Finite Group Databasebeta

← #82483

Subgroups — #82483

Order 384.

468 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 173 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×C3×C4×Q8C211yesmaximal, characteristic
2C2×#870C211yesmaximal, characteristic
3#5230C212yesmaximal
4#4983C212yesmaximal
5#4976C214yesmaximal
6C4×C4⋊Dic3C211yesmaximal, characteristic
7#4928C212yesmaximal
8#4928C212yesmaximal
9#2584—31nomaximal, Sylow
10C22×C3×Q8C2211yescharacteristic
11C3×C4×Q8—24no—
12C2×C3×C4⋊C4C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C42C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×C4⋊Dic3C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×Dic3⋊C4C2214yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2214yes—
23C2×C4×Dic3C2212yes—
24#870—24no—
25C3⋊#95C2214yes—
26C3⋊#95C2214yes—
27C3⋊#95C2212yes—
28C2×#125—31no—
29C2×C4×Q8D311yescharacteristic
30#398—32no—
31#395—32no—
32#389—34no—
33#389—32no—
34#389—32no—
35C4×C4⋊C4—31no—
36C2×C3×Q8—22no—
37C2×C3×Q8—24no—
38C22×C3×C4C2312yes—
39C22×C3×C4C2311yescharacteristic
40C22×C3×C4C2311yescharacteristic
41C22×C3×C4C2312yes—
42C22×C3×C4C2311yescharacteristic
43C22×Dic3C2312yes—
44C22×Dic3C2312yes—
45C22×Dic3C2314yes—
46C3×C4⋊C4—24no—
47C3×C4⋊C4—22no—
48C3×C42—24no—
49C3×C42D414yes—
50C4⋊Dic3—22no—
51C4⋊Dic3—22no—
52C4⋊Dic3Q818yescomplemented
53Dic3⋊C4—24no—
54Dic3⋊C4—24no—
55Dic3⋊C4—28no—
56Dic3⋊C4—44no—
57C4×Dic3—42no—
58C22×Q8C2×D311yescharacteristic
59#125—64no—
60C4×Q8—24no—
61C2×C4⋊C4—32no—
62C2×C4⋊C4—32no—
63C2×C4⋊C4—31no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4—34no—
67C2×C4⋊C4—34no—
68C2×C4⋊C4C2×D312yes—
69C2×C4⋊C4C2×D311yescharacteristic
70C2×C42—32no—
71C2×C42C2×D312yes—
72C2×C42C2×D311yescharacteristic
73#95—32no—
74#95—34no—
75#95—34no—
76C23×C3C2411yescharacteristic
77C3×Q8—28no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4—22no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4C2×D412yes—
87C2×C3×C4C2×D412yes—
88C2×C3×C4C2×D412yes—
89C2×C3×C4C2×Q814yes—
90C2×C3×C4#4114yes—
91C2×Dic3—24no—
92C2×Dic3—24no—
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—24no—
96C2×Dic3—22no—
97C2×Dic3—22no—
98C2×Dic3C2×Q818yes—
99C2×Q8—24no—
100C2×Q8—22no—
101C22×C4—34no—
102C22×C4—32no—
103C22×C4—32no—
104C22×C4C22×D312yes—
105C22×C4C22×D312yes—
106C22×C4C22×D311yescharacteristic
107C22×C4C22×D311yescharacteristic
108C22×C4C22×D311yescharacteristic
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—38no—
112C4⋊C4—68no—
113C4⋊C4—64no—
114C4⋊C4—64no—
115C4⋊C4—62no—
116C4⋊C4—62no—
117C4⋊C4—124no—
118C42—24no—
119C42—122no—
120C42#6714yes—
121C22×C3C22×D411yescharacteristic
122C22×C3C22×Q812yes—
123C22×C3C2×#4112yes—
124C22×C3H(5,𝔽2)11yescharacteristic
125C22×C32−1+411yescharacteristic
126C3×C4—22no—
127C3×C4—24no—
128C3×C4—22no—
129C3×C4—42no—
130C3×C4C4⋊Q814yes—
131Dic3—28no—
132Dic3—42no—
133Dic3—42no—
134C23C23×D311yescharacteristic
135Q8—28no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—38no—
145C2×C4—64no—
146C2×C4—64no—
147C2×C4—64no—
148C2×C4—62no—
149C2×C4—62no—
150C2×C4—62no—
151C2×C4—62no—
152C2×C4#23514yes—
153C2×C4D3×Q814yes—
154C2×C4C2×#6712yes—
155C2×C4C2×#6712yes—
156C2×C4C2×#6712yes—
157C2×C3#52511yescharacteristic
158C2×C3C2×C4⋊Q811yescharacteristic
159C2×C3#53711yescharacteristic
160C2×C3#55412yes—
161C2×C3#55712yes—
162C22C2×#23512yes—
163C22C2×D3×Q812yes—
164C22#100911yescharacteristic
165C22#101011yescharacteristic
166C22C22×#6711yescharacteristic
167C4—24no—
168C4—22no—
169C4—22no—
170C4—42no—
171C4—68no—
172C4—122no—
173C4—122no—
174C4#94414yes—
175C3#258411yescharacteristic, complemented, Sylow
176C2#556712yes—
177C2#557912yes—
178C2#579111yescharacteristic
179C2C2×#94411yescharacteristic
180C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC2×C3×C4×Q8C2
hypercenterC2×C4×Q8D3
hypocenterC3#2584
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
52. C4⋊Dic3Q8semidirect
175. C3#2584semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q82
C2×#8702
#52302
#49832
#49762
C4×C4⋊Dic32
#49282
#25843