Finite Group Databasebeta

← #27158212

Subgroups — #27158212

Order 12416.

672 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 162 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#20261685C218yesmaximal, complemented
2#20261677C212yesmaximal, complemented
3#20261676C212yesmaximal
4C4×C22⋊#87604C212yesmaximal, complemented
5C2×C22⋊C4×D97C211yesmaximal, characteristic
6C2×C4⋊#87604C2214yescomplemented
7C2×C22⋊#87604C2214yes—
8C2×C22⋊#87604C2214yes—
9C2×C22⋊#87604C2214yes—
10C2×C4×#87604C2214yescomplemented
11(C2×#87604)⋊C4C2214yescomplemented
12(C2×#87604)⋊C4C2214yescomplemented
13C24×D97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15C22×C4×D97C2211yescharacteristic
16C2×C22⋊Dic97C411yescharacteristic, complemented
17C2×#13095488C411yescharacteristic, complemented
18C2×#13095488C411yescharacteristic, complemented
19C22⋊C4×D97C414yescomplemented
20C22⋊C4×D97C2214yes—
21C2×C22⋊C4×C97C411yescharacteristic, complemented
22C22×#87604C2318yes—
23C4⋊#87604—44no—
24C22⋊#87604—28no—
25C22⋊#87604—44no—
26C22⋊#87604—44no—
27C4×#87604—44no—
28C23×D97—24no—
29C23×D97—42no—
30C23×D97—42no—
31C23×D97C2×C412yes—
32C23×D97C2×C411yescharacteristic
33C23×D97C2312yes—
34C23×D97C2311yescharacteristic
35C22×Dic97C2×C411yescharacteristic
36C22×Dic97C2×C411yescharacteristic
37C2×C4×D97—42no—
38C2×C4×D97—42no—
39C2×C4×D97C2×C412yes—
40C2×C4×D97C2×C412yes—
41C2×C4×D97C2312yes—
42C2×C4×D97C2312yes—
43C22⋊Dic97C2×C414yes—
44#13095488C2×C414yes—
45#13095488C2×C414yes—
46C24×C97C2×C411yescharacteristic
47C22×C4×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C22⋊C4×C97C2×C414yes—
50C23×C97—22no—
51C23×C97—41no—
52C23×C97—41no—
53C23×C97C22×C412yes—
54C23×C97C22×C411yescharacteristic
55C22×D97—22no—
56C22×D97—22no—
57C22×D97—22no—
58C22×D97—24no—
59C22×D97—24no—
60C22×D97—24no—
61C22×D97—24no—
62C22×D97—42no—
63C22×D97—42no—
64C22×D97—41no—
65C22×D97—41no—
66C22×D97—416no—
67C22×D97C22×C412yes—
68C22×D97C22×C411yescharacteristic
69C22×D97#4114yes—
70C22×D97C2411yescharacteristic
71C2×#87604—28no—
72C2×#87604—28no—
73C2×#87604—28no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97C22×C412yes—
77C2×C4×C97C22×C412yes—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97C22×C412yes—
81C2×Dic97C22×C412yes—
82C4×D97—24no—
83C4×D97—24no—
84C4×D97—42no—
85C4×D97—42no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—22no—
89C22×C97—22no—
90C22×C97—22no—
91C22×C97—44no—
92C22×C97#11712yes—
93C22×C97C23×C411yescharacteristic
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—28no—
97C2×D97—22no—
98C2×D97—22no—
99C2×D97—22no—
100C2×D97—48no—
101C2×D97—44no—
102C2×D97—48no—
103C2×D97#11712yes—
104C2×D97C2×#4112yes—
105C2×D97H(5,𝔽2)12yes—
106C2×D97H(5,𝔽2)12yes—
107#87604—48no—
108C4×C97—22no—
109C4×C97—22no—
110Dic97—22no—
111Dic97—22no—
112C2×C97—22no—
113C2×C97—42no—
114C2×C97C2×#11711yescharacteristic
115C2×C97#51811yescharacteristic
116C2×C97#51811yescharacteristic
117D97—22no—
118D97—42no—
119D97#53814yes—
120#2312—971nomaximal, Sylow
121C97#231211yescharacteristic, complemented, Sylow
122C22×C22⋊C4—971no—
123#388—978no—
124#380—972no—
125#379—972no—
126C4×C22⋊C4—972no—
127C25—971no—
128C23×C4—971no—
129C23×C4—971no—
130C2×C4⋊C4—974no—
131C2×C22⋊C4—971no—
132C2×C22⋊C4—971no—
133C2×C22⋊C4—971no—
134C2×C22⋊C4—974no—
135C2×C22⋊C4—974no—
136C2×C22⋊C4—974no—
137C2×C22⋊C4—974no—
138C2×C22⋊C4—974no—
139C2×C22⋊C4#8760411yescharacteristic, complemented
140C2×C42—974no—
141#95—974no—
142#95—974no—
143C24—971no—
144C24—972no—
145C24—971no—
146C24—972no—
147C24—3882no—
148C24—3882no—
149C24—1944no—
150C24C2×#8760411yescharacteristic
151C22×C4—971no—
152C22×C4—971no—
153C22×C4—972no—
154C22×C4—972no—
155C22×C4—972no—
156C22×C4—972no—
157C22×C4—978no—
158C22×C4—3882no—
159C22×C4—3882no—
160C22×C4C2×#8760411yescharacteristic
161C22×C4C2×#8760411yescharacteristic
162C4⋊C4—3884no—
163C22⋊C4—974no—
164C22⋊C4—974no—
165C22⋊C4—974no—
166C22⋊C4—3884no—
167C22⋊C4—3884no—
168C22⋊C4—1948no—
169C22⋊C4C2×#8760414yes—
170C42—3884no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—971no—
175C23—974no—
176C23—971no—
177C23—972no—
178C23—3881no—
179C23—3881no—
180C23—3882no—
181C23—3882no—
182C23—38816no—
183C23—1944no—
184C23—1944no—
185C23—1944no—
186C23—1944no—
187C23—1942no—
188C23—1942no—
189C23—1942no—
190C23C22×#8760412yes—
191C23C22×#8760411yescharacteristic
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—972no—
195C2×C4—972no—
196C2×C4—3882no—
197C2×C4—3882no—
198C2×C4—3881no—
199C2×C4—3881no—
200C2×C4—1944no—
201C2×C4—1944no—
202C2×C4—1948no—
203C2×C4—1948no—
204C2×C4—1948no—
205C2×C4C22×#8760412yes—
206C2×C4C22×#8760412yes—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—44no—
213C22—972no—
214C22—972no—
215C22—972no—
216C22—972no—
217C22—3884no—
218C22—3888no—
219C22—3888no—
220C22—1948no—
221C22—1942no—
222C22—1942no—
223C22—1942no—
224C22—1941no—
225C22—1941no—
226C22#1569088912yes—
227C22C23×#8760411yescharacteristic
228C4—22no—
229C4—22no—
230C4—3888no—
231C4—1942no—
232C4—1942no—
233C2—22no—
234C2—42no—
235C2—974no—
236C2—3882no—
237C2—1942no—
238C2C2×#1569088911yescharacteristic
239C2#2026180211yescharacteristic
240C2#2026180211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC2×C22⋊C4×C97C4
prime radicalC24×D97C22
cosocleC22×D97C24
hypercenterC2×C22⋊C4#87604
hypocenterC97#2312
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261685C2semidirect
2. #20261677C2semidirect
4. C4×C22⋊#87604C2semidirect
6. C2×C4⋊#87604C22semidirect
10. C2×C4×#87604C22semidirect
11. (C2×#87604)⋊C4C22semidirect
12. (C2×#87604)⋊C4C22semidirect
16. C2×C22⋊Dic97C4semidirect
17. C2×#13095488C4semidirect
18. C2×#13095488C4semidirect
19. C22⋊C4×D97C4semidirect
21. C2×C22⋊C4×C97C4semidirect
121. C97#2312semidirect
139. C2×C22⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616852
#202616772
#202616762
C4×C22⋊#876042
C2×C22⋊C4×D972
#231297