Finite Group Databasebeta

← #68136

Subgroups — #68136

Order 384.

208 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 49 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#5515C211yesmaximal, characteristic
2#4986C211yesmaximal, characteristic, complemented
3#4735C211yesmaximal, characteristic
4#4722C211yesmaximal, characteristic, complemented
5C3×#324C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic
7#4467C211yesmaximal, characteristic, complemented
8#1615—31nomaximal, Sylow
9C2×C3×C4⋊C4C2211yescharacteristic
10#930C2211yescharacteristic
11#929—21no—
12C4×#67—21no—
13C2×C4⋊Dic3C2211yescharacteristic
14#924C2211yescharacteristic
15#898—21no—
16#896—21no—
17#894—21no—
18#889—21no—
19C3×C22⋊C8C2211yescharacteristic
20C3×C22⋊C8C2211yescharacteristic
21C3⋊#95—21no—
22C3⋊#95—21no—
23C3⋊#95C2211yescharacteristic
24#821—21no—
25#818—21no—
26#817—21no—
27#816—21no—
28#539—31no—
29#484—31no—
30#481—31no—
31#400—31no—
32#328—31no—
33#327—31no—
34#324D311yescharacteristic, complemented
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4C2311yescharacteristic
38C2×#67—21no—
39C22×Dic3—21no—
40C22×Dic3—21no—
41C2×D12D411yescharacteristic
42C2×C4×D3—21no—
43C2×Dic6D411yescharacteristic
44C2×C3×C8—21no—
45C2×C3×C8—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4—21no—
48#217—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55C4⋊Dic3D411yescharacteristic
56C4⋊Dic3D411yescharacteristic
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61C4×Dic3—21no—
62#126—61no—
63#125—61no—
64#123—61no—
65#123—61no—
66C22⋊Q8—31no—
67C22⋊Q8—61no—
68#121—31no—
69C4×D4—61no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4C2×D311yescharacteristic
72#107—61no—
73#106—61no—
74#103—61no—
75#102—61no—
76C22⋊C8C2×D311yescharacteristic
77C22⋊C8C2×D311yescharacteristic
78#95—31no—
79#95—61no—
80#95—61no—
81C23×C3C2×D411yescharacteristic
82C22×D3—21no—
83C2×C3×C4—21no—
84C2×C3×C4—21no—
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91#67—41no—
92#67—41no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3—21no—
96C2×Dic3—41no—
97C2×Dic3—41no—
98C2×Dic3—42no—
99C2×Dic3—41no—
100C2×Dic3—41no—
101D12—41no—
102C4×D3—81no—
103Dic6—41no—
104C3×C8—41no—
105C3×C8—41no—
106C2×Q8—31no—
107C2×D4—31no—
108C2×D4—61no—
109C22×C4—61no—
110C22×C4—61no—
111C22×C4—61no—
112C22×C4D1211yescharacteristic
113C22×C4D1211yescharacteristic
114C22×C4C22×D311yescharacteristic
115C2×C8—21no—
116C2×C8—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C42—61no—
133C22×C3—21no—
134C22×C3—21no—
135C22×C3#12011yescharacteristic
136C22×C3#13511yescharacteristic
137C22×C3#13711yescharacteristic
138C2×D3—41no—
139C2×D3—41no—
140C2×D3—41no—
141C3×C4—21no—
142C3×C4—21no—
143C3×C4—41no—
144C3×C4—41no—
145Dic3—41no—
146Dic3—41no—
147Dic3—41no—
148Dic3—81no—
149Dic3—81no—
150C23—61no—
151C23C2×D1211yescharacteristic
152Q8—121no—
153D4—121no—
154D4—121no—
155D4—121no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—121no—
166C2×C4—121no—
167C2×C4—122no—
168C2×C4—121no—
169C2×C4—121no—
170C2×C4—241no—
171C2×C4D3×D411yescharacteristic
172C2×C4D3×D411yescharacteristic
173C8—41no—
174C8—41no—
175C2×C3—21no—
176C2×C3—21no—
177C2×C3#45211yescharacteristic
178C2×C3#45411yescharacteristic
179C2×C3#45811yescharacteristic
180D3—81no—
181C22—21no—
182C22—21no—
183C22—121no—
184C22—121no—
185C22—121no—
186C22#88211yescharacteristic
187C22#90411yescharacteristic
188C22#90911yescharacteristic
189C4—21no—
190C4—21no—
191C4—41no—
192C4—41no—
193C4—121no—
194C4—121no—
195C4—121no—
196C4—241no—
197C4—241no—
198C3#161511yescharacteristic, complemented, Sylow
199C2—21no—
200C2—21no—
201C2—241no—
202C2#473311yescharacteristic
203C2#474211yescharacteristic
204C2#475311yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radical#930C22
hypercenter#324D3
hypocenterC3#1615
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #4986C2semidirect
4. #4722C2semidirect
5. C3×#324C2semidirect
7. #4467C2semidirect
34. #324D3semidirect
198. C3#1615semidirect

Maximal subgroups

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

subgroupindex
#55152
#49862
#47352
#47222
C3×#3242
#44702
#44672
#16153