Finite Group Databasebeta

← #10591632

Subgroups — #10591632

Order 576.

238 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 58 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
2#63919C211yesmaximal, characteristic, complemented
3C3×#835C212yesmaximal, complemented
4#63564C212yesmaximal
5#63555C211yesmaximal, characteristic, complemented
6#4794—32nomaximal
7C2×C32×Q8C2211yescharacteristic
8C2×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×#627—21no—
10#3752—21no—
11#3752—21no—
12C4×C3⋊Dic3C2211yescharacteristic
13C3×C4⋊Dic3C2212yes—
14C2×C3×C3⋊C8C2212yes—
15#3724—21no—
16#947—32no—
17#947—61no—
18#888—32no—
19C3×#103—32no—
20#835D312yescomplemented
21#821—32no—
22#815—32no—
23#809—32no—
24C22×C3⋊D3—21no—
25C32×Q8—22no—
26C2×C32×C4—21no—
27C2×C32×C4C2311yescharacteristic
28C2×C3⋊Dic3D411yescharacteristic
29C2×C3⋊Dic3D411yescharacteristic
30C4⋊(C3⋊D3)—22no—
31C2×C3×Dic3—22no—
32#627—42no—
33C3×C3⋊C8—22no—
34#489—91noSylow
35C2×C3×Q8—21no—
36C2×C3×Q8C2×D312yes—
37C2×D12—32no—
38C2×D12—61no—
39C2×Dic6—62no—
40C2×C3×C8—32no—
41C3×C4⋊C4—32no—
42(C2×D3)⋊C4—62no—
43(C2×D3)⋊C4—62no—
44(C2×D3)⋊C4—62no—
45(C2×D3)⋊C4—62no—
46C4⋊Dic3C2×D312yes—
47Dic3⋊C4—62no—
48C4×Dic3—32no—
49C4×Dic3—61no—
50C2×C3⋊C8C2×D312yes—
51C22×C32C2×D411yescharacteristic
52C2×C3⋊D3—41no—
53C2×C3⋊D3—41no—
54C2×C3⋊D3—41no—
55C32×C4—41no—
56C32×C4#4112yes—
57C3⋊Dic3—21no—
58C3⋊Dic3—21no—
59C3×Dic3—42no—
60C4⋊Q8—91no—
61#124—91no—
62#103—32no—
63#102—92no—
64#97—91no—
65C22×D3—62no—
66C22×D3—62no—
67C3×Q8—22no—
68C3×Q8—22no—
69C3×Q8—42no—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—21no—
73C2×C3×C4—62no—
74C2×C3×C4C22×D312yes—
75C2×Dic3—22no—
76C2×Dic3—32no—
77C2×Dic3—32no—
78C2×Dic3—61no—
79C2×Dic3—61no—
80D12—62no—
81D12—62no—
82D12—122no—
83Dic6—124no—
84C3×C8—62no—
85C3⋊C8—22no—
86C2×C32#12411yescharacteristic
87C2×C32AGL(1,ℤ/8)11yescharacteristic
88C2×C32#13711yescharacteristic
89C3⋊D3—81no—
90C2×Q8—181no—
91C2×Q8D3211yescharacteristic
92C2×D4—91no—
93C2×C8—32no—
94C4⋊C4—32no—
95C4⋊C4—181no—
96C22⋊C4—181no—
97C22⋊C4—181no—
98C42—91no—
99C22×C3—21no—
100C22×C3D3×D412yes—
101C2×D3—122no—
102C2×D3—122no—
103C2×D3—122no—
104C2×D3—122no—
105C2×D3—122no—
106C2×D3—122no—
107C3×C4—22no—
108C3×C4—42no—
109C3×C4—42no—
110C3×C4—122no—
111C3×C4#23512yes—
112C3×C4#23712yes—
113Dic3—42no—
114Dic3—62no—
115Dic3—62no—
116Dic3—121no—
117Dic3—121no—
118C32#48911yescharacteristic, complemented, Sylow
119C23—181no—
120Q8—22no—
121Q8—362no—
122D4—182no—
123C2×C4—21no—
124C2×C4—62no—
125C2×C4—91no—
126C2×C4—91no—
127C2×C4C2×D3211yescharacteristic
128C8—62no—
129C2×C3—21no—
130C2×C3—21no—
131C2×C3—21no—
132C2×C3#88512yes—
133C2×C3#91412yes—
134C2×C3#91812yes—
135D3—242no—
136D3—242no—
137C22—361no—
138C22—361no—
139C22—361no—
140C22#381111yescharacteristic
141C4—41no—
142C4—122no—
143C4—181no—
144C4—181no—
145C4#380412yes—
146C3—21no—
147C3#479412yescomplemented
148C2—721no—
149C2#6394111yescharacteristic
150C2#6394511yescharacteristic
151C2#6395311yescharacteristic
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#489
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. #63919C2semidirect
3. C3×#835C2semidirect
5. #63555C2semidirect
20. #835D3semidirect
118. C32#489semidirect
147. C3#4794semidirect

Maximal subgroups

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

subgroupindex
#641562
#639192
C3×#8352
#635642
#635552
#47943