Finite Group Databasebeta

← #10591629

Subgroups — #10591629

Order 576.

286 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 66 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#64156C211yesmaximal, characteristic
2C2×C3×#216C212yesmaximal, complemented
3#63873C211yesmaximal, characteristic, complemented
4C2×#3717C212yesmaximal
5#63555C211yesmaximal, characteristic, complemented
6#5190—32nomaximal
7C2×C32×Q8C2211yescharacteristic
8C2×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×Dic6C2212yes—
10#3752—22no—
11C4×C3⋊Dic3C2211yescharacteristic
12C3×#216—24no—
13C2×C3×C3⋊C8C2212yes—
14#3724—22no—
15#3717—24no—
16C2×C3×Q16—32no—
17#947—32no—
18#947—61no—
19#944—32no—
20C2×#216D312yescomplemented
21C2×#215—32no—
22C2×#204—32no—
23#815—32no—
24C22×C3⋊D3—21no—
25C32×Q8—22no—
26C2×C32×C4—21no—
27C2×C32×C4C2311yescharacteristic
28C2×C3⋊Dic3D412yes—
29C4⋊(C3⋊D3)—22no—
30C2×C3×Dic3—22no—
31C3×Dic6—22no—
32C3×Dic6—22no—
33C3×C3⋊C8D414yes—
34#497—91noSylow
35C2×C3×Q8—21no—
36C2×C3×Q8—32no—
37C2×C3×Q8C2×D312yes—
38C2×D12—32no—
39C2×D12—61no—
40C2×Dic6C2×D312yes—
41C3×Q16—64no—
42C2×C3×C8—32no—
43#216—24no—
44#215—64no—
45(C2×D3)⋊C4—64no—
46(C2×D3)⋊C4—64no—
47Dic3⋊C4—64no—
48C4×Dic3—32no—
49C4×Dic3—61no—
50C2×C3⋊C8C2×D312yes—
51#204—64no—
52C22×C32C2×D411yescharacteristic
53C2×C3⋊D3—41no—
54C2×C3⋊D3—42no—
55C32×C4—41no—
56C32×C4C2×D412yes—
57C3⋊Dic3—22no—
58C3×Dic3—42no—
59C2×Q16—32no—
60C2×SD16—92no—
61C4⋊Q8—91no—
62#124—91no—
63#97—91no—
64C22×D3—62no—
65C22×D3—62no—
66C3×Q8—22no—
67C3×Q8—22no—
68C3×Q8—42no—
69C3×Q8—62no—
70C3×Q8—62no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—21no—
74C2×C3×C4—62no—
75C2×C3×C4C22×D312yes—
76C2×Dic3—22no—
77C2×Dic3—34no—
78C2×Dic3—62no—
79D12—62no—
80D12—62no—
81D12—122no—
82Dic6—22no—
83Dic6—22no—
84C3×C8—34no—
85C3⋊C8#6714yes—
86C2×C32#12711yescharacteristic
87C2×C32#13712yes—
88C3⋊D3—81no—
89C2×Q8—32no—
90C2×Q8D3211yescharacteristic
91C2×D4—91no—
92Q16—64no—
93SD16—184no—
94C2×C8—32no—
95C4⋊C4—182no—
96C22⋊C4—182no—
97C42—91no—
98C22×C3—21no—
99C22×C3D3×D412yes—
100C2×D3—122no—
101C2×D3—124no—
102C2×D3—124no—
103C2×D3—122no—
104C3×C4—22no—
105C3×C4—42no—
106C3×C4—42no—
107C3×C4—122no—
108C3×C4D3×D412yes—
109C3×C4C2×#6712yes—
110Dic3—42no—
111Dic3—64no—
112Dic3—122no—
113C32#49711yescharacteristic, complemented, Sylow
114C23—181no—
115Q8—22no—
116Q8—62no—
117Q8—62no—
118D4—182no—
119C2×C4—21no—
120C2×C4—62no—
121C2×C4—92no—
122C2×C4C2×D3211yescharacteristic
123C8—34no—
124C2×C3—22no—
125C2×C3—21no—
126C2×C3#91814yes—
127C2×C3#94012yes—
128D3—242no—
129D3—242no—
130C22—362no—
131C22—361no—
132C22#381111yescharacteristic
133C4—41no—
134C4—122no—
135C4—182no—
136C4D3×#6712yes—
137C3—21no—
138C3#519012yescomplemented
139C2—721no—
140C2#6394512yes—
141C2#6397911yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC2×Q8D32
hypocenterC32#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#216C2semidirect
3. #63873C2semidirect
5. #63555C2semidirect
20. C2×#216D3semidirect
113. C32#497semidirect
138. C3#5190semidirect

Maximal subgroups

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

subgroupindex
#641562
C2×C3×#2162
#638732
C2×#37172
#635552
#51903