Finite Group Databasebeta

← #68378

Subgroups — #68378

Order 384.

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

normal subgroups only

This group has 184 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
2#5064C211yesmaximal, characteristic, complemented
3#4714C211yesmaximal, characteristic
4#4703C211yesmaximal, characteristic, complemented
5C3×#330C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic
8#1664—31nomaximal, Sylow
9C3×#125C2211yescharacteristic
10#898—21no—
11#898—21no—
12#897—21no—
13#896—21no—
14#894—21no—
15#888—21no—
16#875C2211yescharacteristic
17#870C2211yescharacteristic
18#869C2211yescharacteristic
19C3×#97C2211yescharacteristic
20C3×#97C2211yescharacteristic
21#821—21no—
22#816—21no—
23#816—21no—
24#816—21no—
25#808—21no—
26#807—21no—
27#803C2211yescharacteristic
28#563—31no—
29#501—31no—
30#491—31no—
31#489—31no—
32#330—31no—
33#330D311yescharacteristic, complemented
34#329—31no—
35C2×D12D411yescharacteristic
36C2×C4×D3—21no—
37C2×Dic6—21no—
38C2×Dic6—21no—
39C2×Dic6D411yescharacteristic
40C2×C3×C8—21no—
41C2×C3×C8—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4D411yescharacteristic
45C3×C4⋊C4D411yescharacteristic
46C3×C42C2311yescharacteristic
47(C2×D3)⋊C4—21no—
48(C2×D3)⋊C4—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55C4⋊Dic3D411yescharacteristic
56C4⋊Dic3D411yescharacteristic
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60C4×Dic3—21no—
61C2×C3⋊C8—21no—
62C4⋊Q8—31no—
63C4⋊Q8—61no—
64#126—61no—
65#126—61no—
66#125—31no—
67#125C2×D311yescharacteristic
68#124—31no—
69#123—61no—
70C22⋊Q8—61no—
71C22⋊Q8—61no—
72#107—61no—
73#106—61no—
74#103—61no—
75#103—61no—
76#103—61no—
77#102—61no—
78#97—31no—
79#97C2×D311yescharacteristic
80#97C2×D311yescharacteristic
81C22×D3—21no—
82C2×C3×C4—21no—
83C2×C3×C4—21no—
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×Dic3—21no—
88C2×Dic3—21no—
89C2×Dic3—21no—
90C2×Dic3—41no—
91D12—41no—
92C4×D3—81no—
93Dic6—41no—
94Dic6—42no—
95Dic6—81no—
96C3×C8—41no—
97C3×C8—41no—
98C3⋊C8—22no—
99C2×Q8—31no—
100C2×Q8—61no—
101C2×Q8—61no—
102C2×D4—31no—
103C22×C4—61no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—61no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—31no—
110C4⋊C4—31no—
111C4⋊C4—61no—
112C4⋊C4—61no—
113C4⋊C4—61no—
114C4⋊C4—61no—
115C4⋊C4—61no—
116C4⋊C4—61no—
117C4⋊C4D1211yescharacteristic
118C4⋊C4D1211yescharacteristic
119C22⋊C4—61no—
120C22⋊C4—61no—
121C22⋊C4—61no—
122C22⋊C4—61no—
123C22⋊C4—61no—
124C42—61no—
125C42C22×D311yescharacteristic
126C22×C3#12011yescharacteristic
127C2×D3—41no—
128C2×D3—41no—
129C2×D3—41no—
130C3×C4—21no—
131C3×C4—21no—
132C3×C4—21no—
133C3×C4—41no—
134C3×C4—41no—
135Dic3—41no—
136Dic3—41no—
137Dic3—41no—
138Dic3—81no—
139C23—61no—
140Q8—121no—
141Q8—122no—
142Q8—241no—
143D4—121no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—61no—
147C2×C4—61no—
148C2×C4—61no—
149C2×C4—121no—
150C2×C4—241no—
151C2×C4C2×D1211yescharacteristic
152C2×C4D3×D411yescharacteristic
153C2×C4D3×D411yescharacteristic
154C8—41no—
155C8—41no—
156C8—62no—
157C2×C3#45411yescharacteristic
158C2×C3#45511yescharacteristic
159C2×C3#45611yescharacteristic
160D3—81no—
161C22—121no—
162C22—121no—
163C22—121no—
164C22#88211yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—121no—
171C4—121no—
172C4—121no—
173C4—241no—
174C3#166411yescharacteristic, complemented, Sylow
175C2—241no—
176C2#475211yescharacteristic
177C2#482511yescharacteristic
178C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3#120
prime radicalC2×D12D4
hypercenter#330D3
hypocenterC3#1664
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #5064C2semidirect
4. #4703C2semidirect
5. C3×#330C2semidirect
6. #4465C2semidirect
33. #330D3semidirect
174. C3#1664semidirect

Maximal subgroups

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

subgroupindex
#56992
#50642
#47142
#47032
C3×#3302
#44652
#44642
#16643