Finite Group Databasebeta

← #10596115

Subgroups — #10596115

Order 576.

750 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 183 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64156C212yesmaximal, complemented
3#64139C211yesmaximal, characteristic, complemented
4C2×#3752C211yesmaximal, characteristic, complemented
5#64107C212yesmaximal, complemented
6#64104C212yesmaximal, complemented
7#64103C212yesmaximal, complemented
8#64091C212yesmaximal, complemented
9#64090C212yesmaximal, complemented
10#5644—34nomaximal
11C23×C3⋊D3C2211yescharacteristic
12C2×C32×Q8C2211yescharacteristic, complemented
13C22×C32×C4C2211yescharacteristic
14C2×#638C2212yes—
15C22×C3⋊Dic3C2211yescharacteristic
16C2×C4⋊(C3⋊D3)C2212yes—
17C2×C4⋊(C3⋊D3)C2212yes—
18C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C2×C4×C3⋊D3C2212yescomplemented
20C32×C4⋊C4C2211yescharacteristic, complemented
21C32×C4⋊C4C2212yescomplemented
22C32×C22⋊C4C2212yes—
23#3752C2212yes—
24#3752C2212yes—
25#3752C2212yes—
26#3752C2212yes—
27#3752C2212yescomplemented
28#3752C2212yescomplemented
29C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
30#3750C2212yescomplemented
31C4×C3⋊Dic3C2212yescomplemented
32C3×C22⋊Q8D314yescomplemented
33#947—38no—
34#930—34no—
35C2×(C2×D3)⋊C4—34no—
36#898—38no—
37#895—38no—
38#894—38no—
39#882—38no—
40#67⋊C4—38no—
41C23×C32C2311yescharacteristic
42C22×C3⋊D3—22no—
43C22×C3⋊D3—41no—
44C22×C3⋊D3—41no—
45C22×C3⋊D3C2312yes—
46C22×C3⋊D3C2312yes—
47C32×Q8—41no—
48C2×C32×C4—41no—
49C2×C32×C4C2312yes—
50C2×C32×C4C2312yes—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2311yescharacteristic
53#638—24no—
54C2×C3⋊Dic3—41no—
55C2×C3⋊Dic3C2312yes—
56C2×C3⋊Dic3C2312yes—
57C4⋊(C3⋊D3)—42no—
58C4⋊(C3⋊D3)—42no—
59C4⋊(C3⋊D3)—41no—
60C4×C3⋊D3—42no—
61#538—91noSylow
62C23×D3—34no—
63C2×C3×Q8C2×D314yescomplemented
64C22×C3×C4C2×D314yes—
65C2×#67—38no—
66C22×Dic3—34no—
67C2×D12—38no—
68C2×D12—38no—
69C2×D12—34no—
70C2×C4×D3—38no—
71C3×C4⋊C4C2×D314yescomplemented
72C3×C4⋊C4C2×D318yescomplemented
73C3×C22⋊C4C2×D318yes—
74(C2×D3)⋊C4—38no—
75(C2×D3)⋊C4—38no—
76(C2×D3)⋊C4—38no—
77(C2×D3)⋊C4—38no—
78(C2×D3)⋊C4—38no—
79(C2×D3)⋊C4—38no—
80C4⋊Dic3—34no—
81Dic3⋊C4—38no—
82C4×Dic3—38no—
83C22×C32—21no—
84C22×C32—21no—
85C22×C32#4112yes—
86C22×C32C2411yescharacteristic
87C2×C3⋊D3—22no—
88C2×C3⋊D3—22no—
89C2×C3⋊D3—22no—
90C2×C3⋊D3—22no—
91C2×C3⋊D3—22no—
92C2×C3⋊D3—22no—
93C2×C3⋊D3—44no—
94C32×C4—22no—
95C32×C4—21no—
96C32×C4—22no—
97C32×C4—21no—
98C3⋊Dic3—22no—
99C3⋊Dic3—22no—
100#126—92no—
101#124—92no—
102#123—92no—
103C22⋊Q8C3⋊D311yescharacteristic, complemented
104#121—91no—
105#121—92no—
106#120—92no—
107C4×D4—92no—
108C2×C22⋊C4—91no—
109C23×C3C22×D314yes—
110C22×D3—38no—
111C22×D3—38no—
112C22×D3—68no—
113C22×D3—124no—
114C22×D3—124no—
115C3×Q8—44no—
116C2×C3×C4—44no—
117C2×C3×C4C22×D318yes—
118C2×C3×C4C22×D318yes—
119C2×C3×C4C22×D314yes—
120C2×C3×C4C22×D314yes—
121#67—616no—
122C2×Dic3—38no—
123C2×Dic3—38no—
124C2×Dic3—124no—
125D12—124no—
126D12—128no—
127D12—128no—
128C4×D3—128no—
129C2×C32—22no—
130C2×C32C2×#4111yescharacteristic
131C2×C32H(5,𝔽2)11yescharacteristic
132C2×C32H(5,𝔽2)11yescharacteristic
133C3⋊D3—42no—
134C3⋊D3—42no—
135C24—91no—
136C2×Q8C2×C3⋊D311yescharacteristic, complemented
137C2×D4—92no—
138C2×D4—92no—
139C2×D4—92no—
140C2×D4—91no—
141C22×C4—91no—
142C22×C4—92no—
143C22×C4C2×C3⋊D311yescharacteristic
144C4⋊C4—92no—
145C4⋊C4—91no—
146C4⋊C4C2×C3⋊D312yescomplemented
147C4⋊C4C2×C3⋊D311yescharacteristic, complemented
148C22⋊C4—92no—
149C22⋊C4—92no—
150C22⋊C4—92no—
151C22⋊C4—92no—
152C22⋊C4—92no—
153C22⋊C4—92no—
154C22⋊C4C2×C3⋊D312yes—
155C42—92no—
156C22×C3—24no—
157C22×C3—24no—
158C22×C3#23918yes—
159C22×C3C23×D314yes—
160C2×D3—68no—
161C2×D3—68no—
162C2×D3—68no—
163C2×D3—68no—
164C2×D3—68no—
165C2×D3—68no—
166C2×D3—1216no—
167C3×C4—24no—
168C3×C4—28no—
169C3×C4—24no—
170C3×C4—28no—
171Dic3—68no—
172Dic3—68no—
173C32#53811yescharacteristic, complemented, Sylow
174C23—92no—
175C23—92no—
176C23—182no—
177C23—361no—
178C23—361no—
179C23C22×C3⋊D311yescharacteristic
180Q8—41no—
181D4—184no—
182D4—361no—
183D4—362no—
184D4—362no—
185C2×C4—41no—
186C2×C4—92no—
187C2×C4—92no—
188C2×C4—362no—
189C2×C4—361no—
190C2×C4C22×C3⋊D312yes—
191C2×C4C22×C3⋊D311yescharacteristic
192C2×C4C22×C3⋊D311yescharacteristic
193C2×C4C22×C3⋊D312yes—
194C2×C3—28no—
195C2×C3#100414yes—
196C2×C3C2×#23914yes—
197C2×C3#100914yes—
198D3—128no—
199D3—128no—
200C22—21no—
201C22—21no—
202C22—182no—
203C22—182no—
204C22—182no—
205C22—182no—
206C22—182no—
207C22—182no—
208C22—364no—
209C22#383212yes—
210C22C23×C3⋊D311yescharacteristic
211C4—21no—
212C4—21no—
213C4—22no—
214C4—22no—
215C4—182no—
216C4—182no—
217C3#564414yescomplemented
218C2—22no—
219C2—362no—
220C2—362no—
221C2#6436111yescharacteristic
222C2C2×#383211yescharacteristic
223C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊Q8C2semidirect
2. #64156C2semidirect
3. #64139C2semidirect
4. C2×#3752C2semidirect
5. #64107C2semidirect
6. #64104C2semidirect
7. #64103C2semidirect
8. #64091C2semidirect
9. #64090C2semidirect
12. C2×C32×Q8C22semidirect
18. C2×C4⋊(C3⋊D3)C22semidirect
19. C2×C4×C3⋊D3C22semidirect
20. C32×C4⋊C4C22semidirect
21. C32×C4⋊C4C22semidirect
27. #3752C22semidirect
28. #3752C22semidirect
29. C4⋊(C3⋊Dic3)C22semidirect
30. #3750C22semidirect
31. C4×C3⋊Dic3C22semidirect
32. C3×C22⋊Q8D3semidirect
63. C2×C3×Q8C2×D3semidirect
71. C3×C4⋊C4C2×D3semidirect
72. C3×C4⋊C4C2×D3semidirect
103. C22⋊Q8C3⋊D3semidirect
136. C2×Q8C2×C3⋊D3semidirect
146. C4⋊C4C2×C3⋊D3semidirect
147. C4⋊C4C2×C3⋊D3semidirect
173. C32#538semidirect
217. C3#5644semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
#641562
#641392
C2×#37522
#641072
#641042
#641032
#640912
#640902
#56443