Finite Group Databasebeta

← #27157599

Subgroups — #27157599

Order 12416.

380 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 78 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×#15690869C212yesmaximal
2#20261677C212yesmaximal, complemented
3#20261613C212yesmaximal, complemented
4C2×#15690772C211yesmaximal, characteristic
5C2×C4⋊#87604—22no—
6C2×C22⋊#87604C2214yes—
7#15690869—24no—
8(C2×#87604)⋊C4—22no—
9C22×D388C2211yescharacteristic
10C22×C4×D97C411yescharacteristic, complemented
11C22×C4×D97C2211yescharacteristic
12#15690772—22no—
13#15690772—22no—
14C2×#13095488C411yescharacteristic, complemented
15C2×#13095488C2211yescharacteristic
16C2×C4×Dic97C411yescharacteristic, complemented
17C2×C4⋊C4×C97C411yescharacteristic, complemented
18C22×#87604—24no—
19C4⋊#87604—44no—
20C22⋊#87604—24no—
21C22⋊#87604—24no—
22C23×D97C2×C411yescharacteristic
23C23×D97C2311yescharacteristic
24C2×D388—22no—
25C2×D388—22no—
26C2×D388D412yes—
27C2×D388Q812yes—
28C22×Dic97C2×C411yescharacteristic
29C22×Dic97C2×C411yescharacteristic
30C2×C4×D97—21no—
31C2×C4×D97—21no—
32C2×C4×D97—42no—
33C2×C4×D97—42no—
34C2×C4×D97D412yes—
35C2×C4×D97D412yes—
36#13095488—22no—
37#13095488—22no—
38C4×Dic97—21no—
39C4×Dic97—21no—
40C22×C4×C97C2×C411yescharacteristic
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97C2×C411yescharacteristic
43C4⋊C4×C97—22no—
44C23×C97C22×C411yescharacteristic
45C22×D97—21no—
46C22×D97—21no—
47C22×D97—41no—
48C22×D97—41no—
49C22×D97—41no—
50C22×D97—41no—
51C22×D97C4⋊C412yescomplemented
52C22×D97C2×D412yes—
53C22×D97C2×D411yescharacteristic
54C22×D97C2×Q811yescharacteristic
55C22×D97#4112yes—
56C2×#87604—44no—
57C2×#87604—44no—
58C2×#87604—44no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97C22⋊C412yes—
65C2×C4×C97C4⋊C412yescomplemented
66C2×Dic97—22no—
67C2×Dic97—21no—
68C2×Dic97—41no—
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C22⋊C411yescharacteristic
71D388—24no—
72D388—24no—
73C4×D97—24no—
74C4×D97—42no—
75C4×D97—42no—
76C4×D97—42no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97C2×C22⋊C411yescharacteristic
80C22×C97C2×C4⋊C411yescharacteristic
81C22×C97#11711yescharacteristic
82C2×D97—22no—
83C2×D97—22no—
84C2×D97—22no—
85C2×D97—41no—
86C2×D97—41no—
87C2×D97—42no—
88C2×D97—42no—
89C2×D97—41no—
90C2×D97—41no—
91C2×D97—42no—
92C2×D97#12112yes—
93C2×D97C22⋊Q812yes—
94#87604—84no—
95C4×C97—22no—
96C4×C97—22no—
97C4×C97—41no—
98Dic97—21no—
99Dic97—41no—
100Dic97#9912yescomplemented
101C2×C97—21no—
102C2×C97—21no—
103C2×C97#38011yescharacteristic
104C2×C97C2×#9911yescharacteristic
105C2×C97#41011yescharacteristic
106D97—22no—
107D97—22no—
108D97—42no—
109#1799—971nomaximal, Sylow
110C97#179911yescharacteristic, complemented, Sylow
111C2×C4×D4—971no—
112C2×#99—972no—
113#380—972no—
114#342—972no—
115C22×D4—971no—
116C23×C4—971no—
117C23×C4—971no—
118C4×D4—1942no—
119C4×D4—1942no—
120C2×C4⋊C4—1942no—
121C2×C4⋊C4#8760411yescharacteristic, complemented
122C2×C22⋊C4—971no—
123C2×C22⋊C4—971no—
124C2×C22⋊C4—974no—
125C2×C42—971no—
126#99—1944no—
127#95—1942no—
128C24—971no—
129C24—971no—
130C2×D4—972no—
131C2×D4—972no—
132C2×D4—1942no—
133C2×D4—1942no—
134C22×C4—971no—
135C22×C4—972no—
136C22×C4—972no—
137C22×C4—971no—
138C22×C4—3882no—
139C22×C4—3882no—
140C22×C4—1941no—
141C22×C4—1941no—
142C22×C4—1944no—
143C22×C4C2×#8760411yescharacteristic
144C22×C4C2×#8760411yescharacteristic
145C22×C4C2×#8760411yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—3884no—
148C22⋊C4—1942no—
149C22⋊C4—1944no—
150C22⋊C4—1944no—
151C22⋊C4—1942no—
152C42—1941no—
153C42—1941no—
154C23—971no—
155C23—971no—
156C23—972no—
157C23—972no—
158C23—972no—
159C23—3881no—
160C23—3881no—
161C23—3881no—
162C23—3881no—
163C23—1941no—
164C23—1941no—
165C23C22×#8760411yescharacteristic
166D4—1944no—
167D4—1944no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—3882no—
176C2×C4—3882no—
177C2×C4—3882no—
178C2×C4—3884no—
179C2×C4—3884no—
180C2×C4—3884no—
181C2×C4—3881no—
182C2×C4—1944no—
183C2×C4—1942no—
184C2×C4—1941no—
185C2×C4C22⋊#8760412yes—
186C2×C4C4⋊#8760412yescomplemented
187C22—21no—
188C22—21no—
189C22—972no—
190C22—972no—
191C22—3882no—
192C22—3882no—
193C22—3881no—
194C22—3881no—
195C22—3882no—
196C22—3881no—
197C22—3881no—
198C22—1942no—
199C22—1942no—
200C22—1942no—
201C22C2×C22⋊#8760411yescharacteristic
202C22C2×C4⋊#8760411yescharacteristic
203C22#1569088911yescharacteristic
204C4—22no—
205C4—22no—
206C4—41no—
207C4—7764no—
208C4—972no—
209C4—3881no—
210C4—1941no—
211C2—21no—
212C2—21no—
213C2—3882no—
214C2—1942no—
215C2—1942no—
216C2#2026167711yescharacteristic
217C2D97⋊#9911yescharacteristic
218C2#2026171611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C22
cosocleC23×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1799
LCSG ▷ C23×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
2. #20261677C2semidirect
3. #20261613C2semidirect
10. C22×C4×D97C4semidirect
14. C2×#13095488C4semidirect
16. C2×C4×Dic97C4semidirect
17. C2×C4⋊C4×C97C4semidirect
51. C22×D97C4⋊C4semidirect
65. C2×C4×C97C4⋊C4semidirect
100. Dic97#99semidirect
110. C97#1799semidirect
121. C2×C4⋊C4#87604semidirect
186. C2×C4C4⋊#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908692
#202616772
#202616132
C2×#156907722
#179997