Finite Group Databasebeta

← C2×C22⋊#3708 #11795957

Subgroups — C2×C22⋊#3708 #11795957

Order 1152.

610 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C23×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C22×#3708C212yesmaximal, complemented
3C22⋊#3708C214yesmaximal, direct product factor
4C23×C32×C4C411yescharacteristic
5C22×C3⋊(C3⋊C8)—24no—
6C22×C3⋊(C3⋊C8)C411yescharacteristic
7C22×C3⋊(C3⋊C8)C412yes—
8C22×C3⋊(C3⋊C8)C2212yes—
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C2×#3708—24no—
11C2×#3708C2214yescomplemented
12C23×C3⋊C8—62no—
13C24×C32C811yescharacteristic
14C22×C32×C4—22no—
15C22×C32×C4—22no—
16C22×C32×C4C811yescharacteristic
17C22×C32×C4C812yes—
18C22×C32×C4C2×C412yes—
19C22×C32×C4C2×C411yescharacteristic
20C2×C3⋊(C3⋊C8)—22no—
21C2×C3⋊(C3⋊C8)—24no—
22C2×C3⋊(C3⋊C8)—24no—
23C2×C3⋊(C3⋊C8)C2×C412yes—
24C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
25C2×C3⋊(C3⋊C8)D414yes—
26C2×C3⋊(C3⋊C8)C2311yescharacteristic
27#3708—24no—
28C2×C22⋊C16—91nomaximal, Sylow
29C23×C3×C4—22no—
30C22×C3⋊C8—64no—
31C22×C3⋊C8—62no—
32C22×C3⋊C8—62no—
33C22×C3⋊C8—64no—
34C22×C3⋊C8—616no—
35C23×C32—22no—
36C23×C32—22no—
37C23×C32C1614yescomplemented
38C23×C32C2×C812yes—
39C23×C32C2×C811yescharacteristic
40C2×C32×C4—22no—
41C2×C32×C4—24no—
42C2×C32×C4—22no—
43C2×C32×C4—22no—
44C2×C32×C4C22⋊C412yes—
45C2×C32×C4C2×C812yes—
46C2×C32×C4C2×C811yescharacteristic
47C2×C32×C4M1612yes—
48C2×C32×C4C22×C411yescharacteristic
49C3⋊(C3⋊C8)—22no—
50C3⋊(C3⋊C8)C22⋊C412yes—
51C3⋊(C3⋊C8)C2×D412yes—
52C23×C8—91no—
53C22×C16—92no—
54C22⋊C16—94no—
55C24×C3—22no—
56C22×C3×C4—28no—
57C22×C3×C4—28no—
58C22×C3×C4—22no—
59C22×C3×C4—24no—
60C22×C3×C4—24no—
61C22×C3×C4—22no—
62C2×C3⋊C8—62no—
63C2×C3⋊C8—64no—
64C2×C3⋊C8—68no—
65C2×C3⋊C8—62no—
66C2×C3⋊C8—616no—
67C2×C3⋊C8—616no—
68C2×C3⋊C8—68no—
69C22×C32—28no—
70C22×C32—22no—
71C22×C32—22no—
72C22×C32C22⋊C812yes—
73C22×C32C2×C1612yes—
74C22×C32C2×C1614yescomplemented
75C22×C32M3212yes—
76C22×C32C22×C811yescharacteristic
77C32×C4—22no—
78C32×C4C22⋊C812yes—
79C32×C4C2×C22⋊C411yescharacteristic
80C32×C4C2×M1611yescharacteristic
81C23×C4#15711yescharacteristic
82C22×C8—91no—
83C22×C8—92no—
84C22×C8—92no—
85C22×C8—91no—
86C22×C8—184no—
87C2×C16—94no—
88C2×C16—184no—
89C23×C3—28no—
90C23×C3—28no—
91C23×C3—28no—
92C23×C3—24no—
93C23×C3—22no—
94C2×C3×C4—28no—
95C2×C3×C4—28no—
96C2×C3×C4—28no—
97C2×C3×C4—216no—
98C2×C3×C4—22no—
99C2×C3×C4—24no—
100C2×C3×C4—22no—
101C2×C3×C4—24no—
102C2×C3×C4—24no—
103C3⋊C8—64no—
104C3⋊C8—64no—
105C3⋊C8—68no—
106C2×C32—24no—
107C2×C32C22⋊C1614yescomplemented
108C2×C32C2×C22⋊C811yescharacteristic
109C2×C32C22×C1611yescharacteristic
110C2×C32C2×M3211yescharacteristic
111C24#62211yescharacteristic
112C22×C4—22no—
113C22×C4—22no—
114C22×C4#62211yescharacteristic
115C22×C4#62212yes—
116C22×C4C2×#15712yes—
117C22×C4C2×#15711yescharacteristic
118C2×C8—91no—
119C2×C8—91no—
120C2×C8—94no—
121C2×C8—92no—
122C2×C8—184no—
123C2×C8—184no—
124C2×C8—182no—
125C16—184no—
126C22×C3—28no—
127C22×C3—232no—
128C22×C3—28no—
129C22×C3—24no—
130C22×C3—28no—
131C22×C3—24no—
132C22×C3—22no—
133C22×C3—24no—
134C3×C4—28no—
135C3×C4—22no—
136C3×C4—22no—
137C3×C4—24no—
138C32C2×C22⋊C1611yescharacteristic, complemented, Sylow
139C23—22no—
140C23—22no—
141C23#370814yescomplemented
142C23C2×#62212yes—
143C23C2×#62211yescharacteristic
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—24no—
148C2×C4C2×#62212yes—
149C2×C4C2×#62211yescharacteristic
150C2×C4#379212yes—
151C2×C4C22⋊#15712yes—
152C2×C4C22×#15711yescharacteristic
153C8—92no—
154C8—92no—
155C8—182no—
156C2×C3—216no—
157C2×C3—22no—
158C2×C3—28no—
159C2×C3—22no—
160C2×C3—22no—
161C22—22no—
162C22—28no—
163C22—22no—
164C22C2×#370812yes—
165C22C2×#370814yescomplemented
166C22#6377312yes—
167C22C22⋊#62212yes—
168C22C22×#62211yescharacteristic
169C4—22no—
170C4C22⋊#62212yes—
171C4C2×#379211yescharacteristic
172C4C2×C22⋊#15711yescharacteristic
173C3—22no—
174C2—24no—
175C2C22⋊#370814yesdirect product factor
176C2C22×#370811yescharacteristic
177C2C2×#6377311yescharacteristic
178C2C2×C22⋊#62211yescharacteristic
subgroupG/H
centerC22×C4C2×#157
commutatorC2×C32C22×C16
FrattiniC2×C4C22×#157
FittingC23×C32×C4C4
socleC23×C32C2×C8
prime radicalC24×C32C8
cosocleC2×C3⋊(C3⋊C8)C23
hypercenterC23×C4#157
hypocenterC32C2×C22⋊C16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×#3708C2semidirect
3. C22⋊#3708C2direct (N × H)
11. C2×#3708C22semidirect
37. C23×C32C16semidirect
74. C22×C32C2×C16semidirect
107. C2×C32C22⋊C16semidirect
138. C32C2×C22⋊C16semidirect
141. C23#3708semidirect
165. C22C2×#3708semidirect
175. C2C22⋊#3708direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊(C3⋊C8)2
C22×#37082
C22⋊#37082
C2×C22⋊C169