Finite Group Databasebeta

← #81740

Subgroups — #81740

Order 384.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#5725C211yesmaximal, characteristic
2D3×#125C211yesmaximal, characteristic, complemented
3C3×#491C211yesmaximal, characteristic, complemented
4#5082C211yesmaximal, characteristic, complemented
5#5067C211yesmaximal, characteristic, complemented
6#4803C214yesmaximal, complemented
7#4800C214yesmaximal, complemented
8#4713C211yesmaximal, characteristic, complemented
9#4707C211yesmaximal, characteristic, complemented
10#3142—31nomaximal, Sylow
11C2×#239C2212yes—
12C3×C4⋊Q8C2211yescharacteristic
13C3×#125C2211yescharacteristic
14#947C2212yes—
15C2×ΓL(1,𝔽25)C2212yes—
16#895—22no—
17#893—22no—
18D3×C4⋊C4—22no—
19D3×C4⋊C4C2212yes—
20#890C2212yes—
21#889—22no—
22#874C2211yescharacteristic
23C42×D3C2211yescharacteristic
24#870C2211yescharacteristic
25C3×#103C2214yes—
26C3×#97C2211yescharacteristic
27#835C2214yes—
28#821C2214yes—
29C8×Dic3C2212yes—
30#809C2214yes—
31#803C2211yescharacteristic
32#532—31no—
33C2×#125—31no—
34#491—31no—
35#491D311yescharacteristic, complemented
36#490—31no—
37#490—31no—
38#488—34no—
39#417—34no—
40C4×M16—31no—
41C2×C3×Q8C2312yes—
42#239—28no—
43C2×D12—22no—
44C2×D12—22no—
45C2×D12C2312yes—
46C2×C4×D3—22no—
47C2×C4×D3—22no—
48C2×C4×D3D412yes—
49C2×C4×D3C2311yescharacteristic
50C2×C3×C8C2312yes—
51C3×C4⋊C4—22no—
52C3×C4⋊C4—22no—
53C3×C4⋊C4C2312yes—
54C3×C42C2311yescharacteristic
55(C2×D3)⋊C4—24no—
56C4⋊Dic3—22no—
57C4⋊Dic3C2312yes—
58Dic3⋊C4—22no—
59Dic3⋊C4—22no—
60Dic3⋊C4—22no—
61C4×Dic3D412yes—
62C4×Dic3C2311yescharacteristic
63C2×C3⋊C8C2312yes—
64ΓL(1,𝔽25)—24no—
65C2×#41—32no—
66C2×M16—32no—
67C4⋊Q8C2×D311yescharacteristic
68#127—31no—
69#125—32no—
70#125—31no—
71#125—62no—
72#125C2×D311yescharacteristic
73#124—32no—
74#121—62no—
75C4×D4—62no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—62no—
78C2×C42—31no—
79#103—34no—
80#103C2×D314yes—
81#102—34no—
82#102—34no—
83#97—31no—
84#97C2×D311yescharacteristic
85C4×C8—32no—
86C22×D3—22no—
87C22×D3C2×D411yescharacteristic
88C3×Q8—24no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4C2×D412yes—
92C2×C3×C4C2411yescharacteristic
93C2×Dic3—22no—
94C2×Dic3C2×D412yes—
95C2×Dic3C2×D411yescharacteristic
96D12—24no—
97D12—44no—
98D12—44no—
99C4×D3—42no—
100C4×D3—44no—
101C4×D3—44no—
102C4×D3#4114yes—
103C3×C8—22no—
104C3⋊C8—22no—
105#41—68no—
106C2×Q8C22×D312yes—
107C2×D4—32no—
108C2×D4—62no—
109C2×D4—62no—
110C22×C4—31no—
111C22×C4—32no—
112C22×C4—62no—
113C22×C4—62no—
114M16—64no—
115C2×C8—32no—
116C2×C8C22×D312yes—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—32no—
120C4⋊C4—62no—
121C4⋊C4—62no—
122C4⋊C4—62no—
123C4⋊C4—62no—
124C4⋊C4C22×D312yes—
125C22⋊C4—64no—
126C42—32no—
127C42—31no—
128C42C22×D311yescharacteristic
129C22×C3C22×D411yescharacteristic
130C2×D3—22no—
131C2×D3—42no—
132C2×D3—44no—
133C2×D3#12412yes—
134C3×C4—22no—
135C3×C4—42no—
136C3×C4—42no—
137C3×C4C2×#4112yes—
138Dic3—22no—
139Dic3—42no—
140Dic3#12412yes—
141C23—31no—
142C23—62no—
143Q8—24no—
144D4—64no—
145D4—124no—
146D4—124no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—31no—
150C2×C4—34no—
151C2×C4—32no—
152C2×C4—62no—
153C2×C4—122no—
154C2×C4—124no—
155C2×C4—124no—
156C2×C4D3×D412yes—
157C2×C4C23×D311yescharacteristic
158C8—22no—
159C8—62no—
160C2×C3C2×#12411yescharacteristic
161C2×C3#57512yes—
162D3—22no—
163D3—82no—
164C22—32no—
165C22—62no—
166C22—122no—
167C22—124no—
168C22C2×D3×D411yescharacteristic
169C4—22no—
170C4—32no—
171C4—42no—
172C4—42no—
173C4—62no—
174C4—122no—
175C4D3×#4112yes—
176C3#314211yescharacteristic, complemented, Sylow
177C2—62no—
178C2—242no—
179C2D3×#12411yescharacteristic
180C2#577712yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#491C2
socleC22×C3C22×D4
prime radical#5725C2
hypercenter#491D3
hypocenterC3#3142
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. D3×#125C2semidirect
3. C3×#491C2semidirect
4. #5082C2semidirect
5. #5067C2semidirect
6. #4803C2semidirect
7. #4800C2semidirect
8. #4713C2semidirect
9. #4707C2semidirect
35. #491D3semidirect
176. C3#3142semidirect

Maximal subgroups

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

subgroupindex
#57252
D3×#1252
C3×#4912
#50822
#50672
#48032
#48002
#47132
#47072
#31423