Finite Group Databasebeta

← #10593499

Subgroups — #10593499

Order 576.

263 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 39 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×#627C211yesmaximal, characteristic
2C2×#3776C212yesmaximal, complemented
3#63982C211yesmaximal, characteristic
4C22⋊#622C211yesmaximal, characteristic
5#63740C212yesmaximal, complemented
6C22×C3⋊Dic3C2211yescharacteristic
7C22×C3×Dic3—21no—
8C2×#627—21no—
9C2×#627—24no—
10C2×#627C2211yescharacteristic
11C2×#627C2212yes—
12C2×#622C2212yes—
13#3776—24no—
14C3×#217—21no—
15#3724C2211yescharacteristic
16Dic3⋊Dic3—21no—
17C22×Dic6—61no—
18#878—61no—
19C23×C32D411yescharacteristic
20C2×C3⋊Dic3—22no—
21C2×C3⋊Dic3D411yescharacteristic
22C2×C3⋊Dic3C2311yescharacteristic
23C2×C3×Dic3—24no—
24C2×C3×Dic3—22no—
25C2×C3×Dic3—21no—
26C2×C3×Dic3—21no—
27#627—24no—
28#627—22no—
29#627—22no—
30#627D414yes—
31#622—22no—
32#451—91nomaximal, Sylow
33C22×C3×C4—61no—
34C22×Dic3—21no—
35C22×Dic3—61no—
36C22×Dic3—61no—
37C2×Dic6—62no—
38C2×Dic6—61no—
39C2×Dic6—62no—
40C2×Dic6—68no—
41C3×C22⋊C4—61no—
42#217—21no—
43C4⋊Dic3—61no—
44Dic3⋊C4—61no—
45Dic3⋊C4—61no—
46C22×C32—21no—
47C22×C32—21no—
48C22×C32Q1612yescomplemented
49C22×C32C2×D411yescharacteristic
50C3⋊Dic3—21no—
51C3⋊Dic3C2×D412yes—
52C3×Dic3—24no—
53C3×Dic3—41no—
54C3×Dic3—41no—
55C22×Q8—91no—
56C2×Q16—92no—
57C22⋊Q8—91no—
58#103—92no—
59C22⋊C8—91no—
60C23×C3—21no—
61C23×C3—21no—
62C2×C3×C4—64no—
63C2×C3×C4—62no—
64C2×C3×C4—61no—
65C2×C3×C4—61no—
66C2×Dic3—24no—
67C2×Dic3—22no—
68C2×Dic3—21no—
69C2×Dic3—21no—
70C2×Dic3—61no—
71C2×Dic3—61no—
72C2×Dic3—61no—
73C2×Dic3—61no—
74C2×Dic3—64no—
75C2×Dic3—122no—
76Dic6—64no—
77Dic6—64no—
78Dic6—68no—
79Dic6—122no—
80C2×C32—22no—
81C2×C32#12011yescharacteristic
82C2×C32C2×Q1611yescharacteristic
83C2×C32#13711yescharacteristic
84C2×Q8—92no—
85C2×Q8—91no—
86C2×Q8—181no—
87C2×Q8—184no—
88C22×C4—61no—
89C22×C4—91no—
90Q16—184no—
91C2×C8—92no—
92C4⋊C4—91no—
93C4⋊C4—181no—
94C22⋊C4—61no—
95C22×C3—22no—
96C22×C3—22no—
97C22×C3—21no—
98C22×C3—21no—
99C22×C3—22no—
100C22×C3—22no—
101C22×C3—41no—
102C22×C3—41no—
103C3×C4—64no—
104C3×C4—121no—
105C3×C4—121no—
106Dic3—24no—
107Dic3—41no—
108Dic3—41no—
109Dic3—62no—
110Dic3—62no—
111Dic3—62no—
112Dic3—121no—
113C32#45111yescharacteristic, complemented, Sylow
114C23GO+(4,𝔽2)11yescharacteristic
115Q8—94no—
116Q8—182no—
117Q8—184no—
118Q8—182no—
119C2×C4—64no—
120C2×C4—62no—
121C2×C4—61no—
122C2×C4—61no—
123C2×C4—91no—
124C2×C4—91no—
125C2×C4—182no—
126C8—182no—
127C2×C3—24no—
128C2×C3—21no—
129C2×C3—21no—
130C2×C3—21no—
131C2×C3—21no—
132C2×C3—21no—
133C2×C3—21no—
134C2×C3—42no—
135C22—21no—
136C22—21no—
137C22#377612yescomplemented
138C22C2×GO+(4,𝔽2)11yescharacteristic
139C4—64no—
140C4—92no—
141C4—121no—
142C4—121no—
143C4—181no—
144C3—21no—
145C3—21no—
146C2—22no—
147C2#6423911yescharacteristic
148C2C2×#377611yescharacteristic
149C2#6424111yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
Fitting, prime radicalC23×C32D4
socleC22×C32C2×D4
hypercenterC23GO+(4,𝔽2)
hypocenterC32#451
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
5. #63740C2semidirect
48. C22×C32Q16semidirect
113. C32#451semidirect
137. C22#3776semidirect

Maximal subgroups

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

subgroupindex
C22×#6272
C2×#37762
#639822
C22⋊#6222
#637402
#4519