Finite Group Databasebeta

← #69707

Subgroups — #69707

Order 384.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#5500C211yesmaximal, characteristic
2C3×#395C211yesmaximal, characteristic
3#5228C212yesmaximal
4#4528C211yesmaximal, characteristic
5#4452C212yesmaximal
6#1605—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C2×C3×C4⋊C4—21no—
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×#216—22no—
11#926—21no—
12#924—21no—
13#924C2212yes—
14#890—22no—
15#888—22no—
16C3×#95—21no—
17C3×#95C2211yescharacteristic
18#835—22no—
19#830C2212yes—
20#544—31no—
21#451—32no—
22#395D311yescharacteristic
23#328—32no—
24#324—31no—
25C2×C3×Q8—22no—
26C2×C3×Q8—42no—
27C22×C3×C4—21no—
28C22×C3×C4—21no—
29C22×C3×C4D411yescharacteristic
30C22×C3×C4D411yescharacteristic
31C22×C3×C4C2311yescharacteristic
32C2×Dic6—21no—
33C2×Dic6D412yes—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—42no—
36#217—22no—
37#216—44no—
38C4⋊Dic3—21no—
39C4⋊Dic3—21no—
40C4⋊Dic3—21no—
41C4⋊Dic3D412yes—
42Dic3⋊C4—22no—
43Dic3⋊C4—22no—
44C4×Dic3—22no—
45C2×C3⋊C8—22no—
46C22×Q8C2×D311yescharacteristic
47C2×Q16—62no—
48C4⋊Q8—62no—
49#125—62no—
50C22⋊Q8—32no—
51C22⋊Q8—61no—
52#117—61no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4C2×D311yescharacteristic
55#103—62no—
56C22⋊C8—32no—
57#95—21no—
58#95C2×D311yescharacteristic
59C23×C3C2×D411yescharacteristic
60C3×Q8—24no—
61C3×Q8—42no—
62C2×C3×C4—21no—
63C2×C3×C4—21no—
64C2×C3×C4—22no—
65C2×C3×C4—22no—
66C2×C3×C4—21no—
67C2×C3×C4—41no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4Q1612yescomplemented
74C2×C3×C4C2×D412yes—
75C2×Dic3—22no—
76C2×Dic3—22no—
77Dic6—42no—
78Dic6—81no—
79C3⋊C8—42no—
80C2×Q8—22no—
81C2×Q8—32no—
82C2×Q8—42no—
83C2×Q8—61no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4#6711yescharacteristic
87C22×C4#6711yescharacteristic
88C22×C4C22×D311yescharacteristic
89Q16—124no—
90C2×C8—62no—
91C4⋊C4—22no—
92C4⋊C4—32no—
93C4⋊C4—42no—
94C4⋊C4—61no—
95C4⋊C4—61no—
96C4⋊C4—61no—
97C4⋊C4—62no—
98C4⋊C4—62no—
99C22⋊C4—62no—
100C42—62no—
101C22×C3—21no—
102C22×C3—21no—
103C22×C3#12011yescharacteristic
104C22×C3C2×Q1611yescharacteristic
105C22×C3#13711yescharacteristic
106C3×C4—22no—
107C3×C4—22no—
108C3×C4—42no—
109C3×C4—41no—
110C3×C4—42no—
111Dic3—42no—
112Dic3—42no—
113C23C2×#6711yescharacteristic
114Q8—24no—
115Q8—42no—
116Q8—122no—
117Q8—241no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—62no—
130C2×C4—62no—
131C2×C4#21612yescomplemented
132C2×C4D3×D412yes—
133C8—122no—
134C2×C3—21no—
135C2×C3—21no—
136C2×C3#45111yescharacteristic
137C2×C3#45611yescharacteristic
138C2×C3B(4,𝔽2)11yescharacteristic
139C22—21no—
140C22—21no—
141C22#93711yescharacteristic
142C22C2×#21611yescharacteristic
143C22#95111yescharacteristic
144C4—22no—
145C4—22no—
146C4—42no—
147C4—42no—
148C4—41no—
149C4—122no—
150C4—122no—
151C3#160511yescharacteristic, complemented, Sylow
152C2—21no—
153C2—21no—
154C2#503211yescharacteristic
155C2#505011yescharacteristic
156C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#395C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#395D3
hypocenterC3#1605
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
73. C2×C3×C4Q16semidirect
131. C2×C4#216semidirect
151. C3#1605semidirect

Maximal subgroups

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

subgroupindex
#55002
C3×#3952
#52282
#45282
#44522
#16053