Finite Group Databasebeta

← #69497

Subgroups — #69497

Order 384.

466 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 69 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#5483C211yesmaximal, characteristic, complemented
2C3×#379C211yesmaximal, characteristic
3#5247C212yesmaximal
4#4527C212yesmaximal, complemented
5#4526C211yesmaximal, characteristic, complemented
6#2029—31nomaximal, Sylow
7C25×C3C2211yescharacteristic
8C2×C3×C22⋊C4—22no—
9C2×C3×C22⋊C4C2211yescharacteristic
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×#217—24no—
12C2×#217C2212yes—
13C2×#217C2212yes—
14#924—22no—
15#878—24no—
16#543—31no—
17#379—32no—
18#379D311yescharacteristic
19#342—31no—
20#342—32no—
21C24×C3—22no—
22C24×C3—21no—
23C24×C3—21no—
24C24×C3—22no—
25C24×C3—44no—
26C24×C3Q812yescomplemented
27C24×C3C2311yescharacteristic
28C22×C3×C4—21no—
29C22×C3×C4D412yes—
30C22×Dic3—22no—
31C22×Dic3D414yes—
32C2×Dic6—22no—
33C3×C22⋊C4—22no—
34C3×C22⋊C4—22no—
35C3×C22⋊C4—44no—
36#217—22no—
37#217—22no—
38#217—22no—
39#217—22no—
40#217—48no—
41C4⋊Dic3—22no—
42Dic3⋊C4—24no—
43C25C2×D311yescharacteristic
44C22⋊Q8—64no—
45C22⋊Q8—62no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—32no—
48C2×C22⋊C4—32no—
49C2×C22⋊C4—64no—
50C2×C22⋊C4C2×D311yescharacteristic
51C2×C22⋊C4C2×D311yescharacteristic
52C23×C3—28no—
53C23×C3—24no—
54C23×C3—24no—
55C23×C3—22no—
56C23×C3—44no—
57C23×C3—42no—
58C23×C3—44no—
59C23×C3—42no—
60C23×C3—44no—
61C23×C3—48no—
62C23×C3—41no—
63C23×C3—42no—
64C23×C3—41no—
65C23×C3C2×D412yes—
66C23×C3C2×D411yescharacteristic
67C23×C3C2×Q811yescharacteristic
68C23×C3#4111yescharacteristic
69C23×C3#4112yes—
70C2×C3×C4—22no—
71C2×C3×C4—42no—
72C2×C3×C4—42no—
73C2×C3×C4—41no—
74C2×Dic3—24no—
75C2×Dic3—44no—
76C2×Dic3—44no—
77C2×Dic3—42no—
78Dic6—82no—
79C24—22no—
80C24—21no—
81C24—21no—
82C24—22no—
83C24—44no—
84C24Dic612yescomplemented
85C24C22×D311yescharacteristic
86C2×Q8—62no—
87C22×C4—21no—
88C22×C4—34no—
89C22×C4—62no—
90C22×C4#6712yes—
91C4⋊C4—64no—
92C4⋊C4—62no—
93C22⋊C4—22no—
94C22⋊C4—22no—
95C22⋊C4—44no—
96C22⋊C4—62no—
97C22⋊C4—62no—
98C22⋊C4—62no—
99C22⋊C4—62no—
100C22⋊C4—128no—
101C22×C3—24no—
102C22×C3—22no—
103C22×C3—28no—
104C22×C3—24no—
105C22×C3—41no—
106C22×C3—41no—
107C22×C3—42no—
108C22×C3—48no—
109C22×C3—44no—
110C22×C3—44no—
111C22×C3—42no—
112C22×C3—42no—
113C22×C3—44no—
114C22×C3#12011yescharacteristic
115C22×C3C22⋊Q812yes—
116C22×C3C22⋊Q811yescharacteristic
117C22×C3#12412yes—
118C22×C3#12411yescharacteristic
119C3×C4—42no—
120C3×C4—81no—
121Dic3—44no—
122Dic3—82no—
123C23—24no—
124C23—28no—
125C23—24no—
126C23—22no—
127C23—44no—
128C23—42no—
129C23—42no—
130C23—44no—
131C23—44no—
132C23—48no—
133C23—42no—
134C23—41no—
135C23—41no—
136C23C2×Dic611yescharacteristic
137C23#23511yescharacteristic
138C23D3×D412yes—
139C23#23712yes—
140C23C2×#6711yescharacteristic
141Q8—242no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—64no—
147C2×C4—124no—
148C2×C4—124no—
149C2×C4—122no—
150C2×C3—22no—
151C2×C3—21no—
152C2×C3—22no—
153C2×C3—21no—
154C2×C3—44no—
155C2×C3#39311yescharacteristic
156C2×C3B(4,𝔽2)12yes—
157C22—22no—
158C22—24no—
159C22—28no—
160C22—24no—
161C22—41no—
162C22—41no—
163C22—42no—
164C22—44no—
165C22—48no—
166C22—44no—
167C22—42no—
168C22—42no—
169C22—44no—
170C22#87812yes—
171C22#88512yes—
172C22#92411yescharacteristic
173C22#93611yescharacteristic
174C22#93711yescharacteristic
175C4—42no—
176C4—81no—
177C4—124no—
178C4—242no—
179C3#202911yescharacteristic, complemented, Sylow
180C2—21no—
181C2—22no—
182C2—22no—
183C2—21no—
184C2—44no—
185C2#494711yescharacteristic
186C2#503212yes—
subgroupG/H
centerC22#937
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
FittingC3×#379C2
socleC22×C3#120
prime radicalC25×C3C22
hypercenter#379D3
hypocenterC3#2029
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #5483C2semidirect
4. #4527C2semidirect
5. #4526C2semidirect
26. C24×C3Q8semidirect
84. C24Dic6semidirect
179. C3#2029semidirect

Maximal subgroups

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

subgroupindex
#54832
C3×#3792
#52472
#45272
#45262
#20293