Finite Group Databasebeta

← #27144667

Subgroups — #27144667

Order 12416.

660 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
1C2×C4×C22⋊D97C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3C22×C4⋊Dic97C211yesmaximal, characteristic, complemented
4#20260627C212yesmaximal, complemented
5#20260622C212yesmaximal, complemented
6#20260612C212yesmaximal
7#20260600C211yesmaximal, characteristic, complemented
8#20260592C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic97C212yesmaximal, complemented
11#380×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97C2211yescharacteristic
13C23×Dic97C2212yes—
14C22×C4×D97C2211yescharacteristic
15C4×C22⋊D97C418yescomplemented
16#15690768—24no—
17C2×C4⋊Dic97—24no—
18C2×C4⋊Dic97C2212yes—
19C2×C4⋊Dic97C2211yescharacteristic
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×C22⋊Dic97C2211yescharacteristic
23C2×#13095488C2212yes—
24C2×#13095488C2212yes—
25C2×#13095488C2211yescharacteristic
26C2×C4×Dic97C2212yes—
27C2×C4×Dic97C2212yes—
28C2×C4×Dic97C2211yescharacteristic
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33C23×C4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2211yescharacteristic
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C22⋊C4×C97C2212yes—
37#95×C97C2212yes—
38C23×D97C2311yescharacteristic
39C2×C22⋊D97—22no—
40C2×C22⋊D97—22no—
41C2×C22⋊D97C2×C414yes—
42C22×Dic97—24no—
43C22×Dic97—24no—
44C22×Dic97—42no—
45C22×Dic97C2312yes—
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2312yes—
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97C2×C414yescomplemented
53C4⋊Dic97—22no—
54C4⋊Dic97—24no—
55C4⋊Dic97D414yescomplemented
56#13095490C2×C414yescomplemented
57C22⋊Dic97C2×C414yes—
58#13095488—24no—
59#13095488—42no—
60#13095488C2×C414yescomplemented
61C4×Dic97—24no—
62C4×Dic97—42no—
63C4×Dic97C2×C414yescomplemented
64C24×C97C2311yescharacteristic
65C22×C4×C97—22no—
66C22×C4×C97—22no—
67C22×C4×C97C2×C414yes—
68C22×C4×C97C2312yes—
69C22×C4×C97C2312yes—
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2311yescharacteristic
72C4⋊C4×C97—22no—
73C4⋊C4×C97—22no—
74C22⋊C4×C97—24no—
75C23×C97—21no—
76C23×C97—21no—
77C23×C97—21no—
78C23×C97—21no—
79C23×C97C22×C412yes—
80C23×C97#4112yes—
81C23×C97#4112yes—
82C23×C97C2411yescharacteristic
83C22×D97—21no—
84C22×D97—21no—
85C22×D97—21no—
86C22×D97—21no—
87C22×D97—21no—
88C22×D97—21no—
89C22×D97C22×C412yes—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97—28no—
93C2×C4×C97—22no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97—22no—
99C2×C4×C97C22×C412yes—
100C2×C4×C97C22×C412yes—
101C2×C4×C97C2×D412yes—
102C2×C4×C97#4112yes—
103C22⋊D97—28no—
104C2×Dic97—21no—
105C2×Dic97—21no—
106C2×Dic97—22no—
107C2×Dic97—21no—
108C2×Dic97—21no—
109C2×Dic97—21no—
110C2×Dic97—22no—
111C2×Dic97—21no—
112C2×Dic97—22no—
113C2×Dic97—22no—
114C2×Dic97—22no—
115C2×Dic97—48no—
116C2×Dic97C22×C412yes—
117C2×Dic97C22×C412yes—
118C2×Dic97C22×C412yes—
119C2×Dic97C2×D414yes—
120C4×D97—28no—
121C22×C97—22no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97—22no—
125C22×C97—22no—
126C22×C97—22no—
127C22×C97#11714yescomplemented
128C22×C97C23×C411yescharacteristic
129C22×C97C22×D411yescharacteristic
130C22×C97C2×#4111yescharacteristic
131C22×C97C2×#4111yescharacteristic
132C22×C97C2×#4111yescharacteristic
133C22×C97H(5,𝔽2)11yescharacteristic
134C22×C972−1+411yescharacteristic
135C2×D97—22no—
136C2×D97—22no—
137C2×D97—22no—
138C2×D97—41no—
139C2×D97—41no—
140C2×D97—41no—
141C2×D97—41no—
142C4×C97—22no—
143C4×C97—42no—
144C4×C97—42no—
145C4×C97C4×D414yes—
146Dic97—22no—
147Dic97—22no—
148Dic97—22no—
149Dic97—24no—
150Dic97—42no—
151C2×C97—24no—
152C2×C97C2×#11711yescharacteristic
153C2×C97C2×C4×D411yescharacteristic
154C2×C97#52011yescharacteristic
155C2×C97#54611yescharacteristic
156C2×C97#54811yescharacteristic
157C2×C97#55211yescharacteristic
158C2×C97#55311yescharacteristic
159D97—42no—
160#2352—971nomaximal, Sylow
161C97#235211yescharacteristic, complemented, Sylow
162C2×C4×D4—971no—
163C2×#117—971no—
164C22×C4⋊C4—971no—
165#388—972no—
166#387—972no—
167#386—972no—
168#382—971no—
169#380—971no—
170#380D9711yescharacteristic, complemented
171C4×C4⋊C4—971no—
172C4×C22⋊C4—972no—
173C22×D4—971no—
174C23×C4—971no—
175C23×C4—972no—
176C23×C4C2×D9711yescharacteristic
177C4×D4—978no—
178#117—1944no—
179C2×C4⋊C4—971no—
180C2×C4⋊C4—971no—
181C2×C4⋊C4—971no—
182C2×C4⋊C4—972no—
183C2×C4⋊C4—1944no—
184C2×C4⋊C4C2×D9711yescharacteristic
185C2×C4⋊C4C2×D9711yescharacteristic
186C2×C22⋊C4—971no—
187C2×C22⋊C4—971no—
188C2×C22⋊C4—972no—
189C2×C22⋊C4—972no—
190C2×C22⋊C4C2×D9712yes—
191C2×C42—971no—
192C2×C42—972no—
193C2×C42—972no—
194#95—972no—
195#95—972no—
196#95—972no—
197#95—972no—
198#95C2×D9712yes—
199C24—971no—
200C24C22×D9711yescharacteristic
201C2×D4—974no—
202C2×D4—1942no—
203C2×D4—1942no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—971no—
207C22×C4—971no—
208C22×C4—971no—
209C22×C4—972no—
210C22×C4—974no—
211C22×C4—972no—
212C22×C4—3882no—
213C22×C4—1944no—
214C22×C4—1944no—
215C22×C4—1942no—
216C22×C4—1942no—
217C22×C4C4×D9714yes—
218C22×C4C22×D9712yes—
219C22×C4C22×D9712yes—
220C22×C4C22×D9711yescharacteristic
221C22×C4C22×D9711yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—974no—
225C4⋊C4—974no—
226C4⋊C4—1944no—
227C4⋊C4—1942no—
228C22⋊C4—24no—
229C22⋊C4—974no—
230C22⋊C4—974no—
231C22⋊C4—3882no—
232C22⋊C4—1944no—
233C42—974no—
234C42—3882no—
235C42—1944no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—972no—
241C23—1941no—
242C23—1941no—
243C23—1941no—
244C23—1941no—
245C23—1941no—
246C23—1941no—
247C23C2×C4×D9712yes—
248C23#1309551012yes—
249C23#1309551212yes—
250C23C23×D9711yescharacteristic
251D4—1948no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—28no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—972no—
262C2×C4—972no—
263C2×C4—972no—
264C2×C4—974no—
265C2×C4—3888no—
266C2×C4—1942no—
267C2×C4—1942no—
268C2×C4—1942no—
269C2×C4—1941no—
270C2×C4—1941no—
271C2×C4—1941no—
272C2×C4—1942no—
273C2×C4—1941no—
274C2×C4—1941no—
275C2×C4—1942no—
276C2×C4—1941no—
277C2×C4—1948no—
278C2×C4C2×C4×D9712yes—
279C2×C4C2×C4×D9712yes—
280C2×C4D4×D9712yes—
281C2×C4#1309551012yes—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—3881no—
289C22—3881no—
290C22—3881no—
291C22—3881no—
292C22—1942no—
293C22—1942no—
294C22—1942no—
295C22#1569076814yescomplemented
296C22C22×C4×D9711yescharacteristic
297C22C2×D4×D9711yescharacteristic
298C22C2×#1309551011yescharacteristic
299C22C2×#1309551011yescharacteristic
300C22C2×#1309551211yescharacteristic
301C22#1569085611yescharacteristic
302C22#1569085711yescharacteristic
303C4—22no—
304C4—42no—
305C4—42no—
306C4—3882no—
307C4—1944no—
308C4—1942no—
309C4—1942no—
310C4—1942no—
311C4#1569077014yes—
312C2—24no—
313C2—3882no—
314C2C2×#1569076811yescharacteristic
315C2C2×#1569077011yescharacteristic
316C2#2026121511yescharacteristic
317C2#2026137611yescharacteristic
318C2#2026139111yescharacteristic
319C2#2026142711yescharacteristic
320C2#2026143711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#380×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenter#380D97
hypocenterC97#2352
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. C22×C4⋊Dic97C2semidirect
4. #20260627C2semidirect
5. #20260622C2semidirect
7. #20260600C2semidirect
8. #20260592C2semidirect
9. C4⋊C4×Dic97C2semidirect
10. C22⋊C4×Dic97C2semidirect
11. #380×C97C2semidirect
15. C4×C22⋊D97C4semidirect
52. C2×C4×D97C2×C4semidirect
55. C4⋊Dic97D4semidirect
56. #13095490C2×C4semidirect
60. #13095488C2×C4semidirect
63. C4×Dic97C2×C4semidirect
127. C22×C97#117semidirect
161. C97#2352semidirect
170. #380D97semidirect
295. C22#15690768semidirect

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×#156907682
C22×C4⋊Dic972
#202606272
#202606222
#202606122
#202606002
#202605922
C4⋊C4×Dic972
C22⋊C4×Dic972
#380×C972
#235297