Finite Group Databasebeta

← #10596206

Subgroups — #10596206

Order 576.

588 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64156C211yesmaximal, characteristic
3#64155C212yesmaximal, complemented
4#64153C211yesmaximal, characteristic, complemented
5#64107C212yesmaximal, complemented
6#64105C212yesmaximal, complemented
7#64103C212yesmaximal
8#64098C212yesmaximal, complemented
9#64085C212yesmaximal, complemented
10#5735—34nomaximal
11C2×C32×Q8C2211yescharacteristic
12C2×C32×Q8C2211yescharacteristic
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C4×C3⋊D3C2212yes—
15C2×#634C2211yescharacteristic
16C32×C4⋊C4C2212yes—
17C32×C4⋊C4C2212yes—
18C32×C42C2211yescharacteristic
19#3752C2212yes—
20#3752C2212yes—
21#3752C2212yes—
22#3752C2212yes—
23#3752C2212yes—
24C4⋊(C3⋊Dic3)C2212yes—
25#3750C2212yes—
26#3750C2212yes—
27#3750C2212yes—
28#3750C2212yes—
29#3750C2212yes—
30C4×C3⋊Dic3C2211yescharacteristic
31C4×C3⋊Dic3C2211yescharacteristic
32C3×C4⋊Q8D314yescomplemented
33#947—34no—
34#946—38no—
35#944—34no—
36#898—38no—
37#896—38no—
38#894—38no—
39#889—38no—
40#876—38no—
41C22×C3⋊D3C2312yes—
42C32×Q8—41no—
43C32×Q8—41no—
44C2×C32×C4C2311yescharacteristic
45C2×C32×C4C2311yescharacteristic
46C2×C32×C4C2312yes—
47C2×C32×C4C2311yescharacteristic
48C2×C32×C4C2312yes—
49C2×C3⋊Dic3C2312yes—
50C2×C3⋊Dic3C2312yes—
51C2×C3⋊Dic3C2312yes—
52C4⋊(C3⋊D3)—41no—
53C4×C3⋊D3—42no—
54#634—41no—
55#563—91noSylow
56C2×C3×Q8C2×D314yes—
57C2×C3×Q8C2×D314yes—
58C2×D12—34no—
59C2×C4×D3—38no—
60C2×Dic6—34no—
61C3×C4⋊C4C2×D318yes—
62C3×C4⋊C4C2×D318yes—
63C3×C42C2×D314yes—
64(C2×D3)⋊C4—38no—
65(C2×D3)⋊C4—38no—
66(C2×D3)⋊C4—38no—
67(C2×D3)⋊C4—38no—
68(C2×D3)⋊C4—38no—
69C4⋊Dic3—38no—
70Dic3⋊C4—38no—
71Dic3⋊C4—38no—
72Dic3⋊C4—38no—
73Dic3⋊C4—38no—
74Dic3⋊C4—38no—
75C4×Dic3—34no—
76C4×Dic3—34no—
77C22×C32C2411yescharacteristic
78C2×C3⋊D3—22no—
79C2×C3⋊D3—24no—
80C32×C4—21no—
81C32×C4—22no—
82C32×C4—21no—
83C32×C4—21no—
84C32×C4—22no—
85C3⋊Dic3—22no—
86C3⋊Dic3—22no—
87C3⋊Dic3—22no—
88C4⋊Q8—91no—
89C4⋊Q8C3⋊D311yescharacteristic, complemented
90#126—92no—
91#126—92no—
92#125—92no—
93#124—91no—
94#123—92no—
95C22⋊Q8—92no—
96C22⋊Q8—92no—
97C22×D3—38no—
98C3×Q8—44no—
99C3×Q8—44no—
100C2×C3×C4C22×D314yes—
101C2×C3×C4C22×D314yes—
102C2×C3×C4C22×D318yes—
103C2×C3×C4C22×D314yes—
104C2×C3×C4C22×D318yes—
105C2×Dic3—38no—
106C2×Dic3—38no—
107C2×Dic3—38no—
108D12—124no—
109C4×D3—128no—
110Dic6—124no—
111C2×C32H(5,𝔽2)11yescharacteristic
112C2×C322−1+412yes—
113C3⋊D3—42no—
114C2×Q8—91no—
115C2×Q8C2×C3⋊D311yescharacteristic
116C2×Q8C2×C3⋊D311yescharacteristic
117C2×D4—91no—
118C22×C4—92no—
119C4⋊C4—92no—
120C4⋊C4—92no—
121C4⋊C4—92no—
122C4⋊C4—92no—
123C4⋊C4—92no—
124C4⋊C4—92no—
125C4⋊C4C2×C3⋊D312yes—
126C4⋊C4C2×C3⋊D312yes—
127C22⋊C4—92no—
128C22⋊C4—92no—
129C22⋊C4—92no—
130C22⋊C4—92no—
131C22⋊C4—92no—
132C42—91no—
133C42—91no—
134C42C2×C3⋊D311yescharacteristic
135C22×C3C23×D314yes—
136C2×D3—68no—
137C2×D3—616no—
138C3×C4—24no—
139C3×C4—28no—
140C3×C4—24no—
141C3×C4—24no—
142C3×C4—28no—
143Dic3—68no—
144Dic3—68no—
145Dic3—68no—
146C32#56311yescharacteristic, complemented, Sylow
147C23—92no—
148Q8—41no—
149Q8—41no—
150Q8—361no—
151D4—361no—
152C2×C4—92no—
153C2×C4—92no—
154C2×C4—92no—
155C2×C4—362no—
156C2×C4C22×C3⋊D311yescharacteristic
157C2×C4C22×C3⋊D311yescharacteristic
158C2×C4C22×C3⋊D312yes—
159C2×C4C22×C3⋊D311yescharacteristic
160C2×C4C22×C3⋊D312yes—
161C2×C3#100414yes—
162C2×C3#100718yes—
163D3—128no—
164C22—182no—
165C22—184no—
166C22C23×C3⋊D311yescharacteristic
167C4—22no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—22no—
172C4—182no—
173C4—182no—
174C4—182no—
175C3#573514yescomplemented
176C2—362no—
177C2#6436111yescharacteristic
178C2#6436412yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊Q8C2
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC4⋊Q8C3⋊D3
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C4⋊Q8C2semidirect
3. #64155C2semidirect
4. #64153C2semidirect
5. #64107C2semidirect
6. #64105C2semidirect
8. #64098C2semidirect
9. #64085C2semidirect
32. C3×C4⋊Q8D3semidirect
89. C4⋊Q8C3⋊D3semidirect
146. C32#563semidirect
175. C3#5735semidirect

Maximal subgroups

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

subgroupindex
C32×C4⋊Q82
#641562
#641552
#641532
#641072
#641052
#641032
#640982
#640852
#57353