Finite Group Databasebeta

← #71784

Subgroups — #71784

Order 384.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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×C3×C4⋊Q8C211yesmaximal, characteristic
2#5812C211yesmaximal, characteristic
3C2×#836C211yesmaximal, characteristic
4#5225C212yesmaximal
5#4523C212yesmaximal
6#1886—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C22×C3×Q8C2211yescharacteristic
9C3×C4⋊Q8—22no—
10C3×C4⋊Q8C414yescomplemented
11C2×C3×C4⋊C4—22no—
12C2×C3×C42C2211yescharacteristic
13Q8×Dic3—24no—
14#926—22no—
15#926C2212yes—
16C2×#208C2212yes—
17#836—22no—
18#835—24no—
19C2×C4⋊Q8D311yescharacteristic
20#519—31no—
21#419—32no—
22C2×#101—31no—
23#339—32no—
24C2×C3×Q8—22no—
25C2×C3×Q8—21no—
26C2×C3×Q8—22no—
27C2×C3×Q8—22no—
28C2×C3×Q8—22no—
29C2×C3×Q8C2×C412yes—
30C2×C3×Q8C2×C412yes—
31C2×C3×Q8D412yes—
32C22×C3×C4—21no—
33C22×C3×C4—21no—
34C22×C3×C4D412yes—
35C22×C3×C4C2311yescharacteristic
36C3×C4⋊C4—28no—
37C3×C42—21no—
38C3×C42C2×C412yes—
39#217—22no—
40C4⋊Dic3—24no—
41C4⋊Dic3D414yes—
42C4×Dic3—24no—
43C4×Dic3—22no—
44#208—22no—
45#208—22no—
46C2×C3⋊C8—22no—
47C22×Q8C2×D311yescharacteristic
48C22×Q8C2×D311yescharacteristic
49C2×M16—32no—
50C4⋊Q8—22no—
51C4⋊Q8Dic314yescomplemented
52C4×Q8—64no—
53#117—32no—
54#117—62no—
55C2×C4⋊C4—22no—
56C2×C42C2×D311yescharacteristic
57#103—64no—
58#101—62no—
59C23×C3C2×D411yescharacteristic
60C3×Q8—24no—
61C3×Q8—24no—
62C3×Q8—24no—
63C3×Q8—42no—
64C2×C3×C4—24no—
65C2×C3×C4—22no—
66C2×C3×C4—22no—
67C2×C3×C4—22no—
68C2×C3×C4—22no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4C22⋊C414yes—
72C2×C3×C4C22×C411yescharacteristic
73C2×C3×C4C2×D412yes—
74C2×C3×C4C2×D411yescharacteristic
75C2×C3×C4#4112yes—
76C2×Dic3—24no—
77C3⋊C8—42no—
78C2×Q8—22no—
79C2×Q8—22no—
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8C2×Dic312yes—
84C2×Q8C2×Dic312yes—
85C2×Q8#6712yes—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4#6712yes—
89C22×C4C22×D311yescharacteristic
90M16—62no—
91M16—62no—
92C2×C8—62no—
93C4⋊C4—28no—
94C4⋊C4—34no—
95C4⋊C4—64no—
96C22⋊C4—62no—
97C42—21no—
98C42—64no—
99C42—62no—
100C42C2×Dic312yes—
101C22×C3—22no—
102C22×C3C2×C22⋊C411yescharacteristic
103C22×C3#12011yescharacteristic
104C22×C3#12311yescharacteristic
105C3×C4—24no—
106C3×C4—24no—
107C3×C4—42no—
108C3×C4C4×D412yes—
109C3×C4#12112yes—
110Dic3—44no—
111C23C2×#6711yescharacteristic
112Q8—24no—
113Q8—24no—
114Q8—24no—
115Q8—42no—
116C2×C4—24no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—64no—
124C2×C4#21714yes—
125C2×C4D3×D412yes—
126C2×C4#23712yes—
127C2×C4C22×Dic311yescharacteristic
128C2×C4C2×#6711yescharacteristic
129C8—122no—
130C2×C3—21no—
131C2×C3—21no—
132C2×C3#38611yescharacteristic
133C2×C3#45612yes—
134C22—22no—
135C22#93511yescharacteristic
136C22#93711yescharacteristic
137C22C2×#21711yescharacteristic
138C4—24no—
139C4—24no—
140C4—42no—
141C4—124no—
142C4D4×Dic312yes—
143C4#93912yes—
144C3#188611yescharacteristic, complemented, Sylow
145C2—21no—
146C2—21no—
147C2#509512yes—
148C2#522011yescharacteristic
subgroupG/H
centerC22#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×C4⋊Q8C2
socleC22×C3#120
prime radicalC23×C3C2×D4
cosocleC22×C3×C4C23
hypercenterC2×C4⋊Q8D3
hypocenterC3#1886
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
10. C3×C4⋊Q8C4semidirect
51. C4⋊Q8Dic3semidirect
144. C3#1886semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
#58122
C2×#8362
#52252
#45232
#18863