Finite Group Databasebeta

← #10595421

Subgroups — #10595421

Order 576.

458 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 148 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#64153C211yesmaximal, characteristic
2C3×#944C212yesmaximal
3#63840C212yesmaximal
4#63839C214yesmaximal
5#63838C212yesmaximal
6#63837C214yesmaximal
7#5722—32nomaximal
8C2×C32×Q8C2211yescharacteristic
9C2×#634C2211yescharacteristic
10C2×C3×Dic6C2212yes—
11C2×#627C2212yes—
12C2×#627C2212yes—
13#3750C2214yes—
14C4×C3⋊Dic3C2211yescharacteristic
15C3×Dic3⋊C4C2214yes—
16C3×Dic3⋊C4C2214yes—
17C3×C4×Dic3C2212yes—
18#3724C2212yes—
19Dic3⋊Dic3C2214yes—
20Dic32C2212yes—
21Dic32C2214yes—
22C3×C4⋊Q8—32no—
23#944—32no—
24#944—34no—
25#944—61no—
26#944D312yes—
27#888—34no—
28#888—34no—
29#887—34no—
30#887—34no—
31C4×Dic6—34no—
32C32×Q8—41no—
33C2×C32×C4C2312yes—
34C2×C32×C4C2311yescharacteristic
35C2×C3⋊Dic3C2312yes—
36C2×C3⋊Dic3C2312yes—
37#634—41no—
38C2×C3×Dic3C2314yes—
39C2×C3×Dic3C2314yes—
40C3×Dic6—42no—
41#627—42no—
42#627Q818yescomplemented
43Q82—91noSylow
44C2×C3×Q8—21no—
45C2×C3×Q8—32no—
46C2×C3×Q8C2×D312yes—
47C2×Dic6—34no—
48C2×Dic6—32no—
49C2×Dic6—34no—
50C2×Dic6—61no—
51C2×Dic6C2×D312yes—
52C3×C4⋊C4—34no—
53C3×C4⋊C4—34no—
54C3×C42—32no—
55C4⋊Dic3—34no—
56Dic3⋊C4—34no—
57Dic3⋊C4—34no—
58Dic3⋊C4—34no—
59Dic3⋊C4—34no—
60Dic3⋊C4—64no—
61Dic3⋊C4C2×D314yes—
62Dic3⋊C4C2×D314yes—
63C4×Dic3—32no—
64C4×Dic3—34no—
65C4×Dic3—34no—
66C4×Dic3—34no—
67C4×Dic3—61no—
68C4×Dic3C2×D312yes—
69C22×C32C2411yescharacteristic
70C32×C4—21no—
71C32×C4—22no—
72C3⋊Dic3—22no—
73C3⋊Dic3C2×Q814yes—
74C3×Dic3—24no—
75C3×Dic3C2×Q818yes—
76C4⋊Q8—32no—
77C4⋊Q8—92no—
78C4⋊Q8—94no—
79C4⋊Q8—91no—
80C4×Q8—94no—
81C4×Q8—92no—
82C3×Q8—42no—
83C3×Q8—42no—
84C3×Q8—122no—
85C2×C3×C4—21no—
86C2×C3×C4—22no—
87C2×C3×C4—34no—
88C2×C3×C4—34no—
89C2×C3×C4C22×D312yes—
90C2×C3×C4C22×D312yes—
91C2×C3×C4C22×D312yes—
92C2×Dic3—34no—
93C2×Dic3—34no—
94C2×Dic3—62no—
95C2×Dic3—62no—
96C2×Dic3C22×D314yes—
97C2×Dic3C22×D314yes—
98Dic6—316no—
99Dic6—42no—
100Dic6—124no—
101Dic6—122no—
102Dic6—122no—
103C2×C32C22×Q812yes—
104C2×C32H(5,𝔽2)11yescharacteristic
105C2×Q8—32no—
106C2×Q8—92no—
107C2×Q8—92no—
108C2×Q8—91no—
109C2×Q8D3211yescharacteristic
110C4⋊C4—34no—
111C4⋊C4—34no—
112C4⋊C4—92no—
113C4⋊C4—94no—
114C4⋊C4—94no—
115C42—32no—
116C42—92no—
117C42—94no—
118C42—91no—
119C22×C3—21no—
120C22×C3C23×D312yes—
121C3×C4—22no—
122C3×C4—22no—
123C3×C4—22no—
124C3×C4—22no—
125C3×C4—38no—
126C3×C4—42no—
127C3×C4—64no—
128Dic3—24no—
129Dic3—38no—
130Dic3—64no—
131Dic3—64no—
132Dic3—122no—
133Dic3D3×Q818yes—
134C32Q8211yescharacteristic, complemented, Sylow
135Q8—41no—
136Q8—98no—
137Q8—122no—
138Q8—361no—
139Q8—362no—
140C2×C4—34no—
141C2×C4—34no—
142C2×C4—92no—
143C2×C4—92no—
144C2×C4C2×D3212yes—
145C2×C4C2×D3211yescharacteristic
146C2×C3—22no—
147C2×C3—21no—
148C2×C3#100412yes—
149C2×C3C2×D3×Q814yes—
150C22C22×D3211yescharacteristic
151C4—22no—
152C4—21no—
153C4—38no—
154C4—64no—
155C4—94no—
156C4—182no—
157C3—21no—
158C3#572212yescomplemented
159C2D32×Q812yes—
160C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, prime radical, cosocleC22×C32C24
FittingC2×C32×Q8C22
hypercenterC2×Q8D32
hypocenterC32Q82
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
42. #627Q8semidirect
134. C32Q82semidirect
158. C3#5722semidirect

Maximal subgroups

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

subgroupindex
#641532
C3×#9442
#638402
#638392
#638382
#638372
#57223