Finite Group Databasebeta

← #74050

Subgroups — #74050

Order 384.

1088 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 245 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
1C23×C4×D3C211yesmaximal, characteristic
2C2×#877C212yesmaximal, complemented
3C3×#381C211yesmaximal, characteristic, complemented
4#5211C211yesmaximal, characteristic, complemented
5#4955C212yesmaximal
6#4667C214yesmaximal, complemented
7#4666C214yesmaximal, complemented
8#2280—31nomaximal, Sylow
9C24×D3C2211yescharacteristic
10C23×C3×C4C2211yescharacteristic
11C23×Dic3C2211yescharacteristic
12C22×C4×D3—28no—
13C22×C4×D3C418yescomplemented
14C22×C4×D3C2212yes—
15C22×C4×D3C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2212yes—
18C2×(C2×D3)⋊C4C2218yes—
19C2×Dic3⋊C4C2214yescomplemented
20C2×C4×Dic3C2214yescomplemented
21#877—28no—
22C3×#95C2214yescomplemented
23C3⋊#95C2214yescomplemented
24C24×C4—31no—
25C2×#117—32no—
26#381—31no—
27#381D311yescharacteristic, complemented
28#380—34no—
29#379—32no—
30C4×C22⋊C4—34no—
31C24×C3C2311yescharacteristic
32C23×D3—24no—
33C23×D3—28no—
34C23×D3C2×C414yes—
35C23×D3C2312yes—
36C22×C3×C4—24no—
37C22×C3×C4C2×C414yes—
38C22×C3×C4C2314yes—
39C22×C3×C4C2312yes—
40C22×Dic3C2×C414yes—
41C22×Dic3D418yes—
42C22×Dic3C2314yes—
43C22×Dic3C2311yescharacteristic
44C22×Dic3C2311yescharacteristic
45C2×C4×D3—232no—
46C2×C4×D3—28no—
47C2×C4×D3—28no—
48C2×C4×D3C2×C418yes—
49C2×C4×D3C2×C418yes—
50C3×C22⋊C4—24no—
51#217—24no—
52(C2×D3)⋊C4—216no—
53Dic3⋊C4—28no—
54C4×Dic3—28no—
55C25—31no—
56C23×C4—31no—
57C23×C4—38no—
58C23×C4—32no—
59C23×C4—32no—
60C23×C4—68no—
61C23×C4C2×D311yescharacteristic
62#117—68no—
63C2×C4⋊C4—34no—
64C2×C22⋊C4—32no—
65C2×C22⋊C4—38no—
66C2×C22⋊C4C2×D312yes—
67C2×C42—34no—
68#95—34no—
69#95C2×D314yescomplemented
70C23×C3—24no—
71C23×C3C22×C412yes—
72C23×C3C2×D414yes—
73C23×C3C2411yescharacteristic
74C22×D3—232no—
75C22×D3—28no—
76C22×D3—24no—
77C22×D3—24no—
78C22×D3—216no—
79C22×D3C22×C414yes—
80C22×D3#4118yes—
81C2×C3×C4—28no—
82C2×C3×C4—24no—
83C2×C3×C4—24no—
84C2×C3×C4—28no—
85C2×C3×C4C22×C414yes—
86C2×Dic3—24no—
87C2×Dic3—28no—
88C2×Dic3C22⋊C4116yescomplemented
89C2×Dic3C22×C412yes—
90C2×Dic3C22×C412yes—
91C2×Dic3C2×D414yes—
92C2×Dic3C2×D414yes—
93C4×D3—216no—
94C4×D3—216no—
95C24—32no—
96C24—34no—
97C24—68no—
98C24—64no—
99C24C22×D311yescharacteristic
100C22×C4—24no—
101C22×C4—34no—
102C22×C4—38no—
103C22×C4—31no—
104C22×C4—31no—
105C22×C4—38no—
106C22×C4—38no—
107C22×C4—34no—
108C22×C4—632no—
109C22×C4—68no—
110C22×C4—68no—
111C22×C4C4×D314yes—
112C22×C4C22×D314yes—
113C22×C4C22×D312yes—
114C4⋊C4—68no—
115C22⋊C4—24no—
116C22⋊C4—616no—
117C22⋊C4—64no—
118C42—68no—
119C22×C3—24no—
120C22×C3—28no—
121C22×C3C2×C22⋊C414yes—
122C22×C3C23×C411yescharacteristic
123C22×C3C22×D412yes—
124C22×C3C2×#4114yes—
125C2×D3—232no—
126C2×D3—24no—
127C2×D3—24no—
128C2×D3—216no—
129C2×D3#11718yescomplemented
130C3×C4—24no—
131C3×C4—44no—
132Dic3—44no—
133Dic3C2×C22⋊C414yes—
134Dic3C2×C22⋊C414yes—
135C23—24no—
136C23—34no—
137C23—38no—
138C23—616no—
139C23—632no—
140C23—68no—
141C23—64no—
142C23—64no—
143C23C2×C4×D312yes—
144C23D3×D414yes—
145C23C23×D311yescharacteristic
146C2×C4—28no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—28no—
150C2×C4—316no—
151C2×C4—32no—
152C2×C4—32no—
153C2×C4—34no—
154C2×C4—34no—
155C2×C4—616no—
156C2×C4—616no—
157C2×C4—64no—
158C2×C4—68no—
159C2×C4C2×C4×D314yes—
160C2×C3—24no—
161C2×C3C22×C22⋊C411yescharacteristic
162C2×C3C2×#11712yes—
163C2×C3#52514yes—
164D3—28no—
165C22—28no—
166C22—24no—
167C22—38no—
168C22—632no—
169C22—616no—
170C22—64no—
171C22—64no—
172C22D3×C22⋊C414yes—
173C22C22×C4×D311yescharacteristic
174C22C2×D3×D412yes—
175C22D3×#4114yes—
176C4—24no—
177C4—34no—
178C4—34no—
179C4—44no—
180C4—124no—
181C3#228011yescharacteristic, complemented, Sylow
182C2—24no—
183C2—68no—
184C2C2×D3×C22⋊C411yescharacteristic
185C2D3×#11712yes—
186C2#565614yes—
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#381C2
socle, cosocleC23×C3C24
prime radicalC24×D3C22
hypercenter#381D3
hypocenterC3#2280
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#877C2semidirect
3. C3×#381C2semidirect
4. #5211C2semidirect
6. #4667C2semidirect
7. #4666C2semidirect
13. C22×C4×D3C4semidirect
19. C2×Dic3⋊C4C22semidirect
20. C2×C4×Dic3C22semidirect
22. C3×#95C22semidirect
23. C3⋊#95C22semidirect
27. #381D3semidirect
69. #95C2×D3semidirect
88. C2×Dic3C22⋊C4semidirect
129. C2×D3#117semidirect
181. C3#2280semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D32
C2×#8772
C3×#3812
#52112
#49552
#46672
#46662
#22803