Finite Group Databasebeta

← #71126

Subgroups — #71126

Order 384.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#5584C211yesmaximal, characteristic
2C3×#475C211yesmaximal, characteristic, complemented
3#5083C211yesmaximal, characteristic
4#5078C211yesmaximal, characteristic, complemented
5#4535C211yesmaximal, characteristic, complemented
6#4486C211yesmaximal, characteristic, complemented
7#4483C211yesmaximal, characteristic
8#1650—31nomaximal, Sylow
9C3×C4⋊Q8C2211yescharacteristic
10C3×C4×Q8C2211yescharacteristic
11#946—21no—
12C2×#215—21no—
13#898—22no—
14#892—21no—
15#875—21no—
16C4×D12C2211yescharacteristic
17#869C2211yescharacteristic
18C4×Dic6—21no—
19C3×#106—21no—
20C3×C4⋊C8C2211yescharacteristic
21C3×#103—21no—
22#835—21no—
23#807—22no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#556—31no—
27#500—31no—
28#475D311yescharacteristic, complemented
29#461—31no—
30#332—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×Q8D411yescharacteristic
35C2×D12D411yescharacteristic
36C2×C4×D3—21no—
37C2×Dic6—21no—
38C2×C3×C8—21no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4D411yescharacteristic
43C3×C42—21no—
44C3×C42C2311yescharacteristic
45#215—42no—
46(C2×D3)⋊C4—22no—
47(C2×D3)⋊C4—22no—
48(C2×D3)⋊C4—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3D412yes—
51C4⋊Dic3D411yescharacteristic
52Dic3⋊C4—22no—
53C4×Dic3—22no—
54C2×C3⋊C8—21no—
55C2×C3⋊C8—21no—
56C2×SD16—61no—
57C4⋊Q8—31no—
58C4⋊Q8C2×D311yescharacteristic
59#126—62no—
60#124—61no—
61C22⋊Q8—61no—
62C4×Q8—61no—
63C4×Q8C2×D311yescharacteristic
64C4×D4—31no—
65#117—61no—
66#107—62no—
67#106—21no—
68C4⋊C8—31no—
69C4⋊C8C2×D311yescharacteristic
70#103—21no—
71#103—61no—
72C4×C8—31no—
73C22×D3—21no—
74C3×Q8—21no—
75C3×Q8—21no—
76C3×Q8—41no—
77C3×Q8—41no—
78C2×C3×C4—21no—
79C2×C3×C4—21no—
80C2×C3×C4—41no—
81C2×C3×C4C2×D411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×Dic3—21no—
85C2×Dic3—22no—
86D12—21no—
87D12—21no—
88C4×D3—41no—
89C4×D3—41no—
90Dic6—41no—
91Dic6—41no—
92C3×C8—41no—
93C3⋊C8—22no—
94C3⋊C8—41no—
95C2×Q8—21no—
96C2×Q8—61no—
97C2×Q8#6711yescharacteristic
98C2×D4—31no—
99C22×C4—61no—
100SD16—122no—
101C2×C8—21no—
102C2×C8—61no—
103C2×C8—61no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—31no—
108C4⋊C4—32no—
109C4⋊C4—62no—
110C4⋊C4—61no—
111C4⋊C4#6711yescharacteristic
112C22⋊C4—62no—
113C22⋊C4—62no—
114C22⋊C4—61no—
115C42—21no—
116C42—62no—
117C42C22×D311yescharacteristic
118C22×C3#12011yescharacteristic
119C2×D3—41no—
120C2×D3—41no—
121C2×D3—41no—
122C3×C4—21no—
123C3×C4—41no—
124C3×C4—41no—
125C3×C4—81no—
126C3×C4#13511yescharacteristic
127C3×C4#13511yescharacteristic
128C3×C4AGL(1,ℤ/8)11yescharacteristic
129C3×C4#13711yescharacteristic
130Dic3—42no—
131Dic3—41no—
132C23—61no—
133Q8—21no—
134Q8—21no—
135Q8—41no—
136Q8—41no—
137Q8—121no—
138Q8—121no—
139D4—61no—
140D4—61no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—61no—
145C2×C4—62no—
146C2×C4—121no—
147C2×C4—121no—
148C2×C4D3×D411yescharacteristic
149C2×C4D3×D411yescharacteristic
150C2×C4C2×#6711yescharacteristic
151C8—41no—
152C8—62no—
153C8—121no—
154C2×C3#44911yescharacteristic
155C2×C3#45211yescharacteristic
156C2×C3#45611yescharacteristic
157D3—81no—
158C22—121no—
159C22—121no—
160C22—121no—
161C22#93711yescharacteristic
162C4—21no—
163C4—41no—
164C4—41no—
165C4—81no—
166C4—122no—
167C4—121no—
168C4#91411yescharacteristic
169C4#91611yescharacteristic
170C4#94211yescharacteristic
171C4#95011yescharacteristic
172C3#165011yescharacteristic, complemented, Sylow
173C2—241no—
174C2#504711yescharacteristic
175C2#517211yescharacteristic
176C2#520511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#475C2
socleC22×C3#120
prime radicalC2×D12D4
hypercenter#475D3
hypocenterC3#1650
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. C3×#475C2semidirect
4. #5078C2semidirect
5. #4535C2semidirect
6. #4486C2semidirect
28. #475D3semidirect
172. C3#1650semidirect

Maximal subgroups

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

subgroupindex
#55842
C3×#4752
#50832
#50782
#45352
#44862
#44832
#16503