Finite Group Databasebeta

← #10590885

Subgroups — #10590885

Order 576.

258 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 62 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#63919C211yesmaximal, characteristic, complemented
2#63914C211yesmaximal, characteristic
3C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#821C212yesmaximal, complemented
5#63562C212yesmaximal, complemented
6#4760—32nomaximal
7C2×C3×D12C2212yes—
8C2×#627—21no—
9C2×#625—21no—
10C2×#625—21no—
11C2×C32×C8C2211yescharacteristic
12C4×C3⋊Dic3C2211yescharacteristic
13C2×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×C4⋊Dic3C2212yes—
15#3724—21no—
16#940—32no—
17#888—32no—
18C3×#102—32no—
19#832—32no—
20#821D312yescomplemented
21C8×Dic3—32no—
22C8×Dic3—61no—
23#809—32no—
24C22×C3×D3—22no—
25C2×C32×C4C2311yescharacteristic
26C2×C3⋊Dic3D411yescharacteristic
27C2×C3⋊Dic3D411yescharacteristic
28C2×C3×Dic3—22no—
29C3×D12—22no—
30C3×D12—22no—
31#627—42no—
32#625—42no—
33#625—42no—
34C32×C8—21no—
35C3⋊(C3⋊C8)—21no—
36#486—91noSylow
37C2×C3×D4—32no—
38C2×#67—62no—
39C2×#67—62no—
40C2×D12C2×D312yes—
41C2×Dic6—62no—
42C2×C3×C8—21no—
43C2×C3×C8C2×D312yes—
44C3×C4⋊C4—32no—
45C4⋊Dic3C2×D312yes—
46Dic3⋊C4—62no—
47C4×Dic3—32no—
48C4×Dic3—61no—
49C2×C3⋊C8—32no—
50C2×C3⋊C8—61no—
51C22×C32C2×D411yescharacteristic
52C2×C3×D3—42no—
53C2×C3×D3—42no—
54C2×C3×D3—42no—
55C32×C4#4111yescharacteristic
56C32×C4#4111yescharacteristic
57C3⋊Dic3D812yes—
58C3⋊Dic3SD1612yes—
59C3×Dic3—42no—
60C4⋊Q8—91no—
61#127—91no—
62#102—32no—
63#102—92no—
64C4×C8—91no—
65C23×C3—62no—
66C22×D3—22no—
67C3×D4—62no—
68C3×D4—62no—
69C2×C3×C4—21no—
70C2×C3×C4—62no—
71C2×C3×C4C22×D312yes—
72#67—124no—
73#67—124no—
74C2×Dic3—22no—
75C2×Dic3—32no—
76C2×Dic3—32no—
77C2×Dic3—61no—
78C2×Dic3—61no—
79D12—22no—
80D12—22no—
81Dic6—124no—
82C3×C8—22no—
83C3×C8—22no—
84C3⋊C8—62no—
85C3⋊C8—62no—
86C2×C32#12411yescharacteristic
87C2×C32C2×D811yescharacteristic
88C2×C32C2×SD1611yescharacteristic
89C3×D3—82no—
90C2×Q8—181no—
91C2×D4—32no—
92C2×D4—181no—
93C2×D4—181no—
94C2×C8—91no—
95C2×C8D3211yescharacteristic
96C4⋊C4—32no—
97C4⋊C4—181no—
98C42—91no—
99C22×C3—21no—
100C22×C3—122no—
101C22×C3—122no—
102C22×C3—122no—
103C22×C3D3×D412yes—
104C2×D3—42no—
105C2×D3—42no—
106C2×D3—42no—
107C3×C4—21no—
108C3×C4—21no—
109C3×C4—122no—
110C3×C4#23512yes—
111C3×C4#23712yes—
112Dic3—34no—
113Dic3—34no—
114Dic3—42no—
115Dic3—62no—
116Dic3—62no—
117C32#48611yescharacteristic, complemented, Sylow
118C23—62no—
119Q8—362no—
120D4—62no—
121D4—62no—
122D4—362no—
123D4—362no—
124C2×C4—62no—
125C2×C4—91no—
126C2×C4—91no—
127C2×C4C2×D3211yescharacteristic
128C8—21no—
129C8—181no—
130C2×C3—21no—
131C2×C3—21no—
132C2×C3—21no—
133C2×C3—242no—
134C2×C3#88512yes—
135C2×C3D3×D812yes—
136C2×C3D3×SD1612yes—
137D3—82no—
138C22—122no—
139C22—122no—
140C22—122no—
141C22#380211yescharacteristic
142C4—92no—
143C4—92no—
144C4—122no—
145C4#379811yescharacteristic
146C4#380511yescharacteristic
147C3—21no—
148C3#476012yescomplemented
149C2—242no—
150C2#6379611yescharacteristic
151C2#6379711yescharacteristic
152C2#6391511yescharacteristic
subgroupG/H
centerC22#3802
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radical#63914C2
hypercenterC2×C8D32
hypocenterC32#486
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #63919C2semidirect
3. C8×C3⋊Dic3C2semidirect
4. C3×#821C2semidirect
5. #63562C2semidirect
20. #821D3semidirect
117. C32#486semidirect
148. C3#4760semidirect

Maximal subgroups

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

subgroupindex
#639192
#639142
C8×C3⋊Dic32
C3×#8212
#635622
#47603