Finite Group Databasebeta

← #27145093

Subgroups — #27145093

Order 12416.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 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
1C23×D388C211yesmaximal, characteristic, complemented
2C2×#15690785C212yesmaximal, complemented
3C2×#15690780C212yesmaximal, complemented
4C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
5#20260653C212yesmaximal, complemented
6#20260645C212yesmaximal, complemented
7#20260636C212yesmaximal, complemented
8#20260590C212yesmaximal, complemented
9#380×C97C211yesmaximal, characteristic, complemented
10C24×D97C2212yes—
11C22×D388—28no—
12C22×D388—24no—
13C22×D388C2212yes—
14C22×D388C2212yes—
15C22×D388C2214yescomplemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic, complemented
18#15690785—28no—
19#15690780—28no—
20C4×C22⋊D97—24no—
21C2×#13095490C2211yescharacteristic, complemented
22C2×C22⋊Dic97C2211yescharacteristic
23C2×#13095488C2214yescomplemented
24C2×#13095488C2211yescharacteristic, complemented
25C2×#13095488C2214yescomplemented
26C2×#13095488C2214yescomplemented
27C2×C4×Dic97C2211yescharacteristic, complemented
28C23×C4×C97C2211yescharacteristic
29C2×C4⋊C4×C97C2212yescomplemented
30C2×C22⋊C4×C97C2212yes—
31#95×C97C2212yescomplemented
32C23×D97—24no—
33C23×D97—28no—
34C23×D97—24no—
35C23×D97—24no—
36C23×D97—44no—
37C23×D97C2314yes—
38C23×D97C2311yescharacteristic
39C2×D388—232no—
40C2×D388—28no—
41C2×D388—24no—
42C2×D388—24no—
43C2×D388—24no—
44C2×D388—28no—
45C2×D388—28no—
46C2×D388—44no—
47C2×D388D418yescomplemented
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97D414yescomplemented
51C22×Dic97C2311yescharacteristic
52C22×Dic97C2311yescharacteristic
53C22×Dic97C2311yescharacteristic
54C2×C4×D97—22no—
55C2×C4×D97—24no—
56#13095490—22no—
57C22⋊Dic97—22no—
58#13095488—28no—
59#13095488—22no—
60#13095488—28no—
61#13095488—44no—
62C4×Dic97—22no—
63C24×C97C2311yescharacteristic
64C22×C4×C97—24no—
65C22×C4×C97D414yescomplemented
66C22×C4×C97C2314yes—
67C22×C4×C97C2311yescharacteristic
68C22×C4×C97C2311yescharacteristic
69C4⋊C4×C97—24no—
70C22⋊C4×C97—24no—
71C23×C97—22no—
72C23×C97—22no—
73C23×C97C2×D412yes—
74C23×C97C2×D412yes—
75C23×C97#4112yes—
76C23×C97C2411yescharacteristic
77C22×D97—216no—
78C22×D97—21no—
79C22×D97—24no—
80C22×D97—22no—
81C22×D97—24no—
82C22×D97—24no—
83C22×D97—21no—
84C22×D97—24no—
85C22×D97—24no—
86C22×D97—22no—
87C22×D97—24no—
88C22×D97—416no—
89C22×D97—416no—
90C22×D97—48no—
91C22×D97—48no—
92C22×D97C2×D418yes—
93C22×D97C2×D412yescomplemented
94C2×C4×C97—22no—
95C2×C4×C97—28no—
96C2×C4×C97—24no—
97C2×C4×C97—24no—
98C2×C4×C97—24no—
99C2×C4×C97C2×D414yes—
100C2×C4×C97C2×D412yes—
101C2×C4×C97C2×D412yes—
102C22⋊D97—28no—
103C2×Dic97—21no—
104C2×Dic97—21no—
105C2×Dic97—21no—
106C2×Dic97—22no—
107C2×Dic97—21no—
108C2×Dic97—22no—
109C2×Dic97C2×D412yescomplemented
110D388—216no—
111D388—216no—
112D388—416no—
113C4×D97—44no—
114C22×C97—22no—
115C22×C97—22no—
116C22×C97—24no—
117C22×C97—24no—
118C22×C97#12114yescomplemented
119C22×C97C22×D412yes—
120C22×C97C22×D411yescharacteristic
121C22×C97C22×D411yescharacteristic
122C22×C97C2×#4111yescharacteristic
123C22×C97H(5,𝔽2)12yes—
124C2×D97—28no—
125C2×D97—22no—
126C2×D97—24no—
127C2×D97—28no—
128C2×D97—28no—
129C2×D97—432no—
130C2×D97—41no—
131C2×D97—44no—
132C2×D97—41no—
133C2×D97—44no—
134C2×D97—44no—
135C2×D97—42no—
136C2×D97—44no—
137C4×C97—22no—
138C4×C97—44no—
139C4×C97#12014yescomplemented
140Dic97—22no—
141Dic97—41no—
142Dic97—41no—
143C2×C97—24no—
144C2×C97C2×#12011yescharacteristic
145C2×C97C2×#12111yescharacteristic
146C2×C97#53511yescharacteristic
147C2×C97D4212yes—
148C2×C97#54612yes—
149D97—42no—
150D97—48no—
151#2408—971nomaximal, Sylow
152C97#240811yescharacteristic, complemented, Sylow
153C23×D4—971no—
154C2×#121—972no—
155C2×#120—972no—
156C2×C4×D4—971no—
157#394—972no—
158#392—972no—
159#390—972no—
160#380D9711yescharacteristic, complemented
161#379—972no—
162C25—972no—
163C22×D4—971no—
164C22×D4—972no—
165C22×D4—974no—
166C22×D4—972no—
167C22×D4—1948no—
168C22×D4—1944no—
169C23×C4—971no—
170C23×C4C2×D9711yescharacteristic
171#121—1948no—
172#120—1948no—
173C4×D4—1944no—
174C2×C4⋊C4—971no—
175C2×C4⋊C4C2×D9712yescomplemented
176C2×C22⋊C4—974no—
177C2×C22⋊C4—971no—
178C2×C22⋊C4—971no—
179C2×C22⋊C4—974no—
180C2×C22⋊C4—974no—
181C2×C22⋊C4C2×D9712yes—
182C2×C42—971no—
183#95C2×D9712yescomplemented
184C24—974no—
185C24—971no—
186C24—3884no—
187C24—1948no—
188C24—1944no—
189C24—1944no—
190C24—1944no—
191C24C22×D9711yescharacteristic
192C2×D4—974no—
193C2×D4—978no—
194C2×D4—3884no—
195C2×D4—1948no—
196C2×D4—1948no—
197C2×D4—1948no—
198C2×D4—19432no—
199C2×D4—1942no—
200C2×D4—1942no—
201C2×D4—1944no—
202C2×D4—1944no—
203C2×D4—1944no—
204C22×C4—24no—
205C22×C4—971no—
206C22×C4—971no—
207C22×C4—971no—
208C22×C4—1944no—
209C22×C4—1942no—
210C22×C4D38814yescomplemented
211C22×C4C22×D9714yes—
212C22×C4C22×D9711yescharacteristic
213C22×C4C22×D9711yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—1942no—
216C22⋊C4—24no—
217C22⋊C4—3884no—
218C22⋊C4—1948no—
219C22⋊C4—1948no—
220C22⋊C4—1942no—
221C22⋊C4—1942no—
222C42—1942no—
223C23—22no—
224C23—22no—
225C23—978no—
226C23—972no—
227C23—3888no—
228C23—38816no—
229C23—38816no—
230C23—3888no—
231C23—19416no—
232C23—1944no—
233C23—1942no—
234C23—1944no—
235C23—1941no—
236C23—1944no—
237C23—1941no—
238C23—1944no—
239C23—1944no—
240C23—1942no—
241C23—1944no—
242C23D4×D9712yes—
243C23C2×D38812yes—
244C23#1309551212yes—
245C23C23×D9711yescharacteristic
246D4—38816no—
247D4—19416no—
248D4—1948no—
249D4—19416no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—972no—
256C2×C4—3884no—
257C2×C4—1942no—
258C2×C4—1941no—
259C2×C4—1941no—
260C2×C4—1942no—
261C2×C4—1941no—
262C2×C4—1941no—
263C2×C4D4×D9714yes—
264C2×C4C2×D38812yes—
265C2×C4C2×D38812yes—
266C22—24no—
267C22—22no—
268C22—22no—
269C22—24no—
270C22—38832no—
271C22—3884no—
272C22—3882no—
273C22—3881no—
274C22—3884no—
275C22—3884no—
276C22—3881no—
277C22—3884no—
278C22—1948no—
279C22—1948no—
280C22—1942no—
281C22—1944no—
282C22—1948no—
283C22#1569078514yescomplemented
284C22C2×D4×D9712yes—
285C22C2×D4×D9711yescharacteristic
286C22C22×D38811yescharacteristic
287C22C2×#1309551211yescharacteristic
288C22#1569085612yes—
289C4—22no—
290C4—44no—
291C4—3881no—
292C4—3881no—
293C4—1942no—
294C4#1569078014yescomplemented
295C2—24no—
296C2—3888no—
297C2—3882no—
298C2C2×#1569078011yescharacteristic
299C2C2×#1569078511yescharacteristic
300C2#2026130411yescharacteristic
301C2#2026136912yes—
302C2#2026137612yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#380×C97C2
hypercenter#380D97
hypocenterC97#2408
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D388C2semidirect
2. C2×#15690785C2semidirect
3. C2×#15690780C2semidirect
4. C2×C4×C22⋊D97C2semidirect
5. #20260653C2semidirect
6. #20260645C2semidirect
7. #20260636C2semidirect
8. #20260590C2semidirect
9. #380×C97C2semidirect
15. C22×D388C22semidirect
17. C22×C4×D97C22semidirect
21. C2×#13095490C22semidirect
23. C2×#13095488C22semidirect
24. C2×#13095488C22semidirect
25. C2×#13095488C22semidirect
26. C2×#13095488C22semidirect
27. C2×C4×Dic97C22semidirect
29. C2×C4⋊C4×C97C22semidirect
31. #95×C97C22semidirect
47. C2×D388D4semidirect
50. C2×C22⋊D97D4semidirect
65. C22×C4×C97D4semidirect
93. C22×D97C2×D4semidirect
109. C2×Dic97C2×D4semidirect
118. C22×C97#121semidirect
139. C4×C97#120semidirect
152. C97#2408semidirect
160. #380D97semidirect
175. C2×C4⋊C4C2×D97semidirect
183. #95C2×D97semidirect
210. C22×C4D388semidirect
283. C22#15690785semidirect
294. C4#15690780semidirect

Maximal subgroups

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

subgroupindex
C23×D3882
C2×#156907852
C2×#156907802
C2×C4×C22⋊D972
#202606532
#202606452
#202606362
#202605902
#380×C972
#240897