Finite Group Databasebeta

← #62862

Subgroups — #62862

Order 256.

3749 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3545C214yesmaximal, complemented
2#3543C214yesmaximal, complemented
3#3542C2116yesmaximal, complemented
4#3534C218yesmaximal, complemented
5#3533C218yesmaximal, complemented
6#3532C218yesmaximal, complemented
7#3491C212yesmaximal, complemented
8#3490C214yesmaximal, complemented
9#3485C212yesmaximal, complemented
10#3484C214yesmaximal, complemented
11C2×#535C211yesmaximal, characteristic, complemented
12C2×#534C212yesmaximal, complemented
13C2×H(5,𝔽2)C2212yescomplemented
14C22×#41C2211yescharacteristic, complemented
15C23×D4C2211yescharacteristic, complemented
16C23×D4C2212yescomplemented
17#563C2214yescomplemented
18#562C22116yescomplemented
19#562C2218yescomplemented
20H(3,𝔽4)C2214yescomplemented
21#560C2218yescomplemented
22#560C22116yescomplemented
23#560C2218yescomplemented
24#559C2214yescomplemented
25#555C2218yescomplemented
26#553C22116yescomplemented
27#552C22116yescomplemented
28#551C22116yescomplemented
29#551C22116yescomplemented
30#551C2218yescomplemented
31#551C22116yescomplemented
32#550C2218yescomplemented
33#547C22116yescomplemented
34#546C22116yescomplemented
35#546C22116yescomplemented
36#546C22116yescomplemented
37#546C22116yescomplemented
38#546C22116yescomplemented
39#546C22116yescomplemented
40D42C22116yescomplemented
41D42C2214yescomplemented
42D42C22116yescomplemented
43#543C2212yescomplemented
44#542C22116yescomplemented
45#540C22116yescomplemented
46#540C22116yescomplemented
47#539C2218yescomplemented
48#539C2218yescomplemented
49#538C22116yescomplemented
50#538C22116yescomplemented
51#538C22116yescomplemented
52#538C2218yescomplemented
53#537C2218yescomplemented
54#536C2214yescomplemented
55#535C2218yescomplemented
56#535C2218yescomplemented
57#535C2218yescomplemented
58#535C22116yescomplemented
59#534C2212yes—
60#534C2218yescomplemented
61#534C2218yescomplemented
62#534C2214yescomplemented
63#534C2218yescomplemented
64#534C2218yescomplemented
65#534C2218yescomplemented
66#534C2218yescomplemented
67#534C2214yescomplemented
68C2×#127C2212yescomplemented
69#529C2218yescomplemented
70C2×#124C2212yescomplemented
71#525C22116yescomplemented
72#525C2218yescomplemented
73#525C2214yescomplemented
74C2×#123C2214yescomplemented
75C2×#123C2214yescomplemented
76C2×#123C2214yescomplemented
77C2×#121C2218yescomplemented
78C2×#121C2214yescomplemented
79C2×#120C2214yescomplemented
80C2×#120C2214yescomplemented
81C2×#120C2214yescomplemented
82C2×#120C2218yescomplemented
83#518C2214yescomplemented
84#518C2212yescomplemented
85C2×#117C2211yescharacteristic, complemented
86C22×C22⋊C4C2212yescomplemented
87C25C2312yes—
88C25C2312yescomplemented
89H(5,𝔽2)—48no—
90C2×#41—44no—
91C2×#41C2318yes—
92C2×#41C2314yes—
93C2×#41C2318yescomplemented
94C2×#41C23116yescomplemented
95C22×Q8C2311yescharacteristic, complemented
96C22×D4—44no—
97C22×D4—48no—
98C22×D4C2312yes—
99C22×D4C2314yes—
100C22×D4C2314yes—
101C22×D4C2318yes—
102C22×D4C2312yes—
103C22×D4C2314yes—
104C22×D4C2314yes—
105C22×D4C2312yes—
106C22×D4C2318yes—
107C22×D4C2318yes—
108C22×D4C2314yescomplemented
109C22×D4C2311yescharacteristic, complemented
110C22×D4C2318yescomplemented
111C22×D4C2318yescomplemented
112C22×D4C23116yescomplemented
113C23×C4C2311yescharacteristic
114C23×C4C2312yescomplemented
115C23×C4C2312yescomplemented
116C23×C4C2314yescomplemented
117C4⋊Q8C2318yescomplemented
118#127C2314yes—
119#127C2318yescomplemented
120#127C2314yescomplemented
121#127C23116yescomplemented
122#126C23116yescomplemented
123#126C23116yescomplemented
124#126C23116yescomplemented
125#126C23116yescomplemented
126#125C2318yescomplemented
127#125C23116yescomplemented
128#124C2314yes—
129#124C2318yescomplemented
130#124C2314yescomplemented
131#124C2318yescomplemented
132#124C23116yescomplemented
133#124C23116yescomplemented
134#124C2318yescomplemented
135#124C23116yescomplemented
136#123C2318yes—
137#123C23116yes—
138#123C2318yes—
139#123C2318yes—
140#123C2318yescomplemented
141#123C2318yescomplemented
142#123C2318yescomplemented
143#123C2318yescomplemented
144#123C2318yescomplemented
145#123C23116yescomplemented
146#123C2318yescomplemented
147#123C23116yescomplemented
148#123C23116yescomplemented
149#123C23116yescomplemented
150#123C23116yescomplemented
151#123C2318yescomplemented
152#123C23116yescomplemented
153#123C23116yescomplemented
154C22⋊Q8C23116yescomplemented
155C22⋊Q8C2318yescomplemented
156C22⋊Q8C23116yescomplemented
157C22⋊Q8C23116yescomplemented
158C22⋊Q8C2318yescomplemented
159C22⋊Q8C23116yescomplemented
160C22⋊Q8C23116yescomplemented
161#121C23116yes—
162#121C23116yes—
163#121C2318yes—
164#121C23116yes—
165#121C2318yes—
166#121C23116yescomplemented
167#121C23116yescomplemented
168#121C2318yescomplemented
169#121C23116yescomplemented
170#121C2318yescomplemented
171#121C23116yescomplemented
172#121C23116yescomplemented
173#121C23116yescomplemented
174#121C23116yescomplemented
175#121C23116yescomplemented
176#121C23116yescomplemented
177#121C23116yescomplemented
178#121C23116yescomplemented
179#121C23116yescomplemented
180#120C2318yes—
181#120C2318yes—
182#120C2318yes—
183#120C2318yes—
184#120C23116yes—
185#120C2318yes—
186#120C23116yes—
187#120C2318yescomplemented
188#120C2318yescomplemented
189#120C2318yescomplemented
190#120C2318yescomplemented
191#120C2318yescomplemented
192#120C23116yescomplemented
193#120C23116yescomplemented
194#120C2318yescomplemented
195#120C2318yescomplemented
196#120C23116yescomplemented
197C4×Q8C2318yescomplemented
198C4×D4C23116yes—
199C4×D4C23116yescomplemented
200C4×D4C23116yescomplemented
201C4×D4C23116yescomplemented
202C4×D4C23116yescomplemented
203C4×D4C23116yescomplemented
204C4×D4C23116yescomplemented
205C4×D4C23116yescomplemented
206C4×D4C2318yescomplemented
207#117C2318yescomplemented
208#117C2318yescomplemented
209#117C2314yescomplemented
210#117C23116yescomplemented
211#117C23116yescomplemented
212C2×C4⋊C4C2312yescomplemented
213C2×C4⋊C4C2318yescomplemented
214C2×C4⋊C4C2318yescomplemented
215C2×C22⋊C4C2314yes—
216C2×C22⋊C4C2318yes—
217C2×C22⋊C4C2312yes—
218C2×C22⋊C4C2318yes—
219C2×C22⋊C4C2314yescomplemented
220C2×C22⋊C4C2318yescomplemented
221C2×C22⋊C4C2318yescomplemented
222C2×C22⋊C4C2314yescomplemented
223C2×C22⋊C4C2318yescomplemented
224C2×C22⋊C4C2318yescomplemented
225C2×C22⋊C4C23116yescomplemented
226C2×C22⋊C4C2318yescomplemented
227C2×C42C2312yescomplemented
228C24—28no—
229C24—44no—
230C24—84no—
231C24—84no—
232C24C2414yes—
233C24C2414yes—
234C24C2414yes—
235C24C2412yes—
236C24C2412yes—
237C24C2412yes—
238C24C2418yes—
239C24C2414yes—
240C24C2411yescharacteristic
241#41—416no—
242#41—416no—
243#41—48no—
244#41—816no—
245C2×Q8—42no—
246C2×Q8C2418yes—
247C2×Q8C2414yes—
248C2×Q8C2414yes—
249C2×D4—216no—
250C2×D4—28no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—48no—
254C2×D4—48no—
255C2×D4—48no—
256C2×D4—42no—
257C2×D4—416no—
258C2×D4—416no—
259C2×D4—48no—
260C2×D4—416no—
261C2×D4—88no—
262C2×D4—88no—
263C2×D4—816no—
264C2×D4C2418yes—
265C2×D4C2418yes—
266C2×D4C2418yes—
267C2×D4C24116yes—
268C2×D4C24116yes—
269C2×D4C2418yes—
270C2×D4C24116yes—
271C2×D4C24116yes—
272C2×D4C24116yes—
273C2×D4C2412yes—
274C2×D4C2412yes—
275C2×D4C2414yes—
276C2×D4C2414yes—
277C2×D4C2414yes—
278C2×D4C2418yes—
279C2×D4C2414yes—
280C2×D4C24116yes—
281C2×D4C2412yes—
282C2×D4C2412yes—
283C22×C4—24no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—84no—
287C22×C4C2411yescharacteristic
288C22×C4C2414yes—
289C22×C4C2412yes—
290C22×C4C2414yes—
291C22×C4C2411yescharacteristic
292C22×C4C2414yes—
293C22×C4C2418yes—
294C22×C4C24116yes—
295C22×C4C2418yes—
296C22×C4C24116yes—
297C22×C4C2418yes—
298C22×C4C2418yes—
299C22×C4C2414yes—
300C22×C4C2418yes—
301C4⋊C4C24116yes—
302C4⋊C4C2418yes—
303C4⋊C4C24116yes—
304C4⋊C4C24116yes—
305C4⋊C4C24116yes—
306C4⋊C4C24116yes—
307C4⋊C4C2418yes—
308C4⋊C4C2418yes—
309C4⋊C4C2418yes—
310C22⋊C4C2418yes—
311C22⋊C4C2418yes—
312C22⋊C4C2418yes—
313C22⋊C4C2418yes—
314C22⋊C4C2418yes—
315C22⋊C4C24116yes—
316C22⋊C4C24116yes—
317C22⋊C4C24116yes—
318C22⋊C4C24116yes—
319C22⋊C4C24116yes—
320C22⋊C4C24116yes—
321C22⋊C4C24116yes—
322C22⋊C4C24116yes—
323C22⋊C4C24116yes—
324C22⋊C4C2418yes—
325C22⋊C4C24116yes—
326C42C2414yes—
327C42C2414yes—
328C42C24116yes—
329C42C2418yes—
330C23—24no—
331C23—24no—
332C23—24no—
333C23—28no—
334C23—28no—
335C23—28no—
336C23—44no—
337C23—42no—
338C23—44no—
339C23—48no—
340C23—44no—
341C23—48no—
342C23—48no—
343C23—48no—
344C23—416no—
345C23—44no—
346C23—48no—
347C23—816no—
348C23—816no—
349C23H(5,𝔽2)14yes—
350C23C2514yes—
351C23C2518yes—
352C23C2514yes—
353C23C2514yes—
354C23C2512yes—
355C23C2511yescharacteristic
356Q8—44no—
357Q8—44no—
358Q8—48no—
359D4—216no—
360D4—216no—
361D4—216no—
362D4—42no—
363D4—44no—
364D4—42no—
365D4—42no—
366D4—44no—
367D4—42no—
368D4—416no—
369D4—48no—
370D4—44no—
371D4—44no—
372D4—416no—
373D4—416no—
374D4—48no—
375D4—416no—
376D4—416no—
377D4—416no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—416no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—416no—
391C2×C4—48no—
392C2×C4C2518yes—
393C2×C4C2512yes—
394C2×C4C25116yes—
395C2×C4C2514yes—
396C2×C4C2518yes—
397C2×C4C2512yes—
398C22—24no—
399C22—22no—
400C22—24no—
401C22—24no—
402C22—28no—
403C22—24no—
404C22—28no—
405C22—28no—
406C22—216no—
407C22—28no—
408C22—44no—
409C22—48no—
410C22—48no—
411C22—84no—
412C22—88no—
413C22—88no—
414C22—88no—
415C22—816no—
416C22—164no—
417C22C2×H(5,𝔽2)14yes—
418C22C2×H(5,𝔽2)12yes—
419C22C2611yescharacteristic
420C4—22no—
421C4—22no—
422C4—28no—
423C4—24no—
424C4—216no—
425C4—28no—
426C2—24no—
427C2—22no—
428C2—44no—
429C2—44no—
430C2—48no—
431C2—44no—
432C2C22×H(5,𝔽2)11yescharacteristic
433C2H(7,𝔽2)12yes—
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. #3545C2semidirect
2. #3543C2semidirect
3. #3542C2semidirect
4. #3534C2semidirect
5. #3533C2semidirect
6. #3532C2semidirect
7. #3491C2semidirect
8. #3490C2semidirect
9. #3485C2semidirect
10. #3484C2semidirect
11. C2×#535C2semidirect
12. C2×#534C2semidirect
13. C2×H(5,𝔽2)C22semidirect
14. C22×#41C22semidirect
15. C23×D4C22semidirect
16. C23×D4C22semidirect
17. #563C22semidirect
18. #562C22semidirect
19. #562C22semidirect
20. H(3,𝔽4)C22semidirect
21. #560C22semidirect
22. #560C22semidirect
23. #560C22semidirect
24. #559C22semidirect
25. #555C22semidirect
26. #553C22semidirect
27. #552C22semidirect
28. #551C22semidirect
29. #551C22semidirect
30. #551C22semidirect
31. #551C22semidirect
32. #550C22semidirect
33. #547C22semidirect
34. #546C22semidirect
35. #546C22semidirect
36. #546C22semidirect
37. #546C22semidirect
38. #546C22semidirect
39. #546C22semidirect
40. D42C22semidirect
41. D42C22semidirect
42. D42C22semidirect
43. #543C22semidirect
44. #542C22semidirect
45. #540C22semidirect
46. #540C22semidirect
47. #539C22semidirect
48. #539C22semidirect
49. #538C22semidirect
50. #538C22semidirect
51. #538C22semidirect
52. #538C22semidirect
53. #537C22semidirect
54. #536C22semidirect
55. #535C22semidirect
56. #535C22semidirect
57. #535C22semidirect
58. #535C22semidirect
60. #534C22semidirect
61. #534C22semidirect
62. #534C22semidirect
63. #534C22semidirect
64. #534C22semidirect
65. #534C22semidirect
66. #534C22semidirect
67. #534C22semidirect
68. C2×#127C22semidirect
69. #529C22semidirect
70. C2×#124C22semidirect
71. #525C22semidirect
72. #525C22semidirect
73. #525C22semidirect
74. C2×#123C22semidirect
75. C2×#123C22semidirect
76. C2×#123C22semidirect
77. C2×#121C22semidirect
78. C2×#121C22semidirect
79. C2×#120C22semidirect
80. C2×#120C22semidirect
81. C2×#120C22semidirect
82. C2×#120C22semidirect
83. #518C22semidirect
84. #518C22semidirect
85. C2×#117C22semidirect
86. C22×C22⋊C4C22semidirect
88. C25C23semidirect
93. C2×#41C23semidirect
94. C2×#41C23semidirect
95. C22×Q8C23semidirect
108. C22×D4C23semidirect
109. C22×D4C23semidirect
110. C22×D4C23semidirect
111. C22×D4C23semidirect
112. C22×D4C23semidirect
114. C23×C4C23semidirect
115. C23×C4C23semidirect
116. C23×C4C23semidirect
117. C4⋊Q8C23semidirect
119. #127C23semidirect
120. #127C23semidirect
121. #127C23semidirect
122. #126C23semidirect
123. #126C23semidirect
124. #126C23semidirect
125. #126C23semidirect
126. #125C23semidirect
127. #125C23semidirect
129. #124C23semidirect
130. #124C23semidirect
131. #124C23semidirect
132. #124C23semidirect
133. #124C23semidirect
134. #124C23semidirect
135. #124C23semidirect
140. #123C23semidirect
141. #123C23semidirect
142. #123C23semidirect
143. #123C23semidirect
144. #123C23semidirect
145. #123C23semidirect
146. #123C23semidirect
147. #123C23semidirect
148. #123C23semidirect
149. #123C23semidirect
150. #123C23semidirect
151. #123C23semidirect
152. #123C23semidirect
153. #123C23semidirect
154. C22⋊Q8C23semidirect
155. C22⋊Q8C23semidirect
156. C22⋊Q8C23semidirect
157. C22⋊Q8C23semidirect
158. C22⋊Q8C23semidirect
159. C22⋊Q8C23semidirect
160. C22⋊Q8C23semidirect
166. #121C23semidirect
167. #121C23semidirect
168. #121C23semidirect
169. #121C23semidirect
170. #121C23semidirect
171. #121C23semidirect
172. #121C23semidirect
173. #121C23semidirect
174. #121C23semidirect
175. #121C23semidirect
176. #121C23semidirect
177. #121C23semidirect
178. #121C23semidirect
179. #121C23semidirect
187. #120C23semidirect
188. #120C23semidirect
189. #120C23semidirect
190. #120C23semidirect
191. #120C23semidirect
192. #120C23semidirect
193. #120C23semidirect
194. #120C23semidirect
195. #120C23semidirect
196. #120C23semidirect
197. C4×Q8C23semidirect
199. C4×D4C23semidirect
200. C4×D4C23semidirect
201. C4×D4C23semidirect
202. C4×D4C23semidirect
203. C4×D4C23semidirect
204. C4×D4C23semidirect
205. C4×D4C23semidirect
206. C4×D4C23semidirect
207. #117C23semidirect
208. #117C23semidirect
209. #117C23semidirect
210. #117C23semidirect
211. #117C23semidirect
212. C2×C4⋊C4C23semidirect
213. C2×C4⋊C4C23semidirect
214. C2×C4⋊C4C23semidirect
219. C2×C22⋊C4C23semidirect
220. C2×C22⋊C4C23semidirect
221. C2×C22⋊C4C23semidirect
222. C2×C22⋊C4C23semidirect
223. C2×C22⋊C4C23semidirect
224. C2×C22⋊C4C23semidirect
225. C2×C22⋊C4C23semidirect
226. C2×C22⋊C4C23semidirect
227. C2×C42C23semidirect

Maximal subgroups

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

subgroupindex
#35452
#35432
#35422
#35342
#35332
#35322
#34912
#34902
#34852
#34842
C2×#5352
C2×#5342