Finite Group Databasebeta

← #71124

Subgroups — #71124

Order 384.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#5577C211yesmaximal, characteristic
2C3×#475C211yesmaximal, characteristic, complemented
3#5088C211yesmaximal, characteristic, complemented
4#5083C211yesmaximal, characteristic, complemented
5#4535C211yesmaximal, characteristic, complemented
6#4485C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1642—31nomaximal, Sylow
9C2×#239—21no—
10C3×C4⋊Q8C2211yescharacteristic
11C3×C4×Q8C2211yescharacteristic
12C2×#215—21no—
13#895—22no—
14#895—21no—
15#874—21no—
16#874C2211yescharacteristic
17C4×D12—21no—
18C4×D12C2211yescharacteristic
19C3×#106—21no—
20C3×C4⋊C8C2211yescharacteristic
21C3×#103—21no—
22#835—21no—
23#809—22no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#550—31no—
27#486—31no—
28#475D311yescharacteristic, complemented
29#461—31no—
30#331—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×Q8D411yescharacteristic
35#239—42no—
36#239—42no—
37C2×D12—22no—
38C2×D12—21no—
39C2×D12—21no—
40C2×D12—22no—
41C2×D12D411yescharacteristic
42C2×D12D412yes—
43C2×C4×D3—22no—
44C2×C4×D3—21no—
45C2×C3×C8—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4D411yescharacteristic
50C3×C42—21no—
51C3×C42C2311yescharacteristic
52#215—42no—
53(C2×D3)⋊C4—22no—
54(C2×D3)⋊C4—21no—
55C4⋊Dic3D411yescharacteristic
56C2×C3⋊C8—21no—
57C2×C3⋊C8—21no—
58C2×#41—61no—
59C2×SD16—61no—
60C4⋊Q8C2×D311yescharacteristic
61#127—31no—
62#127—61no—
63#121—62no—
64#121—61no—
65C4×Q8C2×D311yescharacteristic
66C4×D4—31no—
67C4×D4—61no—
68#106—21no—
69C4⋊C8—31no—
70C4⋊C8C2×D311yescharacteristic
71#103—21no—
72#103—61no—
73#102—62no—
74C4×C8—31no—
75C22×D3—21no—
76C22×D3—22no—
77C3×Q8—21no—
78C3×Q8—41no—
79C3×Q8—41no—
80C3×Q8SD1612yescomplemented
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—41no—
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×Dic3—21no—
88D12—21no—
89D12—21no—
90D12—42no—
91D12—44no—
92D12—41no—
93D12—41no—
94D12—41no—
95D12—41no—
96D12—82no—
97C4×D3—44no—
98C4×D3—41no—
99C4×D3—41no—
100C3×C8—41no—
101C3⋊C8—41no—
102C3⋊C8—41no—
103#41—122no—
104#41—122no—
105C2×Q8—21no—
106C2×Q8#6711yescharacteristic
107C2×D4—31no—
108C2×D4—32no—
109C2×D4—62no—
110C2×D4—62no—
111C2×D4—61no—
112C2×D4—61no—
113C22×C4—62no—
114C22×C4—61no—
115SD16—122no—
116C2×C8—21no—
117C2×C8—61no—
118C2×C8—61no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—31no—
123C4⋊C4#6711yescharacteristic
124C22⋊C4—62no—
125C22⋊C4—61no—
126C42—21no—
127C42C22×D311yescharacteristic
128C22×C3#12011yescharacteristic
129C2×D3—41no—
130C2×D3—42no—
131C2×D3—42no—
132C2×D3—41no—
133C2×D3—42no—
134C2×D3—41no—
135C3×C4—22no—
136C3×C4—21no—
137C3×C4—41no—
138C3×C4—81no—
139C3×C4C2×SD1611yescharacteristic
140C3×C4#13511yescharacteristic
141C3×C4AGL(1,ℤ/8)11yescharacteristic
142C3×C4AGL(1,ℤ/8)11yescharacteristic
143Dic3—41no—
144C23—61no—
145C23—62no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8#21512yescomplemented
150D4—61no—
151D4—61no—
152D4—122no—
153D4—124no—
154D4—121no—
155D4—121no—
156D4—121no—
157D4—121no—
158D4—242no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—61no—
163C2×C4—124no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4D3×D411yescharacteristic
167C2×C4D3×D411yescharacteristic
168C2×C4C2×#6711yescharacteristic
169C8—41no—
170C8—121no—
171C8—121no—
172C2×C3#44911yescharacteristic
173C2×C3#45011yescharacteristic
174C2×C3#45511yescharacteristic
175D3—81no—
176D3—82no—
177C22—121no—
178C22—122no—
179C22—122no—
180C22—121no—
181C22—122no—
182C22—121no—
183C22#93711yescharacteristic
184C4—21no—
185C4—22no—
186C4—41no—
187C4—81no—
188C4—121no—
189C4#91411yescharacteristic
190C4#91611yescharacteristic
191C4C2×#21511yescharacteristic
192C4#94911yescharacteristic
193C3#164211yescharacteristic, complemented, Sylow
194C2—241no—
195C2—242no—
196C2#504611yescharacteristic
197C2#517211yescharacteristic
198C2#520211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#475C2
socleC22×C3#120
prime radical#5577C2
hypercenter#475D3
hypocenterC3#1642
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. C3×#475C2semidirect
3. #5088C2semidirect
4. #5083C2semidirect
5. #4535C2semidirect
6. #4485C2semidirect
7. #4483C2semidirect
28. #475D3semidirect
80. C3×Q8SD16semidirect
149. Q8#215semidirect
193. C3#1642semidirect

Maximal subgroups

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

subgroupindex
#55772
C3×#4752
#50882
#50832
#45352
#44852
#44832
#16423