Finite Group Databasebeta

← #69719

Subgroups — #69719

Order 384.

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

normal subgroups only

This group has 202 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×#400C211yesmaximal, characteristic
3#4970C212yesmaximal
4#4528C211yesmaximal, characteristic
5#4452C212yesmaximal
6#1617—31nomaximal, Sylow
7C2×C3×C4⋊C4C2211yescharacteristic
8C2×C3×C4⋊C4C2211yescharacteristic
9#926—21no—
10#924—21no—
11#924C2212yes—
12#890—22no—
13#888—22no—
14C3×#95—21no—
15C3×#95—21no—
16C3×#95C2211yescharacteristic
17#830C2212yes—
18#810—22no—
19#807—22no—
20#544—31no—
21#484—32no—
22#400D311yescharacteristic
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—
34#217—22no—
35C4⋊Dic3—21no—
36C4⋊Dic3—21no—
37C4⋊Dic3—21no—
38C4⋊Dic3D412yes—
39Dic3⋊C4—22no—
40Dic3⋊C4—22no—
41C4×Dic3—22no—
42C2×C3⋊C8—22no—
43C4⋊Q8—62no—
44#125—62no—
45C22⋊Q8—32no—
46C22⋊Q8—61no—
47#117—61no—
48C2×C4⋊C4C2×D311yescharacteristic
49C2×C4⋊C4C2×D311yescharacteristic
50#107—62no—
51#103—62no—
52C22⋊C8—32no—
53#95—21no—
54#95—21no—
55#95C2×D311yescharacteristic
56C23×C3C2×D411yescharacteristic
57C2×C3×C4—21no—
58C2×C3×C4—21no—
59C2×C3×C4—21no—
60C2×C3×C4—41no—
61C2×C3×C4—41no—
62C2×C3×C4—41no—
63C2×C3×C4—41no—
64C2×C3×C4—42no—
65C2×C3×C4—41no—
66C2×C3×C4—41no—
67C2×C3×C4Q1612yes—
68C2×C3×C4C2×D412yes—
69C2×Dic3—22no—
70C2×Dic3—22no—
71Dic6—42no—
72Dic6—81no—
73C3⋊C8—42no—
74C2×Q8—32no—
75C2×Q8—61no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4#6711yescharacteristic
79C22×C4#6711yescharacteristic
80C22×C4C22×D311yescharacteristic
81C2×C8—62no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—32no—
85C4⋊C4—61no—
86C4⋊C4—61no—
87C4⋊C4—61no—
88C4⋊C4—62no—
89C4⋊C4—62no—
90C22⋊C4—62no—
91C42—62no—
92C22×C3—21no—
93C22×C3—21no—
94C22×C3#12011yescharacteristic
95C22×C3C2×Q1611yescharacteristic
96C22×C3#13711yescharacteristic
97C3×C4—22no—
98C3×C4—22no—
99C3×C4—41no—
100C3×C4—81no—
101C3×C4—81no—
102Dic3—42no—
103Dic3—42no—
104C23C2×#6711yescharacteristic
105Q8—122no—
106Q8—241no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—62no—
118C2×C4—62no—
119C2×C4#21612yes—
120C2×C4D3×D412yes—
121C8—122no—
122C2×C3—21no—
123C2×C3—21no—
124C2×C3#45111yescharacteristic
125C2×C3#45411yescharacteristic
126C2×C3#45811yescharacteristic
127C22—21no—
128C22—21no—
129C22#93711yescharacteristic
130C22C2×#21611yescharacteristic
131C22#95111yescharacteristic
132C4—22no—
133C4—22no—
134C4—41no—
135C4—81no—
136C4—81no—
137C4—122no—
138C4—122no—
139C3#161711yescharacteristic, complemented, Sylow
140C2—21no—
141C2—21no—
142C2#505011yescharacteristic
143C2#505311yescharacteristic
144C2#506211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#400C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#400D3
hypocenterC3#1617
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
139. C3#1617semidirect

Maximal subgroups

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

subgroupindex
#55002
C3×#4002
#49702
#45282
#44522
#16173