Finite Group Databasebeta

← #68978

Subgroups — #68978

Order 384.

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

normal subgroups only

This group has 246 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#4975C211yesmaximal, characteristic
4C3×#343C211yesmaximal, characteristic
5#4555C212yesmaximal
6#2060—31nomaximal, Sylow
7C2×C3×M16C2212yes—
8C2×C3×C4⋊C4C2211yescharacteristic
9#926C2212yes—
10C2×C4⋊Dic3C2211yescharacteristic
11#924—22no—
12C2×Dic3⋊C4—22no—
13C2×C4×Dic3C2211yescharacteristic
14#890—22no—
15#888—22no—
16#818—22no—
17#817—22no—
18#544—31no—
19#428—32no—
20#384—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—
38Dic3⋊C4—44no—
39C4×Dic3—22no—
40C4×Dic3—21no—
41C4×Dic3Q812yescomplemented
42C2×M16C2×D312yes—
43C4⋊Q8—62no—
44#125—62no—
45C22⋊Q8—62no—
46#117—32no—
47C2×C4⋊C4—31no—
48C2×C4⋊C4—62no—
49C2×C4⋊C4C2×D311yescharacteristic
50C2×C42—31no—
51#107—62no—
52#106—62no—
53C23×C3C2×D411yescharacteristic
54C2×C3×C4—22no—
55C2×C3×C4—42no—
56C2×C3×C4C2×D412yes—
57C2×C3×C4C2×Q812yes—
58C2×C3×C4C2×Q811yescharacteristic
59C2×C3×C4#4111yescharacteristic
60C2×Dic3—22no—
61C2×Dic3—22no—
62C2×Dic3—22no—
63C2×Dic3—22no—
64C2×Dic3—22no—
65C2×Dic3—42no—
66C2×Dic3—41no—
67Dic6—82no—
68C3×C8—24no—
69C2×Q8—62no—
70C22×C4—61no—
71C22×C4—61no—
72C22×C4D1212yes—
73C22×C4C22×D311yescharacteristic
74M16—22no—
75M16Dic614yescomplemented
76C2×C8—22no—
77C4⋊C4—22no—
78C4⋊C4—32no—
79C4⋊C4—32no—
80C4⋊C4—62no—
81C4⋊C4—64no—
82C4⋊C4—62no—
83C4⋊C4—124no—
84C22⋊C4—62no—
85C42—32no—
86C42—62no—
87C42—61no—
88C22×C3—21no—
89C22×C3—21no—
90C22×C3#12011yescharacteristic
91C22×C3C22⋊Q811yescharacteristic
92C22×C3C4⋊Q811yescharacteristic
93C3×C4—42no—
94C3×C4C22⋊Q812yes—
95C3×C4C4⋊Q812yes—
96Dic3—24no—
97Dic3—42no—
98Dic3—42no—
99Dic3—81no—
100C23C2×D1211yescharacteristic
101Q8—242no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—62no—
105C2×C4—62no—
106C2×C4—62no—
107C2×C4—62no—
108C2×C4—62no—
109C2×C4—122no—
110C2×C4—121no—
111C2×C4C2×Dic612yes—
112C2×C4D3×D412yes—
113C2×C4#23711yescharacteristic
114C2×C4D3×Q811yescharacteristic
115C8—24no—
116C2×C3—21no—
117C2×C3—21no—
118C2×C3#39511yescharacteristic
119C2×C3#45311yescharacteristic
120C2×C3#45611yescharacteristic
121C22—21no—
122C22—21no—
123C22#86911yescharacteristic
124C22#88211yescharacteristic
125C22#89711yescharacteristic
126C4—42no—
127C4—64no—
128C4—122no—
129C4—122no—
130C4—241no—
131C4#87812yes—
132C4#88812yes—
133C3#206011yescharacteristic, complemented, Sylow
134C2—21no—
135C2—21no—
136C2#465611yescharacteristic
137C2#474711yescharacteristic
138C2#475211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#343C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#343D3
hypocenterC3#2060
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
41. C4×Dic3Q8semidirect
75. M16Dic6semidirect
133. C3#2060semidirect

Maximal subgroups

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

subgroupindex
#55002
#51212
#49752
C3×#3432
#45552
#20603