Finite Group Databasebeta

← #10593508

Subgroups — #10593508

Order 576.

184 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 33 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
1C2×#3792C211yesmaximal, characteristic
2C2×#3776C211yesmaximal, characteristic, complemented
3#63982C212yesmaximal
4#63742C211yesmaximal, characteristic, complemented
5#63740C212yesmaximal, complemented
6C22×C3⋊Dic3C2211yescharacteristic
7C2×#627C2212yes—
8#3792—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#470—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—22no—
36C22×C32—21no—
37C22×C32C2×D411yescharacteristic
38C3⋊Dic3—21no—
39C3⋊Dic3C2×D411yescharacteristic
40C3⋊Dic3#4111yescharacteristic
41C3×Dic3—42no—
42C3×Dic3—42no—
43C2×Q16—91no—
44C2×M16—91no—
45C22⋊Q8—92no—
46#106—91no—
47#103—92no—
48C23×C3—22no—
49C2×C3×C4—62no—
50C2×C3×C4—62no—
51C2×Dic3—22no—
52C2×Dic3—22no—
53C2×Dic3—62no—
54C2×Dic3—62no—
55C2×Dic3—122no—
56C2×Dic3—122no—
57Dic6—122no—
58Dic6—122no—
59C2×C32—41no—
60C2×C32#12111yescharacteristic
61C2×C32#13712yes—
62C2×Q8—92no—
63C22×C4—91no—
64Q16—182no—
65M16—182no—
66C2×C8—91no—
67C2×C8—91no—
68C4⋊C4—92no—
69C4⋊C4—182no—
70C22⋊C4—62no—
71C22×C3—24no—
72C22×C3—22no—
73C22×C3—42no—
74C22×C3—42no—
75C3×C4—122no—
76C3×C4—122no—
77Dic3—42no—
78Dic3—42no—
79Dic3—62no—
80Dic3—62no—
81Dic3—122no—
82C32#47011yescharacteristic, complemented, Sylow
83C23GO+(4,𝔽2)11yescharacteristic
84Q8—182no—
85Q8—182no—
86C2×C4—62no—
87C2×C4—62no—
88C2×C4—91no—
89C2×C4—91no—
90C2×C4—181no—
91C2×C4—181no—
92C8—92no—
93C8—181no—
94C2×C3—22no—
95C2×C3—22no—
96C2×C3—22no—
97C2×C3—44no—
98C22—22no—
99C22—21no—
100C22C2×GO+(4,𝔽2)11yescharacteristic
101C4—91no—
102C4—91no—
103C4—122no—
104C4—122no—
105C4—181no—
106C3—22no—
107C2—41no—
108C2#6423912yes—
109C2#6424211yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
Fitting, prime radicalC23×C32D4
socleC22×C32C2×D4
hypercenterC23GO+(4,𝔽2)
hypocenterC32#470
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. #63742C2semidirect
5. #63740C2semidirect
82. C32#470semidirect

Maximal subgroups

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

subgroupindex
C2×#37922
C2×#37762
#639822
#637422
#637402
#4709