Finite Group Databasebeta

← #10596239

Subgroups — #10596239

Order 576.

786 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 179 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#41×C3⋊D3C212yesmaximal, complemented
2C32×#135C211yesmaximal, characteristic, complemented
3#64159C212yesmaximal, complemented
4#64127C212yesmaximal, complemented
5#64124C212yesmaximal, complemented
6#64123C212yesmaximal, complemented
7#64120C212yesmaximal, complemented
8#64113C211yesmaximal, characteristic, complemented
9C2×#3743C211yesmaximal, characteristic, complemented
10#5768—34nomaximal
11C32×#41C2212yescomplemented
12#3832C2212yes—
13Q8×C3⋊D3C2212yescomplemented
14#3830—22no—
15#3830C2212yes—
16D4×C3⋊D3—22no—
17D4×C3⋊D3C2212yescomplemented
18#3828—22no—
19#3828C2212yescomplemented
20C2×C4×C3⋊D3—22no—
21C2×C4×C3⋊D3C2211yescharacteristic
22C32×Q16C2211yescharacteristic, complemented
23C32×SD16C2212yescomplemented
24C32×D8C2211yescharacteristic, complemented
25C2×C32×C8C2211yescharacteristic, complemented
26#3756C2212yescomplemented
27#3755C2212yescomplemented
28#3754C2212yescomplemented
29#3753C2212yescomplemented
30C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
31#3746C2211yescharacteristic, complemented
32#3745C2211yescharacteristic, complemented
33#3744C2212yescomplemented
34#3743C2212yescomplemented
35#3743C2212yescomplemented
36D3×#41—38no—
37C3×#135D314yescomplemented
38#950—38no—
39#918—38no—
40#915—38no—
41#914—38no—
42#911—38no—
43#904—34no—
44C2×ΓL(1,𝔽25)—34no—
45C22×C3⋊D3—22no—
46C22×C3⋊D3D411yescharacteristic
47C32×Q8C2312yes—
48C32×D4—22no—
49C32×D4C2312yes—
50C2×C32×C4—22no—
51C2×C32×C4C2311yescharacteristic
52#638—22no—
53#638—22no—
54#638—22no—
55C2×C3⋊Dic3—22no—
56C2×C3⋊Dic3D411yescharacteristic
57C4⋊(C3⋊D3)—22no—
58C4⋊(C3⋊D3)C2312yes—
59C4×C3⋊D3—22no—
60C4×C3⋊D3—22no—
61C4×C3⋊D3—22no—
62C4×C3⋊D3D412yes—
63C4×C3⋊D3C2312yes—
64#634—22no—
65#634C2312yes—
66C32×C8C2311yescharacteristic
67C32×C8C2311yescharacteristic
68C3⋊(C3⋊C8)C2312yes—
69#575—91noSylow
70C3×#41C2×D318yescomplemented
71#239—38no—
72D3×Q8—38no—
73#237—38no—
74#237—68no—
75D3×D4—38no—
76D3×D4—68no—
77#235—38no—
78#235—68no—
79C2×C4×D3—34no—
80C2×C4×D3—68no—
81C3×Q16C2×D314yescomplemented
82C3×SD16C2×D318yescomplemented
83C3×D8C2×D314yescomplemented
84C2×C3×C8C2×D314yescomplemented
85#216—38no—
86#215—38no—
87#214—38no—
88#213—38no—
89C2×C3⋊C8—34no—
90Dic12—34no—
91D24—34no—
92#204—38no—
93ΓL(1,𝔽25)—38no—
94ΓL(1,𝔽25)—38no—
95C22×C32—22no—
96C22×C32C2×D411yescharacteristic
97C2×C3⋊D3—22no—
98C2×C3⋊D3—41no—
99C2×C3⋊D3—44no—
100C2×C3⋊D3C2×D412yes—
101C32×C4—22no—
102C32×C4C2×D411yescharacteristic
103C32×C4C2411yescharacteristic
104C3⋊Dic3—22no—
105C3⋊Dic3C2×D412yes—
106C2×#41—92no—
107#137—92no—
108#137—92no—
109AGL(1,ℤ/8)—92no—
110AGL(1,ℤ/8)—92no—
111#135—92no—
112#135—91no—
113#135C3⋊D311yescharacteristic, complemented
114C2×M16—91no—
115C22×D3—34no—
116C22×D3—68no—
117C3×Q8C22×D318yes—
118C3×D4—28no—
119C3×D4C22×D318yes—
120C2×C3×C4—28no—
121C2×C3×C4C22×D314yes—
122#67—68no—
123#67—68no—
124#67—68no—
125C2×Dic3—34no—
126C2×Dic3—68no—
127D12—38no—
128D12—68no—
129C4×D3—38no—
130C4×D3—38no—
131C4×D3—68no—
132C4×D3—68no—
133C4×D3—68no—
134Dic6—38no—
135Dic6—68no—
136C3×C8C22×D314yes—
137C3×C8C22×D314yes—
138C3⋊C8—38no—
139C2×C32—21no—
140C2×C32—42no—
141C2×C32C22×D411yescharacteristic
142C3⋊D3—22no—
143C3⋊D3—42no—
144#41—92no—
145#41—92no—
146#41—92no—
147#41—182no—
148#41—182no—
149#41C2×C3⋊D312yescomplemented
150C2×Q8—92no—
151C2×D4—92no—
152C2×D4—182no—
153C22×C4—91no—
154C22×C4—182no—
155Q16—92no—
156Q16—91no—
157Q16C2×C3⋊D311yescharacteristic, complemented
158SD16—92no—
159SD16—92no—
160SD16—92no—
161SD16C2×C3⋊D312yescomplemented
162D8—92no—
163D8—91no—
164D8C2×C3⋊D311yescharacteristic, complemented
165M16—92no—
166M16—92no—
167C2×C8—91no—
168C2×C8C2×C3⋊D311yescharacteristic, complemented
169C22×C3—28no—
170C22×C3D3×D414yes—
171C2×D3—38no—
172C2×D3—68no—
173C2×D3—124no—
174C2×D3—1216no—
175C3×C4—28no—
176C3×C4D3×D414yes—
177C3×C4C23×D314yes—
178Dic3—38no—
179Dic3—68no—
180C32#57511yescharacteristic, complemented, Sylow
181C23—91no—
182C23—182no—
183Q8—92no—
184Q8—182no—
185Q8C22×C3⋊D312yes—
186D4—22no—
187D4—92no—
188D4—182no—
189D4—182no—
190D4—182no—
191D4—182no—
192D4C22×C3⋊D312yes—
193C2×C4—22no—
194C2×C4—91no—
195C2×C4—92no—
196C2×C4—92no—
197C2×C4—182no—
198C2×C4—182no—
199C2×C4—182no—
200C2×C4—182no—
201C2×C4C22×C3⋊D311yescharacteristic
202C8—92no—
203C8C22×C3⋊D311yescharacteristic
204C8C22×C3⋊D311yescharacteristic
205C2×C3—24no—
206C2×C3—48no—
207C2×C3C2×D3×D414yes—
208D3—68no—
209D3—128no—
210C22—22no—
211C22—92no—
212C22—182no—
213C22—361no—
214C22—364no—
215C22D4×C3⋊D311yescharacteristic
216C4—22no—
217C4—92no—
218C4—182no—
219C4D4×C3⋊D311yescharacteristic
220C4C23×C3⋊D311yescharacteristic
221C3#576814yescomplemented
222C2—21no—
223C2—42no—
224C2—182no—
225C2—362no—
226C2C2×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC4D4×C3⋊D3
commutator, cosocleC32×C4C24
FrattiniC4C23×C3⋊D3
FittingC32×#135C2
socleC2×C32C22×D4
hypercenter#135C3⋊D3
hypocenterC32#575
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×C3⋊D3C2semidirect
2. C32×#135C2semidirect
3. #64159C2semidirect
4. #64127C2semidirect
5. #64124C2semidirect
6. #64123C2semidirect
7. #64120C2semidirect
8. #64113C2semidirect
9. C2×#3743C2semidirect
11. C32×#41C22semidirect
13. Q8×C3⋊D3C22semidirect
17. D4×C3⋊D3C22semidirect
19. #3828C22semidirect
22. C32×Q16C22semidirect
23. C32×SD16C22semidirect
24. C32×D8C22semidirect
25. C2×C32×C8C22semidirect
26. #3756C22semidirect
27. #3755C22semidirect
28. #3754C22semidirect
29. #3753C22semidirect
30. C2×C3⋊(C3⋊C8)C22semidirect
31. #3746C22semidirect
32. #3745C22semidirect
33. #3744C22semidirect
34. #3743C22semidirect
35. #3743C22semidirect
37. C3×#135D3semidirect
70. C3×#41C2×D3semidirect
81. C3×Q16C2×D3semidirect
82. C3×SD16C2×D3semidirect
83. C3×D8C2×D3semidirect
84. C2×C3×C8C2×D3semidirect
113. #135C3⋊D3semidirect
149. #41C2×C3⋊D3semidirect
157. Q16C2×C3⋊D3semidirect
161. SD16C2×C3⋊D3semidirect
164. D8C2×C3⋊D3semidirect
168. C2×C8C2×C3⋊D3semidirect
180. C32#575semidirect
221. C3#5768semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊D32
C32×#1352
#641592
#641272
#641242
#641232
#641202
#641132
C2×#37432
#57683