Finite Group Databasebeta

← #10596248

Subgroups — #10596248

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⋊D3C211yesmaximal, characteristic, complemented
2C2×#3832C211yesmaximal, characteristic, complemented
3C32×#137C211yesmaximal, characteristic, complemented
4#64158C211yesmaximal, characteristic, complemented
5#64151C211yesmaximal, characteristic, complemented
6#64128C212yesmaximal, complemented
7#64127C212yesmaximal, complemented
8#64125C212yesmaximal, complemented
9#64123C212yesmaximal, complemented
10#64117C211yesmaximal, characteristic, complemented
11M16×C3⋊D3C211yesmaximal, characteristic, complemented
12#5777—34nomaximal
13C32×#41C2211yescharacteristic
14C2×C32×Q8C2211yescharacteristic, complemented
15#3832—21no—
16#3832—21no—
17#3832C2212yes—
18#3832C2212yescomplemented
19#3832C2211yescharacteristic, complemented
20Q8×C3⋊D3C2211yescharacteristic, complemented
21#3830—21no—
22#3830C2211yescharacteristic, complemented
23D4×C3⋊D3—21no—
24D4×C3⋊D3C2211yescharacteristic, complemented
25#3828—21no—
26#3828C2211yescharacteristic
27C2×C4⋊(C3⋊D3)—21no—
28C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
29C2×C4×C3⋊D3—21no—
30C2×C4×C3⋊D3—21no—
31C2×C4×C3⋊D3C2211yescharacteristic
32C32×Q16C2212yescomplemented
33C32×SD16C2212yescomplemented
34C32×M16C2211yescharacteristic, complemented
35#3756C2212yescomplemented
36#3755C2212yescomplemented
37#3755C2212yescomplemented
38#3753C2212yescomplemented
39#3748C2211yescharacteristic, complemented
40#3745C2212yescomplemented
41#3744C2212yescomplemented
42#3743C2212yescomplemented
43C8×C3⋊D3C2212yescomplemented
44D3×#41—34no—
45C2×#239—34no—
46C3×#137D314yescomplemented
47#949—34no—
48#942—34no—
49#919—38no—
50#918—38no—
51#916—38no—
52#914—38no—
53#908—34no—
54D3×M16—34no—
55C22×C3⋊D3—21no—
56C22×C3⋊D3—21no—
57C22×C3⋊D3D411yescharacteristic
58C32×Q8—21no—
59C32×Q8C2311yescharacteristic
60C32×Q8C2312yes—
61C32×D4—21no—
62C32×D4C2311yescharacteristic
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4C2311yescharacteristic
66#638—21no—
67#638—21no—
68#638—21no—
69C2×C3⋊Dic3—21no—
70C2×C3⋊Dic3D411yescharacteristic
71C4⋊(C3⋊D3)—22no—
72C4⋊(C3⋊D3)—22no—
73C4⋊(C3⋊D3)—21no—
74C4⋊(C3⋊D3)—21no—
75C4⋊(C3⋊D3)C2312yes—
76C4⋊(C3⋊D3)C2311yescharacteristic
77C4×C3⋊D3—21no—
78C4×C3⋊D3—21no—
79C4×C3⋊D3—21no—
80C4×C3⋊D3—22no—
81C4×C3⋊D3—22no—
82C4×C3⋊D3D411yescharacteristic
83C4×C3⋊D3D411yescharacteristic
84C4×C3⋊D3C2311yescharacteristic
85C4×C3⋊D3C2311yescharacteristic
86#634—21no—
87#634C2311yescharacteristic
88C32×C8C2312yes—
89C3⋊(C3⋊C8)C2312yes—
90#575—91noSylow
91C3×#41C2×D314yes—
92C2×C3×Q8C2×D314yescomplemented
93#239—38no—
94#239—38no—
95#239—34no—
96#239—64no—
97#239—64no—
98D3×Q8—34no—
99#237—34no—
100#237—64no—
101D3×D4—34no—
102D3×D4—64no—
103#235—34no—
104#235—64no—
105C2×D12—34no—
106C2×D12—64no—
107C2×C4×D3—34no—
108C2×C4×D3—64no—
109C2×C4×D3—64no—
110C3×Q16C2×D318yescomplemented
111C3×SD16C2×D318yescomplemented
112C3×M16C2×D314yescomplemented
113#216—38no—
114#215—38no—
115#215—38no—
116#213—38no—
117#208—34no—
118D24—38no—
119#204—38no—
120ΓL(1,𝔽25)—38no—
121D3×C8—38no—
122C22×C32—21no—
123C22×C32C2×D411yescharacteristic
124C2×C3⋊D3—22no—
125C2×C3⋊D3—21no—
126C2×C3⋊D3—41no—
127C2×C3⋊D3—42no—
128C2×C3⋊D3—42no—
129C2×C3⋊D3C2×D411yescharacteristic
130C2×C3⋊D3C2×D411yescharacteristic
131C32×C4—21no—
132C32×C4—22no—
133C32×C4C2×D411yescharacteristic
134C32×C4C2411yescharacteristic
135C3⋊Dic3—21no—
136C3⋊Dic3C2×D411yescharacteristic
137C3⋊Dic3C2×D411yescharacteristic
138C2×#41—91no—
139C2×#41—91no—
140#137—91no—
141#137—92no—
142#137C3⋊D311yescharacteristic, complemented
143AGL(1,ℤ/8)—91no—
144AGL(1,ℤ/8)—92no—
145AGL(1,ℤ/8)—91no—
146#135—92no—
147#135—92no—
148C2×M16—91no—
149C22×D3—34no—
150C22×D3—64no—
151C22×D3—64no—
152C3×Q8—24no—
153C3×Q8C22×D314yes—
154C3×Q8C22×D318yes—
155C3×D4—24no—
156C3×D4C22×D314yes—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4C22×D314yes—
160#67—64no—
161#67—64no—
162#67—64no—
163C2×Dic3—34no—
164C2×Dic3—64no—
165D12—38no—
166D12—34no—
167D12—68no—
168D12—68no—
169D12—64no—
170D12—64no—
171C4×D3—34no—
172C4×D3—34no—
173C4×D3—34no—
174C4×D3—34no—
175C4×D3—64no—
176C4×D3—68no—
177C4×D3—68no—
178C4×D3—64no—
179C4×D3—64no—
180Dic6—34no—
181Dic6—64no—
182C3×C8C22×D318yes—
183C3⋊C8—38no—
184C2×C32—21no—
185C2×C32—41no—
186C2×C32C22×D411yescharacteristic
187C3⋊D3—21no—
188C3⋊D3—21no—
189C3⋊D3—42no—
190C3⋊D3—41no—
191#41—91no—
192#41—92no—
193#41—92no—
194#41—91no—
195#41—91no—
196#41—181no—
197#41—181no—
198#41—181no—
199#41—181no—
200#41C2×C3⋊D311yescharacteristic
201C2×Q8—91no—
202C2×Q8C2×C3⋊D311yescharacteristic, complemented
203C2×D4—91no—
204C2×D4—91no—
205C2×D4—181no—
206C2×D4—181no—
207C22×C4—91no—
208C22×C4—181no—
209C22×C4—181no—
210Q16—92no—
211Q16C2×C3⋊D312yescomplemented
212SD16—92no—
213SD16—92no—
214SD16—92no—
215SD16C2×C3⋊D312yescomplemented
216D8—92no—
217D8—92no—
218M16—91no—
219M16—92no—
220M16C2×C3⋊D311yescharacteristic, complemented
221C2×C8—92no—
222C22×C3—24no—
223C22×C3D3×D414yes—
224C2×D3—34no—
225C2×D3—34no—
226C2×D3—68no—
227C2×D3—64no—
228C2×D3—124no—
229C2×D3—128no—
230C2×D3—128no—
231C3×C4—24no—
232C3×C4—28no—
233C3×C4D3×D414yes—
234C3×C4C23×D314yes—
235Dic3—34no—
236Dic3—34no—
237Dic3—64no—
238C32#57511yescharacteristic, complemented, Sylow
239C23—91no—
240C23—181no—
241C23—181no—
242Q8—21no—
243Q8—91no—
244Q8—181no—
245Q8C22×C3⋊D311yescharacteristic
246Q8C22×C3⋊D312yes—
247D4—21no—
248D4—92no—
249D4—91no—
250D4—181no—
251D4—182no—
252D4—182no—
253D4—181no—
254D4—181no—
255D4—181no—
256D4—181no—
257D4C22×C3⋊D311yescharacteristic
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—181no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4C22×C3⋊D311yescharacteristic
272C8—92no—
273C8C22×C3⋊D312yes—
274C2×C3—24no—
275C2×C3—44no—
276C2×C3C2×D3×D414yes—
277D3—64no—
278D3—64no—
279D3—128no—
280D3—124no—
281C22—21no—
282C22—91no—
283C22—91no—
284C22—182no—
285C22—181no—
286C22—361no—
287C22—362no—
288C22—362no—
289C22D4×C3⋊D311yescharacteristic
290C4—21no—
291C4—22no—
292C4—91no—
293C4—91no—
294C4—181no—
295C4D4×C3⋊D311yescharacteristic
296C4C23×C3⋊D311yescharacteristic
297C3#577714yescomplemented
298C2—21no—
299C2—41no—
300C2—181no—
301C2—181no—
302C2—362no—
303C2—361no—
304C2C2×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊D3
commutator, cosocleC32×C4C24
FrattiniC4C23×C3⋊D3
FittingC32×#137C2
socleC2×C32C22×D4
hypercenter#137C3⋊D3
hypocenterC32#575
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #41×C3⋊D3C2semidirect
2. C2×#3832C2semidirect
3. C32×#137C2semidirect
4. #64158C2semidirect
5. #64151C2semidirect
6. #64128C2semidirect
7. #64127C2semidirect
8. #64125C2semidirect
9. #64123C2semidirect
10. #64117C2semidirect
11. M16×C3⋊D3C2semidirect
14. C2×C32×Q8C22semidirect
18. #3832C22semidirect
19. #3832C22semidirect
20. Q8×C3⋊D3C22semidirect
22. #3830C22semidirect
24. D4×C3⋊D3C22semidirect
28. C2×C4⋊(C3⋊D3)C22semidirect
32. C32×Q16C22semidirect
33. C32×SD16C22semidirect
34. C32×M16C22semidirect
35. #3756C22semidirect
36. #3755C22semidirect
37. #3755C22semidirect
38. #3753C22semidirect
39. #3748C22semidirect
40. #3745C22semidirect
41. #3744C22semidirect
42. #3743C22semidirect
43. C8×C3⋊D3C22semidirect
46. C3×#137D3semidirect
92. C2×C3×Q8C2×D3semidirect
110. C3×Q16C2×D3semidirect
111. C3×SD16C2×D3semidirect
112. C3×M16C2×D3semidirect
142. #137C3⋊D3semidirect
202. C2×Q8C2×C3⋊D3semidirect
211. Q16C2×C3⋊D3semidirect
215. SD16C2×C3⋊D3semidirect
220. M16C2×C3⋊D3semidirect
238. C32#575semidirect
297. C3#5777semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊D32
C2×#38322
C32×#1372
#641582
#641512
#641282
#641272
#641252
#641232
#641172
M16×C3⋊D32
#57773