Finite Group Databasebeta

← #27150080

Subgroups — #27150080

Order 12416.

924 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690806C211yesmaximal, characteristic, complemented
2C2×#15690785C211yesmaximal, characteristic, complemented
3C2×#15690783C214yesmaximal, complemented
4C2×#15690781C211yesmaximal, characteristic, complemented
5C2×C42×D97C211yesmaximal, characteristic, complemented
6#20260637C211yesmaximal, characteristic, complemented
7#20260635C214yesmaximal, complemented
8#20260604C211yesmaximal, characteristic, complemented
9#390×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C22×D388C2212yes—
12C22×C22⋊D97C2214yes—
13C22×C22⋊D97C2212yes—
14C22×C4×D97C2211yescharacteristic
15C22×C4×D97C2212yes—
16#15690806—24no—
17#15690785—24no—
18#15690783—216no—
19#15690781—24no—
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2214yes—
22C2×C22⋊Dic97C2212yes—
23C2×#13095488C2212yes—
24C2×#13095488C2214yes—
25C2×C4×Dic97C2212yes—
26C2×C4×Dic97C2211yescharacteristic
27C42×D97—24no—
28C22×D4×C97C2211yescharacteristic
29C2×C4⋊C4×C97C2211yescharacteristic
30C2×C22⋊C4×C97C2214yes—
31C2×C42×C97C2211yescharacteristic
32C23×D97C2312yes—
33C23×D97C2311yescharacteristic
34C2×D388—28no—
35C2×D388—24no—
36C2×D388—22no—
37C2×D388—22no—
38C2×D388—22no—
39C2×C22⋊D97—24no—
40C2×C22⋊D97—28no—
41C2×C22⋊D97—28no—
42C2×C22⋊D97—28no—
43C2×C22⋊D97—28no—
44C22×Dic97C2312yes—
45C22×Dic97C2311yescharacteristic
46C22×Dic97C2312yes—
47C2×C4×D97—22no—
48C2×C4×D97—22no—
49C2×C4×D97—24no—
50C2×C4×D97—24no—
51C2×C4×D97D414yescomplemented
52C2×C4×D97D418yescomplemented
53C4⋊Dic97—22no—
54#13095490—28no—
55C22⋊Dic97—24no—
56#13095488—28no—
57#13095488—24no—
58C4×Dic97—24no—
59C4×Dic97D414yescomplemented
60C24×C97C2312yes—
61C2×D4×C97—22no—
62C2×D4×C97—22no—
63C2×D4×C97—22no—
64C22×C4×C97C2312yes—
65C22×C4×C97C2312yes—
66C22×C4×C97C2311yescharacteristic
67C4⋊C4×C97—22no—
68C22⋊C4×C97—28no—
69C42×C97—22no—
70C23×C97—24no—
71C23×C97—22no—
72C23×C97—22no—
73C23×C97—22no—
74C23×C97—24no—
75C23×C97C2411yescharacteristic
76C22×D97—21no—
77C22×D97—22no—
78C22×D97—21no—
79C22×D97—24no—
80C22×D97—22no—
81C22×D97—22no—
82C22×D97—24no—
83C22×D97—22no—
84C22×D97C2×D412yes—
85C22×D97C2×D414yes—
86D4×C97—44no—
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97—24no—
90C2×C4×C97—22no—
91C2×C4×C97C2×D414yes—
92C2×C4×C97C2×D412yes—
93C2×C4×C97C2×D412yes—
94C2×C4×C97#4112yes—
95C22⋊D97—48no—
96C22⋊D97—416no—
97C2×Dic97—21no—
98C2×Dic97—21no—
99C2×Dic97—24no—
100C2×Dic97—22no—
101C2×Dic97C2×D412yes—
102C2×Dic97C2×D414yes—
103C2×Dic97C2×D414yes—
104C2×Dic97#4114yes—
105D388—48no—
106D388—44no—
107C4×D97—28no—
108C4×D97—216no—
109C22×C97—44no—
110C22×C97—42no—
111C22×C97—42no—
112C22×C97—44no—
113C22×C97—42no—
114C22×C97C22×D411yescharacteristic
115C22×C97C22×D412yes—
116C22×C97C22×D411yescharacteristic
117C22×C97C2×#4112yes—
118C22×C97C2×#4111yescharacteristic
119C2×D97—24no—
120C2×D97—22no—
121C2×D97—24no—
122C2×D97—22no—
123C2×D97—42no—
124C2×D97—44no—
125C2×D97—42no—
126C2×D97—44no—
127C2×D97—42no—
128C2×D97#12714yescomplemented
129C4×C97—24no—
130C4×C97—42no—
131C4×C97#12114yescomplemented
132Dic97—22no—
133Dic97—42no—
134Dic97#12118yescomplemented
135C2×C97—82no—
136C2×C97C2×#12112yes—
137C2×C97C2×#12111yescharacteristic
138C2×C97C2×#12711yescharacteristic
139C2×C97#53212yes—
140C2×C97#53211yescharacteristic
141D97—24no—
142D97—82no—
143#2856—971nomaximal, Sylow
144C97#285611yescharacteristic, complemented, Sylow
145C2×#127—971no—
146C2×#121—971no—
147C2×#121—971no—
148C2×#121—974no—
149C22×C42—971no—
150#390—974no—
151#390—971no—
152#390D9711yescharacteristic, complemented
153#384—971no—
154C22×D4—972no—
155C22×D4—974no—
156C22×D4—972no—
157C22×D4—971no—
158C22×D4C2×D9711yescharacteristic
159C23×C4—972no—
160C23×C4—971no—
161#127—1944no—
162#121—19416no—
163#121—1944no—
164#121—1944no—
165C2×C4⋊C4—974no—
166C2×C4⋊C4—971no—
167C2×C4⋊C4C2×D9711yescharacteristic
168C2×C22⋊C4—972no—
169C2×C22⋊C4—972no—
170C2×C22⋊C4—974no—
171C2×C22⋊C4C2×D9714yes—
172C2×C42—971no—
173C2×C42—972no—
174C2×C42—1944no—
175C2×C42C2×D9711yescharacteristic
176C24—971no—
177C24—972no—
178C24C22×D9712yes—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—1948no—
183C2×D4—1944no—
184C2×D4—1948no—
185C2×D4—1948no—
186C2×D4—1948no—
187C2×D4—1948no—
188C2×D4—1944no—
189C2×D4—1942no—
190C2×D4—1942no—
191C2×D4—1942no—
192C22×C4—972no—
193C22×C4—971no—
194C22×C4—978no—
195C22×C4—974no—
196C22×C4—972no—
197C22×C4—1944no—
198C22×C4—1944no—
199C22×C4—1942no—
200C22×C4—1942no—
201C22×C4C22×D9712yes—
202C22×C4C22×D9712yes—
203C22×C4C22×D9711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—1948no—
206C4⋊C4—1942no—
207C22⋊C4—28no—
208C22⋊C4—1944no—
209C22⋊C4—1944no—
210C22⋊C4—1948no—
211C42—22no—
212C42—974no—
213C42—1944no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—24no—
218C23—22no—
219C23—974no—
220C23—972no—
221C23—1944no—
222C23—1942no—
223C23—1942no—
224C23—1944no—
225C23—1942no—
226C23—1941no—
227C23—1942no—
228C23—1941no—
229C23C23×D9711yescharacteristic
230D4—44no—
231D4—3888no—
232D4—38816no—
233D4—3888no—
234D4—3884no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—974no—
240C2×C4—974no—
241C2×C4—974no—
242C2×C4—972no—
243C2×C4—19416no—
244C2×C4—1944no—
245C2×C4—1942no—
246C2×C4—1941no—
247C2×C4—1941no—
248C2×C4—1948no—
249C2×C4D4×D9714yes—
250C2×C4D4×D9712yes—
251C2×C4D4×D9712yes—
252C2×C4#1309551112yes—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—974no—
259C22—3884no—
260C22—3882no—
261C22—3882no—
262C22—3884no—
263C22—3882no—
264C22—1944no—
265C22—1942no—
266C22—1944no—
267C22—1942no—
268C22C2×D4×D9711yescharacteristic
269C22C2×D4×D9712yes—
270C22C2×D4×D9711yescharacteristic
271C22#41×D9712yes—
272C22C2×#1309551111yescharacteristic
273C4—24no—
274C4—42no—
275C4—978no—
276C4—3882no—
277C4—1942no—
278C4#1569078314yescomplemented
279C2—82no—
280C2—7762no—
281C2—1944no—
282C2#121×D9712yes—
283C2C2×#1569078311yescharacteristic
284C2#127×D9711yescharacteristic
285C2#2026128212yes—
286C2#2026128411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#390×C97C2
hypercenter#390D97
hypocenterC97#2856
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#15690806C2semidirect
2. C2×#15690785C2semidirect
3. C2×#15690783C2semidirect
4. C2×#15690781C2semidirect
5. C2×C42×D97C2semidirect
6. #20260637C2semidirect
7. #20260635C2semidirect
8. #20260604C2semidirect
9. #390×C97C2semidirect
51. C2×C4×D97D4semidirect
52. C2×C4×D97D4semidirect
59. C4×Dic97D4semidirect
128. C2×D97#127semidirect
131. C4×C97#121semidirect
134. Dic97#121semidirect
144. C97#2856semidirect
152. #390D97semidirect
278. C4#15690783semidirect

Maximal subgroups

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

subgroupindex
C2×#156908062
C2×#156907852
C2×#156907832
C2×#156907812
C2×C42×D972
#202606372
#202606352
#202606042
#390×C972
#285697