Finite Group Databasebeta

← #10596186

Subgroups — #10596186

Order 576.

1002 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×#3830C211yesmaximal, characteristic, complemented
2C2×D4×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#127C211yesmaximal, characteristic, complemented
4#64149C211yesmaximal, characteristic, complemented
5#64148C214yesmaximal, complemented
6#64146C214yesmaximal, complemented
7#64145C211yesmaximal, characteristic, complemented
8#64084C211yesmaximal, characteristic, complemented
9#64081C211yesmaximal, characteristic, complemented
10#5715—34nomaximal
11C23×C3⋊D3C2212yes—
12C2×C32×D4C2211yescharacteristic
13C2×C32×D4C2211yescharacteristic, complemented
14C2×C32×D4C2214yescomplemented
15C2×#638C2212yes—
16C2×#638C2212yes—
17C2×#638C2214yescomplemented
18C22×C3⋊Dic3C2212yescomplemented
19#3830—24no—
20D4×C3⋊D3—24no—
21C2×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic
23C2×#634C2211yescharacteristic, complemented
24C32×C42C2211yescharacteristic, complemented
25#3757C2214yescomplemented
26#3752C2212yes—
27#3752C2214yescomplemented
28#3750C2212yescomplemented
29C4×C3⋊Dic3C2211yescharacteristic, complemented
30C2×#237—34no—
31C2×D3×D4—34no—
32C3×#127D314yescomplemented
33#940—34no—
34#939—316no—
35#937—316no—
36#936—34no—
37#875—34no—
38#872—34no—
39C23×C32C2312yes—
40C23×C32C2312yes—
41C22×C3⋊D3—24no—
42C22×C3⋊D3—44no—
43C22×C3⋊D3C2312yes—
44C22×C3⋊D3C2311yescharacteristic
45C32×D4—22no—
46C32×D4—22no—
47C32×D4—44no—
48C2×C32×C4C2312yes—
49C2×C32×C4C2311yescharacteristic
50#638—24no—
51#638—24no—
52#638—44no—
53C2×C3⋊Dic3—24no—
54C2×C3⋊Dic3C2312yes—
55C2×C3⋊Dic3C2312yes—
56C2×C3⋊Dic3C2311yescharacteristic
57C4⋊(C3⋊D3)—22no—
58C4×C3⋊D3D414yescomplemented
59#634—22no—
60#535—91noSylow
61C23×D3—38no—
62C2×C3×D4C2×D314yes—
63C2×C3×D4C2×D314yescomplemented
64C2×C3×D4C2×D3116yescomplemented
65C2×#67—38no—
66C2×#67—38no—
67C2×#67—316no—
68C22×Dic3—38no—
69#237—616no—
70D3×D4—616no—
71C2×D12—34no—
72C2×C4×D3—34no—
73C2×Dic6—34no—
74C3×C42C2×D314yescomplemented
75#217—316no—
76(C2×D3)⋊C4—38no—
77(C2×D3)⋊C4—316no—
78Dic3⋊C4—38no—
79C4×Dic3—34no—
80C22×C32—22no—
81C22×C32—22no—
82C22×C32—24no—
83C22×C32—24no—
84C22×C32C2411yescharacteristic
85C2×C3⋊D3—22no—
86C2×C3⋊D3—22no—
87C2×C3⋊D3—24no—
88C2×C3⋊D3—48no—
89C2×C3⋊D3C2×D412yes—
90C32×C4—22no—
91C32×C4C2×D412yes—
92C3⋊Dic3—22no—
93C3⋊Dic3—22no—
94C3⋊Dic3C2×D412yes—
95C2×#41—91no—
96C22×D4—91no—
97#127—91no—
98#127C3⋊D311yescharacteristic, complemented
99#124—91no—
100#124—91no—
101#121—94no—
102#120—94no—
103#117—91no—
104C23×C3C22×D318yes—
105C23×C3C22×D318yes—
106C22×D3—34no—
107C22×D3—38no—
108C22×D3—616no—
109C22×D3—1216no—
110C3×D4—28no—
111C3×D4—28no—
112C3×D4—416no—
113C2×C3×C4C22×D318yes—
114C2×C3×C4C22×D314yes—
115#67—616no—
116#67—616no—
117#67—1216no—
118C2×Dic3—34no—
119C2×Dic3—38no—
120C2×Dic3—38no—
121C2×Dic3—616no—
122D12—68no—
123C4×D3—316no—
124Dic6—68no—
125C2×C32—42no—
126C2×C32—42no—
127C2×C32C22×D411yescharacteristic
128C2×C32H(5,𝔽2)12yes—
129C3⋊D3—22no—
130C3⋊D3—42no—
131C24—92no—
132#41—184no—
133C2×Q8—91no—
134C2×D4—92no—
135C2×D4—92no—
136C2×D4—94no—
137C2×D4—91no—
138C2×D4—184no—
139C2×D4C2×C3⋊D311yescharacteristic
140C2×D4C2×C3⋊D311yescharacteristic, complemented
141C2×D4C2×C3⋊D314yescomplemented
142C22×C4—91no—
143C22×C4—92no—
144C4⋊C4—92no—
145C22⋊C4—94no—
146C22⋊C4—92no—
147C22⋊C4—94no—
148C42—91no—
149C42C2×C3⋊D311yescharacteristic, complemented
150C22×C3—28no—
151C22×C3—28no—
152C22×C3—216no—
153C22×C3—216no—
154C22×C3C23×D314yes—
155C2×D3—38no—
156C2×D3—68no—
157C2×D3—68no—
158C2×D3—616no—
159C2×D3—1232no—
160C3×C4—28no—
161C3×C4D3×D418yes—
162Dic3—38no—
163Dic3—68no—
164Dic3—68no—
165C32#53511yescharacteristic, complemented, Sylow
166C23—91no—
167C23—92no—
168C23—184no—
169C23—364no—
170C23C22×C3⋊D312yes—
171C23C22×C3⋊D312yes—
172Q8—182no—
173D4—22no—
174D4—22no—
175D4—44no—
176D4—184no—
177D4—184no—
178D4—182no—
179D4—364no—
180C2×C4—91no—
181C2×C4—94no—
182C2×C4—92no—
183C2×C4—92no—
184C2×C4—184no—
185C2×C4C22×C3⋊D312yes—
186C2×C4C22×C3⋊D311yescharacteristic
187C2×C3—48no—
188C2×C3—48no—
189C2×C3C2×D3×D414yes—
190C2×C3#100418yes—
191D3—68no—
192D3—128no—
193C22—22no—
194C22—22no—
195C22—24no—
196C22—24no—
197C22—92no—
198C22—182no—
199C22—182no—
200C22—184no—
201C22—368no—
202C22C23×C3⋊D311yescharacteristic
203C4—22no—
204C4—92no—
205C4—182no—
206C4—182no—
207C4D4×C3⋊D312yes—
208C3#571514yescomplemented
209C2—42no—
210C2—42no—
211C2—182no—
212C2—362no—
213C2C2×D4×C3⋊D311yescharacteristic
214C2#6436112yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#127C2
hypercenter#127C3⋊D3
hypocenterC32#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#3830C2semidirect
2. C2×D4×C3⋊D3C2semidirect
3. C32×#127C2semidirect
4. #64149C2semidirect
5. #64148C2semidirect
6. #64146C2semidirect
7. #64145C2semidirect
8. #64084C2semidirect
9. #64081C2semidirect
13. C2×C32×D4C22semidirect
14. C2×C32×D4C22semidirect
17. C2×#638C22semidirect
18. C22×C3⋊Dic3C22semidirect
23. C2×#634C22semidirect
24. C32×C42C22semidirect
25. #3757C22semidirect
27. #3752C22semidirect
28. #3750C22semidirect
29. C4×C3⋊Dic3C22semidirect
32. C3×#127D3semidirect
58. C4×C3⋊D3D4semidirect
63. C2×C3×D4C2×D3semidirect
64. C2×C3×D4C2×D3semidirect
74. C3×C42C2×D3semidirect
98. #127C3⋊D3semidirect
140. C2×D4C2×C3⋊D3semidirect
141. C2×D4C2×C3⋊D3semidirect
149. C42C2×C3⋊D3semidirect
165. C32#535semidirect
208. C3#5715semidirect

Maximal subgroups

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

subgroupindex
C2×#38302
C2×D4×C3⋊D32
C32×#1272
#641492
#641482
#641462
#641452
#640842
#640812
#57153