Finite Group Databasebeta

← #10596245

Subgroups — #10596245

Order 576.

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

normal subgroups only

This group has 804 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#64366C211yesmaximal, characteristic, complemented
2#64361C211yesmaximal, characteristic, complemented
3C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#64159C211yesmaximal, characteristic, complemented
5C2×#3753C211yesmaximal, characteristic, complemented
6#64125C212yesmaximal, complemented
7#64123C212yesmaximal, complemented
8#64120C212yesmaximal, complemented
9D8×C3⋊D3C212yesmaximal, complemented
10#64117C211yesmaximal, characteristic, complemented
11#64116C211yesmaximal, characteristic, complemented
12#5774—34nomaximal
13C32×#41C2211yescharacteristic, complemented
14C2×C32×D4C2211yescharacteristic, complemented
15C2×#638—22no—
16#3832C2212yescomplemented
17#3830—21no—
18#3830C2212yescomplemented
19D4×C3⋊D3—22no—
20D4×C3⋊D3—21no—
21D4×C3⋊D3C2212yescomplemented
22D4×C3⋊D3C2212yescomplemented
23#3828—21no—
24#3828C2211yescharacteristic
25#3828C2211yescharacteristic, complemented
26C2×C4⋊(C3⋊D3)—21no—
27C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
28C32×SD16C2212yescomplemented
29C32×D8C2212yescomplemented
30C32×M16C2211yescharacteristic, complemented
31#3756C2211yescharacteristic, complemented
32#3755C2211yescharacteristic, complemented
33#3754C2211yescharacteristic, complemented
34#3753C2211yescharacteristic, complemented
35#3753C2212yescomplemented
36#3753C2212yescomplemented
37C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
38#3745C2212yescomplemented
39#3744C2212yescomplemented
40#3743C2212yescomplemented
41C8×C3⋊D3C2212yescomplemented
42#1009—34no—
43#1004—34no—
44C3×AGL(1,ℤ/8)D314yescomplemented
45#950—34no—
46C2×#213—34no—
47#916—38no—
48#914—38no—
49#911—38no—
50D3×D8—38no—
51#908—34no—
52#907—34no—
53C23×C32—21no—
54C22×C3⋊D3—21no—
55C22×C3⋊D3—22no—
56C22×C3⋊D3—22no—
57C32×Q8C2311yescharacteristic
58C32×D4—21no—
59C32×D4—21no—
60C32×D4C2312yes—
61C32×D4C2311yescharacteristic
62C2×C32×C4—21no—
63C2×C32×C4C2311yescharacteristic
64#638—21no—
65#638—22no—
66#638—22no—
67#638—22no—
68#638D412yes—
69C2×C3⋊Dic3—22no—
70C4⋊(C3⋊D3)—22no—
71C4⋊(C3⋊D3)—21no—
72C4⋊(C3⋊D3)D411yescharacteristic
73C4⋊(C3⋊D3)C2312yes—
74C4⋊(C3⋊D3)C2311yescharacteristic
75C4×C3⋊D3—21no—
76C4×C3⋊D3—22no—
77C4×C3⋊D3C2312yes—
78#634D411yescharacteristic
79#634C2311yescharacteristic
80C32×C8C2312yes—
81C3⋊(C3⋊C8)C2311yescharacteristic
82C3⋊(C3⋊C8)C2311yescharacteristic
83#576—91noSylow
84C3×#41C2×D314yescomplemented
85C2×C3×D4C2×D314yescomplemented
86C2×#67—68no—
87#239—38no—
88#237—38no—
89#237—64no—
90D3×D4—38no—
91D3×D4—38no—
92D3×D4—64no—
93D3×D4—68no—
94#235—34no—
95#235—34no—
96#235—64no—
97C2×D12—34no—
98C2×D12—64no—
99C3×SD16C2×D318yescomplemented
100C3×D8C2×D318yescomplemented
101C3×M16C2×D314yescomplemented
102#216—34no—
103#215—34no—
104#214—34no—
105#213—34no—
106#213—38no—
107#213—38no—
108C2×C3⋊C8—34no—
109D24—38no—
110#204—38no—
111ΓL(1,𝔽25)—38no—
112D3×C8—38no—
113C22×C32—22no—
114C22×C32—21no—
115C22×C32—81no—
116C22×C32C2×D411yescharacteristic
117C2×C3⋊D3—22no—
118C2×C3⋊D3—21no—
119C2×C3⋊D3—82no—
120C2×C3⋊D3—82no—
121C2×C3⋊D3—81no—
122C2×C3⋊D3C2×D412yes—
123C32×C4—21no—
124C32×C4C2×D411yescharacteristic
125C32×C4C2411yescharacteristic
126C3⋊Dic3—21no—
127C3⋊Dic3C2×D412yes—
128H(5,𝔽2)—91no—
129H(5,𝔽2)—91no—
130AGL(1,ℤ/8)—92no—
131AGL(1,ℤ/8)—92no—
132AGL(1,ℤ/8)—91no—
133AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
134#135—91no—
135#135—92no—
136C2×D8—91no—
137C2×D8—92no—
138#131—91no—
139C23×C3—24no—
140C22×D3—68no—
141C22×D3—68no—
142C22×D3—64no—
143C3×Q8C22×D314yes—
144C3×D4—24no—
145C3×D4—24no—
146C3×D4C22×D318yes—
147C3×D4C22×D314yes—
148C2×C3×C4—24no—
149C2×C3×C4C22×D314yes—
150#67—38no—
151#67—68no—
152#67—68no—
153#67—68no—
154#67—64no—
155C2×Dic3—68no—
156D12—34no—
157D12—38no—
158D12—34no—
159D12—68no—
160D12—64no—
161C4×D3—38no—
162C4×D3—68no—
163C4×D3—64no—
164Dic6—34no—
165Dic6—34no—
166C3×C8C22×D318yes—
167C3⋊C8—34no—
168C3⋊C8—34no—
169C2×C32—21no—
170C2×C32—41no—
171C2×C32—42no—
172C2×C32C22×D411yescharacteristic
173C3⋊D3—22no—
174C3⋊D3—42no—
175C3⋊D3—41no—
176#41—91no—
177#41—92no—
178#41—92no—
179#41—91no—
180#41—181no—
181#41—181no—
182#41C2×C3⋊D311yescharacteristic, complemented
183C2×D4—92no—
184C2×D4—92no—
185C2×D4—91no—
186C2×D4—181no—
187C2×D4—182no—
188C2×D4—181no—
189C2×D4—182no—
190C2×D4C2×C3⋊D311yescharacteristic, complemented
191Q16—91no—
192SD16—91no—
193SD16—91no—
194SD16—92no—
195SD16C2×C3⋊D312yescomplemented
196D8—92no—
197D8—91no—
198D8—92no—
199D8—92no—
200D8C2×C3⋊D312yescomplemented
201M16—92no—
202M16C2×C3⋊D311yescharacteristic, complemented
203C2×C8—91no—
204C2×C8—92no—
205C22×C3—28no—
206C22×C3—24no—
207C22×C3—84no—
208C22×C3D3×D414yes—
209C2×D3—38no—
210C2×D3—68no—
211C2×D3—64no—
212C2×D3—248no—
213C2×D3—248no—
214C2×D3—244no—
215C3×C4—24no—
216C3×C4D3×D414yes—
217C3×C4C23×D314yes—
218Dic3—38no—
219Dic3—64no—
220C32#57611yescharacteristic, complemented, Sylow
221C23—21no—
222C23—182no—
223C23—182no—
224C23—181no—
225Q8—91no—
226Q8—91no—
227Q8C22×C3⋊D311yescharacteristic
228D4—21no—
229D4—21no—
230D4—92no—
231D4—92no—
232D4—91no—
233D4—91no—
234D4—182no—
235D4—182no—
236D4—182no—
237D4—182no—
238D4—181no—
239D4—181no—
240D4C22×C3⋊D312yes—
241D4C22×C3⋊D311yescharacteristic
242C2×C4—21no—
243C2×C4—92no—
244C2×C4—182no—
245C2×C4—182no—
246C2×C4—181no—
247C2×C4C22×C3⋊D311yescharacteristic
248C8—91no—
249C8—91no—
250C8C22×C3⋊D312yes—
251C2×C3—24no—
252C2×C3—44no—
253C2×C3—48no—
254C2×C3C2×D3×D414yes—
255D3—68no—
256D3—128no—
257D3—124no—
258C22—21no—
259C22—22no—
260C22—81no—
261C22—92no—
262C22—182no—
263C22—181no—
264C22—721no—
265C22—722no—
266C22—722no—
267C22D4×C3⋊D311yescharacteristic
268C4—21no—
269C4—92no—
270C4—181no—
271C4D4×C3⋊D311yescharacteristic
272C4C23×C3⋊D311yescharacteristic
273C3#577414yescomplemented
274C2—21no—
275C2—42no—
276C2—41no—
277C2—182no—
278C2—362no—
279C2—361no—
280C2C2×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#576
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. #64366C2semidirect
2. #64361C2semidirect
3. C32×AGL(1,ℤ/8)C2semidirect
4. #64159C2semidirect
5. C2×#3753C2semidirect
6. #64125C2semidirect
7. #64123C2semidirect
8. #64120C2semidirect
9. D8×C3⋊D3C2semidirect
10. #64117C2semidirect
11. #64116C2semidirect
13. C32×#41C22semidirect
14. C2×C32×D4C22semidirect
16. #3832C22semidirect
18. #3830C22semidirect
21. D4×C3⋊D3C22semidirect
22. D4×C3⋊D3C22semidirect
25. #3828C22semidirect
27. C2×C4⋊(C3⋊D3)C22semidirect
28. C32×SD16C22semidirect
29. C32×D8C22semidirect
30. C32×M16C22semidirect
31. #3756C22semidirect
32. #3755C22semidirect
33. #3754C22semidirect
34. #3753C22semidirect
35. #3753C22semidirect
36. #3753C22semidirect
37. C2×C3⋊(C3⋊C8)C22semidirect
38. #3745C22semidirect
39. #3744C22semidirect
40. #3743C22semidirect
41. C8×C3⋊D3C22semidirect
44. C3×AGL(1,ℤ/8)D3semidirect
84. C3×#41C2×D3semidirect
85. C2×C3×D4C2×D3semidirect
99. C3×SD16C2×D3semidirect
100. C3×D8C2×D3semidirect
101. C3×M16C2×D3semidirect
133. AGL(1,ℤ/8)C3⋊D3semidirect
182. #41C2×C3⋊D3semidirect
190. C2×D4C2×C3⋊D3semidirect
195. SD16C2×C3⋊D3semidirect
200. D8C2×C3⋊D3semidirect
202. M16C2×C3⋊D3semidirect
220. C32#576semidirect
273. C3#5774semidirect

Maximal subgroups

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

subgroupindex
#643662
#643612
C32×AGL(1,ℤ/8)2
#641592
C2×#37532
#641252
#641232
#641202
D8×C3⋊D32
#641172
#641162
#57743