Finite Group Databasebeta

← #10593504

Subgroups — #10593504

Order 576.

187 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 35 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#63982C211yesmaximal, characteristic
2C2×#3724C211yesmaximal, characteristic
3C22⋊#622C211yesmaximal, characteristic
4#63742C212yesmaximal, complemented
5#63740C212yesmaximal, complemented
6C22×C3⋊Dic3C2211yescharacteristic
7C22×C3×Dic3—21no—
8C2×#627C2211yescharacteristic
9C2×#622C2212yes—
10C3×#217—21no—
11#3724—21no—
12#3724C2212yes—
13#3724C2211yescharacteristic
14Dic3⋊Dic3—21no—
15C2×Dic3⋊C4—61no—
16#878—61no—
17C23×C32D411yescharacteristic
18C2×C3⋊Dic3—22no—
19C2×C3⋊Dic3D411yescharacteristic
20C2×C3⋊Dic3C2311yescharacteristic
21C2×C3×Dic3—21no—
22C2×C3×Dic3—21no—
23C2×C3×Dic3—22no—
24C2×C3×Dic3—42no—
25#627—22no—
26#622—22no—
27#483—91nomaximal, Sylow
28C22×C3×C4—61no—
29C22×Dic3—21no—
30C22×Dic3—61no—
31C22×Dic3—61no—
32C2×Dic6—61no—
33C3×C22⋊C4—61no—
34#217—21no—
35C4⋊Dic3—61no—
36Dic3⋊C4—61no—
37Dic3⋊C4—62no—
38Dic3⋊C4—61no—
39Dic3⋊C4—62no—
40C22×C32—21no—
41C22×C32—21no—
42C22×C32SD1612yes—
43C22×C32C2×D411yescharacteristic
44C3⋊Dic3—21no—
45C3⋊Dic3#4112yes—
46C3×Dic3—41no—
47C3×Dic3—42no—
48C3×Dic3—41no—
49C22⋊Q8—91no—
50C2×C4⋊C4—91no—
51#106—92no—
52#103—92no—
53C22⋊C8—91no—
54C23×C3—21no—
55C23×C3—21no—
56C2×C3×C4—62no—
57C2×C3×C4—61no—
58C2×C3×C4—61no—
59C2×C3×C4—122no—
60C2×Dic3—21no—
61C2×Dic3—22no—
62C2×Dic3—21no—
63C2×Dic3—42no—
64C2×Dic3—61no—
65C2×Dic3—61no—
66C2×Dic3—61no—
67C2×Dic3—61no—
68C2×Dic3—64no—
69C2×Dic3—122no—
70Dic6—122no—
71C2×C32—22no—
72C2×C32#12311yescharacteristic
73C2×C32C2×SD1611yescharacteristic
74C2×C32#13711yescharacteristic
75C2×Q8—91no—
76C22×C4—61no—
77C22×C4—91no—
78C2×C8—92no—
79C4⋊C4—92no—
80C4⋊C4—91no—
81C4⋊C4—181no—
82C4⋊C4—181no—
83C22⋊C4—61no—
84C22×C3—22no—
85C22×C3—22no—
86C22×C3—21no—
87C22×C3—21no—
88C22×C3—22no—
89C22×C3—22no—
90C22×C3—41no—
91C22×C3—41no—
92C3×C4—121no—
93C3×C4—122no—
94C3×C4—121no—
95Dic3—41no—
96Dic3—42no—
97Dic3—41no—
98Dic3—62no—
99Dic3—62no—
100Dic3—62no—
101Dic3—121no—
102C32#48311yescharacteristic, complemented, Sylow
103C23GO+(4,𝔽2)11yescharacteristic
104Q8—182no—
105C2×C4—61no—
106C2×C4—61no—
107C2×C4—62no—
108C2×C4—91no—
109C2×C4—91no—
110C2×C4—122no—
111C2×C4—182no—
112C8—182no—
113C2×C3—24no—
114C2×C3—21no—
115C2×C3—21no—
116C2×C3—21no—
117C2×C3—21no—
118C2×C3—21no—
119C2×C3—21no—
120C2×C3—42no—
121C22—21no—
122C22—21no—
123C22#377512yes—
124C22C2×GO+(4,𝔽2)11yescharacteristic
125C4—92no—
126C4—121no—
127C4—122no—
128C4—121no—
129C4—181no—
130C3—21no—
131C3—21no—
132C2—22no—
133C2#6423311yescharacteristic
134C2C2×#377511yescharacteristic
135C2#6423911yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
Fitting, prime radicalC23×C32D4
socleC22×C32C2×D4
hypercenterC23GO+(4,𝔽2)
hypocenterC32#483
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #63742C2semidirect
5. #63740C2semidirect
102. C32#483semidirect

Maximal subgroups

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

subgroupindex
#639822
C2×#37242
C22⋊#6222
#637422
#637402
#4839