Finite Group Databasebeta

← #19772

Subgroups — #19772

Order 256.

355 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3330C211yesmaximal, characteristic, complemented
2#3294C211yesmaximal, characteristic, complemented
3#3294C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1681C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1654C211yesmaximal, characteristic, complemented
12#1639C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15#1580C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#547C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic, complemented
20D42C2211yescharacteristic
21#498C2211yescharacteristic, complemented
22#494C2211yescharacteristic, complemented
23#487C2211yescharacteristic, complemented
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#484—21no—
28#480—21no—
29#480—21no—
30#474C2211yescharacteristic
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#468—21no—
34#467—21no—
35#466—21no—
36#466—21no—
37#461C2211yescharacteristic
38#461C2211yescharacteristic, complemented
39#461C2211yescharacteristic, complemented
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#452—21no—
43#452—21no—
44#449—21no—
45#447—21no—
46#447—21no—
47C4×D8C2211yescharacteristic
48C4×D8C2211yescharacteristic
49#436C2211yescharacteristic, complemented
50C8×D4C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic, complemented
53#417—21no—
54C2×#102—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#332C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C22×D4—21no—
71C22×D4—21no—
72#135—44no—
73C2×Q16—21no—
74C2×Q16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8D411yescharacteristic
81C2×D8D411yescharacteristic
82C2×M16—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127C2311yescharacteristic
90#123—41no—
91#123—41no—
92#123—41no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—21no—
101#120—41no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#107D411yescharacteristic
113#107D411yescharacteristic
114#106—41no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—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—41no—
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—
146#41—42no—
147#41—42no—
148#41—42no—
149#41—81no—
150#41—81no—
151#41—81no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4C2×D411yescharacteristic
168C2×D4C2×D411yescharacteristic
169C2×D4C2×D411yescharacteristic
170C2×D4C2×D411yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178Q16—42no—
179Q16—81no—
180SD16—42no—
181SD16—42no—
182SD16—81no—
183D8—22no—
184D8—22no—
185D8—81no—
186D8—81no—
187M16—81no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—42no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8C2×D411yescharacteristic
197C2×C8C2×D411yescharacteristic
198C2×C8C2×D411yescharacteristic
199C2×C8C2×D411yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C4⋊C4C2×D411yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C42C2411yescharacteristic
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—82no—
228C23—82no—
229C23—161no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—81no—
247D4—82no—
248D4—82no—
249D4—82no—
250D4—82no—
251D4—82no—
252D4#13512yescomplemented
253D4AGL(1,ℤ/8)12yes—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D411yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C8—21no—
276C8—21no—
277C8—21no—
278C8—21no—
279C8—41no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—41no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—164no—
300C22D4211yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4C2×#13511yescharacteristic
310C4C2×AGL(1,ℤ/8)11yescharacteristic
311C4#57611yescharacteristic
312C4#57611yescharacteristic
313C2—42no—
314C2—42no—
315C2—81no—
316C2—81no—
317C2—161no—
318C2#326411yescharacteristic
319C2#328011yescharacteristic
320C2#328911yescharacteristic
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. #3294C2semidirect
4. #3283C2semidirect
5. #1720C2semidirect
6. #1701C2semidirect
7. #1681C2semidirect
8. #1679C2semidirect
9. #1669C2semidirect
10. #1654C2semidirect
11. #1654C2semidirect
12. #1639C2semidirect
13. #1625C2semidirect
14. #1581C2semidirect
15. #1580C2semidirect
19. #547C22semidirect
21. #498C22semidirect
22. #494C22semidirect
23. #487C22semidirect
38. #461C22semidirect
39. #461C22semidirect
49. #436C22semidirect
52. C8×D4C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
60. #332C22semidirect
64. D4⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect
252. D4#135semidirect

Maximal subgroups

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

subgroupindex
#33302
#32942
#32832
#17202
#17012
#16812
#16792
#16692
#16542
#16392
#16252
#15812
#15802