Finite Group Databasebeta

← #83066

Subgroups — #83066

Order 384.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#125C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic
3#4993C212yesmaximal, complemented
4#4992C212yesmaximal, complemented
5#4991C212yesmaximal
6#4989C214yesmaximal, complemented
7#4981C212yesmaximal, complemented
8#4937C211yesmaximal, characteristic, complemented
9#2633—31nomaximal, Sylow
10C22×D12C2211yescharacteristic
11C22×C4×D3C2212yes—
12C3×#125—24no—
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2214yes—
20C2×(C2×D3)⋊C4C2214yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2214yes—
23C4×D12—24no—
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2214yes—
28C2×#125D311yescharacteristic, complemented
29C2×C4×D4—31no—
30#399—32no—
31#398—32no—
32#397—32no—
33#394—32no—
34#385—34no—
35#383—31no—
36C23×D3C2312yes—
37C22×C3×C4C2312yes—
38C22×C3×C4C2312yes—
39C22×C3×C4C2311yescharacteristic
40C22×C3×C4C2311yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C22×Dic3C2314yes—
43C22×Dic3C2312yes—
44C2×D12—24no—
45C2×D12—22no—
46C2×C4×D3—24no—
47C2×C4×D3—44no—
48C3×C4⋊C4—24no—
49C3×C4⋊C4—22no—
50C3×C4⋊C4—22no—
51C3×C4⋊C4—24no—
52C3×C42D414yes—
53(C2×D3)⋊C4—28no—
54(C2×D3)⋊C4—24no—
55(C2×D3)⋊C4—44no—
56C4⋊Dic3—22no—
57Dic3⋊C4—44no—
58C22×D4—31no—
59C23×C4—32no—
60#125—24no—
61C4×D4—64no—
62C2×C4⋊C4—31no—
63C2×C4⋊C4—34no—
64C2×C4⋊C4C2×D312yes—
65C2×C4⋊C4C2×D312yes—
66C2×C4⋊C4C2×D311yescharacteristic
67C2×C4⋊C4C2×D311yescharacteristic
68C2×C22⋊C4—32no—
69C2×C22⋊C4—34no—
70C2×C22⋊C4—34no—
71C2×C42C2×D311yescharacteristic
72#95—32no—
73#95—32no—
74#95—32no—
75#95—34no—
76C23×C3C2411yescharacteristic
77C22×D3—22no—
78C22×D3—22no—
79C22×D3—24no—
80C22×D3—24no—
81C22×D3#4114yes—
82C2×C3×C4—22no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—24no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4C2×D412yes—
91C2×C3×C4C2×D412yes—
92C2×C3×C4C2×D412yes—
93C2×Dic3—22no—
94C2×Dic3—24no—
95C2×Dic3—24no—
96C2×Dic3—24no—
97C2×Dic3—24no—
98D12—28no—
99C4×D3—48no—
100C24—32no—
101C2×D4—64no—
102C2×D4—62no—
103C22×C4—34no—
104C22×C4—32no—
105C22×C4—64no—
106C22×C4—124no—
107C22×C4C22×D312yes—
108C22×C4C22×D312yes—
109C22×C4C22×D311yescharacteristic
110C22×C4C22×D311yescharacteristic
111C22×C4C22×D311yescharacteristic
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—62no—
117C4⋊C4—124no—
118C22⋊C4—68no—
119C22⋊C4—64no—
120C22⋊C4—124no—
121C42#6714yes—
122C22×C3C22×D411yescharacteristic
123C22×C3C2×#4112yes—
124C22×C3H(5,𝔽2)12yes—
125C22×C32−1+412yes—
126C2×D3—24no—
127C2×D3—28no—
128C2×D3—42no—
129C2×D3—42no—
130C2×D3—44no—
131C3×C4—22no—
132C3×C4—22no—
133C3×C4—22no—
134C3×C4—42no—
135C3×C4—42no—
136Dic3—42no—
137Dic3—44no—
138C23—34no—
139C23—64no—
140C23—62no—
141C23—64no—
142C23—62no—
143C23C23×D311yescharacteristic
144D4—68no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—64no—
154C2×C4—64no—
155C2×C4—64no—
156C2×C4—64no—
157C2×C4—62no—
158C2×C4—128no—
159C2×C4C2×#6712yes—
160C2×C4C2×#6712yes—
161C2×C4C2×#6712yes—
162C2×C3#52511yescharacteristic
163C2×C3#53511yescharacteristic
164C2×C3#53611yescharacteristic
165C2×C3#53914yes—
166D3—44no—
167C22—68no—
168C22—64no—
169C22—124no—
170C22—122no—
171C22—122no—
172C22#100712yes—
173C22D3×#4112yes—
174C22#100912yes—
175C22C22×#6711yescharacteristic
176C4—22no—
177C4—22no—
178C4—22no—
179C4—42no—
180C4—42no—
181C4—124no—
182C4—122no—
183C3#263311yescharacteristic, complemented, Sylow
184C2—124no—
185C2#569214yes—
186C2#581611yescharacteristic
187C2#582811yescharacteristic
188C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#125C2
prime radicalC22×D12C22
hypercenterC2×#125D3
hypocenterC3#2633
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C3×#125C2semidirect
3. #4993C2semidirect
4. #4992C2semidirect
6. #4989C2semidirect
7. #4981C2semidirect
8. #4937C2semidirect
28. C2×#125D3semidirect
183. C3#2633semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1252
C2×C4×D122
#49932
#49922
#49912
#49892
#49812
#49372
#26333