Finite Group Databasebeta

← #10596246

Subgroups — #10596246

Order 576.

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

normal subgroups only

This group has 774 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#64367C211yesmaximal, characteristic, complemented
2#64361C211yesmaximal, characteristic, complemented
3C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#64159C211yesmaximal, characteristic, complemented
5C2×#3754C211yesmaximal, characteristic, complemented
6#64124C212yesmaximal, complemented
7SD16×C3⋊D3C212yesmaximal, complemented
8#64121C212yesmaximal, complemented
9#64120C212yesmaximal, complemented
10#64118C211yesmaximal, characteristic, complemented
11#64116C211yesmaximal, characteristic, complemented
12#5775—34nomaximal
13C32×#41C2211yescharacteristic, complemented
14C2×C32×D4C2211yescharacteristic
15C2×#638—22no—
16Q8×C3⋊D3C2212yescomplemented
17#3830—22no—
18#3830—21no—
19#3830C2212yes—
20#3830C2212yescomplemented
21D4×C3⋊D3—21no—
22D4×C3⋊D3C2212yes—
23#3828—21no—
24#3828C2211yescharacteristic
25#3828C2211yescharacteristic
26C2×#634—21no—
27C2×#634C2211yescharacteristic, complemented
28C32×SD16C2212yescomplemented
29C32×D8C2212yescomplemented
30C32×M16C2211yescharacteristic, complemented
31#3756C2211yescharacteristic, complemented
32#3755C2211yescharacteristic, complemented
33#3754C2212yescomplemented
34#3754C2211yescharacteristic, complemented
35#3754C2212yescomplemented
36#3753C2211yescharacteristic, complemented
37C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
38#3746C2212yescomplemented
39#3744C2212yescomplemented
40#3743C2212yescomplemented
41C8×C3⋊D3C2212yescomplemented
42#1010—34no—
43#1004—34no—
44C3×AGL(1,ℤ/8)D314yescomplemented
45#950—34no—
46C2×#214—34no—
47#915—38no—
48D3×SD16—38no—
49#912—38no—
50#911—38no—
51#909—34no—
52#907—34no—
53C23×C32—21no—
54C22×C3⋊D3—22no—
55C32×Q8C2311yescharacteristic
56C32×D4—21no—
57C32×D4—21no—
58C32×D4C2311yescharacteristic
59C32×D4C2312yes—
60C2×C32×C4—21no—
61C2×C32×C4C2311yescharacteristic
62#638—21no—
63#638—22no—
64#638—22no—
65#638—22no—
66#638D412yes—
67C2×C3⋊Dic3—21no—
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3—22no—
70C4⋊(C3⋊D3)D411yescharacteristic
71C4⋊(C3⋊D3)C2311yescharacteristic
72C4×C3⋊D3—22no—
73C4×C3⋊D3—21no—
74C4×C3⋊D3C2312yes—
75#634—21no—
76#634—22no—
77#634D411yescharacteristic
78#634C2312yes—
79#634C2311yescharacteristic
80C32×C8C2312yes—
81C3⋊(C3⋊C8)C2311yescharacteristic
82C3⋊(C3⋊C8)C2311yescharacteristic
83#577—91noSylow
84C3×#41C2×D314yescomplemented
85C2×C3×D4C2×D314yes—
86C2×#67—68no—
87D3×Q8—38no—
88#237—38no—
89#237—38no—
90#237—64no—
91#237—68no—
92D3×D4—38no—
93D3×D4—64no—
94#235—34no—
95#235—34no—
96#235—64no—
97C2×Dic6—34no—
98C2×Dic6—64no—
99C3×SD16C2×D318yescomplemented
100C3×D8C2×D318yescomplemented
101C3×M16C2×D314yescomplemented
102#216—34no—
103#215—34no—
104#214—38no—
105#214—34no—
106#214—38no—
107#213—34no—
108C2×C3⋊C8—34no—
109Dic12—38no—
110#204—38no—
111ΓL(1,𝔽25)—38no—
112D3×C8—38no—
113C22×C32—21no—
114C22×C32—22no—
115C22×C32—81no—
116C22×C32C2×D411yescharacteristic
117C2×C3⋊D3—21no—
118C2×C3⋊D3—82no—
119C2×C3⋊D3C2×D412yes—
120C32×C4—21no—
121C32×C4C2×D411yescharacteristic
122C32×C4C2411yescharacteristic
123C3⋊Dic3—22no—
124C3⋊Dic3—21no—
125C3⋊Dic3C2×D412yes—
1262−1+4—91no—
127H(5,𝔽2)—91no—
128#137—92no—
129#137—91no—
130AGL(1,ℤ/8)—92no—
131AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
132#135—91no—
133#135—92no—
134C2×SD16—91no—
135C2×SD16—92no—
136#131—91no—
137C23×C3—24no—
138C22×D3—68no—
139C3×Q8C22×D314yes—
140C3×D4—24no—
141C3×D4—24no—
142C3×D4C22×D314yes—
143C3×D4C22×D318yes—
144C2×C3×C4—24no—
145C2×C3×C4C22×D314yes—
146#67—38no—
147#67—68no—
148#67—68no—
149#67—68no—
150#67—64no—
151C2×Dic3—68no—
152C2×Dic3—68no—
153C2×Dic3—64no—
154D12—34no—
155D12—34no—
156C4×D3—38no—
157C4×D3—68no—
158C4×D3—64no—
159Dic6—34no—
160Dic6—38no—
161Dic6—34no—
162Dic6—68no—
163Dic6—64no—
164C3×C8C22×D318yes—
165C3⋊C8—34no—
166C3⋊C8—34no—
167C2×C32—21no—
168C2×C32—42no—
169C2×C32—41no—
170C2×C32C22×D411yescharacteristic
171C3⋊D3—22no—
172C3⋊D3—41no—
173#41—91no—
174#41—92no—
175#41—92no—
176#41—91no—
177#41—181no—
178#41—182no—
179#41—181no—
180#41C2×C3⋊D311yescharacteristic, complemented
181C2×Q8—92no—
182C2×Q8—91no—
183C2×Q8—181no—
184C2×D4—92no—
185C2×D4—181no—
186C2×D4—182no—
187C2×D4C2×C3⋊D311yescharacteristic
188Q16—91no—
189Q16—92no—
190SD16—91no—
191SD16—92no—
192SD16—92no—
193SD16—91no—
194SD16—92no—
195SD16C2×C3⋊D312yescomplemented
196D8—91no—
197D8C2×C3⋊D312yescomplemented
198M16—92no—
199M16C2×C3⋊D311yescharacteristic, complemented
200C2×C8—91no—
201C2×C8—92no—
202C22×C3—24no—
203C22×C3—28no—
204C22×C3—84no—
205C22×C3D3×D414yes—
206C2×D3—38no—
207C2×D3—64no—
208C2×D3—248no—
209C3×C4—24no—
210C3×C4D3×D414yes—
211C3×C4C23×D314yes—
212Dic3—38no—
213Dic3—68no—
214Dic3—64no—
215C32#57711yescharacteristic, complemented, Sylow
216C23—21no—
217C23—182no—
218Q8—91no—
219Q8—92no—
220Q8—91no—
221Q8—182no—
222Q8—181no—
223Q8C22×C3⋊D311yescharacteristic
224D4—21no—
225D4—21no—
226D4—92no—
227D4—91no—
228D4—91no—
229D4—182no—
230D4—182no—
231D4—182no—
232D4—181no—
233D4C22×C3⋊D311yescharacteristic
234D4C22×C3⋊D312yes—
235C2×C4—21no—
236C2×C4—92no—
237C2×C4—182no—
238C2×C4—182no—
239C2×C4—182no—
240C2×C4—181no—
241C2×C4—181no—
242C2×C4C22×C3⋊D311yescharacteristic
243C8—91no—
244C8—91no—
245C8C22×C3⋊D312yes—
246C2×C3—24no—
247C2×C3—48no—
248C2×C3—44no—
249C2×C3C2×D3×D414yes—
250D3—68no—
251D3—124no—
252C22—21no—
253C22—22no—
254C22—81no—
255C22—92no—
256C22—181no—
257C22—722no—
258C22D4×C3⋊D311yescharacteristic
259C4—21no—
260C4—92no—
261C4—181no—
262C4—182no—
263C4D4×C3⋊D311yescharacteristic
264C4C23×C3⋊D311yescharacteristic
265C3#577514yescomplemented
266C2—21no—
267C2—42no—
268C2—41no—
269C2—182no—
270C2—361no—
271C2C2×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊D3
commutator, cosocleC32×C4C24
FrattiniC4C23×C3⋊D3
FittingC32×AGL(1,ℤ/8)C2
socleC2×C32C22×D4
hypercenterAGL(1,ℤ/8)C3⋊D3
hypocenterC32#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #64367C2semidirect
2. #64361C2semidirect
3. C32×AGL(1,ℤ/8)C2semidirect
4. #64159C2semidirect
5. C2×#3754C2semidirect
6. #64124C2semidirect
7. SD16×C3⋊D3C2semidirect
8. #64121C2semidirect
9. #64120C2semidirect
10. #64118C2semidirect
11. #64116C2semidirect
13. C32×#41C22semidirect
16. Q8×C3⋊D3C22semidirect
20. #3830C22semidirect
27. C2×#634C22semidirect
28. C32×SD16C22semidirect
29. C32×D8C22semidirect
30. C32×M16C22semidirect
31. #3756C22semidirect
32. #3755C22semidirect
33. #3754C22semidirect
34. #3754C22semidirect
35. #3754C22semidirect
36. #3753C22semidirect
37. C2×C3⋊(C3⋊C8)C22semidirect
38. #3746C22semidirect
39. #3744C22semidirect
40. #3743C22semidirect
41. C8×C3⋊D3C22semidirect
44. C3×AGL(1,ℤ/8)D3semidirect
84. C3×#41C2×D3semidirect
99. C3×SD16C2×D3semidirect
100. C3×D8C2×D3semidirect
101. C3×M16C2×D3semidirect
131. AGL(1,ℤ/8)C3⋊D3semidirect
180. #41C2×C3⋊D3semidirect
195. SD16C2×C3⋊D3semidirect
197. D8C2×C3⋊D3semidirect
199. M16C2×C3⋊D3semidirect
215. C32#577semidirect
265. C3#5775semidirect

Maximal subgroups

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

subgroupindex
#643672
#643612
C32×AGL(1,ℤ/8)2
#641592
C2×#37542
#641242
SD16×C3⋊D32
#641212
#641202
#641182
#641162
#57753