Finite Group Databasebeta

← #69993

Subgroups — #69993

Order 384.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#5791C212yesmaximal, complemented
2#5748C211yesmaximal, characteristic, complemented
3C3×#407C211yesmaximal, characteristic, complemented
4#5139C212yesmaximal, complemented
5#4557C211yesmaximal, characteristic, complemented
6#2010—31nomaximal, Sylow
7C23×C3×C4C2211yescharacteristic
8C2×#235C2212yes—
9C2×C3×M16C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11#953—22no—
12#930—22no—
13#929—22no—
14C4×#67—24no—
15#926C2212yes—
16#924—22no—
17#909—22no—
18#908—22no—
19#904—22no—
20C3×C22⋊C8—22no—
21#825—24no—
22#575—31no—
23#525—32no—
24#421—32no—
25#407D311yescharacteristic, complemented
26H(3,ℤ/4)—31no—
27C24×C3D411yescharacteristic
28C22×C3×C4—21no—
29C22×C3×C4—22no—
30C22×C3×C4—22no—
31C22×C3×C4—21no—
32C22×C3×C4D411yescharacteristic
33C22×C3×C4C2311yescharacteristic
34C2×#67—24no—
35C2×#67—22no—
36#235—22no—
37#235—22no—
38#235—22no—
39#235—22no—
40C2×D12D412yescomplemented
41C2×C4×D3—22no—
42C2×Dic6D412yescomplemented
43C3×M16—22no—
44C3×M16—22no—
45C2×C3×C8—21no—
46C2×C3×C8D412yescomplemented
47#217—24no—
48#217—22no—
49(C2×D3)⋊C4—24no—
50C4⋊Dic3—22no—
51Dic3⋊C4—24no—
52C4×Dic3D414yescomplemented
53Dic12—22no—
54D24—22no—
55#204—24no—
56C2×#41—32no—
57C23×C4C2×D311yescharacteristic
58#137—62no—
59AGL(1,ℤ/8)—62no—
60#135—62no—
61C2×M16C2×D311yescharacteristic
62C2×M16C2×D311yescharacteristic
63#123—62no—
64C22⋊Q8—62no—
65#121—62no—
66#120—62no—
67C4×D4—64no—
68#117—32no—
69#104—64no—
70C22⋊C8—22no—
71C23×C3—22no—
72C23×C3—21no—
73C23×C3—42no—
74C23×C3C2×D411yescharacteristic
75C22×D3—22no—
76C2×C3×C4—22no—
77C2×C3×C4—21no—
78C2×C3×C4—41no—
79C2×C3×C4—42no—
80C2×C3×C4—41no—
81C2×C3×C4C2×D412yes—
82C2×C3×C4C2×D412yes—
83C2×C3×C4C2×D411yescharacteristic
84C2×C3×C4#4111yescharacteristic
85#67—48no—
86#67—42no—
87#67—42no—
88C2×Dic3—24no—
89C2×Dic3—22no—
90D12—22no—
91D12—22no—
92C4×D3—42no—
93C4×D3—42no—
94Dic6—22no—
95Dic6—22no—
96C3×C8—22no—
97C3×C8—41no—
98C24D1211yescharacteristic
99#41—62no—
100#41—62no—
101#41—62no—
102#41—62no—
103C2×Q8—32no—
104C2×D4—32no—
105C2×D4—64no—
106C2×D4—62no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—62no—
112C22×C4D1211yescharacteristic
113C22×C4C22×D311yescharacteristic
114Q16—62no—
115SD16—64no—
116D8—62no—
117M16—22no—
118M16—22no—
119C2×C8—21no—
120C2×C8#6712yescomplemented
121C4⋊C4—64no—
122C4⋊C4—62no—
123C22⋊C4—64no—
124C22⋊C4—64no—
125C22⋊C4—62no—
126C42—34no—
127C22×C3—24no—
128C22×C3—22no—
129C22×C3—41no—
130C22×C3—42no—
131C22×C3—41no—
132C22×C3—41no—
133C22×C3#12012yes—
134C22×C3#12111yescharacteristic
135C2×D3—44no—
136C2×D3—42no—
137C3×C4—41no—
138C3×C4#12011yescharacteristic
139C3×C4#12112yes—
140C3×C4#12711yescharacteristic
141Dic3—44no—
142Dic3—42no—
143C23—22no—
144C23—21no—
145C23—42no—
146C23—62no—
147C23C2×D1211yescharacteristic
148Q8—62no—
149Q8—62no—
150D4—62no—
151D4—62no—
152D4—128no—
153D4—122no—
154D4—122no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—64no—
161C2×C4—62no—
162C2×C4—122no—
163C2×C4—122no—
164C2×C4#23511yescharacteristic
165C2×C4D3×D412yes—
166C2×C4D3×D412yes—
167C2×C4C2×#6711yescharacteristic
168C8—22no—
169C8—41no—
170C2×C3—22no—
171C2×C3—21no—
172C2×C3—42no—
173C2×C3#39211yescharacteristic
174D3—82no—
175C22—24no—
176C22—22no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—124no—
182C22—122no—
183C22#88212yes—
184C22#93011yescharacteristic
185C4—41no—
186C4—124no—
187C4—122no—
188C4#88412yes—
189C4#93711yescharacteristic
190C4#94011yescharacteristic
191C3#201011yescharacteristic, complemented, Sylow
192C2—22no—
193C2—21no—
194C2—42no—
195C2—242no—
196C2#496011yescharacteristic
subgroupG/H
centerC4#940
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#407C2
socleC2×C3#392
hypercenter#407D3
hypocenterC3#2010
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #5791C2semidirect
2. #5748C2semidirect
3. C3×#407C2semidirect
4. #5139C2semidirect
5. #4557C2semidirect
25. #407D3semidirect
40. C2×D12D4semidirect
42. C2×Dic6D4semidirect
46. C2×C3×C8D4semidirect
52. C4×Dic3D4semidirect
120. C2×C8#67semidirect
191. C3#2010semidirect

Maximal subgroups

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

subgroupindex
#57912
#57482
C3×#4072
#51392
#45572
#20103