Finite Group Databasebeta

← #27158335

Subgroups — #27158335

Order 12416.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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
1#20261802C212yesmaximal, complemented
2#20261715C212yesmaximal, complemented
3#20261714C212yesmaximal, complemented
4D97⋊#99C214yesmaximal, complemented
5C2×#15690868C214yesmaximal, complemented
6C2×#15690855C211yesmaximal, characteristic
7D4×#87604—28no—
8#15690889C2214yescomplemented
9C2×C22⋊#87604C2218yescomplemented
10#15690869C2214yescomplemented
11#15690868C2214yescomplemented
12#15690868C2218yescomplemented
13#15690855—28no—
14C2×#13095511C411yescharacteristic, complemented
15C2×#13095511C2211yescharacteristic
16C2×#13095510C411yescharacteristic, complemented
17C2×#13095510C412yescomplemented
18C2×#13095510C2211yescharacteristic
19C22×C22⋊D97C412yescomplemented
20C22×C22⋊D97C2212yes—
21C2×D4×D97C411yescharacteristic, complemented
22C2×D4×D97C2211yescharacteristic
23C2×D4×D97C2212yes—
24C22×D4×C97C411yescharacteristic, complemented
25C22×#87604—28no—
26C4⋊#87604—24no—
27C22⋊#87604—28no—
28C22⋊#87604C2318yes—
29C4×#87604—24no—
30C23×D97C2×C412yescomplemented
31C23×D97C2312yes—
32#13095511—24no—
33#13095511—24no—
34#13095510—24no—
35#13095510—28no—
36#13095510—24no—
37C2×Dic194C2×C411yescharacteristic, complemented
38C2×Dic194C2×C411yescharacteristic, complemented
39C2×D388C2×C411yescharacteristic
40C2×D388C2311yescharacteristic
41C2×C22⋊D97—28no—
42C2×C22⋊D97—28no—
43C2×C22⋊D97C2×C412yes—
44C2×C22⋊D97C2×C412yescomplemented
45C2×C22⋊D97C2×C412yescomplemented
46C2×C22⋊D97C2×C411yescharacteristic, complemented
47C2×C22⋊D97C2×C412yescomplemented
48C2×C22⋊D97C2312yes—
49C2×C22⋊D97C2311yescharacteristic
50D4×D97—28no—
51D4×D97—24no—
52D4×D97D418yes—
53C22×Dic97C2×C412yescomplemented
54C22×Dic97C2×C412yescomplemented
55C2×C4×D97C2×C411yescharacteristic, complemented
56C2×C4×D97C2×C412yescomplemented
57C2×C4×D97C2311yescharacteristic
58C24×C97C2×C412yescomplemented
59C2×D4×C97—24no—
60C2×D4×C97C2×C411yescharacteristic
61C2×D4×C97C2×C411yescharacteristic, complemented
62C2×D4×C97C2×C412yescomplemented
63C22×C4×C97C2×C411yescharacteristic, complemented
64C23×C97—24no—
65C23×C97—82no—
66C23×C97C22×C411yescharacteristic
67C23×C97C22×C412yes—
68C23×C97C22×C412yes—
69C22×D97—24no—
70C22×D97—82no—
71C22×D97—82no—
72C22×D97C22×C411yescharacteristic
73C22×D97C22×C412yes—
74C22×D97C2×D418yes—
75C22×D97C2411yescharacteristic
76C2×#87604—28no—
77C2×#87604—88no—
78D4×C97—24no—
79D4×C97—22no—
80D4×C97C22⋊C414yes—
81C2×C4×C97—22no—
82C2×C4×C97C22×C411yescharacteristic
83C2×C4×C97C22×C411yescharacteristic
84C22⋊D97—24no—
85C22⋊D97—24no—
86C22⋊D97—24no—
87C22⋊D97—24no—
88C22⋊D97—22no—
89C22⋊D97—22no—
90C22⋊D97C22⋊C418yes—
91C2×Dic97—24no—
92C2×Dic97—24no—
93C2×Dic97C22×C411yescharacteristic
94C2×Dic97C22×C412yes—
95C2×Dic97C22×C411yescharacteristic
96D388—22no—
97D388C22⋊C414yes—
98C4×D97—24no—
99C4×D97—22no—
100C4×D97C2×D414yes—
101Dic194—22no—
102Dic194—22no—
103C22×C97—22no—
104C22×C97—21no—
105C22×C97—42no—
106C22×C97—42no—
107C22×C97—41no—
108C22×C97—84no—
109C22×C97C2×C22⋊C414yes—
110C22×C97C23×C411yescharacteristic
111C2×D97—21no—
112C2×D97—48no—
113C2×D97—42no—
114C2×D97—41no—
115C2×D97—41no—
116C2×D97—84no—
117C2×D97C2×C22⋊C414yes—
118C2×D97C22×D412yes—
119#87604—48no—
120C4×C97—21no—
121C4×C97C2×C22⋊C412yes—
122Dic97—22no—
123Dic97—21no—
124Dic97C2×C22⋊C412yes—
125C2×C97—21no—
126C2×C97—24no—
127C2×C97—42no—
128C2×C97—41no—
129C2×C97C22×C22⋊C411yescharacteristic
130D97—22no—
131D97—24no—
132D97—41no—
133#2883—971nomaximal, Sylow
134C97#288311yescharacteristic, complemented, Sylow
135C2×H(5,𝔽2)—971no—
136#518—972no—
137#410—972no—
138#410—972no—
139C2×#99—974no—
140C2×#99—974no—
141H(5,𝔽2)—1948no—
142C2×#41—971no—
143C2×#41—972no—
144C2×#41—971no—
145C2×#41—971no—
146C2×#41—971no—
147C22×D4—971no—
148C22×D4—972no—
149C22×D4—972no—
150C22×D4—971no—
151C22×D4—972no—
152C22×D4#8760411yescharacteristic, complemented
153C4×D4—1948no—
154#117—974no—
155C2×C22⋊C4—978no—
156#99—974no—
157#99—978no—
158#99—974no—
159C24—972no—
160C24—972no—
161C24C2×#8760412yescomplemented
162#41—1944no—
163#41—1944no—
164#41—1944no—
165#41—1948no—
166#41—1944no—
167C2×Q8—971no—
168C2×Q8—971no—
169C2×D4—24no—
170C2×D4—972no—
171C2×D4—978no—
172C2×D4—971no—
173C2×D4—972no—
174C2×D4—972no—
175C2×D4—972no—
176C2×D4—972no—
177C2×D4—971no—
178C2×D4—971no—
179C2×D4—971no—
180C2×D4—1944no—
181C2×D4—1948no—
182C2×D4—1948no—
183C2×D4—1948no—
184C2×D4C2×#8760411yescharacteristic
185C2×D4C2×#8760412yescomplemented
186C2×D4C2×#8760411yescharacteristic, complemented
187C22×C4—971no—
188C22×C4—972no—
189C22×C4—972no—
190C22×C4—972no—
191C22×C4—971no—
192C22×C4—1948no—
193C22×C4C2×#8760411yescharacteristic, complemented
194C4⋊C4—1944no—
195C22⋊C4—978no—
196C22⋊C4—1948no—
197C42—1944no—
198C23—24no—
199C23—82no—
200C23—7762no—
201C23—7762no—
202C23—978no—
203C23—971no—
204C23—972no—
205C23—971no—
206C23—1944no—
207C23C22×#8760412yes—
208C23C22×#8760411yescharacteristic
209C23C22×#8760412yes—
210Q8—1942no—
211Q8—1942no—
212D4—24no—
213D4—22no—
214D4—978no—
215D4—974no—
216D4—1942no—
217D4—1942no—
218D4—1942no—
219D4—1944no—
220D4—1944no—
221D4—1944no—
222D4—1944no—
223D4C22⋊#8760414yes—
224C2×C4—22no—
225C2×C4—7768no—
226C2×C4—971no—
227C2×C4—974no—
228C2×C4—972no—
229C2×C4—971no—
230C2×C4—1944no—
231C2×C4—1944no—
232C2×C4—1948no—
233C2×C4—1944no—
234C2×C4—1942no—
235C2×C4C22×#8760411yescharacteristic
236C2×C4C22×#8760411yescharacteristic
237C22—21no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22—84no—
243C22—7764no—
244C22—972no—
245C22—974no—
246C22—3881no—
247C22—3888no—
248C22—3881no—
249C22—3882no—
250C22—1941no—
251C22C2×C22⋊#8760414yes—
252C22C23×#8760411yescharacteristic
253C4—21no—
254C4—972no—
255C4—3888no—
256C4—1942no—
257C4—1941no—
258C4C2×C22⋊#8760412yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—42no—
263C2—3881no—
264C2—1942no—
265C2—1944no—
266C2C22×C22⋊#8760411yescharacteristic
subgroupG/H
centerC2C22×C22⋊#87604
commutatorC22×C97C23×C4
FrattiniC22C23×#87604
FittingC22×D4×C97C4
socleC2×C97C22×C22⋊C4
prime radicalC2×#15690855C2
cosocleC22×D97C24
hypercenterC22×D4#87604
hypocenterC97#2883
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×D4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20261802C2semidirect
2. #20261715C2semidirect
3. #20261714C2semidirect
4. D97⋊#99C2semidirect
5. C2×#15690868C2semidirect
8. #15690889C22semidirect
9. C2×C22⋊#87604C22semidirect
10. #15690869C22semidirect
11. #15690868C22semidirect
12. #15690868C22semidirect
14. C2×#13095511C4semidirect
16. C2×#13095510C4semidirect
17. C2×#13095510C4semidirect
19. C22×C22⋊D97C4semidirect
21. C2×D4×D97C4semidirect
24. C22×D4×C97C4semidirect
30. C23×D97C2×C4semidirect
37. C2×Dic194C2×C4semidirect
38. C2×Dic194C2×C4semidirect
44. C2×C22⋊D97C2×C4semidirect
45. C2×C22⋊D97C2×C4semidirect
46. C2×C22⋊D97C2×C4semidirect
47. C2×C22⋊D97C2×C4semidirect
53. C22×Dic97C2×C4semidirect
54. C22×Dic97C2×C4semidirect
55. C2×C4×D97C2×C4semidirect
56. C2×C4×D97C2×C4semidirect
58. C24×C97C2×C4semidirect
61. C2×D4×C97C2×C4semidirect
62. C2×D4×C97C2×C4semidirect
63. C22×C4×C97C2×C4semidirect
134. C97#2883semidirect
152. C22×D4#87604semidirect
161. C24C2×#87604semidirect
185. C2×D4C2×#87604semidirect
186. C2×D4C2×#87604semidirect
193. C22×C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618022
#202617152
#202617142
D97⋊#992
C2×#156908682
C2×#156908552
#288397