Finite Group Databasebeta

← #10593492

Subgroups — #10593492

Order 576.

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

normal subgroups only

This group has 207 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#63980C211yesmaximal, characteristic
2C2×#3724C211yesmaximal, characteristic, complemented
3C22⋊#622C211yesmaximal, characteristic, complemented
4#63743C212yesmaximal, complemented
5#63739C212yesmaximal, complemented
6C22×C3⋊Dic3C2211yescharacteristic
7C2×C3×#67—21no—
8C22×C3×Dic3—21no—
9C2×#625C2211yescharacteristic
10C2×#622C2212yes—
11#3724—21no—
12#3724C2212yes—
13#3724C2211yescharacteristic
14(C2×D3)⋊Dic3—21no—
15#939—61no—
16C2×Dic3⋊C4—61no—
17C23×C32D411yescharacteristic
18C22×C3×D3—21no—
19C2×C3⋊Dic3—22no—
20C2×C3⋊Dic3D411yescharacteristic
21C2×C3⋊Dic3C2311yescharacteristic
22C3×#67—42no—
23C2×C3×Dic3—21no—
24C2×C3×Dic3—22no—
25C2×C3×Dic3—42no—
26#625—22no—
27#622—22no—
28#480—91nomaximal, Sylow
29C2×C3×D4—61no—
30C22×C3×C4—61no—
31C2×#67—21no—
32C2×#67—61no—
33C22×Dic3—21no—
34C22×Dic3—61no—
35C22×Dic3—61no—
36#217—61no—
37(C2×D3)⋊C4—61no—
38Dic3⋊C4—62no—
39Dic3⋊C4—61no—
40Dic3⋊C4—62no—
41C22×C32—21no—
42C22×C32—21no—
43C22×C32D812yes—
44C22×C32C2×D411yescharacteristic
45C2×C3×D3—41no—
46C2×C3×D3—41no—
47C2×C3×D3—41no—
48C3⋊Dic3—21no—
49C3⋊Dic3#4112yes—
50C3×Dic3—42no—
51C3×Dic3—41no—
52#121—91no—
53C2×C4⋊C4—91no—
54#107—92no—
55#102—92no—
56C22⋊C8—91no—
57C23×C3—21no—
58C23×C3—21no—
59C23×C3—61no—
60C22×D3—21no—
61C3×D4—122no—
62C2×C3×C4—62no—
63C2×C3×C4—61no—
64C2×C3×C4—122no—
65#67—42no—
66#67—122no—
67C2×Dic3—22no—
68C2×Dic3—21no—
69C2×Dic3—42no—
70C2×Dic3—61no—
71C2×Dic3—61no—
72C2×Dic3—61no—
73C2×Dic3—61no—
74C2×Dic3—64no—
75C2×Dic3—122no—
76C2×C32—22no—
77C2×C32#12311yescharacteristic
78C2×C32C2×D811yescharacteristic
79C2×C32#13711yescharacteristic
80C3×D3—81no—
81C2×D4—61no—
82C2×D4—91no—
83C22×C4—61no—
84C22×C4—91no—
85C2×C8—92no—
86C4⋊C4—92no—
87C4⋊C4—91no—
88C4⋊C4—181no—
89C22⋊C4—181no—
90C22×C3—22no—
91C22×C3—22no—
92C22×C3—21no—
93C22×C3—21no—
94C22×C3—22no—
95C22×C3—22no—
96C22×C3—41no—
97C22×C3—41no—
98C22×C3—121no—
99C22×C3—121no—
100C22×C3—121no—
101C2×D3—41no—
102C2×D3—41no—
103C2×D3—41no—
104C3×C4—122no—
105C3×C4—121no—
106Dic3—42no—
107Dic3—41no—
108Dic3—62no—
109Dic3—62no—
110Dic3—62no—
111Dic3—121no—
112C32#48011yescharacteristic, complemented, Sylow
113C23—61no—
114C23GO+(4,𝔽2)11yescharacteristic
115D4—122no—
116D4—182no—
117C2×C4—61no—
118C2×C4—62no—
119C2×C4—91no—
120C2×C4—91no—
121C2×C4—122no—
122C2×C4—182no—
123C8—182no—
124C2×C3—24no—
125C2×C3—21no—
126C2×C3—21no—
127C2×C3—21no—
128C2×C3—21no—
129C2×C3—21no—
130C2×C3—21no—
131C2×C3—42no—
132C2×C3—241no—
133D3—81no—
134C22—21no—
135C22—21no—
136C22—121no—
137C22—121no—
138C22—121no—
139C22#377412yes—
140C22C2×GO+(4,𝔽2)11yescharacteristic
141C4—92no—
142C4—122no—
143C4—121no—
144C4—181no—
145C3—21no—
146C3—21no—
147C2—22no—
148C2—241no—
149C2#6423311yescharacteristic
150C2C2×#377411yescharacteristic
151C2#6423911yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23
FittingC23×C32D4
socleC22×C32C2×D4
prime radical#63980C2
hypercenterC23GO+(4,𝔽2)
hypocenterC32#480
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×#3724C2semidirect
3. C22⋊#622C2semidirect
4. #63743C2semidirect
5. #63739C2semidirect
112. C32#480semidirect

Maximal subgroups

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

subgroupindex
#639802
C2×#37242
C22⋊#6222
#637432
#637392
#4809