Finite Group Databasebeta

← #71156

Subgroups — #71156

Order 384.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#5579C211yesmaximal, characteristic
2C3×#477C211yesmaximal, characteristic, complemented
3#5092C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic
5#4535C211yesmaximal, characteristic, complemented
6#4483C211yesmaximal, characteristic
7#4481C211yesmaximal, characteristic, complemented
8#1632—31nomaximal, Sylow
9C3×C4⋊Q8C2211yescharacteristic
10C3×C4×Q8C2211yescharacteristic
11#946—21no—
12C2×#216—22no—
13#896—22no—
14D3×C4⋊C4—21no—
15C4×D12—21no—
16C4×D12C2211yescharacteristic
17#870—21no—
18#869C2211yescharacteristic
19C3×#107—21no—
20C3×C4⋊C8C2211yescharacteristic
21C3×#103—21no—
22#809—21no—
23#808—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#554—31no—
27#494—31no—
28#477D311yescharacteristic, complemented
29#476—31no—
30#333—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×Q8D411yescharacteristic
35C2×D12D411yescharacteristic
36C2×C4×D3—21no—
37C2×C4×D3—22no—
38C2×Dic6D412yes—
39C2×C3×C8—21no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4D411yescharacteristic
44C3×C42—21no—
45C3×C42C2311yescharacteristic
46#216—44no—
47(C2×D3)⋊C4—22no—
48(C2×D3)⋊C4—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3—21no—
51C4⋊Dic3D411yescharacteristic
52Dic3⋊C4—22no—
53Dic3⋊C4—22no—
54C2×C3⋊C8—21no—
55C2×C3⋊C8—21no—
56C2×Q16—62no—
57C4⋊Q8—31no—
58C4⋊Q8C2×D311yescharacteristic
59#125—61no—
60C22⋊Q8—62no—
61C22⋊Q8—61no—
62C4×Q8C2×D311yescharacteristic
63C4×D4—31no—
64C4×D4—61no—
65C2×C4⋊C4—61no—
66#107—21no—
67#106—61no—
68C4⋊C8—31no—
69C4⋊C8C2×D311yescharacteristic
70#103—21no—
71#102—61no—
72C4×C8—31no—
73C22×D3—21no—
74C3×Q8—21no—
75C3×Q8—21no—
76C3×Q8—42no—
77C2×C3×C4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—41no—
80C2×C3×C4C2×D411yescharacteristic
81C2×C3×C4C2×D411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C2×Dic3—21no—
84C2×Dic3—22no—
85D12—21no—
86D12Q1612yescomplemented
87C4×D3—41no—
88C4×D3—41no—
89C4×D3—82no—
90Dic6—42no—
91C3×C8—41no—
92C3⋊C8—22no—
93C3⋊C8—41no—
94C2×Q8—21no—
95C2×Q8—32no—
96C2×Q8#6711yescharacteristic
97C2×D4—31no—
98C22×C4—62no—
99C22×C4—61no—
100Q16—124no—
101C2×C8—21no—
102C2×C8—61no—
103C2×C8—61no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—31no—
108C4⋊C4—62no—
109C4⋊C4—62no—
110C4⋊C4—61no—
111C4⋊C4—61no—
112C4⋊C4#6711yescharacteristic
113C22⋊C4—62no—
114C22⋊C4—61no—
115C42—21no—
116C42C22×D311yescharacteristic
117C22×C3#12011yescharacteristic
118C2×D3—22no—
119C2×D3—41no—
120C2×D3—41no—
121C3×C4—21no—
122C3×C4—41no—
123C3×C4—41no—
124C3×C4—81no—
125C3×C4C2×Q1611yescharacteristic
126C3×C4#13511yescharacteristic
127C3×C4#13711yescharacteristic
128C3×C4#13711yescharacteristic
129Dic3—41no—
130Dic3—42no—
131C23—61no—
132Q8—21no—
133Q8—21no—
134Q8—42no—
135Q8—122no—
136D4—32no—
137D4—61no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—61no—
142C2×C4—62no—
143C2×C4—121no—
144C2×C4—121no—
145C2×C4—242no—
146C2×C4D3×D411yescharacteristic
147C2×C4D3×D411yescharacteristic
148C2×C4C2×#6711yescharacteristic
149C8—41no—
150C8—62no—
151C8—121no—
152C2×C3#45111yescharacteristic
153C2×C3#45211yescharacteristic
154C2×C3#45311yescharacteristic
155D3—42no—
156C22—62no—
157C22—121no—
158C22—121no—
159C22#93711yescharacteristic
160C4—21no—
161C4—41no—
162C4—41no—
163C4—81no—
164C4—122no—
165C4—121no—
166C4D3×Q1611yescharacteristic
167C4#91811yescharacteristic
168C4#94211yescharacteristic
169C4#95011yescharacteristic
170C3#163211yescharacteristic, complemented, Sylow
171C2—122no—
172C2#504711yescharacteristic
173C2#518611yescharacteristic
174C2#519811yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#477C2
socleC22×C3#120
prime radicalC2×D12D4
hypercenter#477D3
hypocenterC3#1632
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. C3×#477C2semidirect
3. #5092C2semidirect
5. #4535C2semidirect
7. #4481C2semidirect
28. #477D3semidirect
86. D12Q16semidirect
170. C3#1632semidirect

Maximal subgroups

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

subgroupindex
#55792
C3×#4772
#50922
#50842
#45352
#44832
#44812
#16323