Finite Group Databasebeta

← #27157423

Subgroups — #27157423

Order 12416.

448 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 138 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×D97⋊M16C211yesmaximal, characteristic
2C2×#15690889C211yesmaximal, characteristic
3#20261608C214yesmaximal, complemented
4C2×C4⋊C4×D97C211yesmaximal, characteristic
5C2×#13095521—21no—
6D97⋊M16—22no—
7D97⋊M16C414yescomplemented
8C2×#13095520—21no—
9C2×#13095519C2212yes—
10#15690889—22no—
11#15690889C414yescomplemented
12C2×C4⋊#87604C2212yes—
13C2×C22⋊#87604—21no—
14C2×C4×#87604—21no—
15C22×C4×D97—21no—
16C22×C4×D97C2211yescharacteristic
17C2×C4⋊Dic97C411yescharacteristic, complemented
18C2×#13095490—21no—
19C4⋊C4×D97—21no—
20C4⋊C4×D97—21no—
21C4⋊C4×D97C412yescomplemented
22C4⋊C4×D97C2212yes—
23C2×C4⋊C4×C97C411yescharacteristic, complemented
24C22×#87604—22no—
25#13095521—24no—
26#13095520—24no—
27#13095519—22no—
28#13095519C2×C414yescomplemented
29C2×#11715897—22no—
30C4⋊#87604—22no—
31C4⋊#87604C2×C414yescomplemented
32C22⋊#87604—24no—
33C4×#87604—24no—
34C23×D97D411yescharacteristic
35C22×Dic97—21no—
36C22×Dic97D411yescharacteristic
37C2×C4×D97—22no—
38C2×C4×D97—22no—
39C2×C4×D97—44no—
40C2×C4×D97C2×C411yescharacteristic
41C2×C4×D97C2×C412yes—
42C2×C4×D97C2×C412yes—
43C2×C4×D97D411yescharacteristic
44C2×C4×D97D411yescharacteristic
45C2×C4×D97D412yes—
46C2×C4×D97Q812yes—
47C2×C4×D97C2311yescharacteristic
48C4⋊Dic97—21no—
49C4⋊Dic97C2×C412yes—
50#13095490—22no—
51#13095490—22no—
52C22×C4×C97—21no—
53C22×C4×C97C2×C411yescharacteristic
54C4⋊C4×C97—21no—
55C4⋊C4×C97C2×C412yes—
56C23×C97C22⋊C411yescharacteristic
57C22×D97—22no—
58C22×D97—22no—
59C22×D97C22⋊C411yescharacteristic
60C22×D97C22⋊C412yes—
61C22×D97C4⋊C412yes—
62C22×D97C2×D411yescharacteristic
63C2×#87604—24no—
64C2×#87604—42no—
65C2×#87604—42no—
66C2×C4×C97—22no—
67C2×C4×C97—42no—
68C2×C4×C97C4212yes—
69C2×C4×C97C22⋊C411yescharacteristic
70C2×C4×C97C4⋊C412yes—
71C2×C4×C97C22×C411yescharacteristic
72C2×Dic97—22no—
73C2×Dic97—42no—
74C2×Dic97C22⋊C411yescharacteristic
75C2×Dic97C22⋊C412yes—
76C2×Dic97C4⋊C412yes—
77C2×Dic97C2×D411yescharacteristic
78C4×D97—44no—
79C4×D97—44no—
80C4×D97C4212yes—
81C4×D97C22⋊C412yes—
82C4×D97C22⋊C412yes—
83C4×D97C4⋊C412yes—
84C4×D97C4⋊C412yes—
85C4×D97C4⋊C412yes—
86C4×D97C22×C411yescharacteristic
87C4×D97C22×C411yescharacteristic
88C4×D97C2×D411yescharacteristic
89C4×D97C2×Q811yescharacteristic
90#11715897—24no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97#9512yes—
94C22×C97C2×C22⋊C411yescharacteristic
95C2×D97—21no—
96C2×D97—21no—
97C2×D97—24no—
98C2×D97—24no—
99C2×D97#9512yes—
100C2×D97C2×C22⋊C411yescharacteristic
101C2×D97C2×C4⋊C411yescharacteristic
102C2×D97AGL(1,ℤ/8)12yes—
103C2×D97#13712yes—
104#87604—44no—
105C4×C97—42no—
106C4×C97#9512yes—
107C4×C97C2×C4211yescharacteristic
108C4×C97C2×C4⋊C411yescharacteristic
109Dic97—42no—
110Dic97#9512yes—
111Dic97C2×C22⋊C411yescharacteristic
112Dic97C2×C4⋊C411yescharacteristic
113C2×C97—22no—
114C2×C97C2×#9511yescharacteristic
115C2×C97#41811yescharacteristic
116C2×C97#41911yescharacteristic
117D97—22no—
118D97#41712yes—
119D97#42812yes—
120#1736—971nomaximal, Sylow
121C97#173611yescharacteristic, complemented, Sylow
122C22×M16—971no—
123C2×#117—971no—
124C22×C4⋊C4—971no—
125#340—974no—
126C23×C4—971no—
127C23×C4—1941no—
128C2×M16—974no—
129C2×M16—1941no—
130C2×M16—1942no—
131C2×M16—1941no—
132C22×C8—972no—
133#117—974no—
134#117—1942no—
135C2×C4⋊C4—972no—
136C2×C4⋊C4—971no—
137C2×C4⋊C4—972no—
138C2×C4⋊C4—972no—
139C2×C4⋊C4—1941no—
140C2×C4⋊C4—1941no—
141C2×C4⋊C4—1941no—
142C2×C4⋊C4#8760411yescharacteristic, complemented
143C2×C22⋊C4—1941no—
144C2×C42—1941no—
145C24—971no—
146C22×C4—21no—
147C22×C4—971no—
148C22×C4—972no—
149C22×C4—972no—
150C22×C4—971no—
151C22×C4—971no—
152C22×C4—972no—
153C22×C4—972no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—3884no—
157C22×C4—1942no—
158C22×C4—1942no—
159C22×C4—1942no—
160C22×C4—1941no—
161C22×C4C2×#8760411yescharacteristic
162M16—1944no—
163M16—1944no—
164C2×C8—974no—
165C2×C8—1942no—
166C2×C8—1942no—
167C4⋊C4—21no—
168C4⋊C4—974no—
169C4⋊C4—972no—
170C4⋊C4—1942no—
171C4⋊C4—1942no—
172C4⋊C4—1942no—
173C4⋊C4—1941no—
174C4⋊C4C2×#8760412yes—
175C22⋊C4—1944no—
176C42—1944no—
177C23—971no—
178C23—972no—
179C23—972no—
180C23—971no—
181C23—1942no—
182C23—1942no—
183C23C22⋊#8760411yescharacteristic
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—972no—
187C2×C4—972no—
188C2×C4—971no—
189C2×C4—971no—
190C2×C4—972no—
191C2×C4—972no—
192C2×C4—972no—
193C2×C4—972no—
194C2×C4—971no—
195C2×C4—971no—
196C2×C4—972no—
197C2×C4—971no—
198C2×C4—972no—
199C2×C4—971no—
200C2×C4—3884no—
201C2×C4—3884no—
202C2×C4—3882no—
203C2×C4—3882no—
204C2×C4—3882no—
205C2×C4—1944no—
206C2×C4—1942no—
207C2×C4C4×#8760412yes—
208C2×C4C22⋊#8760411yescharacteristic
209C2×C4C4⋊#8760412yes—
210C2×C4C22×#8760411yescharacteristic
211C8—1944no—
212C22—21no—
213C22—21no—
214C22—971no—
215C22—971no—
216C22—972no—
217C22—972no—
218C22—972no—
219C22—1944no—
220C22—1944no—
221C22—1941no—
222C22—1941no—
223C22(C2×#87604)⋊C412yes—
224C22C2×C22⋊#8760411yescharacteristic
225C4—42no—
226C4—971no—
227C4—972no—
228C4—971no—
229C4—3884no—
230C4—3882no—
231C4(C2×#87604)⋊C412yes—
232C4C2×C4×#8760411yescharacteristic
233C4C2×C4⋊#8760411yescharacteristic
234C2—22no—
235C2—972no—
236C2—972no—
237C2—1942no—
238C2C2×(C2×#87604)⋊C411yescharacteristic
239C2#2026172711yescharacteristic
240C2#2026172811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C2×C42
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1736
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261608C2semidirect
7. D97⋊M16C4semidirect
11. #15690889C4semidirect
17. C2×C4⋊Dic97C4semidirect
21. C4⋊C4×D97C4semidirect
23. C2×C4⋊C4×C97C4semidirect
28. #13095519C2×C4semidirect
31. C4⋊#87604C2×C4semidirect
121. C97#1736semidirect
142. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
C2×D97⋊M162
C2×#156908892
#202616082
C2×C4⋊C4×D972
#173697