Finite Group Databasebeta

← #20038

Subgroups — #20038

Order 256.

344 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3384C211yesmaximal, characteristic, complemented
2#3331C211yesmaximal, characteristic, complemented
3#3294C211yesmaximal, characteristic
4D4×SD16C211yesmaximal, characteristic, complemented
5#1714C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1685C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1671C211yesmaximal, characteristic, complemented
10#1649C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C22×SD16—21no—
17#554C2211yescharacteristic, complemented
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20D42C2211yescharacteristic
21#501—21no—
22#500C2211yescharacteristic, complemented
23#498C2211yescharacteristic, complemented
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#487C2211yescharacteristic, complemented
27#486C2211yescharacteristic
28#483—21no—
29#480—21no—
30#478—21no—
31#476—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic, complemented
36#470—21no—
37#466—21no—
38#465—21no—
39#462C2211yescharacteristic, complemented
40#461C2211yescharacteristic
41#460C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#450—21no—
45#448—21no—
46#447—21no—
47#446C2211yescharacteristic, complemented
48#442—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic, complemented
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#417—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
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—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—41no—
77C2×SD16—44no—
78C2×SD16D411yescharacteristic
79C2×D8—21no—
80C2×D8D411yescharacteristic
81C2×M16—21no—
82C22×C8—21no—
83C22×C8—21no—
84C4⋊Q8—21no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127C2311yescharacteristic
89#125—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96#121—21no—
97#121—21no—
98#121—21no—
99#120—41no—
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108#107—21no—
109#107—21no—
110#107—41no—
111#107D411yescharacteristic
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106D411yescharacteristic
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103D411yescharacteristic
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
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—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×Q8—81no—
154C2×Q8C2×D411yescharacteristic
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—81no—
164C2×D4—81no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174Q16—81no—
175SD16—22no—
176SD16—42no—
177SD16—42no—
178SD16—42no—
179SD16—81no—
180SD16—82no—
181D8—22no—
182D8—81no—
183M16—81no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—42no—
190C2×C8—81no—
191C2×C8C2×D411yescharacteristic
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42—21no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—82no—
224C23—82no—
225C23—161no—
226Q8—21no—
227Q8—21no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—81no—
233Q8—82no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—81no—
243D4—82no—
244D4—82no—
245D4—82no—
246D4AGL(1,ℤ/8)12yes—
247D4#13712yes—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—161no—
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C8—22no—
269C8—21no—
270C8—21no—
271C8—21no—
272C8—21no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—164no—
290C22D4211yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4C2×AGL(1,ℤ/8)11yescharacteristic
301C4C2×#13711yescharacteristic
302C4#57611yescharacteristic
303C4#57711yescharacteristic
304C2—42no—
305C2—42no—
306C2—81no—
307C2—161no—
308C2#326411yescharacteristic
309C2#326611yescharacteristic
310C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3294C2
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. #3384C2semidirect
2. #3331C2semidirect
4. D4×SD16C2semidirect
5. #1714C2semidirect
6. #1702C2semidirect
7. #1685C2semidirect
8. #1683C2semidirect
9. #1671C2semidirect
10. #1649C2semidirect
11. #1647C2semidirect
12. #1629C2semidirect
13. #1622C2semidirect
14. #1586C2semidirect
15. #1585C2semidirect
17. #554C22semidirect
22. #500C22semidirect
23. #498C22semidirect
26. #487C22semidirect
35. #474C22semidirect
39. #462C22semidirect
47. #446C22semidirect
51. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
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
#33842
#33312
#32942
D4×SD162
#17142
#17022
#16852
#16832
#16712
#16492
#16472
#16292
#16222
#15862
#15852