Finite Group Databasebeta

← #68134

Subgroups — #68134

Order 384.

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

normal subgroups only

This group has 216 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#4981C211yesmaximal, characteristic, complemented
3#4736C211yesmaximal, characteristic
4#4721C211yesmaximal, characteristic, complemented
5C3×#324C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic
7#4467C211yesmaximal, characteristic, complemented
8#1609—31nomaximal, Sylow
9C2×C3×C4⋊C4C2211yescharacteristic
10#930C2211yescharacteristic
11#929—21no—
12C4×#67—21no—
13C2×C4⋊Dic3C2211yescharacteristic
14#924C2211yescharacteristic
15C2×Dic3⋊C4—21no—
16#898—21no—
17#896—21no—
18#894—21no—
19#889—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×C22⋊C8C2211yescharacteristic
22C3⋊#95—21no—
23C3⋊#95C2211yescharacteristic
24#821—21no—
25#818—21no—
26#817—21no—
27#816—21no—
28#539—31no—
29#483—31no—
30#480—31no—
31#398—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—
61Dic3⋊C4—42no—
62C4×Dic3—21no—
63#126—61no—
64#125—61no—
65#123—61no—
66#123—61no—
67C22⋊Q8—31no—
68C22⋊Q8—61no—
69#121—31no—
70C4×D4—61no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—61no—
73C2×C4⋊C4C2×D311yescharacteristic
74#107—61no—
75#106—61no—
76#103—61no—
77#102—61no—
78C22⋊C8C2×D311yescharacteristic
79C22⋊C8C2×D311yescharacteristic
80#95—31no—
81#95—61no—
82C23×C3C2×D411yescharacteristic
83C22×D3—21no—
84C2×C3×C4—21no—
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—21no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4C2×D411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92#67—41no—
93#67—41no—
94C2×Dic3—22no—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3—41no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103D12—41no—
104C4×D3—81no—
105Dic6—41no—
106C3×C8—41no—
107C3×C8—41no—
108C2×Q8—31no—
109C2×D4—31no—
110C2×D4—61no—
111C22×C4—61no—
112C22×C4—61no—
113C22×C4—61no—
114C22×C4D1211yescharacteristic
115C22×C4D1211yescharacteristic
116C22×C4C22×D311yescharacteristic
117C2×C8—21no—
118C2×C8—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—31no—
122C4⋊C4—31no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C4⋊C4—61no—
129C4⋊C4—122no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C22⋊C4—61no—
135C42—61no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3#12011yescharacteristic
139C22×C3#13511yescharacteristic
140C22×C3AGL(1,ℤ/8)11yescharacteristic
141C2×D3—41no—
142C2×D3—41no—
143C2×D3—41no—
144C3×C4—21no—
145C3×C4—21no—
146C3×C4—41no—
147C3×C4—41no—
148Dic3—42no—
149Dic3—41no—
150Dic3—41no—
151Dic3—41no—
152Dic3—81no—
153C23—61no—
154C23C2×D1211yescharacteristic
155Q8—121no—
156D4—121no—
157D4—121no—
158D4—121no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—62no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—121no—
170C2×C4—121no—
171C2×C4—121no—
172C2×C4—121no—
173C2×C4—121no—
174C2×C4—241no—
175C2×C4D3×D411yescharacteristic
176C2×C4D3×D411yescharacteristic
177C8—41no—
178C8—41no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3#44911yescharacteristic
182C2×C3#45611yescharacteristic
183C2×C3#45811yescharacteristic
184D3—81no—
185C22—21no—
186C22—21no—
187C22—121no—
188C22—121no—
189C22—121no—
190C22#88211yescharacteristic
191C22#90411yescharacteristic
192C22#90811yescharacteristic
193C4—21no—
194C4—21no—
195C4—41no—
196C4—41no—
197C4—122no—
198C4—121no—
199C4—121no—
200C4—121no—
201C4—241no—
202C3#160911yescharacteristic, complemented, Sylow
203C2—21no—
204C2—21no—
205C2—241no—
206C2#473411yescharacteristic
207C2#474211yescharacteristic
208C2#475211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3#120
prime radical#930C22
hypercenter#324D3
hypocenterC3#1609
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. #4981C2semidirect
4. #4721C2semidirect
5. C3×#324C2semidirect
7. #4467C2semidirect
34. #324D3semidirect
202. C3#1609semidirect

Maximal subgroups

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

subgroupindex
#55152
#49812
#47362
#47212
C3×#3242
#44702
#44672
#16093