Finite Group Databasebeta

← #10592524

Subgroups — #10592524

Order 576.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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#64366C211yesmaximal, characteristic
2#64160C211yesmaximal, characteristic, complemented
3#64118C211yesmaximal, characteristic, complemented
4#64084C211yesmaximal, characteristic, complemented
5C32×#104C211yesmaximal, characteristic, complemented
6#63650C211yesmaximal, characteristic, complemented
7#63632C211yesmaximal, characteristic, complemented
8#4825—34nomaximal
9C32×#41C2211yescharacteristic
10#3832—21no—
11D4×C3⋊D3—22no—
12D4×C3⋊D3—21no—
13#3828—21no—
14#3828C2211yescharacteristic
15C2×C4⋊(C3⋊D3)—21no—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×#634C2211yescharacteristic, complemented
18C32×M16C2211yescharacteristic, complemented
19C32×C42C2211yescharacteristic, complemented
20#3756—21no—
21#3754—21no—
22#3752—22no—
23#3748C2211yescharacteristic, complemented
24#3746—21no—
25#3744—21no—
26#1009—34no—
27#951—34no—
28#909—34no—
29#875—34no—
30C3×#104D314yescomplemented
31#823—34no—
32#805—34no—
33C22×C3⋊D3—22no—
34C22×C3⋊D3D412yes—
35C32×Q8D411yescharacteristic
36C32×D4—21no—
37C32×D4D411yescharacteristic
38C2×C32×C4—21no—
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41#638—22no—
42#638—21no—
43C2×C3⋊Dic3—21no—
44C4⋊(C3⋊D3)—22no—
45C4⋊(C3⋊D3)—21no—
46C4⋊(C3⋊D3)—21no—
47C4⋊(C3⋊D3)D411yescharacteristic
48C4×C3⋊D3—22no—
49C4×C3⋊D3—21no—
50#634—21no—
51#634—21no—
52#634D411yescharacteristic
53C32×C8—21no—
54C3⋊(C3⋊C8)—21no—
55#455—91noSylow
56C3×#41C2×D314yes—
57#239—64no—
58D3×D4—68no—
59D3×D4—64no—
60#235—34no—
61#235—64no—
62C2×D12—34no—
63C2×D12—64no—
64C2×Dic6—34no—
65C3×M16C2×D314yescomplemented
66C3×C42C2×D314yescomplemented
67#216—64no—
68#214—64no—
69(C2×D3)⋊C4—68no—
70#208—34no—
71Dic12—64no—
72#204—64no—
73C22×C32—21no—
74C22×C32C2×D411yescharacteristic
75C2×C3⋊D3—22no—
76C2×C3⋊D3—21no—
77C2×C3⋊D3—42no—
78C2×C3⋊D3—82no—
79C32×C4—21no—
80C32×C4—41no—
81C32×C4C2×D411yescharacteristic
82C32×C4C2×D411yescharacteristic
83C3⋊Dic3—21no—
84C3⋊Dic3—41no—
85H(5,𝔽2)—91no—
86#137—91no—
87#137—91no—
88#124—91no—
89#104—91no—
90#104C3⋊D311yescharacteristic, complemented
91#100—91no—
92C22×D3—38no—
93C22×D3—68no—
94C3×Q8D1214yes—
95C3×D4—24no—
96C3×D4D1214yes—
97C2×C3×C4—24no—
98C2×C3×C4—24no—
99C2×C3×C4C22×D314yes—
100#67—68no—
101#67—64no—
102C2×Dic3—64no—
103D12—34no—
104D12—68no—
105D12—64no—
106D12—64no—
107C4×D3—68no—
108C4×D3—64no—
109Dic6—34no—
110Dic6—64no—
111Dic6—64no—
112C3×C8—24no—
113C3⋊C8—64no—
114C2×C32—21no—
115C2×C32—41no—
116C2×C32#12011yescharacteristic
117C3⋊D3—42no—
118C3⋊D3—41no—
119#41—91no—
120#41—181no—
121#41—181no—
122#41C2×C3⋊D311yescharacteristic
123C2×Q8—91no—
124C2×D4—91no—
125C2×D4—181no—
126C2×D4—182no—
127C2×D4—181no—
128Q16—181no—
129Q16—181no—
130SD16—181no—
131SD16—181no—
132M16—91no—
133M16C2×C3⋊D311yescharacteristic, complemented
134C22⋊C4—182no—
135C42C2×C3⋊D311yescharacteristic, complemented
136C22×C3—24no—
137C22×C3D3×D414yes—
138C2×D3—64no—
139C2×D3—68no—
140C2×D3—128no—
141C2×D3—248no—
142C3×C4—24no—
143C3×C4—44no—
144C3×C4C2×D1214yes—
145C3×C4D3×D414yes—
146Dic3—64no—
147Dic3—124no—
148C32#45511yescharacteristic, complemented, Sylow
149C23—92no—
150C23—182no—
151Q8—91no—
152Q8—181no—
153Q8—181no—
154Q8C4⋊(C3⋊D3)11yescharacteristic
155D4—21no—
156D4—91no—
157D4—182no—
158D4—182no—
159D4—181no—
160D4—181no—
161D4—181no—
162D4C4⋊(C3⋊D3)11yescharacteristic
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—182no—
166C2×C4—181no—
167C2×C4—181no—
168C2×C4C22×C3⋊D311yescharacteristic
169C8—21no—
170C8—181no—
171C2×C3—24no—
172C2×C3—44no—
173C2×C3#88214yes—
174D3—128no—
175D3—124no—
176C22—21no—
177C22—181no—
178C22—182no—
179C22—362no—
180C22—722no—
181C22D4×C3⋊D311yescharacteristic
182C4—21no—
183C4—41no—
184C4—181no—
185C4—361no—
186C4C2×C4⋊(C3⋊D3)11yescharacteristic
187C4D4×C3⋊D311yescharacteristic
188C3#482514yescomplemented
189C2—21no—
190C2—41no—
191C2—362no—
192C2—361no—
193C2#6409111yescharacteristic
subgroupG/H
centerC2#64091
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
FittingC32×#104C2
socleC2×C32#120
prime radical#64366C2
hypercenter#104C3⋊D3
hypocenterC32#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #64160C2semidirect
3. #64118C2semidirect
4. #64084C2semidirect
5. C32×#104C2semidirect
6. #63650C2semidirect
7. #63632C2semidirect
17. C2×#634C22semidirect
18. C32×M16C22semidirect
19. C32×C42C22semidirect
23. #3748C22semidirect
30. C3×#104D3semidirect
65. C3×M16C2×D3semidirect
66. C3×C42C2×D3semidirect
90. #104C3⋊D3semidirect
133. M16C2×C3⋊D3semidirect
135. C42C2×C3⋊D3semidirect
148. C32#455semidirect
188. C3#4825semidirect

Maximal subgroups

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

subgroupindex
#643662
#641602
#641182
#640842
C32×#1042
#636502
#636322
#48253