Finite Group Databasebeta

← #27158213

Subgroups — #27158213

Order 12416.

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

normal subgroups only

This group has 512 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#20261684C218yesmaximal, complemented
2#20261678C212yesmaximal
3#20261677C212yesmaximal
4C4×C22⋊#87604C212yesmaximal
5C2×C4⋊C4×D97C211yesmaximal, characteristic
6C2×C4⋊#87604C2214yes—
7C2×C22⋊#87604C2214yes—
8C2×C4×#87604C2214yes—
9(C2×#87604)⋊C4C2214yes—
10(C2×#87604)⋊C4C2214yes—
11(C2×#87604)⋊C4C2218yes—
12C22×C4×D97C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15C2×C4⋊Dic97C411yescharacteristic, complemented
16C2×#13095490C411yescharacteristic, complemented
17C2×#13095490C411yescharacteristic, complemented
18C4⋊C4×D97C414yescomplemented
19C4⋊C4×D97C2214yes—
20C2×C4⋊C4×C97C411yescharacteristic, complemented
21C22×#87604C2318yes—
22C4⋊#87604—44no—
23C22⋊#87604—28no—
24C4×#87604—44no—
25C23×D97C2311yescharacteristic
26C22×Dic97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2×C411yescharacteristic
29C2×C4×D97—24no—
30C2×C4×D97—42no—
31C2×C4×D97—42no—
32C2×C4×D97C2×C412yes—
33C2×C4×D97C2×C412yes—
34C2×C4×D97C2×C412yes—
35C2×C4×D97C2312yes—
36C2×C4×D97C2312yes—
37C2×C4×D97C2312yes—
38C4⋊Dic97C2×C414yes—
39#13095490C2×C414yes—
40#13095490C2×C414yes—
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97C2×C411yescharacteristic
43C22×C4×C97C2×C411yescharacteristic
44C4⋊C4×C97C2×C414yes—
45C23×C97C22×C411yescharacteristic
46C22×D97—22no—
47C22×D97—22no—
48C22×D97C22×C411yescharacteristic
49C22×D97#4114yes—
50C22×D97C2411yescharacteristic
51C2×#87604—28no—
52C2×#87604—28no—
53C2×#87604—28no—
54C2×C4×C97—22no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C22×C412yes—
58C2×C4×C97C22×C412yes—
59C2×C4×C97C22×C412yes—
60C2×Dic97—22no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63C2×Dic97C22×C412yes—
64C2×Dic97C22×C412yes—
65C2×Dic97C22×C412yes—
66C4×D97—24no—
67C4×D97—24no—
68C4×D97—24no—
69C4×D97—24no—
70C4×D97—42no—
71C4×D97—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97#11712yes—
75C22×C97C23×C411yescharacteristic
76C2×D97—21no—
77C2×D97—21no—
78C2×D97—28no—
79C2×D97#11712yes—
80C2×D97C2×#4112yes—
81C2×D97H(5,𝔽2)12yes—
82C2×D972−1+412yes—
83#87604—48no—
84C4×C97—22no—
85C4×C97—22no—
86C4×C97—22no—
87Dic97—22no—
88Dic97—22no—
89Dic97—22no—
90C2×C97—22no—
91C2×C97C2×#11711yescharacteristic
92C2×C97#51811yescharacteristic
93C2×C97#51911yescharacteristic
94D97—22no—
95D97#53914yes—
96#2313—971nomaximal, Sylow
97C97#231311yescharacteristic, complemented, Sylow
98C22×C4⋊C4—971no—
99#387—978no—
100#381—972no—
101#380—972no—
102C4×C22⋊C4—972no—
103C23×C4—971no—
104C23×C4—971no—
105C23×C4—971no—
106C2×C4⋊C4—971no—
107C2×C4⋊C4—971no—
108C2×C4⋊C4—971no—
109C2×C4⋊C4—974no—
110C2×C4⋊C4—974no—
111C2×C4⋊C4—974no—
112C2×C4⋊C4#8760411yescharacteristic, complemented
113C2×C22⋊C4—974no—
114C2×C42—974no—
115#95—974no—
116#95—974no—
117#95—978no—
118C24—971no—
119C22×C4—971no—
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—972no—
123C22×C4—972no—
124C22×C4—972no—
125C22×C4—972no—
126C22×C4—972no—
127C22×C4—972no—
128C22×C4—978no—
129C22×C4—3882no—
130C22×C4—3882no—
131C22×C4—1944no—
132C22×C4C2×#8760411yescharacteristic
133C22×C4C2×#8760411yescharacteristic
134C22×C4C2×#8760411yescharacteristic
135C4⋊C4—974no—
136C4⋊C4—974no—
137C4⋊C4—974no—
138C4⋊C4—3884no—
139C4⋊C4C2×#8760414yes—
140C22⋊C4—1948no—
141C42—3884no—
142C23—971no—
143C23—974no—
144C23—971no—
145C23—1942no—
146C23—1942no—
147C23C22×#8760411yescharacteristic
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—972no—
152C2×C4—972no—
153C2×C4—972no—
154C2×C4—3882no—
155C2×C4—3882no—
156C2×C4—3881no—
157C2×C4—3881no—
158C2×C4—1944no—
159C2×C4—1944no—
160C2×C4—1944no—
161C2×C4—1944no—
162C2×C4—1948no—
163C2×C4—1948no—
164C2×C4—1948no—
165C2×C4—1942no—
166C2×C4C22×#8760412yes—
167C2×C4C22×#8760412yes—
168C2×C4C22×#8760412yes—
169C22—21no—
170C22—21no—
171C22—972no—
172C22—972no—
173C22—972no—
174C22—972no—
175C22—1948no—
176C22—1941no—
177C22—1941no—
178C22#1569088912yes—
179C22C23×#8760411yescharacteristic
180C4—22no—
181C4—22no—
182C4—22no—
183C4—3888no—
184C4—1942no—
185C4—1942no—
186C4—1942no—
187C2—22no—
188C2—974no—
189C2—1942no—
190C2C2×#1569088911yescharacteristic
191C2#2026180211yescharacteristic
192C2#2026180311yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC2×C4⋊C4×C97C4
prime radicalC23×D97C23
cosocleC22×D97C24
hypercenterC2×C4⋊C4#87604
hypocenterC97#2313
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261684C2semidirect
15. C2×C4⋊Dic97C4semidirect
16. C2×#13095490C4semidirect
17. C2×#13095490C4semidirect
18. C4⋊C4×D97C4semidirect
20. C2×C4⋊C4×C97C4semidirect
97. C97#2313semidirect
112. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616842
#202616782
#202616772
C4×C22⋊#876042
C2×C4⋊C4×D972
#231397