Finite Group Databasebeta

← #68979

Subgroups — #68979

Order 384.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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#5121C212yesmaximal
3#4977C211yesmaximal, characteristic
4C3×#343C211yesmaximal, characteristic
5#4555C212yesmaximal
6#2061—31nomaximal, Sylow
7C2×C3×M16C2212yes—
8C2×C3×C4⋊C4C2211yescharacteristic
9#926C2212yes—
10C2×C4⋊Dic3C2211yescharacteristic
11#924—22no—
12C2×C4×Dic3C2211yescharacteristic
13#890—22no—
14#888—22no—
15C3⋊#95—22no—
16#818—22no—
17#817—22no—
18#544—31no—
19#428—32no—
20#382—31no—
21#343D311yescharacteristic
22#339—32no—
23C22×C3×C4D412yes—
24C22×C3×C4C2311yescharacteristic
25C22×Dic3—21no—
26C22×Dic3—21no—
27C2×Dic6—22no—
28C3×M16—22no—
29C3×M16Q814yescomplemented
30C2×C3×C8—22no—
31C3×C4⋊C4—22no—
32#217—22no—
33C4⋊Dic3—22no—
34C4⋊Dic3—22no—
35C4⋊Dic3D412yes—
36C4⋊Dic3D412yes—
37Dic3⋊C4—24no—
38C4×Dic3—22no—
39C4×Dic3—21no—
40C4×Dic3Q812yescomplemented
41C2×M16C2×D312yes—
42C4⋊Q8—62no—
43#125—62no—
44C22⋊Q8—62no—
45#117—32no—
46C2×C4⋊C4—31no—
47C2×C4⋊C4C2×D311yescharacteristic
48C2×C42—31no—
49#107—62no—
50#106—62no—
51#95—62no—
52C23×C3C2×D411yescharacteristic
53C2×C3×C4—22no—
54C2×C3×C4—42no—
55C2×C3×C4C2×D412yes—
56C2×C3×C4C2×Q812yes—
57C2×C3×C4C2×Q811yescharacteristic
58C2×C3×C4#4111yescharacteristic
59C2×Dic3—22no—
60C2×Dic3—22no—
61C2×Dic3—22no—
62C2×Dic3—41no—
63C2×Dic3—41no—
64C2×Dic3—42no—
65C2×Dic3—41no—
66Dic6—82no—
67C3×C8—24no—
68C2×Q8—62no—
69C22×C4—61no—
70C22×C4—61no—
71C22×C4D1212yes—
72C22×C4C22×D311yescharacteristic
73M16—22no—
74M16Dic614yescomplemented
75C2×C8—22no—
76C4⋊C4—22no—
77C4⋊C4—32no—
78C4⋊C4—32no—
79C4⋊C4—62no—
80C4⋊C4—64no—
81C4⋊C4—62no—
82C22⋊C4—62no—
83C42—32no—
84C42—62no—
85C42—61no—
86C22×C3—21no—
87C22×C3—21no—
88C22×C3#12011yescharacteristic
89C22×C3C22⋊Q811yescharacteristic
90C22×C3C4⋊Q811yescharacteristic
91C3×C4—42no—
92C3×C4C22⋊Q812yes—
93C3×C4C4⋊Q812yes—
94Dic3—42no—
95Dic3—42no—
96Dic3—42no—
97Dic3—81no—
98C23C2×D1211yescharacteristic
99Q8—242no—
100C2×C4—22no—
101C2×C4—42no—
102C2×C4—62no—
103C2×C4—62no—
104C2×C4—62no—
105C2×C4—121no—
106C2×C4—121no—
107C2×C4—122no—
108C2×C4—121no—
109C2×C4C2×Dic612yes—
110C2×C4D3×D412yes—
111C2×C4#23711yescharacteristic
112C2×C4D3×Q811yescharacteristic
113C8—24no—
114C2×C3—21no—
115C2×C3—21no—
116C2×C3#39511yescharacteristic
117C2×C3#45411yescharacteristic
118C2×C3#45511yescharacteristic
119C22—21no—
120C22—21no—
121C22#86911yescharacteristic
122C22#88211yescharacteristic
123C22#89711yescharacteristic
124C4—42no—
125C4—122no—
126C4—122no—
127C4—122no—
128C4—241no—
129C4#87812yes—
130C4#88812yes—
131C3#206111yescharacteristic, complemented, Sylow
132C2—21no—
133C2—21no—
134C2#465611yescharacteristic
135C2#474611yescharacteristic
136C2#475311yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#343C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#343D3
hypocenterC3#2061
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
29. C3×M16Q8semidirect
40. C4×Dic3Q8semidirect
74. M16Dic6semidirect
131. C3#2061semidirect

Maximal subgroups

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

subgroupindex
#55002
#51212
#49772
C3×#3432
#45552
#20613