Finite Group Databasebeta

← #68371

Subgroups — #68371

Order 384.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#5699C211yesmaximal, characteristic, complemented
2#5067C211yesmaximal, characteristic, complemented
3#4713C211yesmaximal, characteristic, complemented
4#4712C211yesmaximal, characteristic
5C3×#330C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic, complemented
8#1657—31nomaximal, Sylow
9C3×#125C2211yescharacteristic
10C2×D24—21no—
11C2×#204—21no—
12#898—21no—
13#898—21no—
14#897—21no—
15#896—21no—
16#894—21no—
17#888—21no—
18#875C2211yescharacteristic
19#874C2211yescharacteristic
20#870C2211yescharacteristic
21C3×#97C2211yescharacteristic
22C3×#97C2211yescharacteristic
23#821—21no—
24#821—21no—
25#809—21no—
26#809—21no—
27#803C2211yescharacteristic
28#563—31no—
29#496—31no—
30#490—31no—
31#490—31no—
32#330—31no—
33#330D311yescharacteristic, complemented
34#329—31no—
35C2×D12—21no—
36C2×D12—21no—
37C2×D12—21no—
38C2×D12D411yescharacteristic
39C2×C4×D3—21no—
40C2×Dic6—21no—
41C2×Dic6D411yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4D411yescharacteristic
47C3×C4⋊C4D411yescharacteristic
48C3×C42C2311yescharacteristic
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53(C2×D3)⋊C4—21no—
54C4⋊Dic3—21no—
55C4⋊Dic3D411yescharacteristic
56C4⋊Dic3D411yescharacteristic
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60C4×Dic3—21no—
61C2×C3⋊C8—21no—
62D24—42no—
63#204—42no—
64C2×SD16—61no—
65C2×D8—61no—
66C4⋊Q8—61no—
67#127—31no—
68#126—61no—
69#126—61no—
70#125—31no—
71#125C2×D311yescharacteristic
72#124—31no—
73#123—61no—
74C22⋊Q8—61no—
75C22⋊Q8—61no—
76#102—61no—
77#102—61no—
78#102—61no—
79#102—61no—
80#97—31no—
81#97C2×D311yescharacteristic
82#97C2×D311yescharacteristic
83C22×D3—21no—
84C22×D3—41no—
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×Dic3—21no—
91C2×Dic3—21no—
92C2×Dic3—21no—
93D12—41no—
94D12—41no—
95D12—41no—
96D12—42no—
97D12—42no—
98C4×D3—81no—
99Dic6—41no—
100Dic6—81no—
101C3×C8—22no—
102C3×C8—41no—
103C3⋊C8—41no—
104C2×Q8—31no—
105C2×Q8—61no—
106C2×D4—31no—
107C2×D4—61no—
108C2×D4—61no—
109C2×D4—61no—
110C22×C4—61no—
111SD16—122no—
112D8—122no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—61no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4D1211yescharacteristic
125C4⋊C4D1211yescharacteristic
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C42—61no—
132C42C22×D311yescharacteristic
133C22×C3#12011yescharacteristic
134C2×D3—41no—
135C2×D3—41no—
136C2×D3—41no—
137C2×D3—81no—
138C2×D3—81no—
139C2×D3—81no—
140C3×C4—21no—
141C3×C4—21no—
142C3×C4—21no—
143C3×C4—41no—
144C3×C4—41no—
145Dic3—41no—
146Dic3—41no—
147Dic3—41no—
148C23—61no—
149C23—121no—
150Q8—121no—
151Q8—241no—
152D4—121no—
153D4—121no—
154D4—121no—
155D4—122no—
156D4—122no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—241no—
163C2×C4C2×D1211yescharacteristic
164C2×C4D3×D411yescharacteristic
165C2×C4D3×D411yescharacteristic
166C8—22no—
167C8—41no—
168C8—121no—
169C2×C3#45311yescharacteristic
170C2×C3#45411yescharacteristic
171C2×C3#45411yescharacteristic
172D3—81no—
173D3—161no—
174C22—121no—
175C22—121no—
176C22—121no—
177C22—241no—
178C22—241no—
179C22—241no—
180C22#88211yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—121no—
187C4—121no—
188C4—121no—
189C3#165711yescharacteristic, complemented, Sylow
190C2—241no—
191C2—481no—
192C2#475311yescharacteristic
193C2#482211yescharacteristic
194C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3#120
prime radical#4712C2
hypercenter#330D3
hypocenterC3#1657
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #5699C2semidirect
2. #5067C2semidirect
3. #4713C2semidirect
5. C3×#330C2semidirect
6. #4465C2semidirect
7. #4464C2semidirect
33. #330D3semidirect
189. C3#1657semidirect

Maximal subgroups

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

subgroupindex
#56992
#50672
#47132
#47122
C3×#3302
#44652
#44642
#16573