Finite Group Databasebeta

← #68998

Subgroups — #68998

Order 384.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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#5500C211yesmaximal, characteristic
2#5120C212yesmaximal
3#4973C211yesmaximal, characteristic
4C3×#343C211yesmaximal, characteristic
5#4555C212yesmaximal
6#2028—31nomaximal, Sylow
7C22×Dic6C2211yescharacteristic
8C2×C3×M16C2212yes—
9C2×C3×C4⋊C4C2211yescharacteristic
10#926C2212yes—
11#924—22no—
12C2×C4×Dic3C2211yescharacteristic
13#890—22no—
14#888—22no—
15C3⋊#95—22no—
16#816—22no—
17#816—22no—
18#544—31no—
19#419—32no—
20#391—31no—
21#343D311yescharacteristic
22#339—32no—
23C22×C3×C4D412yes—
24C22×C3×C4C2311yescharacteristic
25C22×Dic3—21no—
26C22×Dic3—21no—
27C2×Dic6—22no—
28C2×Dic6—22no—
29C2×Dic6—21no—
30C2×Dic6—22no—
31C2×Dic6Q812yescomplemented
32C3×M16—22no—
33C3×M16—22no—
34C2×C3×C8—22no—
35C3×C4⋊C4—22no—
36#217—22no—
37C4⋊Dic3—22no—
38C4⋊Dic3D412yes—
39C4⋊Dic3D412yes—
40Dic3⋊C4—24no—
41C4×Dic3—22no—
42C4×Dic3—22no—
43C22×Q8—31no—
44C2×M16C2×D312yes—
45C4⋊Q8—62no—
46#125—62no—
47C22⋊Q8—62no—
48#117—32no—
49C2×C4⋊C4C2×D311yescharacteristic
50C2×C42—31no—
51#103—62no—
52#103—62no—
53#95—62no—
54C23×C3C2×D411yescharacteristic
55C2×C3×C4—22no—
56C2×C3×C4—42no—
57C2×C3×C4C2×D412yes—
58C2×C3×C4C2×Q811yescharacteristic
59C2×C3×C4#4112yes—
60C2×C3×C4#4111yescharacteristic
61C2×Dic3—22no—
62C2×Dic3—22no—
63C2×Dic3—22no—
64C2×Dic3—22no—
65C2×Dic3—41no—
66C2×Dic3—41no—
67C2×Dic3—41no—
68C2×Dic3—41no—
69Dic6—24no—
70Dic6—42no—
71Dic6—82no—
72C3×C8—42no—
73C2×Q8—32no—
74C2×Q8—62no—
75C2×Q8—61no—
76C2×Q8—62no—
77C2×Q8—62no—
78C22×C4—61no—
79C22×C4—61no—
80C22×C4D1212yes—
81C22×C4C22×D311yescharacteristic
82M16—22no—
83M16—22no—
84C2×C8—22no—
85C4⋊C4—22no—
86C4⋊C4—32no—
87C4⋊C4—32no—
88C4⋊C4—62no—
89C4⋊C4—64no—
90C22⋊C4—62no—
91C42—62no—
92C42—62no—
93C22×C3—21no—
94C22×C3—21no—
95C22×C3#12011yescharacteristic
96C22×C3C22⋊Q811yescharacteristic
97C22×C3#12411yescharacteristic
98C3×C4—42no—
99C3×C4C22⋊Q812yes—
100C3×C4#12412yes—
101Dic3—42no—
102Dic3—42no—
103Dic3—42no—
104Dic3—42no—
105C23C2×D1211yescharacteristic
106Q8—64no—
107Q8—122no—
108Q8—242no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—62no—
112C2×C4—62no—
113C2×C4—62no—
114C2×C4—62no—
115C2×C4—121no—
116C2×C4—121no—
117C2×C4—121no—
118C2×C4—121no—
119C2×C4#23512yes—
120C2×C4D3×D412yes—
121C2×C4#23711yescharacteristic
122C2×C4D3×Q811yescharacteristic
123C8—42no—
124C2×C3—21no—
125C2×C3—21no—
126C2×C3#39311yescharacteristic
127C2×C3#45511yescharacteristic
128C2×C3#45611yescharacteristic
129C22—21no—
130C22—21no—
131C22#87511yescharacteristic
132C22#88211yescharacteristic
133C22#89711yescharacteristic
134C4—42no—
135C4—122no—
136C4—122no—
137C4—122no—
138C4—122no—
139C4#88512yes—
140C4#89612yes—
141C3#202811yescharacteristic, complemented, Sylow
142C2—21no—
143C2—21no—
144C2#467311yescharacteristic
145C2#474611yescharacteristic
146C2#475211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#343C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#343D3
hypocenterC3#2028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
31. C2×Dic6Q8semidirect
141. C3#2028semidirect

Maximal subgroups

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

subgroupindex
#55002
#51202
#49732
C3×#3432
#45552
#20283