Finite Group Databasebeta

← #20022

Subgroups — #20022

Order 256.

384 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3330C211yesmaximal, characteristic, complemented
2#3294C211yesmaximal, characteristic, complemented
3D4×D8C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1687C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1639C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17C22×D8—21no—
18#547C2211yescharacteristic, complemented
19#547C2211yescharacteristic, complemented
20D42C2211yescharacteristic
21D42C2211yescharacteristic
22#493C2211yescharacteristic
23#492C2211yescharacteristic, complemented
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic, complemented
26#486C2211yescharacteristic, complemented
27#486C2211yescharacteristic, complemented
28#484—21no—
29#480—21no—
30#474C2211yescharacteristic
31#474C2211yescharacteristic, complemented
32#468—21no—
33#468—21no—
34#466—21no—
35#466—21no—
36#461C2211yescharacteristic
37#461C2211yescharacteristic, complemented
38#459C2211yescharacteristic
39#459C2211yescharacteristic
40#459C2211yescharacteristic, complemented
41#459C2211yescharacteristic, complemented
42#450—21no—
43#449—21no—
44#449—21no—
45#447—21no—
46#447—21no—
47#447—21no—
48C4×D8C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic, complemented
51#436C2211yescharacteristic, complemented
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic, complemented
54#417—21no—
55C2×#102—21no—
56#334C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#332C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic, complemented
61#331C2211yescharacteristic, complemented
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73AGL(1,ℤ/8)—44no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—21no—
84C2×D8—44no—
85C2×D8D411yescharacteristic
86C2×D8D411yescharacteristic
87C2×M16—21no—
88C2×M16—21no—
89C22×C8—21no—
90C22×C8—21no—
91C4⋊Q8C2311yescharacteristic
92C4⋊Q8C2311yescharacteristic
93#127C2311yescharacteristic
94#127C2311yescharacteristic
95#123—41no—
96#123—41no—
97C22⋊Q8—21no—
98C22⋊Q8—21no—
99#121—21no—
100#121—21no—
101#121—21no—
102#121—21no—
103#121—21no—
104#121—21no—
105#120—41no—
106#120—41no—
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C4×D4C2311yescharacteristic
110C4×D4C2311yescharacteristic
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113#107—21no—
114#107—21no—
115#107D411yescharacteristic
116#107D411yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103—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—41no—
134#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138#102D411yescharacteristic
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C4×C8C2311yescharacteristic
146C24—41no—
147C24—41no—
148#41—42no—
149#41—42no—
150#41—81no—
151#41—81no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4C2×D411yescharacteristic
175C2×D4C2×D411yescharacteristic
176C2×D4C2×D411yescharacteristic
177C2×D4C2×D411yescharacteristic
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184SD16—42no—
185SD16—42no—
186SD16—81no—
187SD16—81no—
188D8—22no—
189D8—22no—
190D8—42no—
191D8—42no—
192D8—42no—
193D8—42no—
194D8—81no—
195D8—81no—
196M16—42no—
197M16—81no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—42no—
204C2×C8—81no—
205C2×C8C2×D411yescharacteristic
206C2×C8C2×D411yescharacteristic
207C2×C8C2×D411yescharacteristic
208C2×C8C2×D411yescharacteristic
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4C2×D411yescharacteristic
216C4⋊C4C2×D411yescharacteristic
217C4⋊C4C2×D411yescharacteristic
218C4⋊C4C2×D411yescharacteristic
219C22⋊C4—21no—
220C22⋊C4—21no—
221C22⋊C4—21no—
222C22⋊C4—21no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C42C2411yescharacteristic
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23—82no—
235C23—82no—
236C23—82no—
237C23—82no—
238C23—161no—
239C23—161no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244D4—21no—
245D4—21no—
246D4—21no—
247D4—21no—
248D4—21no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—81no—
259D4—81no—
260D4—82no—
261D4—82no—
262D4—82no—
263D4—82no—
264D4—82no—
265D4—82no—
266D4AGL(1,ℤ/8)12yes—
267D4AGL(1,ℤ/8)12yescomplemented
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C22×D411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C8—21no—
288C8—21no—
289C8—21no—
290C8—21no—
291C8—41no—
292C8—41no—
293C8—41no—
294C8—41no—
295C8—41no—
296C22—22no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—164no—
315C22—164no—
316C22D4211yescharacteristic
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4C2×AGL(1,ℤ/8)11yescharacteristic
325C4C2×AGL(1,ℤ/8)11yescharacteristic
326C4#57611yescharacteristic
327C4#57611yescharacteristic
328C2—42no—
329C2—42no—
330C2—81no—
331C2—81no—
332C2—161no—
333C2—161no—
334C2#326411yescharacteristic
335C2#326411yescharacteristic
336C2#329011yescharacteristic
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. #3330C2semidirect
2. #3294C2semidirect
3. D4×D8C2semidirect
4. #3272C2semidirect
5. #1717C2semidirect
6. #1701C2semidirect
7. #1687C2semidirect
8. #1679C2semidirect
9. #1665C2semidirect
10. #1639C2semidirect
11. #1639C2semidirect
12. #1625C2semidirect
13. #1625C2semidirect
14. #1586C2semidirect
15. #1586C2semidirect
18. #547C22semidirect
19. #547C22semidirect
23. #492C22semidirect
24. #487C22semidirect
25. #486C22semidirect
26. #486C22semidirect
27. #486C22semidirect
31. #474C22semidirect
37. #461C22semidirect
40. #459C22semidirect
41. #459C22semidirect
50. #436C22semidirect
51. #436C22semidirect
53. C8×D4C22semidirect
56. #334C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
60. #331C22semidirect
61. #331C22semidirect
63. D4⋊C8C22semidirect
64. D4⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect
267. D4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#33302
#32942
D4×D82
#32722
#17172
#17012
#16872
#16792
#16652
#16392
#16252
#15862