Finite Group Databasebeta

← #20159

Subgroups — #20159

Order 256.

364 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3325C211yesmaximal, characteristic, complemented
2#3325C211yesmaximal, characteristic, complemented
3#3294C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1677C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1635C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1619C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#555C2211yescharacteristic
18#547C2211yescharacteristic
19D42C2211yescharacteristic, complemented
20D42C2211yescharacteristic, complemented
21#493C2211yescharacteristic
22#492C2211yescharacteristic, complemented
23#487C2211yescharacteristic, complemented
24#486C2211yescharacteristic, complemented
25#486C2211yescharacteristic, complemented
26#486C2211yescharacteristic, complemented
27#482—21no—
28#482—21no—
29#480—21no—
30#476C2211yescharacteristic, complemented
31#476C2211yescharacteristic, complemented
32#474C2211yescharacteristic, complemented
33#469—21no—
34#469—21no—
35#468—21no—
36#466—21no—
37#461C2211yescharacteristic, complemented
38#459C2211yescharacteristic
39#459C2211yescharacteristic
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#450—21no—
43#449—21no—
44#447—21no—
45#447—21no—
46#447—21no—
47C4×D8C2211yescharacteristic
48C4×D8C2211yescharacteristic
49#436C2211yescharacteristic, complemented
50#436C2211yescharacteristic, complemented
51#436C2211yescharacteristic, complemented
52C8×D4C2211yescharacteristic, complemented
53#418—21no—
54#418—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic, complemented
61#331C2211yescharacteristic, complemented
62D4⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72AGL(1,ℤ/8)—44no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—41no—
81C2×D8D411yescharacteristic
82C2×D8D411yescharacteristic
83C2×M16—21no—
84C2×M16—21no—
85C2×M16—21no—
86C22×C8—21no—
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127C2311yescharacteristic
90#127C2311yescharacteristic
91#124—41no—
92#123—41no—
93C22⋊Q8—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—21no—
101#120—41no—
102#120—41no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107#117—21no—
108#117—21no—
109C2×C4⋊C4—21no—
110#107—21no—
111#107D411yescharacteristic
112#107D411yescharacteristic
113#106—21no—
114#106—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—41no—
121#102—21no—
122#102—21no—
123#102—21no—
124#102—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146C24—41no—
147#41—42no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4C2×D411yescharacteristic
172C2×D4C2×D411yescharacteristic
173C2×D4C2×D411yescharacteristic
174C2×D4C2×D411yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180SD16—42no—
181SD16—42no—
182SD16—81no—
183D8—22no—
184D8—41no—
185D8—42no—
186D8—81no—
187D8—81no—
188D8—81no—
189D8—82no—
190M16—42no—
191M16—81no—
192M16—81no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—81no—
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C2×C8C2×D411yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C42—41no—
222C42C2411yescharacteristic
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23—82no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—161no—
234C23—161no—
235Q8—41no—
236Q8—41no—
237Q8—81no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—21no—
243D4—21no—
244D4—21no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—81no—
255D4—82no—
256D4—82no—
257D4—82no—
258D4—82no—
259D4—82no—
260D4—161no—
261D4AGL(1,ℤ/8)12yes—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4C22×D411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C8—21no—
281C8—21no—
282C8—21no—
283C8—21no—
284C8—41no—
285C8—41no—
286C8—41no—
287C8—41no—
288C8—41no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—164no—
308C22—164no—
309C22D4211yescharacteristic
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4C2×AGL(1,ℤ/8)11yescharacteristic
318C4#57511yescharacteristic
319C4#57611yescharacteristic
320C4#57611yescharacteristic
321C2—42no—
322C2—81no—
323C2—81no—
324C2—81no—
325C2—161no—
326C2—161no—
327C2#326411yescharacteristic
328C2#327311yescharacteristic
329C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3325C2semidirect
2. #3325C2semidirect
3. #3294C2semidirect
4. #3272C2semidirect
5. #1717C2semidirect
6. #1701C2semidirect
7. #1697C2semidirect
8. #1685C2semidirect
9. #1677C2semidirect
10. #1646C2semidirect
11. #1635C2semidirect
12. #1619C2semidirect
13. #1619C2semidirect
14. #1594C2semidirect
15. #1581C2semidirect
19. D42C22semidirect
20. D42C22semidirect
22. #492C22semidirect
23. #487C22semidirect
24. #486C22semidirect
25. #486C22semidirect
26. #486C22semidirect
30. #476C22semidirect
31. #476C22semidirect
32. #474C22semidirect
37. #461C22semidirect
49. #436C22semidirect
50. #436C22semidirect
51. #436C22semidirect
52. C8×D4C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #331C22semidirect
61. #331C22semidirect
62. D4⋊C8C22semidirect
63. D4⋊C8C22semidirect
64. D4⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect

Maximal subgroups

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

subgroupindex
#33252
#32942
#32722
#17172
#17012
#16972
#16852
#16772
#16462
#16352
#16192
#15942
#15812