Finite Group Databasebeta

← #68953

Subgroups — #68953

Order 384.

300 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 77 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×#877C211yesmaximal, characteristic
2#4943C212yesmaximal
3C3×#342C211yesmaximal, characteristic
4#4527C212yesmaximal
5#4526C211yesmaximal, characteristic
6#1820—31nomaximal, Sylow
7C23×Dic3C2211yescharacteristic
8C2×C3×C22⋊C4C2212yes—
9C2×C3×C22⋊C4C2211yescharacteristic
10C2×#217C2212yes—
11C2×#217C2211yescharacteristic
12C2×Dic3⋊C4C412yescomplemented
13C2×C4×Dic3C412yescomplemented
14#877—24no—
15C3⋊#95—24no—
16C2×#117—31no—
17#381—32no—
18#342—31no—
19#342—32no—
20#342D311yescharacteristic
21C24×C3C2311yescharacteristic
22C22×C3×C4—22no—
23C22×C3×C4C2×C412yes—
24C22×Dic3—22no—
25C22×Dic3—21no—
26C22×Dic3—21no—
27C22×Dic3—22no—
28C22×Dic3C2×C411yescharacteristic
29C22×Dic3C2×C411yescharacteristic
30C22×Dic3C2×C412yes—
31C22×Dic3D414yes—
32C3×C22⋊C4—22no—
33C3×C22⋊C4—22no—
34C3×C22⋊C4—22no—
35#217—22no—
36#217—22no—
37#217—22no—
38Dic3⋊C4—24no—
39C4×Dic3—24no—
40C23×C4—31no—
41#117—64no—
42C2×C4⋊C4—32no—
43C2×C22⋊C4—31no—
44C2×C22⋊C4—32no—
45C2×C22⋊C4C2×D312yes—
46C2×C22⋊C4C2×D311yescharacteristic
47C2×C42—32no—
48#95—64no—
49C23×C3—42no—
50C23×C3C22×C411yescharacteristic
51C23×C3C2×D412yes—
52C23×C3#4112yes—
53C23×C3#4111yescharacteristic
54C23×C3#4111yescharacteristic
55C2×C3×C4—22no—
56C2×C3×C4—42no—
57C2×C3×C4—44no—
58C2×C3×C4—42no—
59C2×Dic3—21no—
60C2×Dic3—21no—
61C2×Dic3—21no—
62C2×Dic3—21no—
63C2×Dic3—24no—
64C2×Dic3—24no—
65C2×Dic3—22no—
66C2×Dic3—42no—
67C2×Dic3—44no—
68C2×Dic3—42no—
69C2×Dic3C22⋊C412yes—
70C2×Dic3C22⋊C412yes—
71C24C22×D311yescharacteristic
72C22×C4—22no—
73C22×C4—31no—
74C22×C4—31no—
75C22×C4—34no—
76C22×C4—32no—
77C22×C4—62no—
78C22×C4—61no—
79C22×C4—61no—
80C22×C4—62no—
81C22×C4C4×D312yes—
82C4⋊C4—64no—
83C22⋊C4—22no—
84C22⋊C4—22no—
85C22⋊C4—22no—
86C22⋊C4—62no—
87C22⋊C4—62no—
88C22⋊C4—62no—
89C42—64no—
90C22×C3—22no—
91C22×C3—24no—
92C22×C3—42no—
93C22×C3—41no—
94C22×C3—41no—
95C22×C3C2×C22⋊C411yescharacteristic
96C22×C3#11711yescharacteristic
97C22×C3#11711yescharacteristic
98C22×C3#12312yes—
99C22×C3#12312yes—
100C3×C4—42no—
101C3×C4—82no—
102Dic3—22no—
103Dic3—22no—
104Dic3—42no—
105Dic3—82no—
106C23—42no—
107C23C2×C4×D311yescharacteristic
108C23#23512yes—
109C23D3×D412yes—
110C23#23711yescharacteristic
111C23#23911yescharacteristic
112C2×C4—22no—
113C2×C4—32no—
114C2×C4—32no—
115C2×C4—42no—
116C2×C4—44no—
117C2×C4—42no—
118C2×C4—61no—
119C2×C4—61no—
120C2×C4—64no—
121C2×C4—61no—
122C2×C4—61no—
123C2×C4—64no—
124C2×C4—62no—
125C2×C4—122no—
126C2×C4—124no—
127C2×C4—122no—
128C2×C3—22no—
129C2×C3—22no—
130C2×C3—21no—
131C2×C3—21no—
132C2×C3#38111yescharacteristic
133C2×C3#41012yes—
134C22—22no—
135C22—24no—
136C22—42no—
137C22—41no—
138C22—41no—
139C22#87211yescharacteristic
140C22D3×C22⋊C411yescharacteristic
141C22#88312yes—
142C22#89211yescharacteristic
143C22#89412yes—
144C4—42no—
145C4—62no—
146C4—62no—
147C4—82no—
148C4—122no—
149C4—242no—
150C3#182011yescharacteristic, complemented, Sylow
151C2—22no—
152C2—22no—
153C2—21no—
154C2—21no—
155C2#466411yescharacteristic
156C2#474012yes—
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#342C2
socleC22×C3C2×C22⋊C4
prime radical, cosocleC24×C3C23
hypercenter#342D3
hypocenterC3#1820
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
12. C2×Dic3⋊C4C4semidirect
13. C2×C4×Dic3C4semidirect
150. C3#1820semidirect

Maximal subgroups

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

subgroupindex
C2×#8772
#49432
C3×#3422
#45272
#45262
#18203