Finite Group Databasebeta

← #69713

Subgroups — #69713

Order 384.

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

normal subgroups only

This group has 210 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×#398C211yesmaximal, characteristic
3#4971C212yesmaximal
4#4528C211yesmaximal, characteristic
5#4452C212yesmaximal
6#1611—31nomaximal, Sylow
7C2×C3×C4⋊C4—21no—
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×C3×C4⋊C4C2211yescharacteristic
10#926—21no—
11#924—21no—
12#924C2212yes—
13#890—22no—
14#888—22no—
15C3×#95—21no—
16C3×#95C2211yescharacteristic
17#830C2212yes—
18#810—22no—
19#808—22no—
20#544—31no—
21#483—32no—
22#398D311yescharacteristic
23#328—32no—
24#324—31no—
25C22×C3×C4—21no—
26C22×C3×C4—21no—
27C22×C3×C4D411yescharacteristic
28C22×C3×C4D411yescharacteristic
29C22×C3×C4C2311yescharacteristic
30C2×Dic6—21no—
31C2×Dic6D412yes—
32C3×C4⋊C4—22no—
33C3×C4⋊C4—22no—
34C3×C4⋊C4—42no—
35#217—22no—
36C4⋊Dic3—21no—
37C4⋊Dic3—21no—
38C4⋊Dic3—21no—
39C4⋊Dic3D412yes—
40Dic3⋊C4—22no—
41Dic3⋊C4—22no—
42C4×Dic3—22no—
43C2×C3⋊C8—22no—
44C4⋊Q8—62no—
45#125—62no—
46C22⋊Q8—32no—
47C22⋊Q8—61no—
48#117—61no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4C2×D311yescharacteristic
51C2×C4⋊C4C2×D311yescharacteristic
52#106—62no—
53#103—62no—
54C22⋊C8—32no—
55#95—21no—
56#95C2×D311yescharacteristic
57C23×C3C2×D411yescharacteristic
58C2×C3×C4—21no—
59C2×C3×C4—21no—
60C2×C3×C4—22no—
61C2×C3×C4—21no—
62C2×C3×C4—41no—
63C2×C3×C4—41no—
64C2×C3×C4—41no—
65C2×C3×C4—41no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4—41no—
69C2×C3×C4SD1612yes—
70C2×C3×C4C2×D412yes—
71C2×Dic3—22no—
72C2×Dic3—22no—
73Dic6—42no—
74Dic6—81no—
75C3⋊C8—42no—
76C2×Q8—32no—
77C2×Q8—61no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4#6711yescharacteristic
81C22×C4#6711yescharacteristic
82C22×C4C22×D311yescharacteristic
83C2×C8—62no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—32no—
87C4⋊C4—42no—
88C4⋊C4—61no—
89C4⋊C4—61no—
90C4⋊C4—61no—
91C4⋊C4—62no—
92C4⋊C4—62no—
93C22⋊C4—62no—
94C42—62no—
95C22×C3—21no—
96C22×C3—21no—
97C22×C3#12011yescharacteristic
98C22×C3C2×SD1611yescharacteristic
99C22×C3#13711yescharacteristic
100C3×C4—22no—
101C3×C4—22no—
102C3×C4—42no—
103C3×C4—41no—
104C3×C4—81no—
105Dic3—42no—
106Dic3—42no—
107C23C2×#6711yescharacteristic
108Q8—122no—
109Q8—241no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—62no—
122C2×C4—62no—
123C2×C4#21412yes—
124C2×C4D3×D412yes—
125C8—122no—
126C2×C3—21no—
127C2×C3—21no—
128C2×C3#44811yescharacteristic
129C2×C3#45611yescharacteristic
130C2×C3#45811yescharacteristic
131C22—21no—
132C22—21no—
133C22C2×#21411yescharacteristic
134C22#93711yescharacteristic
135C22#95111yescharacteristic
136C4—22no—
137C4—22no—
138C4—42no—
139C4—41no—
140C4—81no—
141C4—122no—
142C4—122no—
143C3#161111yescharacteristic, complemented, Sylow
144C2—21no—
145C2—21no—
146C2#504011yescharacteristic
147C2#505311yescharacteristic
148C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#398C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#398D3
hypocenterC3#1611
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
143. C3#1611semidirect

Maximal subgroups

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

subgroupindex
#55002
C3×#3982
#49712
#45282
#44522
#16113