Finite Group Databasebeta

← #20166

Subgroups — #20166

Order 256.

347 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3358C211yesmaximal, characteristic, complemented
2#3329C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic, complemented
4#3274C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1687C211yesmaximal, characteristic, complemented
9#1678C211yesmaximal, characteristic, complemented
10#1639C211yesmaximal, characteristic, complemented
11#1631C211yesmaximal, characteristic, complemented
12#1627C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#555C2211yescharacteristic
18#550C2211yescharacteristic
19D4×Q8C2211yescharacteristic, complemented
20D42C2211yescharacteristic
21#493C2211yescharacteristic
22#492C2211yescharacteristic, complemented
23#487C2211yescharacteristic, complemented
24#486—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic, complemented
28#485—21no—
29#482—21no—
30#477C2211yescharacteristic, complemented
31#476C2211yescharacteristic
32#474—21no—
33#474C2211yescharacteristic
34#469—21no—
35#469—21no—
36#468—21no—
37#461C2211yescharacteristic
38#460C2211yescharacteristic
39#459—21no—
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic, complemented
43#449—21no—
44#448—21no—
45#447—21no—
46#447—21no—
47C8×Q8C2211yescharacteristic, complemented
48C4×SD16C2211yescharacteristic
49C4×D8—21no—
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic, complemented
54#419—21no—
55#418—21no—
56#334C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×Q8—21no—
69C22×D4—21no—
70C22×D4—21no—
71AGL(1,ℤ/8)—44no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—41no—
75C2×SD16D411yescharacteristic
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8D411yescharacteristic
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—21no—
85C4⋊Q8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127—21no—
89#127C2311yescharacteristic
90#127C2311yescharacteristic
91#124—41no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—41no—
100#120—41no—
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4—21no—
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107#117—21no—
108#117—21no—
109#107—21no—
110#107—21no—
111#107D411yescharacteristic
112#106—21no—
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103D411yescharacteristic
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#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—41no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8C2×D411yescharacteristic
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—41no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4C2×D411yescharacteristic
170C2×D4C2×D411yescharacteristic
171C2×D4C2×D411yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177SD16—22no—
178SD16—42no—
179SD16—81no—
180SD16—82no—
181D8—41no—
182D8—42no—
183D8—42no—
184D8—42no—
185D8—81no—
186D8—81no—
187M16—42no—
188M16—81no—
189M16—81no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C2×C8C2×D411yescharacteristic
198C2×C8C2×D411yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42—21no—
218C42—41no—
219C42C2411yescharacteristic
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23—82no—
226C23—82no—
227C23—161no—
228Q8—21no—
229Q8—41no—
230Q8—41no—
231Q8—82no—
232Q8—81no—
233Q8AGL(1,ℤ/8)12yes—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—82no—
249D4—81no—
250D4—81no—
251D4—82no—
252D4—82no—
253D4—161no—
254D4—161no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
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—82no—
269C2×C4—82no—
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4H(5,𝔽2)11yescharacteristic
273C8—21no—
274C8—21no—
275C8—21no—
276C8—21no—
277C8—41no—
278C8—41no—
279C8—41no—
280C8—41no—
281C8—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—164no—
298C22D4211yescharacteristic
299C4—22no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—81no—
306C4—81no—
307C4C2×AGL(1,ℤ/8)11yescharacteristic
308C4#57511yescharacteristic
309C4#57611yescharacteristic
310C4#57711yescharacteristic
311C2—81no—
312C2—81no—
313C2—81no—
314C2—161no—
315C2—161no—
316C2#326411yescharacteristic
317C2#327511yescharacteristic
318C2#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. #3358C2semidirect
2. #3329C2semidirect
3. #3325C2semidirect
4. #3274C2semidirect
5. #1717C2semidirect
6. #1703C2semidirect
7. #1701C2semidirect
8. #1687C2semidirect
9. #1678C2semidirect
10. #1639C2semidirect
11. #1631C2semidirect
12. #1627C2semidirect
13. #1624C2semidirect
14. #1594C2semidirect
15. #1583C2semidirect
19. D4×Q8C22semidirect
22. #492C22semidirect
23. #487C22semidirect
27. #486C22semidirect
30. #477C22semidirect
42. #459C22semidirect
47. C8×Q8C22semidirect
53. #436C22semidirect
56. #334C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
61. #331C22semidirect
62. Q8⋊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
#33582
#33292
#33252
#32742
#17172
#17032
#17012
#16872
#16782
#16392
#16312
#16272
#16242
#15942
#15832