Finite Group Databasebeta

← #61649

Subgroups — #61649

Order 256.

1021 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3538C211yesmaximal, characteristic, complemented
2#3485C211yesmaximal, characteristic
3#3323C214yesmaximal, complemented
4#3315C214yesmaximal, complemented
5#3276C214yesmaximal, complemented
6#3266C214yesmaximal, complemented
7#3223C212yesmaximal, complemented
8#3202C212yesmaximal, complemented
9#3196C212yesmaximal, complemented
10#3123C212yesmaximal, complemented
11#3009C212yesmaximal, complemented
12#3007C212yesmaximal, complemented
13#2978C211yesmaximal, characteristic, complemented
14C2×H(5,𝔽2)C2211yescharacteristic
15C22×#41C2211yescharacteristic
16C2×#137C2214yescomplemented
17C2×#137C2212yescomplemented
18C22×SD16C2212yescomplemented
19#563—22no—
20#552C2214yescomplemented
21#551—22no—
22D4×Q8C2214yescomplemented
23#547—22no—
24#547C2214yes—
25#546—22no—
26#546C2214yes—
27#544—22no—
28#541C2212yescomplemented
29#539—22no—
30#536C2211yescharacteristic
31#536C2211yescharacteristic
32#536C2212yes—
33#534—22no—
34#525C2211yescharacteristic
35#525C2211yescharacteristic
36C2×#123—22no—
37C2×C22⋊Q8C2212yescomplemented
38#518C2211yescharacteristic
39#497C2212yescomplemented
40#494C2212yescomplemented
41#491C2212yescomplemented
42#488C2212yescomplemented
43#485C2214yescomplemented
44#485C2214yescomplemented
45#484C2212yescomplemented
46#481C2212yescomplemented
47#478C2214yescomplemented
48#470C2214yescomplemented
49#470C2214yescomplemented
50#468C2214yescomplemented
51#467C2214yescomplemented
52#464C2214yescomplemented
53#462C2214yescomplemented
54#461C2214yescomplemented
55#452C2214yes—
56#452C2214yescomplemented
57#452C2214yescomplemented
58#452C2214yescomplemented
59#451C2212yescomplemented
60#451C2214yescomplemented
61#451C2214yescomplemented
62#450C2214yes—
63#450C2214yescomplemented
64#448C2212yescomplemented
65#440C2214yescomplemented
66C4×SD16C2214yescomplemented
67#436C2214yescomplemented
68#435C2214yescomplemented
69#433C2212yescomplemented
70#418C2212yescomplemented
71#417C2214yescomplemented
72#416C2212yescomplemented
73#408C2212yes—
74#407C2212yescomplemented
75H(5,𝔽2)—44no—
76C2×#41—21no—
77C2×#41—22no—
78C2×#41—44no—
79C2×#41C2314yes—
80C2×#41C2312yes—
81C2×#41C2311yescharacteristic
82C2×#41C2311yescharacteristic
83C22×Q8C2312yes—
84C22×Q8C2311yescharacteristic
85C22×Q8C2312yes—
86C22×D4—21no—
87C22×D4—22no—
88C22×D4—21no—
89C22×D4C2311yescharacteristic
90C22×D4C2312yes—
91C23×C4—21no—
92C23×C4C2311yescharacteristic
93#137—48no—
94#137—44no—
95C2×Q16C2314yes—
96C2×Q16C2314yes—
97C2×Q16C2314yes—
98C2×SD16—44no—
99C2×SD16C2314yes—
100C2×SD16C2314yes—
101C2×SD16C2314yes—
102C2×M16C2314yes—
103C2×M16C2312yes—
104C22×C8C2312yes—
105C4⋊Q8—22no—
106C4⋊Q8—21no—
107C4⋊Q8—24no—
108C4⋊Q8C2312yes—
109C4⋊Q8C2312yes—
110#126—24no—
111#126—24no—
112#125—24no—
113#125—24no—
114#125C2312yes—
115#124—21no—
116#124—22no—
117#124C2312yes—
118#123—22no—
119#123—22no—
120#123—22no—
121#123—24no—
122#123—21no—
123#123—24no—
124#123—22no—
125#123—24no—
126#123—24no—
127#123D412yes—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—24no—
132C22⋊Q8—24no—
133C22⋊Q8D412yes—
134C22⋊Q8C2314yes—
135C22⋊Q8C2314yes—
136C22⋊Q8C2312yes—
137C22⋊Q8C2314yes—
138C22⋊Q8C2314yes—
139#121—24no—
140#121—24no—
141#121D412yes—
142#121C2312yes—
143#120—21no—
144#120—24no—
145#120D412yes—
146C4×Q8C2314yes—
147C4×D4—24no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4C2314yes—
151C4×D4C2314yes—
152#117—24no—
153#117C2311yescharacteristic
154#117C2312yes—
155#117C2311yescharacteristic
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4C2312yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162#107C2314yes—
163#106C2314yes—
164C4⋊C8C2314yes—
165#103C2314yes—
166#103C2314yes—
167#103C2314yes—
168#103C2314yes—
169#103C2314yes—
170#102C2314yes—
171#102C2314yes—
172#102C2314yes—
173C22⋊C8C2314yes—
174C22⋊C8C2312yes—
175C22⋊C8C2312yes—
176C22⋊C8C2314yes—
177#97C2312yes—
178C4×C8C2312yes—
179C24—21no—
180C24—22no—
181C24C2×D411yescharacteristic
182#41—28no—
183#41—24no—
184#41—42no—
185#41—44no—
186#41—42no—
187#41—42no—
188#41—48no—
189#41—44no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—44no—
196C2×Q8—42no—
197C2×Q8C2×D411yescharacteristic
198C2×Q8C2412yes—
199C2×Q8C2411yescharacteristic
200C2×Q8C2414yes—
201C2×D4—24no—
202C2×D4—24no—
203C2×D4—24no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—48no—
213C2×D4C2×D414yes—
214C2×D4C2×D412yes—
215C2×D4C2×D411yescharacteristic
216C2×D4C2411yescharacteristic
217C2×D4C2412yes—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4C2×D412yes—
229C22×C4C2×D411yescharacteristic
230C22×C4C2411yescharacteristic
231C22×C4C2412yes—
232C22×C4C2412yes—
233Q16—44no—
234Q16—44no—
235Q16—44no—
236SD16—28no—
237SD16—28no—
238SD16—44no—
239M16—44no—
240M16—42no—
241C2×C8—42no—
242C2×C8C2414yes—
243C2×C8C2414yes—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4C2×D414yes—
255C4⋊C4C2×D412yes—
256C4⋊C4C2414yes—
257C4⋊C4C2412yes—
258C22⋊C4—21no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4C2×D412yes—
269C22⋊C4C2×D414yes—
270C22⋊C4C2×D414yes—
271C42—22no—
272C42—24no—
273C42—21no—
274C42C2412yes—
275C23—24no—
276C23—22no—
277C23—21no—
278C23—41no—
279C23—42no—
280C23—44no—
281C23—82no—
282C23—84no—
283C23C22×D411yescharacteristic
284C23C22×D412yes—
285C23C22×D412yes—
286Q8—24no—
287Q8—24no—
288Q8—24no—
289Q8—22no—
290Q8—41no—
291Q8—42no—
292Q8—82no—
293Q8—82no—
294D4—24no—
295D4—22no—
296D4—22no—
297D4—41no—
298D4—42no—
299D4—44no—
300D4—44no—
301D4—44no—
302D4—48no—
303D4—42no—
304D4—82no—
305D4H(5,𝔽2)14yes—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—84no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4C22×D414yes—
326C2×C4C22×D412yes—
327C2×C4C22×D411yescharacteristic
328C2×C4C22×D412yes—
329C2×C4C2511yescharacteristic
330C8—24no—
331C8—24no—
332C22—22no—
333C22—24no—
334C22—22no—
335C22—22no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—24no—
340C22—44no—
341C22—42no—
342C22—41no—
343C22—84no—
344C22—82no—
345C22—164no—
346C22C23×D411yescharacteristic
347C4—24no—
348C4—22no—
349C4—21no—
350C4—22no—
351C4—44no—
352C4—42no—
353C4—41no—
354C4—42no—
355C4—44no—
356C4C2×H(5,𝔽2)12yes—
357C2—42no—
358C2—41no—
359C2—42no—
360C2—44no—
361C2—81no—
362C2#348411yescharacteristic
363C2#358612yes—
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3485C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #525 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3538C2semidirect
3. #3323C2semidirect
4. #3315C2semidirect
5. #3276C2semidirect
6. #3266C2semidirect
7. #3223C2semidirect
8. #3202C2semidirect
9. #3196C2semidirect
10. #3123C2semidirect
11. #3009C2semidirect
12. #3007C2semidirect
13. #2978C2semidirect
16. C2×#137C22semidirect
17. C2×#137C22semidirect
18. C22×SD16C22semidirect
20. #552C22semidirect
22. D4×Q8C22semidirect
28. #541C22semidirect
37. C2×C22⋊Q8C22semidirect
39. #497C22semidirect
40. #494C22semidirect
41. #491C22semidirect
42. #488C22semidirect
43. #485C22semidirect
44. #485C22semidirect
45. #484C22semidirect
46. #481C22semidirect
47. #478C22semidirect
48. #470C22semidirect
49. #470C22semidirect
50. #468C22semidirect
51. #467C22semidirect
52. #464C22semidirect
53. #462C22semidirect
54. #461C22semidirect
56. #452C22semidirect
57. #452C22semidirect
58. #452C22semidirect
59. #451C22semidirect
60. #451C22semidirect
61. #451C22semidirect
63. #450C22semidirect
64. #448C22semidirect
65. #440C22semidirect
66. C4×SD16C22semidirect
67. #436C22semidirect
68. #435C22semidirect
69. #433C22semidirect
70. #418C22semidirect
71. #417C22semidirect
72. #416C22semidirect
74. #407C22semidirect

Maximal subgroups

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

subgroupindex
#35382
#34852
#33232
#33152
#32762
#32662
#32232
#32022
#31962
#31232
#30092
#30072
#29782