Finite Group Databasebeta

← #60734

Subgroups — #60734

Order 256.

1197 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C212yesmaximal, complemented
2#3191C214yesmaximal, complemented
3#3190C214yesmaximal, complemented
4#3189C211yesmaximal, characteristic, complemented
5#3188C214yesmaximal, complemented
6#3187C212yesmaximal, complemented
7#3186C211yesmaximal, characteristic, complemented
8#3091C214yesmaximal, complemented
9#3050C214yesmaximal, complemented
10#3002C214yesmaximal, complemented
11C22×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#41C2212yescomplemented
13C24×C4C2211yescharacteristic
14C2×#135C2214yescomplemented
15C23×C8C2212yescomplemented
16#525—22no—
17#525—22no—
18#525—22no—
19#525—22no—
20#525C2214yes—
21#525C2214yes—
22#525C2214yescomplemented
23#525C2214yescomplemented
24C2×#123—22no—
25C2×C22⋊Q8C2212yes—
26C2×C22⋊Q8C2212yescomplemented
27C2×#121C2212yes—
28C2×#121C2212yescomplemented
29C2×#120—22no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×#117C2212yescomplemented
33#485C2218yescomplemented
34#484C2214yescomplemented
35#483C2214yescomplemented
36#482C2218yescomplemented
37#481C2214yescomplemented
38#480C2214yescomplemented
39#467C2214yescomplemented
40#467C2214yescomplemented
41#466C2214yescomplemented
42#466C2214yescomplemented
43#465C2218yescomplemented
44#465C2218yescomplemented
45#452C2218yescomplemented
46#451C2214yescomplemented
47#450C2214yescomplemented
48#449C2218yescomplemented
49#448C2214yescomplemented
50#447C2214yescomplemented
51#427C2214yescomplemented
52#416C2218yescomplemented
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55C2×C22⋊C8C2214yescomplemented
56C2×C22⋊C8C2214yescomplemented
57C2×C22⋊C8C2212yescomplemented
58C25D411yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—48no—
62C2×#41C2314yes—
63C2×#41C2314yes—
64C22×Q8C2312yes—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4C2312yes—
69C23×C4—22no—
70C23×C4—44no—
71C23×C4D411yescharacteristic
72C23×C4D412yes—
73C23×C4D412yes—
74C23×C4D411yescharacteristic
75C23×C4D411yescharacteristic
76C23×C4C2312yes—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2312yes—
80C23×C4C2311yescharacteristic
81#135—48no—
82C2×Q16C2314yescomplemented
83C2×SD16C2318yescomplemented
84C2×D8C2314yescomplemented
85C22×C8—44no—
86C22×C8C2314yes—
87C22×C8C2312yes—
88C22×C8C2312yes—
89C22×C8C2314yes—
90#123—24no—
91#123—24no—
92#123—24no—
93#123—24no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8C2314yes—
99C22⋊Q8C2314yes—
100C22⋊Q8C2314yes—
101C22⋊Q8C2314yes—
102#121—22no—
103#121—22no—
104#121—22no—
105#121—22no—
106#121C2314yes—
107#121C2314yes—
108#121C2314yes—
109#121C2314yes—
110#120—24no—
111#120—24no—
112#120—24no—
113#120—24no—
114C4×D4—24no—
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—24no—
118C4×D4—24no—
119C4×D4—24no—
120C4×D4—24no—
121C4×D4—24no—
122#117—22no—
123#117—22no—
124#117C2314yes—
125#117C2314yes—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C42—22no—
136#107C2314yescomplemented
137#107C2314yescomplemented
138#106C2318yescomplemented
139#103C2318yescomplemented
140#103C2318yescomplemented
141#102C2318yescomplemented
142#102C2318yescomplemented
143C22⋊C8C2314yes—
144C22⋊C8C2314yes—
145C22⋊C8C2318yes—
146C24—24no—
147C24—22no—
148C24—42no—
149C24—42no—
150C24C2×D411yescharacteristic
151C24C2×D412yes—
152C24C2×D412yes—
153C24C2×D411yescharacteristic
154C24C2×D411yescharacteristic
155#41—216no—
156#41—48no—
157#41—48no—
158#41—48no—
159C2×Q8—22no—
160C2×Q8—44no—
161C2×Q8C2414yes—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—22no—
169C2×D4—44no—
170C2×D4—48no—
171C2×D4—48no—
172C2×D4—48no—
173C2×D4C2414yes—
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—48no—
190C22×C4C2×D412yes—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D412yes—
193C22×C4C2×D412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D412yes—
197C22×C4C2×D412yes—
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D412yes—
203C22×C4C2412yes—
204C22×C4C2412yes—
205C22×C4C2411yescharacteristic
206C22×C4C2411yescharacteristic
207C22×C4C2411yescharacteristic
208Q16—44no—
209SD16—48no—
210D8—44no—
211C2×C8—28no—
212C2×C8—42no—
213C2×C8—44no—
214C2×C8—42no—
215C2×C8C2414yes—
216C2×C8C2414yes—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4C2414yes—
224C4⋊C4C2414yes—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C42—24no—
236C42—24no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—22no—
244C23—24no—
245C23—24no—
246C23—41no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23—416no—
253C23—44no—
254C23—44no—
255C23—44no—
256C23#13514yescomplemented
257C23C22×D412yes—
258C23C22×D411yescharacteristic
259C23C22×D411yescharacteristic
260C23C22×D412yes—
261C23C22×D411yescharacteristic
262Q8—28no—
263Q8—44no—
264D4—28no—
265D4—44no—
266D4—48no—
267D4—48no—
268D4—48no—
269D4—48no—
270D4—48no—
271D4—48no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4C22×D412yes—
298C2×C4C22×D411yescharacteristic
299C2×C4C22×D411yescharacteristic
300C2×C4C22×D412yes—
301C2×C4C22×D411yescharacteristic
302C2×C4H(5,𝔽2)14yes—
303C2×C4C2511yescharacteristic
304C8—24no—
305C8—24no—
306C22—28no—
307C22—22no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—44no—
319C22—48no—
320C22—48no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—88no—
326C22#53412yes—
327C22C2×#13514yes—
328C22C2×#13512yes—
329C22#57512yes—
330C22C23×D411yescharacteristic
331C4—22no—
332C4—21no—
333C4—21no—
334C4—22no—
335C4—44no—
336C4—44no—
337C4—44no—
338C4#53412yes—
339C4C2×H(5,𝔽2)12yes—
340C2—22no—
341C2—22no—
342C2—22no—
343C2—24no—
344C2—42no—
345C2—84no—
346C2C2×#53411yescharacteristic
347C2C22×#13511yescharacteristic
348C2C2×#57511yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutator, Frattini, cosocleC2×C4C25
socleC22C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#525C2semidirect
2. #3191C2semidirect
3. #3190C2semidirect
4. #3189C2semidirect
5. #3188C2semidirect
6. #3187C2semidirect
7. #3186C2semidirect
8. #3091C2semidirect
9. #3050C2semidirect
10. #3002C2semidirect
11. C22×C22⋊C8C2semidirect
12. C22×#41C22semidirect
14. C2×#135C22semidirect
15. C23×C8C22semidirect
22. #525C22semidirect
23. #525C22semidirect
26. C2×C22⋊Q8C22semidirect
28. C2×#121C22semidirect
32. C2×#117C22semidirect
33. #485C22semidirect
34. #484C22semidirect
35. #483C22semidirect
36. #482C22semidirect
37. #481C22semidirect
38. #480C22semidirect
39. #467C22semidirect
40. #467C22semidirect
41. #466C22semidirect
42. #466C22semidirect
43. #465C22semidirect
44. #465C22semidirect
45. #452C22semidirect
46. #451C22semidirect
47. #450C22semidirect
48. #449C22semidirect
49. #448C22semidirect
50. #447C22semidirect
51. #427C22semidirect
52. #416C22semidirect
55. C2×C22⋊C8C22semidirect
56. C2×C22⋊C8C22semidirect
57. C2×C22⋊C8C22semidirect
82. C2×Q16C23semidirect
83. C2×SD16C23semidirect
84. C2×D8C23semidirect
136. #107C23semidirect
137. #107C23semidirect
138. #106C23semidirect
139. #103C23semidirect
140. #103C23semidirect
141. #102C23semidirect
142. #102C23semidirect
256. C23#135semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
#31912
#31902
#31892
#31882
#31872
#31862
#30912
#30502
#30022
C22×C22⋊C82