Finite Group Databasebeta

← #62826

Subgroups — #62826

Order 256.

3335 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3564C2112yesmaximal, complemented
2#3560C2112yesmaximal, complemented
3#3521C2124yesmaximal, complemented
4#3492C216yesmaximal, complemented
5#3490C216yesmaximal, complemented
6#3444C212yesmaximal, complemented
7C22×#126C211yesmaximal, characteristic, complemented
8#563C22124yescomplemented
9#560C2218yescomplemented
10#556C22124yescomplemented
11#554C22124yescomplemented
12#553C22124yescomplemented
13#553C22124yescomplemented
14#552C22124yescomplemented
15#552C22124yescomplemented
16#551C22124yescomplemented
17#551C22124yescomplemented
18#551C22124yescomplemented
19#547C22124yescomplemented
20#546C22124yescomplemented
21#546C22124yescomplemented
22#544C2216yescomplemented
23#543C2216yescomplemented
24#542C22124yescomplemented
25#541C22124yescomplemented
26#539C22112yescomplemented
27#539C22112yescomplemented
28#539C22124yescomplemented
29#538C22112yes—
30#538C22112yes—
31#538C22124yescomplemented
32#537C2216yescomplemented
33#534C2216yescomplemented
34#533C2216yescomplemented
35#532C2216yescomplemented
36#529C22112yescomplemented
37#529C22112yescomplemented
38#529C22124yescomplemented
39C2×#126C22112yescomplemented
40C2×#126C22112yescomplemented
41C2×#126C2218yescomplemented
42#525C22112yescomplemented
43#525C22112yescomplemented
44C2×#120C2214yes—
45#520C22112yescomplemented
46#518C22112yescomplemented
47C4×#41C22112yescomplemented
48C22×C4⋊C4C2213yescomplemented
49C22×C22⋊C4C2213yes—
50C22×C42C2211yescharacteristic, complemented
51C25C2311yescharacteristic
52C2×#41C23112yes—
53C2×#41C23112yes—
54C22×D4C23112yes—
55C23×C4C2313yes—
56C23×C4C2313yes—
57C4⋊Q8C2316yescomplemented
58C4⋊Q8C23124yescomplemented
59#127C2312yescomplemented
60#126C2318yes—
61#126C23124yes—
62#126C23124yes—
63#126C2318yes—
64#126C23124yes—
65#126C23124yes—
66#126C2318yescomplemented
67#126C23124yescomplemented
68#126C2318yescomplemented
69#126C23124yescomplemented
70#125C23112yescomplemented
71#125C23124yescomplemented
72#125C23124yescomplemented
73#125C23124yescomplemented
74#124C23124yes—
75#124C23124yes—
76#124C23112yescomplemented
77#123C23124yes—
78#123C23124yes—
79#123C23124yes—
80#123C23124yes—
81#123C23124yes—
82#123C23124yescomplemented
83#123C23124yescomplemented
84C22⋊Q8C23112yes—
85C22⋊Q8C23112yes—
86C22⋊Q8C23112yes—
87C22⋊Q8C23112yes—
88C22⋊Q8C23124yes—
89C22⋊Q8C23124yes—
90C22⋊Q8C23124yescomplemented
91C22⋊Q8C23124yescomplemented
92C22⋊Q8C23124yescomplemented
93#121C23112yes—
94#121C23112yes—
95#121C23112yes—
96#121C23112yes—
97#121C23124yes—
98#121C23124yes—
99#121C23124yescomplemented
100#120C2318yes—
101#120C23124yes—
102#120C23124yes—
103C4×Q8C23112yescomplemented
104C4×Q8C23112yescomplemented
105C4×Q8C23124yescomplemented
106C4×D4C23124yes—
107C4×D4C23124yes—
108C4×D4C23124yes—
109C4×D4C23124yes—
110C4×D4C23124yes—
111C4×D4C23124yes—
112C4×D4C23112yescomplemented
113C4×D4C23112yescomplemented
114C4×D4C23124yescomplemented
115#117C23112yes—
116#117C23124yes—
117#117C23112yes—
118#117C23124yescomplemented
119#117C23124yescomplemented
120C2×C4⋊C4C2316yes—
121C2×C4⋊C4C2316yes—
122C2×C4⋊C4C23112yes—
123C2×C4⋊C4C23112yes—
124C2×C4⋊C4C23124yescomplemented
125C2×C22⋊C4C2316yes—
126C2×C22⋊C4C2316yes—
127C2×C22⋊C4C23112yes—
128C2×C22⋊C4C23112yes—
129C2×C22⋊C4C23112yes—
130C2×C22⋊C4C23124yes—
131C2×C42C2316yes—
132C2×C42C2316yes—
133C2×C42C2318yescomplemented
134C24—46no—
135C24C2411yescharacteristic
136C24C2416yes—
137C24C2414yes—
138#41—424no—
139#41—424no—
140C2×Q8C24112yes—
141C2×Q8C24112yes—
142C2×D4—424no—
143C2×D4C24124yes—
144C2×D4C24124yes—
145C2×D4C24112yes—
146C2×D4C24112yes—
147C22×C4—46no—
148C22×C4—46no—
149C22×C4C2416yes—
150C22×C4C2416yes—
151C22×C4C2413yes—
152C22×C4C2416yes—
153C22×C4C2413yes—
154C22×C4C24124yes—
155C22×C4C2416yes—
156C22×C4C2416yes—
157C22×C4C24112yes—
158C22×C4C24124yes—
159C4⋊C4C24124yes—
160C4⋊C4C24124yes—
161C4⋊C4C24112yes—
162C4⋊C4C24124yes—
163C4⋊C4C24124yes—
164C4⋊C4C24112yes—
165C4⋊C4C24124yes—
166C4⋊C4C24112yes—
167C4⋊C4C24112yes—
168C4⋊C4C24124yes—
169C22⋊C4C24124yes—
170C22⋊C4C24124yes—
171C22⋊C4C24124yes—
172C22⋊C4C24124yes—
173C22⋊C4C24124yes—
174C22⋊C4C24124yes—
175C22⋊C4C24124yes—
176C22⋊C4C24124yes—
177C42C2412yes—
178C42C2416yes—
179C42C2416yes—
180C42C2412yes—
181C42C24124yes—
182C42C24124yes—
183C23—26no—
184C23—212no—
185C23—46no—
186C23—46no—
187C23—412no—
188C23—88no—
189C23C2514yes—
190C23C2518yes—
191C23C2513yes—
192Q8—412no—
193Q8—412no—
194D4—248no—
195D4—424no—
196D4—412no—
197D4—412no—
198C2×C4—212no—
199C2×C4—212no—
200C2×C4—43no—
201C2×C4—46no—
202C2×C4—46no—
203C2×C4—46no—
204C2×C4—43no—
205C2×C4—412no—
206C2×C4—424no—
207C2×C4—424no—
208C2×C4C2516yes—
209C2×C4C2516yes—
210C2×C4C25124yes—
211C2×C4C25112yes—
212C22—28no—
213C22—23no—
214C22—23no—
215C22—224no—
216C22—212no—
217C22—812no—
218C22—164no—
219C22#58516yes—
220C22C2611yescharacteristic
221C4—26no—
222C4—212no—
223C4—26no—
224C4—224no—
225C2—26no—
226C2—44no—
227C2—48no—
228C2C2×#58513yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3564C2semidirect
2. #3560C2semidirect
3. #3521C2semidirect
4. #3492C2semidirect
5. #3490C2semidirect
6. #3444C2semidirect
7. C22×#126C2semidirect
8. #563C22semidirect
9. #560C22semidirect
10. #556C22semidirect
11. #554C22semidirect
12. #553C22semidirect
13. #553C22semidirect
14. #552C22semidirect
15. #552C22semidirect
16. #551C22semidirect
17. #551C22semidirect
18. #551C22semidirect
19. #547C22semidirect
20. #546C22semidirect
21. #546C22semidirect
22. #544C22semidirect
23. #543C22semidirect
24. #542C22semidirect
25. #541C22semidirect
26. #539C22semidirect
27. #539C22semidirect
28. #539C22semidirect
31. #538C22semidirect
32. #537C22semidirect
33. #534C22semidirect
34. #533C22semidirect
35. #532C22semidirect
36. #529C22semidirect
37. #529C22semidirect
38. #529C22semidirect
39. C2×#126C22semidirect
40. C2×#126C22semidirect
41. C2×#126C22semidirect
42. #525C22semidirect
43. #525C22semidirect
45. #520C22semidirect
46. #518C22semidirect
47. C4×#41C22semidirect
48. C22×C4⋊C4C22semidirect
50. C22×C42C22semidirect
57. C4⋊Q8C23semidirect
58. C4⋊Q8C23semidirect
59. #127C23semidirect
66. #126C23semidirect
67. #126C23semidirect
68. #126C23semidirect
69. #126C23semidirect
70. #125C23semidirect
71. #125C23semidirect
72. #125C23semidirect
73. #125C23semidirect
76. #124C23semidirect
82. #123C23semidirect
83. #123C23semidirect
90. C22⋊Q8C23semidirect
91. C22⋊Q8C23semidirect
92. C22⋊Q8C23semidirect
99. #121C23semidirect
103. C4×Q8C23semidirect
104. C4×Q8C23semidirect
105. C4×Q8C23semidirect
112. C4×D4C23semidirect
113. C4×D4C23semidirect
114. C4×D4C23semidirect
118. #117C23semidirect
119. #117C23semidirect
124. C2×C4⋊C4C23semidirect
133. C2×C42C23semidirect

Maximal subgroups

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

subgroupindex
#35642
#35602
#35212
#34922
#34902
#34442
C22×#1262