Finite Group Databasebeta

← #71050

Subgroups — #71050

Order 384.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic6C211yesmaximal, characteristic
2C3×#461C211yesmaximal, characteristic
3#5094C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#4486C211yesmaximal, characteristic, complemented
7#4484C211yesmaximal, characteristic, complemented
8#1651—31nomaximal, Sylow
9C22×Dic6—21no—
10C2×C3×SD16—21no—
11C3×C4⋊Q8C2211yescharacteristic
12C3×C4×D4C2211yescharacteristic
13D4×Dic3—21no—
14#924—21no—
15#888—21no—
16#878—22no—
17#869C2211yescharacteristic
18C4×Dic6C2211yescharacteristic
19C3×C4⋊C8C2211yescharacteristic
20C3×#103—21no—
21#810—21no—
22#808—22no—
23#808—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26D4×Q8—31no—
27#498—31no—
28#475—31no—
29#461D311yescharacteristic
30#332—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×D4D411yescharacteristic
35C22×C3×C4—21no—
36C22×Dic3—22no—
37C2×Dic6—22no—
38C2×Dic6—21no—
39C2×Dic6—42no—
40C2×Dic6—42no—
41C2×Dic6D411yescharacteristic
42C3×SD16—42no—
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4D411yescharacteristic
47C3×C22⋊C4—21no—
48C3×C42C2311yescharacteristic
49#217—22no—
50C4⋊Dic3—21no—
51C4⋊Dic3D411yescharacteristic
52C4⋊Dic3D412yes—
53Dic3⋊C4—22no—
54Dic3⋊C4—21no—
55C4×Dic3—21no—
56C2×C3⋊C8—21no—
57C2×C3⋊C8—21no—
58C22×Q8—61no—
59C2×SD16—21no—
60C4⋊Q8—31no—
61C4⋊Q8—61no—
62C4⋊Q8C2×D311yescharacteristic
63C22⋊Q8—62no—
64C22⋊Q8—61no—
65C4×Q8—31no—
66C4×D4—61no—
67C4×D4C2×D311yescharacteristic
68#106—61no—
69#106—62no—
70C4⋊C8—31no—
71C4⋊C8C2×D311yescharacteristic
72#103—21no—
73#103—61no—
74C4×C8—31no—
75C23×C3—21no—
76C3×Q8—41no—
77C3×Q8—41no—
78C3×D4—21no—
79C3×D4SD1612yes—
80C2×C3×C4—21no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×Dic3—21no—
88C2×Dic3—21no—
89C2×Dic3—22no—
90C2×Dic3—44no—
91Dic6—21no—
92Dic6—44no—
93Dic6—41no—
94Dic6—41no—
95Dic6SD1612yescomplemented
96C3×C8—41no—
97C3⋊C8—22no—
98C3⋊C8—41no—
99C2×Q8—21no—
100C2×Q8—31no—
101C2×Q8—62no—
102C2×Q8—61no—
103C2×Q8—122no—
104C2×Q8—122no—
105C2×D4#6711yescharacteristic
106C22×C4—21no—
107C22×C4—62no—
108SD16—42no—
109C2×C8—21no—
110C2×C8—61no—
111C2×C8—61no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—31no—
115C4⋊C4—32no—
116C4⋊C4—62no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4#6711yescharacteristic
120C22⋊C4—21no—
121C22⋊C4—62no—
122C42—61no—
123C42C22×D311yescharacteristic
124C22×C3—22no—
125C22×C3—41no—
126C22×C3—41no—
127C22×C3#12011yescharacteristic
128C3×C4—21no—
129C3×C4—41no—
130C3×C4—81no—
131C3×C4C2×SD1611yescharacteristic
132C3×C4C2×SD1611yescharacteristic
133C3×C4AGL(1,ℤ/8)11yescharacteristic
134C3×C4#13711yescharacteristic
135Dic3—22no—
136Dic3—42no—
137Dic3—41no—
138C23—21no—
139Q8—32no—
140Q8—41no—
141Q8—41no—
142Q8—61no—
143Q8—124no—
144Q8—121no—
145Q8—121no—
146D4—21no—
147D4#21412yes—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—61no—
153C2×C4—61no—
154C2×C4—62no—
155C2×C4—124no—
156C2×C4D3×D411yescharacteristic
157C2×C4D3×D411yescharacteristic
158C2×C4C2×#6711yescharacteristic
159C8—41no—
160C8—62no—
161C8—121no—
162C2×C3—42no—
163C2×C3#44811yescharacteristic
164C2×C3#45011yescharacteristic
165C2×C3#45611yescharacteristic
166C22—22no—
167C22—41no—
168C22—41no—
169C22#93711yescharacteristic
170C4—21no—
171C4—41no—
172C4—62no—
173C4—81no—
174C4—122no—
175C4—121no—
176C4D3×SD1611yescharacteristic
177C4#91411yescharacteristic
178C4C2×#21411yescharacteristic
179C4#95111yescharacteristic
180C3#165111yescharacteristic, complemented, Sylow
181C2—42no—
182C2#504011yescharacteristic
183C2#516911yescharacteristic
184C2#520511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#461C2
socleC22×C3#120
prime radicalC2×C3×D4D4
hypercenter#461D3
hypocenterC3#1651
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
3. #5094C2semidirect
4. #5080C2semidirect
6. #4486C2semidirect
7. #4484C2semidirect
95. Dic6SD16semidirect
180. C3#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic62
C3×#4612
#50942
#50802
D4⋊(C3⋊C8)2
#44862
#44842
#16513