Finite Group Databasebeta

← #27156962

Subgroups — #27156962

Order 12416.

456 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 84 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#20261703C212yesmaximal, complemented
2#20261579C212yesmaximal, complemented
3#20261578C212yesmaximal, complemented
4C2×C22⋊C4×D97C211yesmaximal, characteristic
5C2×#13095519—22no—
6#15690867—22no—
7C22⋊#11715897C2214yescomplemented
8C24×D97C411yescharacteristic
9C22×C4×D97C411yescharacteristic
10C22×C4×D97C2211yescharacteristic
11C2×C22⋊Dic97C2211yescharacteristic
12C2×#13095488C411yescharacteristic
13C2×#13095488C2211yescharacteristic
14C22⋊C4×D97—22no—
15C22⋊C4×D97—22no—
16C2×C22⋊C4×C97C411yescharacteristic
17#13095519—44no—
18C2×#11715897—24no—
19C23×D97—21no—
20C23×D97—21no—
21C23×D97—22no—
22C23×D97—42no—
23C23×D97—42no—
24C23×D97C814yescomplemented
25C23×D97C2×C411yescharacteristic
26C23×D97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2311yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97—42no—
32C2×C4×D97—42no—
33C2×C4×D97D412yes—
34C2×C4×D97D412yes—
35C22⋊Dic97—22no—
36#13095488—22no—
37#13095488—22no—
38C24×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2×C411yescharacteristic
41C22⋊C4×C97—22no—
42C23×C97—21no—
43C23×C97—21no—
44C23×C97—41no—
45C23×C97—41no—
46C23×C97C2×C812yescomplemented
47C23×C97C22×C411yescharacteristic
48C22×D97—22no—
49C22×D97—22no—
50C22×D97—21no—
51C22×D97—21no—
52C22×D97—24no—
53C22×D97—24no—
54C22×D97—42no—
55C22×D97—42no—
56C22×D97—48no—
57C22×D97—48no—
58C22×D97—42no—
59C22×D97—42no—
60C22×D97—41no—
61C22×D97—41no—
62C22×D97C22⋊C411yescharacteristic
63C22×D97C22⋊C411yescharacteristic
64C22×D97C2×C812yes—
65C22×D97C2×C812yescomplemented
66C22×D97M1612yes—
67C2×C4×C97—21no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97C22⋊C412yes—
71C2×Dic97—21no—
72C2×Dic97—41no—
73C2×Dic97—41no—
74C2×Dic97C2×D412yes—
75C4×D97—24no—
76C4×D97—42no—
77C4×D97—42no—
78C4×D97—42no—
79#11715897—44no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—22no—
83C22×C97—42no—
84C22×C97—42no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97C2×C22⋊C411yescharacteristic
88C22×C97C22×C811yescharacteristic
89C22×C97C2×M1611yescharacteristic
90C2×D97—21no—
91C2×D97—21no—
92C2×D97—24no—
93C2×D97—24no—
94C2×D97—22no—
95C2×D97—48no—
96C2×D97—42no—
97C2×D97—41no—
98C2×D97—41no—
99C2×D97—48no—
100C2×D97—42no—
101C2×D97C22⋊C812yes—
102C2×D97C22⋊C812yes—
103C2×D97#9912yes—
104C2×D97#10012yes—
105C4×C97—22no—
106C4×C97—41no—
107Dic97—22no—
108Dic97—41no—
109C2×C97—21no—
110C2×C97—21no—
111C2×C97—42no—
112C2×C97C2×C22⋊C811yescharacteristic
113C2×C97C2×#9911yescharacteristic
114C2×C97C2×#10011yescharacteristic
115D97—22no—
116D97—42no—
117D97#32314yescomplemented
118C2×#323—971nomaximal, Sylow
119C97C2×#32311yescharacteristic, complemented, Sylow
120C22×C22⋊C4—971no—
121C2×C22⋊C8—972no—
122#323—972no—
123#323—972no—
124C25—971no—
125C23×C4—971no—
126C23×C4—971no—
127C22×C8—1942no—
128C2×C22⋊C4—971no—
129C2×C22⋊C4—971no—
130C2×C22⋊C4—971no—
131C2×C22⋊C4—1942no—
132C2×C22⋊C4—1942no—
133C2×C22⋊C4#8760411yescharacteristic
134C22⋊C8—974no—
135C22⋊C8—1942no—
136C24—971no—
137C24—974no—
138C24—971no—
139C24—3882no—
140C24—3882no—
141C24—1941no—
142C24—1941no—
143C24—1942no—
144C24C2×#8760411yescharacteristic
145C22×C4—971no—
146C22×C4—972no—
147C22×C4—972no—
148C22×C4—971no—
149C22×C4—3882no—
150C22×C4—3882no—
151C22×C4—1941no—
152C22×C4—1941no—
153C22×C4C2×#8760411yescharacteristic
154C22×C4C2×#8760411yescharacteristic
155C2×C8—3884no—
156C2×C8—1944no—
157C22⋊C4—22no—
158C22⋊C4—1942no—
159C22⋊C4—1942no—
160C22⋊C4—1942no—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23—971no—
166C23—971no—
167C23—972no—
168C23—972no—
169C23—972no—
170C23—3881no—
171C23—3881no—
172C23—3882no—
173C23—3882no—
174C23—3888no—
175C23—3882no—
176C23—3882no—
177C23—3888no—
178C23—1944no—
179C23—1944no—
180C23—1942no—
181C23—1942no—
182C23—1941no—
183C23—1941no—
184C23#1309551912yescomplemented
185C23C22×#8760411yescharacteristic
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—972no—
190C2×C4—3882no—
191C2×C4—3882no—
192C2×C4—3882no—
193C2×C4—3881no—
194C2×C4—3881no—
195C2×C4—1944no—
196C2×C4—1941no—
197C2×C4C22⋊#8760412yes—
198C8—3884no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—972no—
207C22—972no—
208C22—972no—
209C22—972no—
210C22—3882no—
211C22—3882no—
212C22—3888no—
213C22—3888no—
214C22—3881no—
215C22—3881no—
216C22—1944no—
217C22—1944no—
218C22—1942no—
219C22—1941no—
220C22—1941no—
221C22C2×C22⋊#8760411yescharacteristic
222C22C2×#1309551911yescharacteristic
223C22D97⋊M1611yescharacteristic
224C4—22no—
225C4—41no—
226C4—3881no—
227C4—1942no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2—974no—
232C2—3882no—
233C2—1942no—
234C2#2026170311yescharacteristic
235C2D97⋊#9911yescharacteristic
236C2D97⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC22×C97C22×C8
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC24×D97C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97C2×#323
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261703C2semidirect
2. #20261579C2semidirect
3. #20261578C2semidirect
7. C22⋊#11715897C22semidirect
24. C23×D97C8semidirect
46. C23×C97C2×C8semidirect
65. C22×D97C2×C8semidirect
117. D97#323semidirect
119. C97C2×#323semidirect
184. C23#13095519semidirect

Maximal subgroups

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

subgroupindex
#202617032
#202615792
#202615782
C2×C22⋊C4×D972
C2×#32397