Finite Group Databasebeta

← #10596162

Subgroups — #10596162

Order 576.

702 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C32×#125C211yesmaximal, characteristic, complemented
2#64107C212yesmaximal, complemented
3#64106C211yesmaximal, characteristic, complemented
4#64104C211yesmaximal, characteristic
5#64103C212yesmaximal, complemented
6#64102C211yesmaximal, characteristic, complemented
7#64102C212yesmaximal, complemented
8#64101C212yesmaximal, complemented
9#64096C211yesmaximal, characteristic, complemented
10#64084C211yesmaximal, characteristic, complemented
11C42×C3⋊D3C211yesmaximal, characteristic, complemented
12#5691—34nomaximal
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C4⋊(C3⋊D3)C2212yes—
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×C4×C3⋊D3C2212yes—
19C2×#634C2211yescharacteristic
20C32×C4⋊C4C2211yescharacteristic
21C32×C4⋊C4C2212yes—
22C32×C4⋊C4C2211yescharacteristic
23C32×C4⋊C4C2212yes—
24C32×C42C2211yescharacteristic
25#3752C2212yes—
26#3752C2212yes—
27#3752C2212yes—
28#3752C2212yes—
29#3752C2212yes—
30C4⋊(C3⋊Dic3)C2212yes—
31#3750C2212yes—
32C4×C3⋊Dic3C2211yescharacteristic
33C4×C3⋊Dic3C2211yescharacteristic
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×#125D314yescomplemented
37#898—38no—
38#897—34no—
39#895—34no—
40#894—38no—
41#893—34no—
42#893—38no—
43#892—38no—
44#887—34no—
45#875—34no—
46C42×D3—34no—
47C22×C3⋊D3C2312yes—
48C22×C3⋊D3C2311yescharacteristic
49C2×C32×C4C2312yes—
50C2×C32×C4C2311yescharacteristic
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C3⋊Dic3C2311yescharacteristic
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2311yescharacteristic
58C4⋊(C3⋊D3)—24no—
59C4⋊(C3⋊D3)—22no—
60C4×C3⋊D3—22no—
61C4×C3⋊D3—22no—
62C4×C3⋊D3—22no—
63C4×C3⋊D3—24no—
64#634—22no—
65#529—91noSylow
66C2×D12—34no—
67C2×D12—38no—
68C2×C4×D3—34no—
69C2×C4×D3—34no—
70C2×C4×D3—34no—
71C2×C4×D3—38no—
72C2×Dic6—34no—
73C3×C4⋊C4C2×D314yes—
74C3×C4⋊C4C2×D318yes—
75C3×C4⋊C4C2×D314yes—
76C3×C4⋊C4C2×D318yes—
77C3×C42C2×D314yes—
78(C2×D3)⋊C4—38no—
79(C2×D3)⋊C4—38no—
80(C2×D3)⋊C4—38no—
81(C2×D3)⋊C4—38no—
82(C2×D3)⋊C4—38no—
83C4⋊Dic3—38no—
84Dic3⋊C4—38no—
85C4×Dic3—34no—
86C4×Dic3—34no—
87C4×Dic3—34no—
88C4×Dic3—38no—
89C22×C32C2411yescharacteristic
90C2×C3⋊D3—21no—
91C2×C3⋊D3—21no—
92C2×C3⋊D3—22no—
93C2×C3⋊D3—22no—
94C2×C3⋊D3—22no—
95C2×C3⋊D3#4112yes—
96C32×C4—21no—
97C32×C4—21no—
98C32×C4—22no—
99C32×C4—22no—
100C32×C4#4112yes—
101C3⋊Dic3—22no—
102C3⋊Dic3—21no—
103C3⋊Dic3—21no—
104C3⋊Dic3#4112yes—
105#126—92no—
106#125C3⋊D311yescharacteristic, complemented
107#124—91no—
108#123—92no—
109C22⋊Q8—91no—
110#121—91no—
111C4×Q8—91no—
112C4×D4—91no—
113C4×D4—92no—
114#117—92no—
115C2×C42—91no—
116C22×D3—34no—
117C22×D3—38no—
118C2×C3×C4C22×D318yes—
119C2×C3×C4C22×D314yes—
120C2×C3×C4C22×D314yes—
121C2×C3×C4C22×D318yes—
122C2×C3×C4C22×D314yes—
123C2×Dic3—34no—
124C2×Dic3—34no—
125C2×Dic3—34no—
126C2×Dic3—38no—
127D12—616no—
128D12—68no—
129C4×D3—68no—
130C4×D3—616no—
131C4×D3—68no—
132C4×D3—68no—
133Dic6—68no—
134C2×C32C2×#4111yescharacteristic
135C2×C32C2×#4111yescharacteristic
136C2×C32C2×#4111yescharacteristic
137C3⋊D3—22no—
138C3⋊D3—42no—
139C2×Q8—91no—
140C2×D4—92no—
141C2×D4—91no—
142C22×C4—91no—
143C22×C4—91no—
144C22×C4—91no—
145C22×C4—92no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4C2×C3⋊D312yes—
149C4⋊C4C2×C3⋊D311yescharacteristic
150C4⋊C4C2×C3⋊D312yes—
151C4⋊C4C2×C3⋊D311yescharacteristic
152C22⋊C4—92no—
153C22⋊C4—92no—
154C22⋊C4—92no—
155C22⋊C4—92no—
156C22⋊C4—92no—
157C42—91no—
158C42—91no—
159C42—91no—
160C42—92no—
161C42C2×C3⋊D311yescharacteristic
162C22×C3C23×D314yes—
163C2×D3—38no—
164C2×D3—64no—
165C2×D3—64no—
166C2×D3—68no—
167C2×D3—68no—
168C2×D3—68no—
169C3×C4—24no—
170C3×C4—28no—
171C3×C4—24no—
172C3×C4—28no—
173C3×C4#23918yes—
174Dic3—38no—
175Dic3—68no—
176Dic3—64no—
177Dic3—64no—
178C32#52911yescharacteristic, complemented, Sylow
179C23—91no—
180C23—92no—
181Q8—182no—
182D4—184no—
183D4—182no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—182no—
189C2×C4—184no—
190C2×C4—182no—
191C2×C4—182no—
192C2×C4C22×C3⋊D311yescharacteristic
193C2×C4C22×C3⋊D311yescharacteristic
194C2×C4C22×C3⋊D312yes—
195C2×C4C22×C3⋊D312yes—
196C2×C4C22×C3⋊D311yescharacteristic
197C2×C3C2×#23914yes—
198C2×C3D3×#4114yes—
199C2×C3D3×#4114yes—
200D3—68no—
201D3—128no—
202C22—92no—
203C22—181no—
204C22—181no—
205C22—182no—
206C22—182no—
207C22—182no—
208C22C23×C3⋊D311yescharacteristic
209C4—21no—
210C4—22no—
211C4—21no—
212C4—22no—
213C4—92no—
214C4—181no—
215C4—182no—
216C4—181no—
217C4#383212yes—
218C3#569114yescomplemented
219C2—182no—
220C2—362no—
221C2C2×#383211yescharacteristic
222C2#41×C3⋊D311yescharacteristic
223C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#125C2
prime radical#64104C2
hypercenter#125C3⋊D3
hypocenterC32#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C32×#125C2semidirect
2. #64107C2semidirect
3. #64106C2semidirect
5. #64103C2semidirect
6. #64102C2semidirect
7. #64102C2semidirect
8. #64101C2semidirect
9. #64096C2semidirect
10. #64084C2semidirect
11. C42×C3⋊D3C2semidirect
36. C3×#125D3semidirect
106. #125C3⋊D3semidirect
178. C32#529semidirect
218. C3#5691semidirect

Maximal subgroups

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

subgroupindex
C32×#1252
#641072
#641062
#641042
#641032
#641022
#641012
#640962
#640842
C42×C3⋊D32
#56913