Finite Group Databasebeta

← #10595521

Subgroups — #10595521

Order 576.

550 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 160 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×#3757C211yesmaximal, characteristic
2C2×C3×#217C212yesmaximal
3#63982C214yesmaximal, complemented
4#63964C218yesmaximal, complemented
5#5483—32nomaximal
6C24×C32C2211yescharacteristic
7C22×C3⋊Dic3C2212yes—
8C22×C3×Dic3C2214yes—
9C2×#627C2214yescomplemented
10#3757C2214yes—
11C3×#217C2214yes—
12C3×#217C2214yes—
13#3724C2214yescomplemented
14Dic3⋊Dic3C2218yescomplemented
15C2×C3×C22⋊C4—32no—
16C2×#217—32no—
17C2×#217—61no—
18C2×#217D312yes—
19#924—38no—
20#878—38no—
21#878—38no—
22C23×C32—42no—
23C23×C32Q814yescomplemented
24C23×C32C2311yescharacteristic
25C23×C32C2312yes—
26C2×C3⋊Dic3—42no—
27C2×C3⋊Dic3C2314yes—
28C2×C3×Dic3—44no—
29C2×C3×Dic3C2318yes—
30#627—44no—
31#543—91noSylow
32C24×C3—21no—
33C24×C3C2×D312yes—
34C22×C3×C4—34no—
35C22×Dic3—34no—
36C22×Dic3—62no—
37C22×Dic3C2×D314yes—
38C2×Dic6—38no—
39C3×C22⋊C4—34no—
40C3×C22⋊C4—34no—
41#217—34no—
42#217—34no—
43#217—64no—
44#217C2×D314yes—
45#217C2×D314yes—
46C4⋊Dic3—38no—
47Dic3⋊C4—38no—
48Dic3⋊C4—38no—
49C22×C32—22no—
50C22×C32—24no—
51C22×C32—44no—
52C22×C32C2×Q812yes—
53C22×C32C2×Q814yes—
54C22×C32C2411yescharacteristic
55C3⋊Dic3—24no—
56C3×Dic3—28no—
57C22⋊Q8—94no—
58C22⋊Q8—98no—
59C2×C22⋊C4—32no—
60C2×C22⋊C4—91no—
61C23×C3—24no—
62C23×C3—21no—
63C23×C3—22no—
64C23×C3—44no—
65C23×C3—44no—
66C23×C3Dic618yescomplemented
67C23×C3C22×D312yes—
68C23×C3C22×D312yes—
69C23×C3C22×D312yes—
70C2×C3×C4—38no—
71C2×C3×C4—124no—
72C2×Dic3—38no—
73C2×Dic3—44no—
74C2×Dic3—64no—
75C2×Dic3—124no—
76C2×Dic3—124no—
77C2×Dic3C22×D318yes—
78Dic6—128no—
79C2×C32—22no—
80C2×C32—24no—
81C2×C32C22×Q811yescharacteristic
82C2×C32H(5,𝔽2)12yes—
83C24D3211yescharacteristic
84C2×Q8—94no—
85C22×C4—34no—
86C22×C4—92no—
87C4⋊C4—94no—
88C4⋊C4—98no—
89C22⋊C4—34no—
90C22⋊C4—34no—
91C22⋊C4—94no—
92C22×C3—24no—
93C22×C3—24no—
94C22×C3—24no—
95C22×C3—24no—
96C22×C3—21no—
97C22×C3—22no—
98C22×C3—24no—
99C22×C3—44no—
100C22×C3—48no—
101C22×C3—48no—
102C22×C3C2×Dic614yes—
103C22×C3C2×Dic614yes—
104C22×C3C2×Dic614yes—
105C22×C3C23×D312yes—
106C3×C4—68no—
107Dic3—28no—
108Dic3—68no—
109Dic3—124no—
110C32#54311yescharacteristic, complemented, Sylow
111C23—42no—
112C23#62714yescomplemented
113C23C2×D3211yescharacteristic
114C23C2×D3212yes—
115Q8—364no—
116C2×C4—38no—
117C2×C4—94no—
118C2×C4—124no—
119C2×C4—362no—
120C2×C3—24no—
121C2×C3—24no—
122C2×C3—24no—
123C2×C3—24no—
124C2×C3—21no—
125C2×C3—22no—
126C2×C3—44no—
127C2×C3C22×Dic612yes—
128C2×C3#100414yes—
129C22—24no—
130C22—22no—
131C22—44no—
132C22C2×#62712yes—
133C22C2×#62714yes—
134C22C22×D3211yescharacteristic
135C4—68no—
136C4—184no—
137C3—21no—
138C3#548312yescomplemented
139C2—22no—
140C2—24no—
141C2C22×#62711yescharacteristic
142C2#6433012yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
Fitting, prime radicalC24×C32C22
hypercenterC24D32
hypocenterC32#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
3. #63982C2semidirect
4. #63964C2semidirect
9. C2×#627C22semidirect
13. #3724C22semidirect
14. Dic3⋊Dic3C22semidirect
23. C23×C32Q8semidirect
66. C23×C3Dic6semidirect
110. C32#543semidirect
112. C23#627semidirect
138. C3#5483semidirect

Maximal subgroups

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

subgroupindex
C2×#37572
C2×C3×#2172
#639822
#639642
#54833