Finite Group Databasebeta

← C22×#63742 #11858387

Subgroups — C22×#63742 #11858387

Order 1152.

1018 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 287 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×#622C211yesmaximal, characteristic
2C22×#3724C212yesmaximal
3C2×#63742C2112yesmaximal, direct product factor
4C23×C3⋊Dic3C2211yescharacteristic
5C23×C3×Dic3—22no—
6C22×#622C418yescomplemented
7C22×#622C2216yes—
8C2×#3724—26no—
9C2×#3724C22112yes—
10#63742C22116yesdirect product factor
11C22×Dic3⋊C4—62no—
12C24×C32D411yescharacteristic
13C22×C3⋊Dic3C2×C414yes—
14C22×C3⋊Dic3D413yes—
15C22×C3⋊Dic3Q814yes—
16C22×C3⋊Dic3C2313yes—
17C22×C3×Dic3—212no—
18C22×C3×Dic3—48no—
19C2×#622C2×C4124yescomplemented
20C2×#622C2314yes—
21#3724—212no—
22#3724C2318yes—
23C22×#106—91nomaximal, Sylow
24C23×C3×C4—62no—
25C23×Dic3—22no—
26C23×Dic3—62no—
27C2×Dic3⋊C4—612no—
28C2×Dic3⋊C4—612no—
29C23×C32C4⋊C414yes—
30C23×C32SD1618yes—
31C23×C32C2×D413yes—
32C2×C3⋊Dic3C4⋊C4112yes—
33C2×C3⋊Dic3C22×C416yes—
34C2×C3⋊Dic3C2×D413yes—
35C2×C3⋊Dic3C2×Q816yes—
36C2×C3⋊Dic3C2411yescharacteristic
37C2×C3×Dic3—28no—
38C2×C3×Dic3—424no—
39#622C22×C418yescomplemented
40C23×C8—91no—
41C22×C4⋊C4—92no—
42C2×#106—912no—
43C24×C3—22no—
44C22×C3×C4—612no—
45C22×C3×C4—128no—
46C22×Dic3—212no—
47C22×Dic3—48no—
48C22×Dic3—66no—
49C22×Dic3—66no—
50C22×Dic3—68no—
51C22×Dic3—68no—
52Dic3⋊C4—68no—
53Dic3⋊C4—624no—
54C22×C32#106116yescomplemented
55C22×C32C2×C4⋊C416yes—
56C22×C32C2×SD16112yes—
57C22×C32C22×D411yescharacteristic
58C3⋊Dic3C2×C4⋊C416yes—
59C3⋊Dic3C23×C411yescharacteristic
60C3⋊Dic3C22×Q811yescharacteristic
61C3×Dic3—48no—
62C23×C4—62no—
63C23×C4—91no—
64C22×C8—98no—
65C22×C8—96no—
66C2×C4⋊C4—912no—
67C2×C4⋊C4—186no—
68#106—916no—
69C23×C3—26no—
70C23×C3—28no—
71C23×C3—28no—
72C23×C3—28no—
73C2×C3×C4—68no—
74C2×C3×C4—1224no—
75C2×Dic3—28no—
76C2×Dic3—424no—
77C2×Dic3—624no—
78C2×Dic3—612no—
79C2×Dic3—66no—
80C2×Dic3—612no—
81C2×Dic3—62no—
82C2×C32C2×#106112yescomplemented
83C2×C32C22×C4⋊C411yescharacteristic
84C2×C32C22×SD1612yes—
85C24GO+(4,𝔽2)11yescharacteristic
86C22×C4—612no—
87C22×C4—93no—
88C22×C4—93no—
89C22×C4—94no—
90C22×C4—94no—
91C22×C4—128no—
92C2×C8—94no—
93C2×C8—924no—
94C4⋊C4—98no—
95C4⋊C4—1812no—
96C22×C3—22no—
97C22×C3—212no—
98C22×C3—212no—
99C22×C3—212no—
100C22×C3—232no—
101C3×C4—128no—
102Dic3—48no—
103Dic3—612no—
104Dic3—62no—
105Dic3—62no—
106C32C22×#10611yescharacteristic, complemented, Sylow
107C23#377314yes—
108C23#377518yes—
109C23C2×GO+(4,𝔽2)13yes—
110C2×C4—68no—
111C2×C4—91no—
112C2×C4—93no—
113C2×C4—96no—
114C2×C4—96no—
115C2×C4—912no—
116C2×C4—1224no—
117C8—98no—
118C2×C3—22no—
119C2×C3—22no—
120C2×C3—22no—
121C2×C3—224no—
122C22#63742116yesdirect product factor
123C22C2×#377316yes—
124C22C2×#3775112yes—
125C22C22×GO+(4,𝔽2)11yescharacteristic
126C4—91no—
127C4—91no—
128C4—96no—
129C4—128no—
130C3—22no—
131C2C2×#63742112yesdirect product factor
132C2C22×#377311yescharacteristic
133C2C22×#377512yes—
subgroupG/H
center, hypercenterC24GO+(4,𝔽2)
commutatorC3⋊Dic3C23×C4
FrattiniC22C22×GO+(4,𝔽2)
Fitting, socle, prime radicalC24×C32D4
cosocleC2×C3⋊Dic3C24
hypocenterC32C22×#106
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#63742C2direct (N × H)
6. C22×#622C4semidirect
10. #63742C22direct (N × H)
19. C2×#622C2×C4semidirect
39. #622C22×C4semidirect
54. C22×C32#106semidirect
82. C2×C32C2×#106semidirect
106. C32C22×#106semidirect
122. C22#63742direct (N × H)
131. C2C2×#63742direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#6222
C22×#37242
C2×#637422
C22×#1069