Finite Group Databasebeta

← #10590922

Subgroups — #10590922

Order 576.

228 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C3×#901C212yesmaximal, complemented
2#63822C211yesmaximal, characteristic
3#63642C211yesmaximal, characteristic, complemented
4#63578C211yesmaximal, characteristic, complemented
5#63568C212yesmaximal, complemented
6#5145—32nomaximal
7C3×#235C2212yes—
8C2×C32×C8C2211yescharacteristic
9C4×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊(C3⋊C8)C2211yescharacteristic
11C3×#208C2212yes—
12C3×ΓL(1,𝔽25)—22no—
13C3×D3×C8—22no—
14#3718C411yescharacteristic, complemented
15#3715C412yescomplemented
16#3713C411yescharacteristic, complemented
17C3×#131—32no—
18#950—32no—
19#901D312yescomplemented
20#837—32no—
21#825—32no—
22#822—32no—
23#815—32no—
24#815—61no—
25C2×C32×C4C2311yescharacteristic
26C2×C3⋊Dic3—21no—
27C3×#67—22no—
28C3×D12C2×C412yes—
29C3×C4×D3—22no—
30C3×Dic6C2×C412yes—
31C32×C8D412yes—
32C3⋊(C3⋊C8)C2×C411yescharacteristic
33C3⋊(C3⋊C8)C2×C411yescharacteristic
34C3×C3⋊C8—22no—
35#444—91noSylow
36C3×#41—32no—
37#235C2×D312yes—
38C3×M16—32no—
39C3×M16—62no—
40C2×C3×C8—21no—
41C2×C3×C8—62no—
42C2×C3×C8C2×D312yes—
43#216—32no—
44#215—32no—
45#214—32no—
46#213—32no—
47C4×Dic3—32no—
48C4×Dic3—61no—
49#208C2×D312yes—
50C2×C3⋊C8—32no—
51C2×C3⋊C8—61no—
52ΓL(1,𝔽25)—22no—
53D3×C8—22no—
54C22×C32#4111yescharacteristic
55C2×C3×D3—22no—
56C32×C4C22×C411yescharacteristic
57C32×C4C2×D411yescharacteristic
58C3⋊Dic3—41no—
59C3×Dic3—22no—
60#135—91no—
61#131—32no—
62#108—91no—
63#104—92no—
64#97—91no—
65C3×Q8—32no—
66C3×D4—32no—
67C3×D4—62no—
68C2×C3×C4—21no—
69C2×C3×C4—62no—
70C2×C3×C4C22×D312yes—
71#67—22no—
72C2×Dic3—62no—
73C2×Dic3—62no—
74D12C4×D312yes—
75C4×D3—22no—
76Dic6C4×D312yes—
77C3×C8—22no—
78C3×C8—62no—
79C3×C8#6712yes—
80C3×C8#6712yes—
81C3⋊C8—22no—
82C3⋊C8—32no—
83C3⋊C8—32no—
84C3⋊C8—61no—
85C3⋊C8—61no—
86C2×C32—21no—
87C2×C32C4×D411yescharacteristic
88C3×D3—42no—
89#41—32no—
90Q16—91no—
91SD16—92no—
92D8—91no—
93M16—32no—
94M16—62no—
95C2×C8—62no—
96C2×C8—91no—
97C2×C8D3211yescharacteristic
98C42—91no—
99C22×C3—21no—
100C22×C3—62no—
101C22×C3#23512yes—
102C2×D3—22no—
103C3×C4—21no—
104C3×C4—21no—
105C3×C4—62no—
106C3×C4C2×C4×D312yes—
107C3×C4C2×#6712yes—
108Dic3—22no—
109Dic3—122no—
110Dic3—122no—
111C32#44411yescharacteristic, complemented, Sylow
112Q8—32no—
113D4—32no—
114D4—62no—
115C2×C4—62no—
116C2×C4—181no—
117C2×C4C2×D3211yescharacteristic
118C8—62no—
119C8—91no—
120C8—91no—
121C8#62512yes—
122C2×C3—22no—
123C2×C3—22no—
124C2×C3—21no—
125C2×C3—122no—
126C2×C3C4×#6712yes—
127D3—42no—
128C22—62no—
129C22#379811yescharacteristic
130C4—62no—
131C4—361no—
132C4C4×D3211yescharacteristic
133C4C2×#62511yescharacteristic
134C3—21no—
135C3#514512yescomplemented
136C2—21no—
137C2—122no—
138C2C4×#62511yescharacteristic
subgroupG/H
centerC4C2×#625
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC2×C32C4×D4
prime radical#63822C2
cosocleC2×C32×C4C23
hypercenterC2×C8D32
hypocenterC32#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C3×#901C2semidirect
3. #63642C2semidirect
4. #63578C2semidirect
5. #63568C2semidirect
14. #3718C4semidirect
15. #3715C4semidirect
16. #3713C4semidirect
19. #901D3semidirect
111. C32#444semidirect
135. C3#5145semidirect

Maximal subgroups

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

subgroupindex
C3×#9012
#638222
#636422
#635782
#635682
#51453