Finite Group Databasebeta

← #27150994

Subgroups — #27150994

Order 12416.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#13095511C211yesmaximal, characteristic
2C2×#15690841C212yesmaximal, complemented
3#20260895C214yesmaximal, complemented
4#20260890C212yesmaximal, complemented
5#20260885C212yesmaximal, complemented
6#20260734C212yesmaximal, complemented
7#20260699C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
9C2×#13095511—28no—
10C2×#13095511—21no—
11C2×#13095511—21no—
12C2×#13095511C2212yes—
13C2×#13095511C2212yes—
14C22×Dic194C2211yescharacteristic
15C22×D388C2211yescharacteristic
16C22×C22⋊D97—21no—
17C22×C4×D97—21no—
18#15690841—28no—
19C2×#13095501C2214yescomplemented
20C2×#13095498C2214yescomplemented
21C2×#13095495C2212yescomplemented
22#15690733C2214yescomplemented
23#15690730C2214yescomplemented
24C22⋊#11715895C2212yes—
25C22⋊#11715895C2212yescomplemented
26C22×Q8×C97C2211yescharacteristic, complemented
27C23×C4×C97C2211yescharacteristic
28C22⋊Q8×C97—21no—
29C22⋊Q8×C97—21no—
30C22⋊Q8×C97C2212yes—
31C22⋊Q8×C97C2212yescomplemented
32C2×C4⋊C4×C97—21no—
33C2×C4⋊C4×C97C2211yescharacteristic, complemented
34C2×C22⋊C4×C97—21no—
35C23×D97—21no—
36#13095511—28no—
37#13095511—28no—
38#13095511—24no—
39#13095511—24no—
40#13095511—24no—
41#13095511D418yes—
42C2×Dic194—21no—
43C2×Dic194—24no—
44C2×Dic194C2312yes—
45C2×D388—21no—
46C2×D388—24no—
47C2×D388C2312yes—
48C2×C22⋊D97—28no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—22no—
51C22×Dic97—21no—
52C2×C4×D97—28no—
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55#13095501—28no—
56#13095498—28no—
57#13095495—24no—
58C2×#11715895C2314yes—
59C24×C97D411yescharacteristic
60C2×Q8×C97—21no—
61C2×Q8×C97—24no—
62C2×Q8×C97C2312yes—
63C22×C4×C97—24no—
64C22×C4×C97—21no—
65C22×C4×C97—21no—
66C22×C4×C97D411yescharacteristic
67C22×C4×C97D411yescharacteristic
68C22×C4×C97D411yescharacteristic
69C22×C4×C97C2311yescharacteristic
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2311yescharacteristic
72C4⋊C4×C97—22no—
73C4⋊C4×C97—22no—
74C4⋊C4×C97—21no—
75C4⋊C4×C97C2312yes—
76C22⋊C4×C97—22no—
77C22⋊C4×C97—22no—
78C23×C97—22no—
79C23×C97—22no—
80C23×C97—41no—
81C23×C97C2×D411yescharacteristic
82C23×C97C2×D411yescharacteristic
83C23×C97C2×D411yescharacteristic
84C22×D97—24no—
85C22×D97—22no—
86C22×D97—42no—
87C22×D97—42no—
88Q8×C97—24no—
89Q8×C97—42no—
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—22no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—42no—
98C2×C4×C97—42no—
99C2×C4×C97C2×D414yes—
100C2×C4×C97C2×D411yescharacteristic
101C2×C4×C97C2×D411yescharacteristic
102C2×C4×C97C2×D411yescharacteristic
103C2×C4×C97C2411yescharacteristic
104C22⋊D97—28no—
105C22⋊D97—28no—
106C2×Dic97—24no—
107C2×Dic97—22no—
108D388—24no—
109D388—22no—
110D388C2×D414yes—
111C4×D97—28no—
112C4×D97—28no—
113Dic194—24no—
114Dic194—22no—
115Dic194C2×D414yes—
116#11715895—24no—
117C22×C97—21no—
118C22×C97—21no—
119C22×C97—21no—
120C22×C97—21no—
121C22×C97—21no—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—44no—
125C22×C97#12012yes—
126C22×C97#13712yescomplemented
127C22×C97C22×D411yescharacteristic
128C2×D97—24no—
129C2×D97—48no—
130C2×D97—42no—
131C2×D97—42no—
132C4×C97—21no—
133C4×C97—21no—
134C4×C97—42no—
135C4×C97—42no—
136C4×C97#12012yes—
137C4×C97C22×D412yes—
138Dic97—24no—
139C2×C97—22no—
140C2×C97—22no—
141C2×C97—41no—
142C2×C97C2×#12011yescharacteristic
143C2×C97C2×#13711yescharacteristic
144C2×C97#57511yescharacteristic
145D97—44no—
146#3007—971nomaximal, Sylow
147C97#300711yescharacteristic, complemented, Sylow
148C22×#41—971no—
149C2×#137—972no—
150C2×C22⋊Q8D9711yescharacteristic, complemented
151#452—974no—
152#451—972no—
153#450—972no—
154#417—972no—
155#407—971no—
156C2×#41—972no—
157C2×#41—972no—
158C2×#41—1948no—
159C2×#41—1941no—
160C2×#41—1941no—
161C22×Q8—971no—
162C22×Q8C2×D9711yescharacteristic, complemented
163C22×D4—971no—
164C22×D4—1941no—
165C23×C4—1941no—
166C23×C4C2×D9711yescharacteristic
167#137—1948no—
168C2×Q16—974no—
169C2×SD16—974no—
170C2×M16—972no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D9712yes—
174C22⋊Q8C2×D9712yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D9711yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—974no—
179#102—974no—
180C22⋊C8—972no—
181C22⋊C8—972no—
182C24—1941no—
183C24C22⋊D9711yescharacteristic
184#41—978no—
185#41—1944no—
186#41—1944no—
187#41—1948no—
188#41—1948no—
189#41—1944no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—972no—
193C2×Q8—1944no—
194C2×Q8—1941no—
195C2×Q8C22×D9712yes—
196C2×D4—972no—
197C2×D4—1944no—
198C2×D4—1948no—
199C2×D4—1942no—
200C2×D4—1942no—
201C2×D4—1941no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—1948no—
206C22×C4—1941no—
207C22×C4—1942no—
208C22×C4—1942no—
209C22×C4C22⋊D9711yescharacteristic
210C22×C4C22⋊D9711yescharacteristic
211C22×C4C22⋊D9711yescharacteristic
212C22×C4C22×D9711yescharacteristic
213C22×C4C22×D9711yescharacteristic
214C22×C4C22×D9711yescharacteristic
215Q16—1948no—
216SD16—1948no—
217M16—1944no—
218C2×C8—974no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D9712yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3882no—
229C23—3882no—
230C23—1944no—
231C23—1942no—
232C23C2×C22⋊D9711yescharacteristic
233C23C2×C22⋊D9711yescharacteristic
234C23C2×C22⋊D9711yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—974no—
238Q8—1944no—
239Q8—1942no—
240D4—974no—
241D4—1944no—
242D4—1948no—
243D4—1948no—
244D4—1942no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1948no—
255C2×C4—1944no—
256C2×C4—1942no—
257C2×C4—1948no—
258C2×C4D4×D9714yes—
259C2×C4C2×C22⋊D9711yescharacteristic
260C2×C4C2×C22⋊D9711yescharacteristic
261C2×C4C2×C22⋊D9711yescharacteristic
262C2×C4C23×D9711yescharacteristic
263C8—1944no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—3888no—
273C22—3882no—
274C22—3882no—
275C22—1944no—
276C22#1569077812yes—
277C22#1569084112yescomplemented
278C22C22×C22⋊D9711yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—1944no—
284C4#1569077812yes—
285C4C2×D4×D9712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—3884no—
290C2C2×#1569077811yescharacteristic
291C2C2×#1569084111yescharacteristic
292C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C22⋊Q8×C97C2
socleC22×C97C22×D4
prime radicalC22×#13095511C2
hypercenterC2×C22⋊Q8D97
hypocenterC97#3007
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15690841C2semidirect
3. #20260895C2semidirect
4. #20260890C2semidirect
5. #20260885C2semidirect
6. #20260734C2semidirect
7. #20260699C2semidirect
8. C2×C22⋊Q8×C97C2semidirect
19. C2×#13095501C22semidirect
20. C2×#13095498C22semidirect
21. C2×#13095495C22semidirect
22. #15690733C22semidirect
23. #15690730C22semidirect
25. C22⋊#11715895C22semidirect
26. C22×Q8×C97C22semidirect
31. C22⋊Q8×C97C22semidirect
33. C2×C4⋊C4×C97C22semidirect
126. C22×C97#137semidirect
147. C97#3007semidirect
150. C2×C22⋊Q8D97semidirect
162. C22×Q8C2×D97semidirect
174. C22⋊Q8C2×D97semidirect
176. C2×C4⋊C4C2×D97semidirect
277. C22#15690841semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156908412
#202608952
#202608902
#202608852
#202607342
#202606992
C2×C22⋊Q8×C972
#300797