Finite Group Databasebeta

← #27144632

Subgroups — #27144632

Order 12416.

776 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D97C211yesmaximal, characteristic
2C2×#15690770C212yesmaximal
3C22×C4⋊Dic97C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
5#20260632C212yesmaximal, complemented
6#20260609C212yesmaximal, complemented
7#20260607C212yesmaximal, complemented
8#20260604C211yesmaximal, characteristic, complemented
9#20260592C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
11#380×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97C2211yescharacteristic
13C23×Dic97C2212yes—
14C22×C4×D97C2212yes—
15C22×C4×D97C2211yescharacteristic
16C4×C22⋊D97C418yescomplemented
17#15690770—28no—
18C2×C4⋊Dic97—24no—
19C2×C4⋊Dic97C2212yes—
20C2×C4⋊Dic97C2211yescharacteristic
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490C2212yes—
23C2×#13095490C2212yes—
24C2×#13095490C2212yes—
25C2×#13095490C2211yescharacteristic
26C4⋊C4×D97—24no—
27C2×C22⋊Dic97C2211yescharacteristic
28C2×#13095488C2212yes—
29C2×#13095488C2211yescharacteristic
30C2×C4×Dic97C2212yes—
31C2×C4×Dic97C2211yescharacteristic
32#15690717C2212yes—
33#15690717C2212yes—
34C23×C4×C97C2211yescharacteristic
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C4⋊C4×C97C2211yescharacteristic
37C2×C22⋊C4×C97C2212yes—
38#95×C97C2212yes—
39C23×D97C2311yescharacteristic
40C2×C22⋊D97C2×C414yes—
41C2×C22⋊D97D414yes—
42C2×C22⋊D97Q814yes—
43C22×Dic97—24no—
44C22×Dic97—24no—
45C22×Dic97—42no—
46C22×Dic97C2312yes—
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2312yes—
50C22×Dic97C2311yescharacteristic
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97—24no—
54C2×C4×D97—24no—
55C2×C4×D97—42no—
56C2×C4×D97C2×C414yescomplemented
57C4⋊Dic97—22no—
58C4⋊Dic97—24no—
59C4⋊Dic97D414yescomplemented
60#13095490—24no—
61#13095490—24no—
62#13095490—42no—
63#13095490C2×C414yescomplemented
64C22⋊Dic97C2×C414yescomplemented
65#13095488—24no—
66#13095488C2×C414yescomplemented
67C4×Dic97—24no—
68C4×Dic97C2×C414yescomplemented
69C24×C97C2311yescharacteristic
70C22×C4×C97—22no—
71C22×C4×C97—22no—
72C22×C4×C97C2×C414yescomplemented
73C22×C4×C97C2312yes—
74C22×C4×C97C2312yes—
75C22×C4×C97C2311yescharacteristic
76C22×C4×C97C2311yescharacteristic
77C4⋊C4×C97—22no—
78C4⋊C4×C97—22no—
79C22⋊C4×C97—24no—
80C23×C97—21no—
81C23×C97—21no—
82C23×C97—21no—
83C23×C97—21no—
84C23×C97C22×C412yes—
85C23×C97C2×D412yes—
86C23×C97C2×Q812yes—
87C23×C97C2411yescharacteristic
88C22×D97—21no—
89C22×D97—21no—
90C22×D97—21no—
91C22×D97—21no—
92C22×D97C22×C412yes—
93C22×D97C2×D412yes—
94C22×D97C2×Q812yes—
95C2×C4×C97—21no—
96C2×C4×C97—21no—
97C2×C4×C97—28no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—22no—
102C2×C4×C97—22no—
103C2×C4×C97—22no—
104C2×C4×C97C22×C412yes—
105C2×C4×C97C22×C412yes—
106C2×C4×C97C2×D412yes—
107C2×C4×C97#4112yes—
108C22⋊D97C4⋊C4116yescomplemented
109C2×Dic97—22no—
110C2×Dic97—21no—
111C2×Dic97—21no—
112C2×Dic97—21no—
113C2×Dic97—22no—
114C2×Dic97—21no—
115C2×Dic97—22no—
116C2×Dic97—22no—
117C2×Dic97—22no—
118C2×Dic97—48no—
119C2×Dic97C22×C412yes—
120C2×Dic97C22×C412yes—
121C2×Dic97C22×C412yes—
122C2×Dic97C2×D414yes—
123C2×Dic97C2×D412yes—
124C2×Dic97C2×Q812yes—
125C4×D97—28no—
126C4×D97—48no—
127C22×C97—22no—
128C22×C97—22no—
129C22×C97—22no—
130C22×C97—22no—
131C22×C97—22no—
132C22×C97—22no—
133C22×C97C2×C4⋊C414yescomplemented
134C22×C97C23×C411yescharacteristic
135C22×C97C22×D411yescharacteristic
136C22×C97C22×D411yescharacteristic
137C22×C97C22×Q811yescharacteristic
138C22×C97C2×#4111yescharacteristic
139C22×C97H(5,𝔽2)11yescharacteristic
140C22×C972−1+411yescharacteristic
141C2×D97—22no—
142C2×D97—22no—
143C2×D97—22no—
144C2×D97—22no—
145C2×D97—22no—
146C2×D97—22no—
147C2×D97C2×C4⋊C414yescomplemented
148C4×C97—22no—
149C4×C97—42no—
150C4×C97—42no—
151C4×C97C4×D414yescomplemented
152Dic97—22no—
153Dic97—22no—
154Dic97—24no—
155Dic97—42no—
156Dic97C2×C4⋊C414yescomplemented
157C2×C97—24no—
158C2×C97C22×C4⋊C411yescharacteristic
159C2×C97C2×C4×D411yescharacteristic
160C2×C97#52011yescharacteristic
161C2×C97D4211yescharacteristic
162C2×C97#54711yescharacteristic
163C2×C97D4×Q811yescharacteristic
164C2×C97#55411yescharacteristic
165D97—24no—
166D4×C4⋊C4—971nomaximal, Sylow
167C97D4×C4⋊C411yescharacteristic, complemented, Sylow
168C2×C4×D4—971no—
169C2×C4×D4—972no—
170C22×C4⋊C4—971no—
171C22×C4⋊C4—971no—
172#389—972no—
173#385—972no—
174#385—972no—
175#384—971no—
176#380—971no—
177#380D9711yescharacteristic, complemented
178C4×C4⋊C4—971no—
179C22×D4—971no—
180C23×C4—971no—
181C23×C4—972no—
182C23×C4—972no—
183C23×C4C2×D9711yescharacteristic
184C4×D4—978no—
185C4×D4—1948no—
186C2×C4⋊C4—971no—
187C2×C4⋊C4—971no—
188C2×C4⋊C4—972no—
189C2×C4⋊C4—972no—
190C2×C4⋊C4—971no—
191C2×C4⋊C4—972no—
192C2×C4⋊C4—972no—
193C2×C4⋊C4—1944no—
194C2×C4⋊C4—1944no—
195C2×C4⋊C4C2×D9711yescharacteristic
196C2×C4⋊C4C2×D9711yescharacteristic
197C2×C22⋊C4—971no—
198C2×C22⋊C4—971no—
199C2×C22⋊C4—972no—
200C2×C22⋊C4C2×D9712yes—
201C2×C42—971no—
202C2×C42—972no—
203#95—972no—
204#95—972no—
205#95C2×D9712yes—
206C24—971no—
207C24C22×D9711yescharacteristic
208C2×D4—974no—
209C2×D4—974no—
210C2×D4—974no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—971no—
214C22×C4—971no—
215C22×C4—971no—
216C22×C4—972no—
217C22×C4—974no—
218C22×C4—972no—
219C22×C4—3882no—
220C22×C4—3882no—
221C22×C4—1944no—
222C22×C4—1944no—
223C22×C4—1944no—
224C22×C4—1944no—
225C22×C4—1942no—
226C22×C4—1942no—
227C22×C4C4×D9714yescomplemented
228C22×C4C22×D9712yes—
229C22×C4C22×D9712yes—
230C22×C4C22×D9711yescharacteristic
231C22×C4C22×D9711yescharacteristic
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—974no—
235C4⋊C4—974no—
236C4⋊C4—3882no—
237C4⋊C4—1944no—
238C4⋊C4—1944no—
239C4⋊C4—1944no—
240C4⋊C4—1942no—
241C22⋊C4—24no—
242C22⋊C4—974no—
243C22⋊C4—974no—
244C22⋊C4—1944no—
245C42—974no—
246C42—1944no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—972no—
252C23—972no—
253C23—972no—
254C23—1941no—
255C23—1941no—
256C23—1941no—
257C23—1941no—
258C23C2×C4×D9712yes—
259C23D4×D9712yes—
260C23Q8×D9712yes—
261C23C23×D9711yescharacteristic
262D4—9716no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—28no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—972no—
273C2×C4—972no—
274C2×C4—972no—
275C2×C4—972no—
276C2×C4—972no—
277C2×C4—974no—
278C2×C4—3888no—
279C2×C4—3888no—
280C2×C4—1942no—
281C2×C4—1942no—
282C2×C4—1942no—
283C2×C4—1942no—
284C2×C4—1941no—
285C2×C4—1941no—
286C2×C4—1941no—
287C2×C4—1942no—
288C2×C4—1941no—
289C2×C4—1948no—
290C2×C4C2×C4×D9712yes—
291C2×C4C2×C4×D9712yes—
292C2×C4D4×D9712yes—
293C2×C4#1309551012yes—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—974no—
301C22—1942no—
302C22—1942no—
303C22—1942no—
304C22—1942no—
305C22—1942no—
306C22—1942no—
307C22C4⋊C4×D9714yescomplemented
308C22C22×C4×D9711yescharacteristic
309C22C2×D4×D9711yescharacteristic
310C22C2×D4×D9711yescharacteristic
311C22C2×Q8×D9711yescharacteristic
312C22C2×#1309551011yescharacteristic
313C22#1569085611yescharacteristic
314C22#1569085711yescharacteristic
315C4—22no—
316C4—42no—
317C4—42no—
318C4—974no—
319C4—3882no—
320C4—1942no—
321C4—1942no—
322C4—1944no—
323C4#1569077014yescomplemented
324C2—24no—
325C2—1944no—
326C2C2×C4⋊C4×D9711yescharacteristic
327C2C2×#1569077011yescharacteristic
328C2#2026121511yescharacteristic
329C2#2026136911yescharacteristic
330C2#2026138511yescharacteristic
331C2#2026139811yescharacteristic
332C2#2026144311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#380×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenter#380D97
hypocenterC97D4×C4⋊C4
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊Dic97C2semidirect
4. C2×C4⋊C4×D97C2semidirect
5. #20260632C2semidirect
6. #20260609C2semidirect
7. #20260607C2semidirect
8. #20260604C2semidirect
9. #20260592C2semidirect
10. C4⋊C4×Dic97C2semidirect
11. #380×C97C2semidirect
16. C4×C22⋊D97C4semidirect
56. C2×C4×D97C2×C4semidirect
59. C4⋊Dic97D4semidirect
63. #13095490C2×C4semidirect
64. C22⋊Dic97C2×C4semidirect
66. #13095488C2×C4semidirect
68. C4×Dic97C2×C4semidirect
72. C22×C4×C97C2×C4semidirect
108. C22⋊D97C4⋊C4semidirect
133. C22×C97C2×C4⋊C4semidirect
147. C2×D97C2×C4⋊C4semidirect
151. C4×C97C4×D4semidirect
156. Dic97C2×C4⋊C4semidirect
167. C97D4×C4⋊C4semidirect
177. #380D97semidirect
227. C22×C4C4×D97semidirect
307. C22C4⋊C4×D97semidirect
323. C4#15690770semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D972
C2×#156907702
C22×C4⋊Dic972
C2×C4⋊C4×D972
#202606322
#202606092
#202606072
#202606042
#202605922
C4⋊C4×Dic972
#380×C972
D4×C4⋊C497