Finite Group Databasebeta

← #68439

Subgroups — #68439

Order 384.

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

normal subgroups only

This group has 240 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#5726C211yesmaximal, characteristic
2#5080C211yesmaximal, characteristic, complemented
3#4846C211yesmaximal, characteristic, complemented
4#4841C211yesmaximal, characteristic, complemented
5C3×#332C211yesmaximal, characteristic, complemented
6#4483C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1654—31nomaximal, Sylow
9C2×#239—21no—
10C3×C4⋊Q8C2211yescharacteristic
11#946—21no—
12C2×#204—21no—
13#895—21no—
14#894—22no—
15D3×C4⋊C4—21no—
16C4×D12C2211yescharacteristic
17C4×D12C2211yescharacteristic
18#869C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21#821—21no—
22#818—21no—
23#816—21no—
24#808—22no—
25C4×C3⋊C8C2211yescharacteristic
26#547—31no—
27#498—31no—
28#474—31no—
29#461—31no—
30#332D311yescharacteristic, complemented
31D4⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34#239—42no—
35#239—42no—
36C2×D12—22no—
37C2×D12D411yescharacteristic
38C2×D12D411yescharacteristic
39C2×C4×D3—22no—
40C2×C4×D3—21no—
41C2×C4×D3—21no—
42C2×Dic6—21no—
43C2×C3×C8—21no—
44C2×C3×C8—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4D412yes—
47C3×C42C2311yescharacteristic
48(C2×D3)⋊C4—22no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51C4⋊Dic3—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3D411yescharacteristic
54C4⋊Dic3D411yescharacteristic
55Dic3⋊C4—22no—
56C2×C3⋊C8—21no—
57#204—42no—
58C2×#41—61no—
59C2×SD16—61no—
60C4⋊Q8—31no—
61C4⋊Q8C2×D311yescharacteristic
62#123—62no—
63C22⋊Q8—61no—
64#121—61no—
65C4×D4—31no—
66C4×D4—31no—
67C2×C4⋊C4—61no—
68#107—61no—
69#106—62no—
70C4⋊C8C2×D311yescharacteristic
71C4⋊C8C2×D311yescharacteristic
72#103—61no—
73#102—61no—
74C4×C8—31no—
75C22×D3—21no—
76C22×D3—21no—
77C3×Q8—41no—
78C3×Q8—41no—
79C2×C3×C4—22no—
80C2×C3×C4C2×D411yescharacteristic
81C2×C3×C4C2×D411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C2×Dic3—21no—
84C2×Dic3—21no—
85C2×Dic3—41no—
86D12—21no—
87D12—21no—
88D12—21no—
89D12—44no—
90D12SD1612yescomplemented
91C4×D3—44no—
92C4×D3—41no—
93C4×D3—41no—
94C4×D3—41no—
95C4×D3—41no—
96Dic6—41no—
97Dic6—41no—
98C3×C8—41no—
99C3×C8—41no—
100C3⋊C8—22no—
101#41—122no—
102#41—122no—
103C2×Q8—21no—
104C2×Q8—61no—
105C2×D4—31no—
106C2×D4—31no—
107C2×D4—62no—
108C22×C4—62no—
109C22×C4—61no—
110C22×C4—61no—
111SD16—122no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—61no—
115C4⋊C4—21no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—62no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4D1212yes—
122C22⋊C4—62no—
123C22⋊C4—61no—
124C22⋊C4—61no—
125C42C22×D311yescharacteristic
126C22×C3#12011yescharacteristic
127C2×D3—22no—
128C2×D3—41no—
129C2×D3—41no—
130C2×D3—41no—
131C2×D3—41no—
132C2×D3—41no—
133C3×C4—21no—
134C3×C4—42no—
135C3×C4C2×SD1611yescharacteristic
136C3×C4#13511yescharacteristic
137C3×C4AGL(1,ℤ/8)11yescharacteristic
138C3×C4AGL(1,ℤ/8)11yescharacteristic
139Dic3—41no—
140Dic3—41no—
141Dic3—81no—
142C23—61no—
143C23—61no—
144Q8—41no—
145Q8—41no—
146Q8—121no—
147Q8—121no—
148D4—32no—
149D4—61no—
150D4—61no—
151D4—61no—
152D4—124no—
153C2×C4—22no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—124no—
157C2×C4—121no—
158C2×C4—121no—
159C2×C4—121no—
160C2×C4—121no—
161C2×C4—121no—
162C2×C4C2×D1211yescharacteristic
163C2×C4D3×D411yescharacteristic
164C2×C4D3×D411yescharacteristic
165C8—41no—
166C8—41no—
167C8—62no—
168C2×C3#44911yescharacteristic
169C2×C3#45011yescharacteristic
170C2×C3#45611yescharacteristic
171D3—42no—
172D3—81no—
173C22—62no—
174C22—121no—
175C22—121no—
176C22—121no—
177C22—121no—
178C22—121no—
179C22#88211yescharacteristic
180C4—21no—
181C4—42no—
182C4—121no—
183C4—121no—
184C4—241no—
185C4#91111yescharacteristic
186C4D3×SD1611yescharacteristic
187C4#91411yescharacteristic
188C4#91911yescharacteristic
189C3#165411yescharacteristic, complemented, Sylow
190C2—122no—
191C2—241no—
192C2#475211yescharacteristic
193C2#477611yescharacteristic
194C2#481011yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#332C2
socleC22×C3#120
prime radical#5726C2
hypercenter#332D3
hypocenterC3#1654
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #5080C2semidirect
3. #4846C2semidirect
4. #4841C2semidirect
5. C3×#332C2semidirect
6. #4483C2semidirect
7. #4483C2semidirect
30. #332D3semidirect
90. D12SD16semidirect
189. C3#1654semidirect

Maximal subgroups

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

subgroupindex
#57262
#50802
#48462
#48412
C3×#3322
#44832
#16543