Finite Group Databasebeta

← #68393

Subgroups — #68393

Order 384.

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

normal subgroups only

This group has 250 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#5718C211yesmaximal, characteristic
2#5070C211yesmaximal, characteristic, complemented
3#4851C211yesmaximal, characteristic, complemented
4#4841C211yesmaximal, characteristic, complemented
5C3×#331C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1642—31nomaximal, Sylow
9C2×#237—21no—
10C3×#127C2211yescharacteristic
11#940—21no—
12#939—22no—
13#938—21no—
14D4×Dic3—21no—
15C2×#204—21no—
16C4×D12C2211yescharacteristic
17#869C2211yescharacteristic
18C4×Dic6C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21#832—22no—
22#817—21no—
23#816—21no—
24#816—21no—
25C4×C3⋊C8C2211yescharacteristic
26#550—31no—
27#486—31no—
28#475—31no—
29#461—31no—
30#331D311yescharacteristic, complemented
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4D412yes—
36C2×#67—22no—
37C2×#67—22no—
38C22×Dic3—22no—
39#237—42no—
40#237—42no—
41C2×D12D411yescharacteristic
42C2×C4×D3—21no—
43C2×Dic6—21no—
44C2×Dic6D411yescharacteristic
45C2×C3×C8—21no—
46C2×C3×C8—21no—
47C3×C42C2311yescharacteristic
48#217—22no—
49(C2×D3)⋊C4—21no—
50C4⋊Dic3—21no—
51C4⋊Dic3—21no—
52C4⋊Dic3D411yescharacteristic
53C4⋊Dic3D411yescharacteristic
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C2×C3⋊C8—21no—
57#204—42no—
58C2×#41—61no—
59C2×SD16—61no—
60C4⋊Q8—31no—
61#127—61no—
62#127C2×D311yescharacteristic
63#121—61no—
64#121—62no—
65C4×Q8—31no—
66C4×D4—31no—
67C4×D4—61no—
68#106—61no—
69C4⋊C8C2×D311yescharacteristic
70C4⋊C8C2×D311yescharacteristic
71#103—61no—
72#103—61no—
73#102—62no—
74C4×C8—31no—
75C23×C3—22no—
76C22×D3—21no—
77C3×D4—42no—
78C3×D4—41no—
79C3×D4—41no—
80C3×D4—41no—
81C3×D4—41no—
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85#67—44no—
86#67—82no—
87C2×Dic3—21no—
88C2×Dic3—21no—
89C2×Dic3—21no—
90C2×Dic3—44no—
91C2×Dic3—41no—
92D12—21no—
93D12—21no—
94C4×D3—41no—
95C4×D3—41no—
96Dic6—21no—
97Dic6—41no—
98Dic6—41no—
99Dic6SD1612yescomplemented
100C3×C8—41no—
101C3×C8—41no—
102C3⋊C8—41no—
103#41—122no—
104#41—122no—
105C2×Q8—31no—
106C2×Q8—61no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—31no—
110C2×D4—62no—
111C2×D4—62no—
112C2×D4D1212yes—
113C22×C4—62no—
114C22×C4—61no—
115SD16—122no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—61no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C22⋊C4—62no—
125C22⋊C4—61no—
126C42—61no—
127C42C22×D311yescharacteristic
128C22×C3—42no—
129C22×C3—42no—
130C22×C3—42no—
131C22×C3#12011yescharacteristic
132C2×D3—41no—
133C2×D3—41no—
134C2×D3—41no—
135C3×C4—21no—
136C3×C4C2×SD1611yescharacteristic
137C3×C4#13511yescharacteristic
138C3×C4AGL(1,ℤ/8)11yescharacteristic
139C3×C4AGL(1,ℤ/8)11yescharacteristic
140Dic3—22no—
141Dic3—41no—
142Dic3—41no—
143Dic3—81no—
144C23—22no—
145C23—61no—
146Q8—32no—
147Q8—61no—
148Q8—121no—
149Q8—121no—
150D4—42no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—61no—
156D4—61no—
157D4—124no—
158D4—242no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—124no—
163C2×C4—121no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4C2×D1211yescharacteristic
167C2×C4D3×D411yescharacteristic
168C2×C4D3×D411yescharacteristic
169C8—41no—
170C8—41no—
171C8—121no—
172C2×C3—82no—
173C2×C3#44911yescharacteristic
174C2×C3#45011yescharacteristic
175C2×C3#45511yescharacteristic
176D3—81no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—121no—
181C22—121no—
182C22—121no—
183C22#88211yescharacteristic
184C4—21no—
185C4—62no—
186C4—121no—
187C4—121no—
188C4—241no—
189C4#91111yescharacteristic
190C4#91111yescharacteristic
191C4D3×SD1611yescharacteristic
192C4#91611yescharacteristic
193C3#164211yescharacteristic, complemented, Sylow
194C2—82no—
195C2—241no—
196C2#474611yescharacteristic
197C2#477611yescharacteristic
198C2#478111yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#331C2
socleC22×C3#120
prime radical#5718C2
hypercenter#331D3
hypocenterC3#1642
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #5070C2semidirect
3. #4851C2semidirect
4. #4841C2semidirect
5. C3×#331C2semidirect
6. #4484C2semidirect
7. #4483C2semidirect
30. #331D3semidirect
99. Dic6SD16semidirect
193. C3#1642semidirect

Maximal subgroups

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

subgroupindex
#57182
#50702
#48512
#48412
C3×#3312
#44842
#44832
#16423