Finite Group Databasebeta

← #68141

Subgroups — #68141

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
2#4986C211yesmaximal, characteristic
3#4722C212yesmaximal
4C3×#324C211yesmaximal, characteristic
5#4467C212yesmaximal
6#1617—31nomaximal, Sylow
7C2×C3×C4⋊C4C2211yescharacteristic
8#926—21no—
9C2×C4⋊Dic3C2211yescharacteristic
10#924—21no—
11#924C2212yes—
12#890—22no—
13#888—22no—
14C3×C22⋊C8C2212yes—
15C3⋊#95—21no—
16C3⋊#95—21no—
17C3⋊#95C2211yescharacteristic
18#818—22no—
19#816—22no—
20#544—31no—
21#484—32no—
22#400—31no—
23#328—32no—
24#324D311yescharacteristic
25C22×C3×C4D411yescharacteristic
26C22×C3×C4D411yescharacteristic
27C22×C3×C4C2311yescharacteristic
28C22×Dic3—21no—
29C22×Dic3—21no—
30C2×Dic6—21no—
31C2×Dic6D412yes—
32C2×C3×C8—22no—
33C3×C4⋊C4—22no—
34#217—22no—
35C4⋊Dic3—21no—
36C4⋊Dic3—21no—
37C4⋊Dic3—21no—
38C4⋊Dic3—22no—
39C4⋊Dic3D412yes—
40Dic3⋊C4—22no—
41Dic3⋊C4—22no—
42C4×Dic3—22no—
43C4⋊Q8—62no—
44#125—62no—
45C22⋊Q8—32no—
46C22⋊Q8—61no—
47#117—61no—
48C2×C4⋊C4—31no—
49C2×C4⋊C4C2×D311yescharacteristic
50#107—62no—
51#103—62no—
52C22⋊C8C2×D312yes—
53#95—31no—
54#95—61no—
55#95—61no—
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×C4Q1612yes—
63C2×C3×C4C2×D412yes—
64C2×Dic3—22no—
65C2×Dic3—22no—
66C2×Dic3—41no—
67C2×Dic3—41no—
68C2×Dic3—42no—
69C2×Dic3—41no—
70C2×Dic3—41no—
71Dic6—42no—
72Dic6—81no—
73C3×C8—42no—
74C2×Q8—32no—
75C2×Q8—61no—
76C22×C4—61no—
77C22×C4—61no—
78C22×C4D1211yescharacteristic
79C22×C4D1211yescharacteristic
80C22×C4C22×D311yescharacteristic
81C2×C8—22no—
82C4⋊C4—22no—
83C4⋊C4—32no—
84C4⋊C4—61no—
85C4⋊C4—61no—
86C4⋊C4—61no—
87C4⋊C4—62no—
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—
100Dic3—42no—
101Dic3—42no—
102Dic3—81no—
103Dic3—81no—
104C23C2×D1211yescharacteristic
105Q8—122no—
106Q8—241no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—62no—
113C2×C4—62no—
114C2×C4—121no—
115C2×C4—121no—
116C2×C4—122no—
117C2×C4—121no—
118C2×C4—121no—
119C2×C4Dic1212yes—
120C2×C4D3×D412yes—
121C8—42no—
122C2×C3—21no—
123C2×C3—21no—
124C2×C3#45111yescharacteristic
125C2×C3#45411yescharacteristic
126C2×C3#45811yescharacteristic
127C22—21no—
128C22—21no—
129C22#88211yescharacteristic
130C22C2×Dic1211yescharacteristic
131C22#90911yescharacteristic
132C4—22no—
133C4—22no—
134C4—41no—
135C4—122no—
136C4—122no—
137C4—241no—
138C4—241no—
139C3#161711yescharacteristic, complemented, Sylow
140C2—21no—
141C2—21no—
142C2#473911yescharacteristic
143C2#474211yescharacteristic
144C2#475311yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#324D3
hypocenterC3#1617
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

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
#49862
#47222
C3×#3242
#44672
#16173