Finite Group Databasebeta

← #10593527

Subgroups — #10593527

Order 576.

188 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 37 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
1C22×#622C211yesmaximal, characteristic
2C2×#3776C211yesmaximal, characteristic, complemented
3#63982C212yesmaximal
4#63743C211yesmaximal, characteristic, complemented
5#63740C212yesmaximal, complemented
6C22×C3⋊Dic3C2211yescharacteristic
7C2×#627C2212yes—
8C2×#622—22no—
9C2×#622C2211yescharacteristic
10C2×#622C2211yescharacteristic
11#3776—22no—
12C3×#217—22no—
13#3724C2212yes—
14Dic3⋊Dic3—22no—
15#878—62no—
16C23×C32D411yescharacteristic
17C2×C3⋊Dic3—21no—
18C2×C3⋊Dic3—21no—
19C2×C3⋊Dic3D411yescharacteristic
20C2×C3⋊Dic3C2311yescharacteristic
21C2×C3×Dic3—22no—
22C2×C3×Dic3—22no—
23#627—22no—
24#627—22no—
25#622—21no—
26#622D412yes—
27#467—91nomaximal, Sylow
28C22×Dic3—62no—
29C2×Dic6—62no—
30C3×C22⋊C4—62no—
31#217—22no—
32C4⋊Dic3—62no—
33Dic3⋊C4—62no—
34Dic3⋊C4—62no—
35C22×C32—21no—
36C22×C32—21no—
37C22×C32Q1612yescomplemented
38C22×C32C2×D411yescharacteristic
39C3⋊Dic3—21no—
40C3⋊Dic3C2×D411yescharacteristic
41C3⋊Dic3#4111yescharacteristic
42C3×Dic3—42no—
43C3×Dic3—42no—
44C2×Q16—91no—
45C22×C8—91no—
46C22⋊Q8—92no—
47#107—91no—
48#103—92no—
49C23×C3—22no—
50C2×C3×C4—62no—
51C2×C3×C4—62no—
52C2×Dic3—22no—
53C2×Dic3—22no—
54C2×Dic3—62no—
55C2×Dic3—62no—
56C2×Dic3—122no—
57C2×Dic3—122no—
58Dic6—122no—
59Dic6—122no—
60C2×C32—22no—
61C2×C32#12111yescharacteristic
62C2×C32C2×Q1611yescharacteristic
63C2×C32#13511yescharacteristic
64C2×Q8—92no—
65C22×C4—91no—
66Q16—182no—
67C2×C8—91no—
68C2×C8—91no—
69C2×C8—182no—
70C4⋊C4—92no—
71C4⋊C4—182no—
72C22⋊C4—62no—
73C22×C3—22no—
74C22×C3—24no—
75C22×C3—42no—
76C22×C3—42no—
77C3×C4—122no—
78C3×C4—122no—
79Dic3—42no—
80Dic3—42no—
81Dic3—62no—
82Dic3—62no—
83Dic3—122no—
84C32#46711yescharacteristic, complemented, Sylow
85C23GO+(4,𝔽2)11yescharacteristic
86Q8—182no—
87Q8—182no—
88C2×C4—62no—
89C2×C4—62no—
90C2×C4—91no—
91C2×C4—91no—
92C2×C4—181no—
93C2×C4—181no—
94C8—92no—
95C8—181no—
96C2×C3—22no—
97C2×C3—22no—
98C2×C3—22no—
99C2×C3—44no—
100C22—21no—
101C22—21no—
102C22#377612yescomplemented
103C22C2×GO+(4,𝔽2)11yescharacteristic
104C4—91no—
105C4—91no—
106C4—122no—
107C4—122no—
108C4—181no—
109C3—22no—
110C2—22no—
111C2#6423711yescharacteristic
112C2C2×#377611yescharacteristic
113C2#6424211yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
Fitting, prime radicalC23×C32D4
socleC22×C32C2×D4
hypercenterC23GO+(4,𝔽2)
hypocenterC32#467
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#3776C2semidirect
4. #63743C2semidirect
5. #63740C2semidirect
37. C22×C32Q16semidirect
84. C32#467semidirect
102. C22#3776semidirect

Maximal subgroups

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

subgroupindex
C22×#6222
C2×#37762
#639822
#637432
#637402
#4679