Finite Group Databasebeta

← #71017

Subgroups — #71017

Order 384.

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

normal subgroups only

This group has 240 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#5563C211yesmaximal, characteristic, complemented
2C3×#459C211yesmaximal, characteristic, complemented
3#5076C211yesmaximal, characteristic, complemented
4#5072C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#4483C211yesmaximal, characteristic, complemented
7#4481C211yesmaximal, characteristic, complemented
8#1624—31nomaximal, Sylow
9C2×C3×D8—21no—
10C3×#127C2211yescharacteristic
11C3×C4×D4C2211yescharacteristic
12#938—21no—
13C2×#214—22no—
14C2×#213—21no—
15#930—21no—
16#926—21no—
17#892—21no—
18#885—22no—
19C4×D12C2211yescharacteristic
20#869C2211yescharacteristic
21C3×C4⋊C8C2211yescharacteristic
22C3×#102—21no—
23#832—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C3⋊C8C2211yescharacteristic
26#555—31no—
27#492—31no—
28#459—31no—
29#459D311yescharacteristic, complemented
30#333—31no—
31D4⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4D411yescharacteristic
37C22×C3×C4—21no—
38C2×#67—22no—
39C2×D12D411yescharacteristic
40C2×C4×D3—21no—
41C2×Dic6D412yes—
42C3×D8—42no—
43C2×C3×C8—21no—
44C3×C4⋊C4D411yescharacteristic
45C3×C22⋊C4—21no—
46C3×C42C2311yescharacteristic
47#217—22no—
48#214—44no—
49#213—42no—
50(C2×D3)⋊C4—22no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3D411yescharacteristic
55C4×Dic3—22no—
56C2×C3⋊C8—21no—
57C2×C3⋊C8—21no—
58C2×SD16—62no—
59C2×D8—21no—
60C2×D8—61no—
61C4⋊Q8—31no—
62#127C2×D311yescharacteristic
63#124—62no—
64#121—61no—
65#121—61no—
66C4×D4—31no—
67C4×D4C2×D311yescharacteristic
68#117—61no—
69#117—61no—
70C4⋊C8—31no—
71C4⋊C8C2×D311yescharacteristic
72#102—21no—
73#102—61no—
74C4×C8—31no—
75C23×C3—21no—
76C23×C3—41no—
77C22×D3—21no—
78C3×D4—21no—
79C3×D4—21no—
80C3×D4—42no—
81C3×D4—41no—
82C3×D4—41no—
83C3×D4—41no—
84C3×D4—41no—
85C2×C3×C4—21no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91#67—82no—
92C2×Dic3—21no—
93C2×Dic3—22no—
94D12—21no—
95D12—21no—
96C4×D3—41no—
97C4×D3—41no—
98Dic6—42no—
99C3×C8—41no—
100C3⋊C8—22no—
101C3⋊C8—41no—
102C2×Q8—32no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—31no—
107C2×D4—62no—
108C2×D4#6711yescharacteristic
109C22×C4—21no—
110C22×C4—61no—
111SD16—124no—
112D8—42no—
113D8—122no—
114C2×C8—21no—
115C2×C8—61no—
116C2×C8—61no—
117C4⋊C4—31no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4#6711yescharacteristic
121C22⋊C4—21no—
122C22⋊C4—62no—
123C22⋊C4—62no—
124C22⋊C4—61no—
125C42—62no—
126C42C22×D311yescharacteristic
127C22×C3—41no—
128C22×C3—41no—
129C22×C3—41no—
130C22×C3—81no—
131C22×C3—81no—
132C22×C3—81no—
133C22×C3#12011yescharacteristic
134C2×D3—41no—
135C2×D3—41no—
136C2×D3—41no—
137C3×C4—21no—
138C3×C4—41no—
139C3×C4#13511yescharacteristic
140C3×C4#13511yescharacteristic
141C3×C4AGL(1,ℤ/8)11yescharacteristic
142C3×C4AGL(1,ℤ/8)11yescharacteristic
143Dic3—41no—
144Dic3—42no—
145C23—21no—
146C23—41no—
147C23—61no—
148Q8—122no—
149D4—21no—
150D4—21no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—61no—
157D4—61no—
158D4—242no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—61no—
163C2×C4—62no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4D3×D411yescharacteristic
167C2×C4D3×D411yescharacteristic
168C2×C4C2×#6711yescharacteristic
169C8—41no—
170C8—62no—
171C8—121no—
172C2×C3—81no—
173C2×C3—161no—
174C2×C3#44911yescharacteristic
175C2×C3#44911yescharacteristic
176C2×C3#45311yescharacteristic
177D3—81no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—121no—
185C22—121no—
186C22—121no—
187C22#93711yescharacteristic
188C4—21no—
189C4—41no—
190C4—122no—
191C4—121no—
192C4#91111yescharacteristic
193C4#91211yescharacteristic
194C4#93211yescharacteristic
195C4#95011yescharacteristic
196C3#162411yescharacteristic, complemented, Sylow
197C2—81no—
198C2—161no—
199C2—241no—
200C2#503711yescharacteristic
201C2#515811yescharacteristic
202C2#519811yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#459C2
socleC22×C3#120
hypercenter#459D3
hypocenterC3#1624
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5563C2semidirect
2. C3×#459C2semidirect
3. #5076C2semidirect
4. #5072C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4483C2semidirect
7. #4481C2semidirect
29. #459D3semidirect
196. C3#1624semidirect

Maximal subgroups

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

subgroupindex
#55632
C3×#4592
#50762
#50722
D4⋊(C3⋊C8)2
#44832
#44812
#16243