Finite Group Databasebeta

← #10593985

Subgroups — #10593985

Order 576.

468 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 120 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#64079C211yesmaximal, characteristic, complemented
2C3×#875C212yesmaximal, complemented
3#63891C214yesmaximal, complemented
4#63887C214yesmaximal, complemented
5#63885C214yesmaximal, complemented
6#5684—32nomaximal
7C2×C3×D12C2212yes—
8C2×C3×Dic6C2212yes—
9C2×#625C2214yes—
10C2×D3×Dic3C2214yes—
11C32×C42C2211yescharacteristic
12C4⋊(C3⋊Dic3)C2212yes—
13#3750C2214yes—
14C3×(C2×D3)⋊C4C2214yes—
15C3×(C2×D3)⋊C4C2214yes—
16Dic3⋊Dic3C2214yes—
17(C2×D3)⋊Dic3C2214yes—
18C3×#124—32no—
19#946—34no—
20#939—34no—
21#886—34no—
22#884—34no—
23#883—34no—
24#878—34no—
25#875D312yescomplemented
26#870—32no—
27#870—61no—
28C22×C3×D3C2314yes—
29C2×C32×C4C2311yescharacteristic
30C2×C32×C4C2312yes—
31C2×C3⋊Dic3C2314yes—
32C2×C3×Dic3C2314yes—
33C3×D12—42no—
34C3×Dic6—42no—
35#625—44no—
36D3×Dic3—44no—
37#562—91noSylow
38C2×C3×Q8—32no—
39C2×C3×D4—32no—
40C2×#67—34no—
41C2×#67—34no—
42C22×Dic3—34no—
43C2×D12C2×D312yes—
44C2×C4×D3—34no—
45C2×Dic6C2×D312yes—
46C3×C22⋊C4—34no—
47C3×C22⋊C4—34no—
48C3×C42—21no—
49C3×C42C2×D312yes—
50#217—34no—
51(C2×D3)⋊C4—34no—
52(C2×D3)⋊C4C2×D314yes—
53(C2×D3)⋊C4C2×D314yes—
54C4⋊Dic3—32no—
55C4⋊Dic3—32no—
56C4⋊Dic3—34no—
57C4⋊Dic3—62no—
58Dic3⋊C4—34no—
59Dic3⋊C4—34no—
60Dic3⋊C4—34no—
61Dic3⋊C4—64no—
62C22×C32C2411yescharacteristic
63C2×C3×D3—24no—
64C2×C3×D3—24no—
65C2×C3×D3—24no—
66C32×C4—21no—
67C32×C4—22no—
68C3⋊Dic3—24no—
69C3×Dic3—24no—
70#125—91no—
71#124—32no—
72#123—94no—
73C22⋊Q8—94no—
74#121—94no—
75C23×C3—34no—
76C22×D3C22×D314yes—
77C3×Q8—122no—
78C3×D4—122no—
79C2×C3×C4—21no—
80C2×C3×C4—22no—
81C2×C3×C4—34no—
82C2×C3×C4C22×D312yes—
83C2×C3×C4C22×D312yes—
84C2×C3×C4C22×D312yes—
85#67—124no—
86#67—124no—
87C2×Dic3—34no—
88C2×Dic3—34no—
89C2×Dic3—64no—
90C2×Dic3—124no—
91C2×Dic3C22×D314yes—
92D12—42no—
93C4×D3—124no—
94Dic6—42no—
95C2×C32H(5,𝔽2)12yes—
96C2×C322−1+411yescharacteristic
97C3×D3—44no—
98C2×Q8—32no—
99C2×D4—32no—
100C2×D4—94no—
101C22×C4—94no—
102C4⋊C4—94no—
103C4⋊C4—94no—
104C4⋊C4—92no—
105C22⋊C4—34no—
106C22⋊C4—34no—
107C22⋊C4—94no—
108C42D3211yescharacteristic
109C22×C3—21no—
110C22×C3—64no—
111C22×C3—64no—
112C22×C3—64no—
113C22×C3C23×D312yes—
114C2×D3—24no—
115C2×D3—24no—
116C2×D3—24no—
117C3×C4—22no—
118C3×C4—22no—
119C3×C4—22no—
120C3×C4—22no—
121C3×C4—42no—
122C3×C4—64no—
123Dic3—24no—
124Dic3—64no—
125Dic3—64no—
126Dic3—124no—
127C32#56211yescharacteristic, complemented, Sylow
128C23—34no—
129Q8—122no—
130D4—122no—
131D4—364no—
132C2×C4—34no—
133C2×C4—94no—
134C2×C4—364no—
135C2×C4C2×D3211yescharacteristic
136C2×C4C2×D3212yes—
137C2×C3—21no—
138C2×C3—22no—
139C2×C3—124no—
140C2×C3#100412yes—
141C2×C3#100912yes—
142C2×C3#101012yes—
143D3—44no—
144C22—64no—
145C22—64no—
146C22—64no—
147C22C22×D3211yescharacteristic
148C4—21no—
149C4—22no—
150C4—64no—
151C4—184no—
152C3—21no—
153C3#568412yescomplemented
154C2—124no—
155C2#6429811yescharacteristic
156C2#6430712yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64079C2semidirect
2. C3×#875C2semidirect
3. #63891C2semidirect
4. #63887C2semidirect
5. #63885C2semidirect
25. #875D3semidirect
127. C32#562semidirect
153. C3#5684semidirect

Maximal subgroups

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

subgroupindex
#640792
C3×#8752
#638912
#638872
#638852
#56843