Finite Group Databasebeta

← #27150195

Subgroups — #27150195

Order 12416.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#15690856C211yesmaximal, characteristic
2#20261209C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic, complemented
4#20260708C212yesmaximal, complemented
5#99×D97C218yesmaximal, complemented
6#410×C97C211yesmaximal, characteristic, complemented
7#15690856—24no—
8#15690856—24no—
9C2×#13095512C412yescomplemented
10C2×#13095511C412yescomplemented
11C2×#13095511C2211yescharacteristic
12C22×D388C412yescomplemented
13C22×D388C2211yescharacteristic
14C2×D4×D97C412yescomplemented
15C2×D4×D97C2214yes—
16C4×D388—24no—
17#15690772—24no—
18#15690769C2212yescomplemented
19C22⋊C4×D97C2218yescomplemented
20#15690725C2214yescomplemented
21#15690724C2218yescomplemented
22C2×#41×C97C2211yescharacteristic
23#117×C97C2212yescomplemented
24#99×C97C2214yescomplemented
25C23×D97C2×C414yescomplemented
26C23×D97C2312yes—
27#13095512—28no—
28#13095511—22no—
29#13095511—22no—
30#13095511—24no—
31#13095511—24no—
32C2×Dic194C2×C411yescharacteristic, complemented
33C2×D388—24no—
34C2×D388—24no—
35C2×D388C2×C412yes—
36C2×D388C2×C412yes—
37C2×D388C2×C414yescomplemented
38C2×D388C2×C411yescharacteristic, complemented
39C2×D388D414yes—
40C2×D388D414yes—
41C2×C22⋊D97C2×C414yescomplemented
42C2×C22⋊D97C2312yes—
43D4×D97—216no—
44D4×D97—28no—
45C2×C4×D97—28no—
46C2×C4×D97C2×C412yescomplemented
47C2×C4×D97C2×C414yescomplemented
48C4⋊Dic97—22no—
49C22⋊Dic97C2314yes—
50#13095488—28no—
51C4×Dic97—22no—
52#41×C97—22no—
53#41×C97—22no—
54C2×Q8×C97C2×C411yescharacteristic, complemented
55C2×D4×C97C2×C411yescharacteristic, complemented
56C2×D4×C97C2312yes—
57C22×C4×C97C2×C412yescomplemented
58C22×C4×C97C2311yescharacteristic
59C4⋊C4×C97—22no—
60C22⋊C4×C97C2314yes—
61C42×C97—22no—
62C23×C97C22×C412yes—
63C23×C97C2411yescharacteristic
64C22×D97—28no—
65C22×D97—82no—
66C22×D97—84no—
67C22×D97C22×C412yes—
68C22×D97C22×C414yes—
69C22×D97C2×D418yes—
70Q8×C97—22no—
71D4×C97—24no—
72D4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—24no—
76C2×C4×C97C22×C412yes—
77C2×C4×C97C22×C411yescharacteristic
78C2×C4×C97C22×C411yescharacteristic
79C2×C4×C97C2×D412yes—
80C2×C4×C97C2×D412yes—
81C22⋊D97—24no—
82C22⋊D97—28no—
83C2×Dic97—24no—
84C2×Dic97C22×C412yes—
85D388—28no—
86D388—21no—
87D388—21no—
88D388C22⋊C418yes—
89D388C22⋊C414yes—
90D388C22⋊C414yes—
91C4×D97—28no—
92C4×D97—24no—
93C4×D97—88no—
94Dic194—21no—
95Dic194—21no—
96C22×C97—22no—
97C22×C97—41no—
98C22×C97—42no—
99C22×C97C23×C411yescharacteristic
100C22×C97C22×D412yes—
101C2×D97—22no—
102C2×D97—48no—
103C2×D97—44no—
104C2×D97—42no—
105C2×D97—88no—
106C2×D97C2×C22⋊C418yes—
107C4×C97—22no—
108C4×C97—44no—
109C4×C97C2×C22⋊C411yescharacteristic
110C4×C97C2×C22⋊C411yescharacteristic
111C4×C97C2×C22⋊C412yes—
112Dic97—22no—
113Dic97—44no—
114C2×C97—22no—
115C2×C97—21no—
116C2×C97—42no—
117C2×C97C22×C22⋊C411yescharacteristic
118D97—28no—
119D97—42no—
120#2883—971nomaximal, Sylow
121C97#288311yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—971no—
123#518—972no—
124#410—971no—
125#410—972no—
126#410D9711yescharacteristic, complemented
127C2×#99—978no—
128H(5,𝔽2)—1944no—
129H(5,𝔽2)—1944no—
130C2×#41—972no—
131C2×#41—972no—
132C2×#41—971no—
133C2×#41C2×D9711yescharacteristic
134C22×D4—972no—
135C22×D4—974no—
136C22×D4—972no—
137C22×D4—971no—
138C4×D4—1944no—
139C4×D4—1944no—
140#117—972no—
141#117C2×D9712yescomplemented
142C2×C22⋊C4—978no—
143#99—974no—
144#99—978no—
145#99C2×D9714yescomplemented
146C24—972no—
147C24—974no—
148#41—22no—
149#41—22no—
150#41—1944no—
151#41—1944no—
152#41—1948no—
153#41—1942no—
154#41—1942no—
155C2×Q8—971no—
156C2×Q8C4×D9711yescharacteristic, complemented
157C2×D4—974no—
158C2×D4—972no—
159C2×D4—972no—
160C2×D4—972no—
161C2×D4—974no—
162C2×D4—971no—
163C2×D4—974no—
164C2×D4—974no—
165C2×D4—1944no—
166C2×D4—1944no—
167C2×D4—1948no—
168C2×D4—19416no—
169C2×D4C4×D9711yescharacteristic, complemented
170C2×D4C22×D9712yes—
171C22×C4—974no—
172C22×C4—972no—
173C22×C4—1948no—
174C22×C4C4×D9712yescomplemented
175C22×C4C22×D9711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—1942no—
178C22⋊C4—974no—
179C22⋊C4—1948no—
180C22⋊C4C22×D9714yes—
181C42—22no—
182C42—1942no—
183C23—7764no—
184C23—7762no—
185C23—974no—
186C23—978no—
187C23—972no—
188C23—1948no—
189C23C2×C4×D9712yes—
190C23C23×D9711yescharacteristic
191Q8—22no—
192Q8—1941no—
193Q8—1941no—
194D4—24no—
195D4—22no—
196D4—974no—
197D4—974no—
198D4—978no—
199D4—1948no—
200D4—1948no—
201D4—1944no—
202D4—1941no—
203D4—1941no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—7768no—
208C2×C4—972no—
209C2×C4—1948no—
210C2×C4—1944no—
211C2×C4—1944no—
212C2×C4C2×C4×D9712yes—
213C2×C4C2×C4×D9711yescharacteristic
214C2×C4C2×C4×D9711yescharacteristic
215C2×C4D4×D9712yes—
216C2×C4D4×D9712yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—7768no—
221C22—978no—
222C22—3882no—
223C22—3888no—
224C22—3884no—
225C22—1942no—
226C22C22×C4×D9711yescharacteristic
227C22C2×D4×D9712yes—
228C4—22no—
229C4—44no—
230C4—3884no—
231C4—1942no—
232C4C22⋊C4×D9711yescharacteristic
233C4C22⋊C4×D9711yescharacteristic
234C4C22⋊C4×D9712yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—3882no—
239C2—1948no—
240C2C2×C22⋊C4×D9711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D97
commutatorC22×C97C23×C4
FrattiniC23C23×D97
Fitting#410×C97C2
socleC2×C97C22×C22⋊C4
prime radicalC2×#15690856C2
cosocleC23×C97C24
hypercenter#410D97
hypocenterC97#2883
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20261209C2semidirect
3. #20260709C2semidirect
4. #20260708C2semidirect
5. #99×D97C2semidirect
6. #410×C97C2semidirect
9. C2×#13095512C4semidirect
10. C2×#13095511C4semidirect
12. C22×D388C4semidirect
14. C2×D4×D97C4semidirect
18. #15690769C22semidirect
19. C22⋊C4×D97C22semidirect
20. #15690725C22semidirect
21. #15690724C22semidirect
23. #117×C97C22semidirect
24. #99×C97C22semidirect
25. C23×D97C2×C4semidirect
32. C2×Dic194C2×C4semidirect
37. C2×D388C2×C4semidirect
38. C2×D388C2×C4semidirect
41. C2×C22⋊D97C2×C4semidirect
46. C2×C4×D97C2×C4semidirect
47. C2×C4×D97C2×C4semidirect
54. C2×Q8×C97C2×C4semidirect
55. C2×D4×C97C2×C4semidirect
57. C22×C4×C97C2×C4semidirect
121. C97#2883semidirect
126. #410D97semidirect
141. #117C2×D97semidirect
145. #99C2×D97semidirect
156. C2×Q8C4×D97semidirect
169. C2×D4C4×D97semidirect
174. C22×C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908562
#202612092
#202607092
#202607082
#99×D972
#410×C972
#288397