Finite Group Databasebeta

← #68354

Subgroups — #68354

Order 384.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#5672C211yesmaximal, characteristic, complemented
2#5058C211yesmaximal, characteristic
3#4713C212yesmaximal, complemented
4C3×#329C211yesmaximal, characteristic, complemented
5#4465C212yesmaximal, complemented
6#1657—31nomaximal, Sylow
7C3×#124C2211yescharacteristic
8#944—21no—
9C2×#215—21no—
10#935—21no—
11C2×#213—21no—
12#879—22no—
13#878—22no—
14#874C2211yescharacteristic
15#870C2212yes—
16C3×#97C2212yes—
17#821—22no—
18#821—22no—
19#803C2211yescharacteristic
20#563—31no—
21#496—31no—
22#490—32no—
23#330—32no—
24#329D311yescharacteristic, complemented
25C2×C3×Q8D411yescharacteristic
26C2×C3×D4D411yescharacteristic
27C22×Dic3—21no—
28C2×D12—22no—
29C2×D12—21no—
30C2×Dic6—21no—
31C2×C3×C8—22no—
32C3×C22⋊C4—22no—
33C3×C42C2311yescharacteristic
34#217—22no—
35#217—21no—
36#215—42no—
37#213—42no—
38C4⋊Dic3D412yes—
39C4⋊Dic3D412yes—
40Dic3⋊C4—22no—
41Dic3⋊C4—22no—
42Dic3⋊C4—22no—
43C4×Dic3—21no—
44C2×C3⋊C8—21no—
45C2×SD16—61no—
46C2×D8—61no—
47C4⋊Q8—61no—
48#127—31no—
49#126—62no—
50#125—32no—
51#124C2×D311yescharacteristic
52#123—61no—
53C22⋊Q8—62no—
54#102—62no—
55#102—62no—
56#97—31no—
57#97C2×D312yes—
58C23×C3—21no—
59C22×D3—41no—
60C3×Q8—41no—
61C3×D4—41no—
62C2×C3×C4—21no—
63C2×C3×C4C2×D411yescharacteristic
64C2×C3×C4C2×D412yes—
65C2×Dic3—22no—
66C2×Dic3—22no—
67C2×Dic3—81no—
68D12—41no—
69D12—41no—
70D12—44no—
71Dic6—81no—
72C3×C8—42no—
73C3⋊C8—22no—
74C2×Q8—61no—
75C2×Q8D1211yescharacteristic
76C2×D4—61no—
77C2×D4—62no—
78C2×D4D1211yescharacteristic
79C22×C4—61no—
80SD16—122no—
81D8—122no—
82C2×C8—22no—
83C2×C8—61no—
84C4⋊C4—32no—
85C4⋊C4—32no—
86C4⋊C4—62no—
87C4⋊C4—62no—
88C4⋊C4—62no—
89C22⋊C4—22no—
90C22⋊C4—62no—
91C22⋊C4—61no—
92C42—61no—
93C42C22×D311yescharacteristic
94C22×C3—41no—
95C22×C3—42no—
96C22×C3#12011yescharacteristic
97C2×D3—81no—
98C2×D3—82no—
99C3×C4—21no—
100C3×C4—22no—
101C3×C4—41no—
102Dic3—42no—
103Dic3—42no—
104C23—21no—
105C23—121no—
106Q8—41no—
107Q8—241no—
108D4—41no—
109D4—121no—
110D4—121no—
111D4—124no—
112C2×C4—21no—
113C2×C4—62no—
114C2×C4—62no—
115C2×C4—241no—
116C2×C4C2×D1211yescharacteristic
117C2×C4D3×D412yes—
118C8—42no—
119C8—62no—
120C2×C3—81no—
121C2×C3#45311yescharacteristic
122C2×C3#45412yes—
123D3—161no—
124C22—41no—
125C22—42no—
126C22—241no—
127C22—242no—
128C22#88211yescharacteristic
129C4—21no—
130C4—22no—
131C4—41no—
132C4—122no—
133C4—122no—
134C3#165711yescharacteristic, complemented, Sylow
135C2—81no—
136C2—481no—
137C2#474711yescharacteristic
138C2#482712yes—
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3#120
prime radical#5058C2
hypercenter#329D3
hypocenterC3#1657
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #5672C2semidirect
3. #4713C2semidirect
4. C3×#329C2semidirect
5. #4465C2semidirect
24. #329D3semidirect
134. C3#1657semidirect

Maximal subgroups

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

subgroupindex
#56722
#50582
#47132
C3×#3292
#44652
#16573