411yescharacteristic
174C2×C97#54411yescharacteristic
175C2×C97#56011yescharacteristic
176C2×C97#56211yescharacteristic
177C2×C97#56211yescharacteristic
178C2×C97#56311yescharacteristic
179#2841—971nomaximal, Sylow
180C97#284111yescharacteristic, complemented, Sylow
181#400—971no—
182#400—971no—
183#399—971no—
184#398—971no—
185#398—971no—
186#398—971no—
187#398—971no—
188#396—971no—
189#396D9711yescharacteristic
190#395—971no—
191#395—971no—
192#393—971no—
193#381—971no—
194#380—971no—
195#380—971no—
196C22×Q8—971no—
197C23×C4—971no—
198C2×C4⋊C4—971no—
199C2×C4⋊C4—971no—
200C2×C4⋊C4—971no—
201C2×C4⋊C4—971no—
202C2×C4⋊C4—971no—
203C2×C4⋊C4—971no—
204C2×C4⋊C4—971no—
205C2×C4⋊C4—971no—
206C2×C4⋊C4—971no—
207C2×C4⋊C4—971no—
208C2×C4⋊C4C2×D9711yescharacteristic
209C2×C4⋊C4C2×D9711yescharacteristic
210C2×C4⋊C4C2×D9711yescharacteristic
211C2×C22⋊C4—971no—
212C2×C22⋊C4—971no—
213C2×C22⋊C4—971no—
214C2×C22⋊C4C2×D9711yescharacteristic
215C2×C22⋊C4C2×D9711yescharacteristic
216C2×C22⋊C4C2×D9711yescharacteristic
217#95—971no—
218#95—971no—
219#95—971no—
220#95—971no—
221#95—971no—
222#95—971no—
223#95—971no—
224#95—971no—
225#95—971no—
226#95—971no—
227#95—971no—
228#95—971no—
229#95—971no—
230#95C2×D9711yescharacteristic
231C24C22×D9711yescharacteristic
232C2×Q8—3881no—
233C2×Q8—3881no—
234C2×Q8—3881no—
235C22×C4—974no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—971no—
241C22×C4—971no—
242C22×C4—971no—
243C22×C4—971no—
244C22×C4—3881no—
245C22×C4—3881no—
246C22×C4C22×D9711yescharacteristic
247C22×C4C22×D9711yescharacteristic
248C22×C4C22×D9711yescharacteristic
249C22×C4C22×D9711yescharacteristic
250C22×C4C22×D9711yescharacteristic
251C22×C4C22×D9711yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—3881no—
256C4⋊C4—3881no—
257C4⋊C4—3881no—
258C4⋊C4—3881no—
259C4⋊C4—3881no—
260C4⋊C4—3881no—
261C4⋊C4—3881no—
262C4⋊C4—3881no—
263C4⋊C4—3881no—
264C4⋊C4—3881no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—3881no—
269C22⋊C4—3881no—
270C22⋊C4—3881no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23Q8×D9712yes—
278C23C23×D9711yescharacteristic
279Q8—3884no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—972no—
299C2×C4—972no—
300C2×C4—3884no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C2×Q8×D9711yescharacteristic
331C22#1569085511yescharacteristic
332C22#1569085511yescharacteristic
333C22#1569085511yescharacteristic
334C22#1569085611yescharacteristic
335C22#1569085711yescharacteristic
336C22#1569085711yescharacteristic
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3881no—
344C4—3881no—
345C4—3881no—
346C4—3881no—
347C4—3881no—
348C4—3881no—
349C4—1942no—
350C4—1942no—
351C2—42no—
352C2#2026136011yescharacteristic
353C2#2026136411yescharacteristic
354C2#2026136411yescharacteristic
355C2#2026147311yescharacteristic
356C2#2026148011yescharacteristic
357C2#2026148011yescharacteristic
358C2#2026148711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#396×C97C2
prime radicalC24×C97C23
hypercenter#396D97
hypocenterC97#2841
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #20260676C2semidirect
2. #20260673C2semidirect
4. #20260668C2semidirect
5. #20260666C2semidirect
6. #20260666C2semidirect
7. #20260665C2semidirect
9. #20260656C2semidirect
10. #20260655C2semidirect
56. C22×Dic97Q8semidirect
180. C97#2841semidirect

Maximal subgroups

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

subgroupindex
#202606762
#202606732
#202606712
#202606682
#202606662
#202606652
#202606582
#202606562
#202606552
#202606472
#202605962
#202605932
#396×C972
#284197