Finite Group Databasebeta

← #20028

Subgroups — #20028

Order 256.

349 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#3397C211yesmaximal, characteristic, complemented
2#3302C211yesmaximal, characteristic, complemented
3#3294C211yesmaximal, characteristic, complemented
4#3274C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1667C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1641C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1628C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#556C2211yescharacteristic, complemented
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20D42C2211yescharacteristic
21#496—21no—
22#493C2211yescharacteristic
23#492C2211yescharacteristic, complemented
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic, complemented
28#485—21no—
29#480—21no—
30#474C2211yescharacteristic
31#474C2211yescharacteristic
32#469—21no—
33#467—21no—
34#466—21no—
35#463—21no—
36#462C2211yescharacteristic, complemented
37#461C2211yescharacteristic
38#460—21no—
39#460C2211yescharacteristic
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic, complemented
43#452—21no—
44#449—21no—
45#447—21no—
46#447—21no—
47#446C2211yescharacteristic, complemented
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic, complemented
54#416—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#332C2211yescharacteristic, 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—
68C2×#41—21no—
69C22×D4—21no—
70C22×D4—21no—
71AGL(1,ℤ/8)—44no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8D411yescharacteristic
84C2×M16—21no—
85C22×C8—21no—
86C22×C8—21no—
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127—21no—
90#127C2311yescharacteristic
91#127C2311yescharacteristic
92#126—41no—
93#124—21no—
94#123—41no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—21no—
101#121—41no—
102#120—41no—
103C4×Q8—21no—
104C4×Q8C2311yescharacteristic
105C4×D4—21no—
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109#117—21no—
110C2×C4⋊C4—21no—
111#107—21no—
112#107—21no—
113#107D411yescharacteristic
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103D411yescharacteristic
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
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—42no—
148#41—81no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4C2×D411yescharacteristic
169C2×D4C2×D411yescharacteristic
170C2×D4C2×D411yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176Q16—81no—
177SD16—22no—
178SD16—42no—
179SD16—42no—
180SD16—81no—
181SD16—81no—
182D8—41no—
183D8—42no—
184D8—42no—
185D8—81no—
186D8—81no—
187D8—81no—
188M16—42no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
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—41no—
206C4⋊C4—41no—
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C42—21no—
220C42—41no—
221C42C2411yescharacteristic
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—82no—
228C23—82no—
229C23—161no—
230Q8—21no—
231Q8—21no—
232Q8—41no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—81no—
250D4—81no—
251D4—82no—
252D4—82no—
253D4—82no—
254D4—82no—
255D4—161no—
256D4AGL(1,ℤ/8)12yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
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—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—164no—
300C22D4211yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4C2×AGL(1,ℤ/8)11yescharacteristic
310C4#57511yescharacteristic
311C4#57611yescharacteristic
312C4#57611yescharacteristic
313C2—42no—
314C2—81no—
315C2—81no—
316C2—161no—
317C2—161no—
318C2#326411yescharacteristic
319C2#326811yescharacteristic
320C2#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. #3397C2semidirect
2. #3302C2semidirect
3. #3294C2semidirect
4. #3274C2semidirect
5. #1717C2semidirect
6. #1702C2semidirect
7. #1699C2semidirect
8. #1667C2semidirect
9. #1665C2semidirect
10. #1641C2semidirect
11. #1639C2semidirect
12. #1628C2semidirect
13. #1625C2semidirect
14. #1586C2semidirect
15. #1585C2semidirect
17. #556C22semidirect
23. #492C22semidirect
24. #487C22semidirect
27. #486C22semidirect
36. #462C22semidirect
42. #459C22semidirect
47. #446C22semidirect
53. C8×D4C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #332C22semidirect
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
#33972
#33022
#32942
#32742
#17172
#17022
#16992
#16672
#16652
#16412
#16392
#16282
#16252
#15862
#15852