Finite Group Databasebeta

← #68138

Subgroups — #68138

Order 384.

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

normal subgroups only

This group has 266 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, complemented
2#4991C211yesmaximal, characteristic
3#4731C212yesmaximal
4C3×#324C211yesmaximal, characteristic, complemented
5#4467C212yesmaximal, complemented
6#1599—31nomaximal, Sylow
7C22×D12C2211yescharacteristic
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×(C2×D3)⋊C4—21no—
10C2×(C2×D3)⋊C4—21no—
11#926—21no—
12#924—21no—
13#924C2212yes—
14C2×#204—22no—
15#890—22no—
16#888—22no—
17C3×C22⋊C8C2212yes—
18C3⋊#95C2211yescharacteristic
19#821—22no—
20#544—31no—
21#450—32no—
22#394—31no—
23#328—32no—
24#324D311yescharacteristic, complemented
25C23×D3—21no—
26C22×C3×C4D411yescharacteristic
27C22×C3×C4D411yescharacteristic
28C22×C3×C4C2311yescharacteristic
29C22×Dic3—21no—
30C2×D12—22no—
31C2×D12—42no—
32C2×Dic6—21no—
33C2×Dic6D412yes—
34C2×C3×C8—22no—
35C3×C4⋊C4—22no—
36#217—22no—
37(C2×D3)⋊C4—42no—
38(C2×D3)⋊C4—42no—
39C4⋊Dic3—21no—
40C4⋊Dic3—21no—
41C4⋊Dic3—21no—
42C4⋊Dic3D412yes—
43Dic3⋊C4—22no—
44Dic3⋊C4—22no—
45C4×Dic3—22no—
46#204—44no—
47C22×D4—31no—
48C2×SD16—62no—
49C4⋊Q8—62no—
50#125—62no—
51C22⋊Q8—32no—
52C22⋊Q8—61no—
53#117—61no—
54C2×C4⋊C4C2×D311yescharacteristic
55C2×C22⋊C4—61no—
56C2×C22⋊C4—61no—
57#102—62no—
58C22⋊C8C2×D312yes—
59#95—31no—
60C23×C3C2×D411yescharacteristic
61C22×D3—22no—
62C22×D3—41no—
63C22×D3—41no—
64C22×D3—41no—
65C22×D3—41no—
66C22×D3—41no—
67C22×D3—41no—
68C2×C3×C4—21no—
69C2×C3×C4—21no—
70C2×C3×C4—21no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4SD1612yescomplemented
74C2×C3×C4C2×D412yes—
75C2×Dic3—22no—
76C2×Dic3—22no—
77C2×Dic3—41no—
78C2×Dic3—41no—
79C2×Dic3—41no—
80D12—24no—
81D12—42no—
82Dic6—42no—
83Dic6—81no—
84C3×C8—42no—
85C24—61no—
86C2×Q8—32no—
87C2×Q8—61no—
88C2×D4—62no—
89C2×D4—122no—
90C22×C4—61no—
91C22×C4D1211yescharacteristic
92C22×C4D1211yescharacteristic
93C22×C4C22×D311yescharacteristic
94SD16—124no—
95C2×C8—22no—
96C4⋊C4—22no—
97C4⋊C4—32no—
98C4⋊C4—61no—
99C4⋊C4—61no—
100C4⋊C4—61no—
101C4⋊C4—62no—
102C4⋊C4—62no—
103C22⋊C4—62no—
104C22⋊C4—122no—
105C22⋊C4—122no—
106C42—62no—
107C22×C3—21no—
108C22×C3—21no—
109C22×C3#12011yescharacteristic
110C22×C3C2×SD1611yescharacteristic
111C22×C3#13711yescharacteristic
112C2×D3—42no—
113C2×D3—42no—
114C2×D3—42no—
115C2×D3—81no—
116C2×D3—81no—
117C2×D3—81no—
118C2×D3—81no—
119C3×C4—22no—
120C3×C4—22no—
121C3×C4—41no—
122Dic3—42no—
123Dic3—42no—
124Dic3—81no—
125C23—62no—
126C23—121no—
127C23—121no—
128C23—121no—
129C23—121no—
130C23—121no—
131C23—121no—
132C23C2×D1211yescharacteristic
133Q8—122no—
134Q8—241no—
135D4—64no—
136D4—122no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—62no—
143C2×C4—62no—
144C2×C4—121no—
145C2×C4—121no—
146C2×C4—121no—
147C2×C4#20412yescomplemented
148C2×C4D3×D412yes—
149C8—42no—
150C2×C3—21no—
151C2×C3—21no—
152C2×C3#44811yescharacteristic
153C2×C3#45411yescharacteristic
154C2×C3B(4,𝔽2)11yescharacteristic
155D3—82no—
156C22—21no—
157C22—21no—
158C22—122no—
159C22—122no—
160C22—122no—
161C22—241no—
162C22—241no—
163C22—241no—
164C22—241no—
165C22#88211yescharacteristic
166C22C2×#20411yescharacteristic
167C22#90911yescharacteristic
168C4—22no—
169C4—22no—
170C4—41no—
171C4—122no—
172C4—122no—
173C4—241no—
174C3#159911yescharacteristic, complemented, Sylow
175C2—21no—
176C2—21no—
177C2—242no—
178C2#473811yescharacteristic
179C2#474111yescharacteristic
180C2#475311yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radicalC22×D12C22
hypercenter#324D3
hypocenterC3#1599
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
1. #5500C2semidirect
4. C3×#324C2semidirect
5. #4467C2semidirect
24. #324D3semidirect
73. C2×C3×C4SD16semidirect
147. C2×C4#204semidirect
174. C3#1599semidirect

Maximal subgroups

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

subgroupindex
#55002
#49912
#47312
C3×#3242
#44672
#15993