Finite Group Databasebeta

← #10596201

Subgroups — #10596201

Order 576.

684 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 203 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#64155C214yesmaximal, complemented
3#64106C212yesmaximal, complemented
4#64102C212yesmaximal
5C4⋊C4×C3⋊D3C212yesmaximal, complemented
6#64099C212yesmaximal, complemented
7C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
8C4×#634C211yesmaximal, characteristic, complemented
9#5730—34nomaximal
10C2×C32×Q8C2212yes—
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C4×C3⋊D3C2212yes—
13C2×C4×C3⋊D3C2214yes—
14C2×#634C2211yescharacteristic
15C32×C4⋊C4C2212yes—
16C32×C4⋊C4C2212yes—
17C32×C42C2211yescharacteristic
18#3752C2212yes—
19#3752C2214yes—
20C4⋊(C3⋊Dic3)C2211yescharacteristic
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2214yes—
23#3750C2212yes—
24#3750C2214yes—
25C4×C3⋊Dic3C2212yes—
26C3×C4⋊Q8D314yescomplemented
27#946—316no—
28#897—38no—
29#893—38no—
30D3×C4⋊C4—38no—
31#890—38no—
32C4×D12—34no—
33C4×Dic6—34no—
34C22×C3⋊D3C2312yes—
35C32×Q8—42no—
36C2×C32×C4C2311yescharacteristic
37C2×C32×C4C2314yes—
38C2×C32×C4C2311yescharacteristic
39C2×C32×C4C2311yescharacteristic
40C2×C3⋊Dic3C2312yes—
41C2×C3⋊Dic3C2312yes—
42C2×C3⋊Dic3C2312yes—
43C4⋊(C3⋊D3)Q814yescomplemented
44C4×C3⋊D3—24no—
45C4×C3⋊D3—44no—
46#634—22no—
47#554—91noSylow
48C2×C3×Q8C2×D318yes—
49C2×D12—34no—
50C2×C4×D3—38no—
51C2×C4×D3—316no—
52C2×Dic6—34no—
53C3×C4⋊C4C2×D318yes—
54C3×C4⋊C4C2×D318yes—
55C3×C42C2×D314yes—
56(C2×D3)⋊C4—38no—
57(C2×D3)⋊C4—316no—
58C4⋊Dic3—34no—
59C4⋊Dic3—34no—
60C4⋊Dic3—316no—
61Dic3⋊C4—38no—
62Dic3⋊C4—316no—
63C4×Dic3—38no—
64C22×C32C2411yescharacteristic
65C2×C3⋊D3—22no—
66C2×C3⋊D3—22no—
67C2×C3⋊D3C2×Q814yes—
68C32×C4—24no—
69C32×C4—21no—
70C32×C4C2×Q812yes—
71C32×C4#4112yes—
72C3⋊Dic3—22no—
73C3⋊Dic3—22no—
74C3⋊Dic3—22no—
75C4⋊Q8C3⋊D311yescharacteristic, complemented
76#125—92no—
77C22⋊Q8—94no—
78C22⋊Q8—92no—
79C4×Q8—91no—
80C4×D4—92no—
81C4×D4—91no—
82C2×C4⋊C4—92no—
83C22×D3—38no—
84C3×Q8—48no—
85C2×C3×C4C22×D314yes—
86C2×C3×C4C22×D3116yes—
87C2×C3×C4C22×D314yes—
88C2×C3×C4C22×D314yes—
89C2×Dic3—38no—
90C2×Dic3—38no—
91C2×Dic3—38no—
92D12—316no—
93C4×D3—616no—
94C4×D3—1216no—
95Dic6—68no—
96C2×C32C22×Q811yescharacteristic
97C2×C32C2×#4111yescharacteristic
98C2×C32H(5,𝔽2)11yescharacteristic
99C3⋊D3—24no—
100C2×Q8—91no—
101C2×Q8C2×C3⋊D312yes—
102C2×D4—91no—
103C22×C4—92no—
104C22×C4—94no—
105C4⋊C4—91no—
106C4⋊C4—94no—
107C4⋊C4—91no—
108C4⋊C4—92no—
109C4⋊C4—94no—
110C4⋊C4C2×C3⋊D312yes—
111C4⋊C4C2×C3⋊D312yes—
112C22⋊C4—94no—
113C22⋊C4—92no—
114C42—92no—
115C42C2×C3⋊D311yescharacteristic
116C22×C3C23×D314yes—
117C2×D3—316no—
118C2×D3—68no—
119C2×D3—68no—
120C3×C4—216no—
121C3×C4—24no—
122C3×C4D3×Q818yes—
123C3×C4#23918yes—
124Dic3—68no—
125Dic3—68no—
126Dic3—68no—
127C32#55411yescharacteristic, complemented, Sylow
128C23—92no—
129Q8—42no—
130Q8—182no—
131D4—94no—
132C2×C4—92no—
133C2×C4—92no—
134C2×C4—92no—
135C2×C4—184no—
136C2×C4—364no—
137C2×C4C22×C3⋊D311yescharacteristic
138C2×C4C22×C3⋊D314yes—
139C2×C4C22×C3⋊D311yescharacteristic
140C2×C4C22×C3⋊D311yescharacteristic
141C2×C3#100414yes—
142C2×C3C2×D3×Q814yes—
143C2×C3C2×#23914yes—
144D3—616no—
145C22—94no—
146C22—182no—
147C22—182no—
148C22C23×C3⋊D311yescharacteristic
149C4—24no—
150C4—21no—
151C4—182no—
152C4—182no—
153C4—182no—
154C4Q8×C3⋊D312yes—
155C4#383212yes—
156C3#573014yescomplemented
157C2—184no—
158C2#6436111yescharacteristic
159C2C2×Q8×C3⋊D311yescharacteristic
160C2C2×#383211yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊Q8C2
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC4⋊Q8C3⋊D3
hypocenterC32#554
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
2. #64155C2semidirect
3. #64106C2semidirect
5. C4⋊C4×C3⋊D3C2semidirect
6. #64099C2semidirect
8. C4×#634C2semidirect
26. C3×C4⋊Q8D3semidirect
43. C4⋊(C3⋊D3)Q8semidirect
75. C4⋊Q8C3⋊D3semidirect
127. C32#554semidirect
156. C3#5730semidirect

Maximal subgroups

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

subgroupindex
C32×C4⋊Q82
#641552
#641062
#641022
C4⋊C4×C3⋊D32
#640992
C4×C4⋊(C3⋊D3)2
C4×#6342
#57303