Finite Group Databasebeta

← #68434

Subgroups — #68434

Order 384.

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

normal subgroups only

This group has 234 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#5093C211yesmaximal, characteristic, complemented
3#4842C211yesmaximal, characteristic, complemented
4#4842C211yesmaximal, characteristic, complemented
5C3×#332C211yesmaximal, characteristic, complemented
6#4483C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic, complemented
8#1635—31nomaximal, Sylow
9C2×#239—21no—
10C3×C4⋊Q8C2211yescharacteristic
11#946—21no—
12#895—21no—
13#894—22no—
14D3×C4⋊C4—21no—
15C4×D12C2211yescharacteristic
16C4×D12C2211yescharacteristic
17#869C2211yescharacteristic
18C3×C4⋊C8C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20#821—21no—
21#821—21no—
22#817—21no—
23#817—21no—
24#810—22no—
25C4×C3⋊C8C2211yescharacteristic
26#547—31no—
27#487—31no—
28#476—31no—
29#476—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—
57C2×#41—61no—
58C4⋊Q8—31no—
59C4⋊Q8C2×D311yescharacteristic
60#123—62no—
61C22⋊Q8—61no—
62#121—61no—
63C4×D4—31no—
64C4×D4—31no—
65C2×C4⋊C4—61no—
66#106—61no—
67#106—61no—
68C4⋊C8C2×D311yescharacteristic
69C4⋊C8C2×D311yescharacteristic
70#103—62no—
71#102—61no—
72#102—61no—
73C4×C8—31no—
74C22×D3—21no—
75C22×D3—21no—
76C3×Q8—41no—
77C3×Q8—41no—
78C2×C3×C4—22no—
79C2×C3×C4C2×D411yescharacteristic
80C2×C3×C4C2×D411yescharacteristic
81C2×C3×C4C2×D411yescharacteristic
82C2×Dic3—21no—
83C2×Dic3—21no—
84C2×Dic3—41no—
85D12—21no—
86D12—21no—
87D12—21no—
88D12—44no—
89D12SD1612yes—
90C4×D3—44no—
91C4×D3—41no—
92C4×D3—41no—
93C4×D3—41no—
94C4×D3—41no—
95Dic6—42no—
96C3×C8—41no—
97C3×C8—41no—
98C3⋊C8—41no—
99#41—122no—
100#41—122no—
101C2×Q8—21no—
102C2×Q8—61no—
103C2×D4—31no—
104C2×D4—31no—
105C2×D4—62no—
106C22×C4—62no—
107C22×C4—61no—
108C22×C4—61no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—61no—
112C4⋊C4—21no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—62no—
116C4⋊C4—61no—
117C4⋊C4—61no—
118C4⋊C4D1212yes—
119C22⋊C4—62no—
120C22⋊C4—61no—
121C22⋊C4—61no—
122C42C22×D311yescharacteristic
123C22×C3#12011yescharacteristic
124C2×D3—22no—
125C2×D3—41no—
126C2×D3—41no—
127C2×D3—41no—
128C2×D3—41no—
129C2×D3—41no—
130C3×C4—21no—
131C3×C4—42no—
132C3×C4C2×SD1611yescharacteristic
133C3×C4#13511yescharacteristic
134C3×C4AGL(1,ℤ/8)11yescharacteristic
135C3×C4#13711yescharacteristic
136Dic3—41no—
137Dic3—41no—
138Dic3—81no—
139C23—61no—
140C23—61no—
141Q8—41no—
142Q8—41no—
143Q8—122no—
144D4—32no—
145D4—61no—
146D4—61no—
147D4—61no—
148D4—124no—
149C2×C4—22no—
150C2×C4—61no—
151C2×C4—61no—
152C2×C4—124no—
153C2×C4—121no—
154C2×C4—121no—
155C2×C4—121no—
156C2×C4—121no—
157C2×C4—121no—
158C2×C4C2×D1211yescharacteristic
159C2×C4D3×D411yescharacteristic
160C2×C4D3×D411yescharacteristic
161C8—41no—
162C8—41no—
163C8—121no—
164C2×C3#45011yescharacteristic
165C2×C3#45211yescharacteristic
166C2×C3#45411yescharacteristic
167D3—42no—
168D3—81no—
169C22—62no—
170C22—121no—
171C22—121no—
172C22—121no—
173C22—121no—
174C22—121no—
175C22#88211yescharacteristic
176C4—21no—
177C4—42no—
178C4—121no—
179C4—121no—
180C4—241no—
181C4#91111yescharacteristic
182C4D3×SD1611yescharacteristic
183C4#91611yescharacteristic
184C4#91811yescharacteristic
185C3#163511yescharacteristic, complemented, Sylow
186C2—122no—
187C2—241no—
188C2#475311yescharacteristic
189C2#477611yescharacteristic
190C2#480811yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#332C2
socleC22×C3#120
prime radical#5726C2
hypercenter#332D3
hypocenterC3#1635
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. #5093C2semidirect
3. #4842C2semidirect
4. #4842C2semidirect
5. C3×#332C2semidirect
6. #4483C2semidirect
7. #4483C2semidirect
30. #332D3semidirect
185. C3#1635semidirect

Maximal subgroups

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

subgroupindex
#57262
#50932
#48422
C3×#3322
#44832
#16353