Finite Group Databasebeta

← #78641

Subgroups — #78641

Order 384.

514 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 125 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×#930C212yesmaximal, complemented
2C3×#407C211yesmaximal, characteristic, complemented
3#5135C214yesmaximal, complemented
4#4735C214yesmaximal, complemented
5#4732C214yesmaximal, complemented
6#3192—31nomaximal, Sylow
7C23×C3×C4C2211yescharacteristic
8C22×#67—22no—
9C22×D12C2212yescomplemented
10C2×C3×M16C2212yescomplemented
11#930—22no—
12#930—22no—
13#930C2214yes—
14#930C2214yescomplemented
15C2×(C2×D3)⋊C4—22no—
16C2×C4⋊Dic3C2212yescomplemented
17C2×D24C2214yescomplemented
18C3×C22⋊C8C2214yescomplemented
19#821C2218yescomplemented
20#817C2214yescomplemented
21C2×#121—32no—
22#481—34no—
23#469—34no—
24#447—34no—
25#407D311yescharacteristic, complemented
26C24×C3D411yescharacteristic
27C23×D3—22no—
28C22×C3×C4—42no—
29C22×C3×C4D412yes—
30C22×C3×C4D411yescharacteristic
31C22×C3×C4C2312yes—
32C22×C3×C4C2311yescharacteristic
33C2×#67—24no—
34C2×#67—24no—
35C2×#67—48no—
36C22×Dic3—22no—
37C2×D12—22no—
38C2×D12—44no—
39C2×D12C2314yes—
40C3×M16—42no—
41C2×C3×C8C2314yes—
42(C2×D3)⋊C4—24no—
43(C2×D3)⋊C4—24no—
44C4⋊Dic3—22no—
45C4⋊Dic3C2314yes—
46D24—44no—
47C22×D4—32no—
48C22×D4—62no—
49C23×C4C2×D311yescharacteristic
50C2×D8—34no—
51C2×M16C2×D312yescomplemented
52#121—34no—
53#121—34no—
54#121—62no—
55#121—62no—
56C2×C4⋊C4—32no—
57C2×C22⋊C4—62no—
58#106—34no—
59#102—38no—
60C22⋊C8C2×D314yescomplemented
61C23×C3—24no—
62C23×C3—41no—
63C23×C3C2×D411yescharacteristic
64C23×C3C2×D412yes—
65C22×D3—24no—
66C22×D3—44no—
67C22×D3—44no—
68C22×D3—44no—
69C2×C3×C4—22no—
70C2×C3×C4—24no—
71C2×C3×C4—41no—
72C2×C3×C4—42no—
73C2×C3×C4C2×D412yes—
74C2×C3×C4C2×D411yescharacteristic
75C2×C3×C4C2411yescharacteristic
76#67—48no—
77#67—48no—
78C2×Dic3—24no—
79C2×Dic3—44no—
80D12—28no—
81D12—44no—
82C3×C8—24no—
83C24—62no—
84C24D1211yescharacteristic
85C2×D4—34no—
86C2×D4—64no—
87C2×D4—64no—
88C2×D4—62no—
89C2×D4—124no—
90C2×D4—128no—
91C22×C4—42no—
92C22×C4—62no—
93C22×C4D1212yes—
94C22×C4D1211yescharacteristic
95C22×C4C22×D312yes—
96C22×C4C22×D311yescharacteristic
97D8—124no—
98M16—42no—
99C2×C8C22×D314yes—
100C4⋊C4—34no—
101C4⋊C4—62no—
102C22⋊C4—64no—
103C22⋊C4—64no—
104C22×C3—22no—
105C22×C3—22no—
106C22×C3—21no—
107C22×C3—22no—
108C22×C3—44no—
109C22×C3AGL(1,ℤ/8)14yes—
110C22×C3C22×D411yescharacteristic
111C2×D3—44no—
112C2×D3—44no—
113C2×D3—44no—
114C2×D3—88no—
115C3×C4—22no—
116C3×C4—21no—
117C3×C4H(5,𝔽2)12yes—
118Dic3—44no—
119C23—24no—
120C23—41no—
121C23—64no—
122C23—124no—
123C23—124no—
124C23—124no—
125C23C2×D1211yescharacteristic
126C23C2×D1212yes—
127D4—68no—
128D4—124no—
129D4—128no—
130D4—128no—
131C2×C4—22no—
132C2×C4—24no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—64no—
136C2×C4—124no—
137C2×C4C2×D1211yescharacteristic
138C2×C4C2×D1212yes—
139C2×C4C23×D311yescharacteristic
140C8—24no—
141C2×C3—24no—
142C2×C3—41no—
143C2×C3#53411yescharacteristic
144C2×C3C2×AGL(1,ℤ/8)12yes—
145D3—84no—
146C22—22no—
147C22—22no—
148C22—22no—
149C22—21no—
150C22—44no—
151C22—124no—
152C22—124no—
153C22—124no—
154C22—248no—
155C22#90814yes—
156C22C22×D1211yescharacteristic
157C4—22no—
158C4—21no—
159C4—124no—
160C4#100412yes—
161C3#319211yescharacteristic, complemented, Sylow
162C2—24no—
163C2—41no—
164C2—244no—
165C2#549311yescharacteristic
166C2C2×#90812yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#407C2
socleC22×C3C22×D4
hypercenter#407D3
hypocenterC3#3192
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C2×#930C2semidirect
2. C3×#407C2semidirect
3. #5135C2semidirect
4. #4735C2semidirect
5. #4732C2semidirect
9. C22×D12C22semidirect
10. C2×C3×M16C22semidirect
14. #930C22semidirect
16. C2×C4⋊Dic3C22semidirect
17. C2×D24C22semidirect
18. C3×C22⋊C8C22semidirect
19. #821C22semidirect
20. #817C22semidirect
25. #407D3semidirect
51. C2×M16C2×D3semidirect
60. C22⋊C8C2×D3semidirect
161. C3#3192semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C3×#4072
#51352
#47352
#47322
#31923