Finite Group Databasebeta

← #20246

Subgroups — #20246

Order 256.

312 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3360C211yesmaximal, characteristic, complemented
2#3357C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic
4#3324C211yesmaximal, characteristic, complemented
5#1719C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1695C211yesmaximal, characteristic, complemented
9#1678C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1638C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#554C2211yescharacteristic, complemented
19D42C2211yescharacteristic
20#500C2211yescharacteristic, complemented
21#492C2211yescharacteristic
22#488—21no—
23#487C2211yescharacteristic, complemented
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#482—21no—
30#480—21no—
31#479—21no—
32#477C2211yescharacteristic, complemented
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#474C2211yescharacteristic
37#469—21no—
38#468—21no—
39#468—21no—
40#463—21no—
41#461C2211yescharacteristic, complemented
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#450—21no—
45#447—21no—
46C8×Q8C2211yescharacteristic, complemented
47C4×SD16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic, complemented
53#428—21no—
54#418—21no—
55#418—21no—
56#334C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C22×D4—21no—
68C22×D4—21no—
69C2×SD16—21no—
70C2×SD16—21no—
71C2×SD16—41no—
72C2×SD16D411yescharacteristic
73C2×D8—21no—
74C2×D8D411yescharacteristic
75C2×M16—21no—
76C2×M16—21no—
77C2×M16—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#127C2311yescharacteristic
82#126—41no—
83#125—21no—
84#124—21no—
85#124—41no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89#121—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#120—41no—
94C4×Q8—21no—
95C4×Q8C2311yescharacteristic
96C4×D4—21no—
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100#117—21no—
101#117—21no—
102#117—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—41no—
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#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C24—41no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—41no—
157C2×D4—81no—
158C2×D4—81no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166SD16—41no—
167SD16—42no—
168SD16—81no—
169SD16—82no—
170D8—41no—
171D8—81no—
172M16—42no—
173M16—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—41no—
209C42—41no—
210C42C2411yescharacteristic
211C23—21no—
212C23—21no—
213C23—21no—
214C23—41no—
215C23—82no—
216C23—82no—
217C23—161no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—81no—
223Q8—81no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236D4—82no—
237D4—161no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—161no—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—21no—
257C8—21no—
258C8—21no—
259C8—21no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—164no—
278C22D4211yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4#57511yescharacteristic
289C4#57511yescharacteristic
290C4#57711yescharacteristic
291C4#57711yescharacteristic
292C2—81no—
293C2—81no—
294C2—81no—
295C2—161no—
296C2#326711yescharacteristic
297C2#327511yescharacteristic
298C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3325C2
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. #3360C2semidirect
2. #3357C2semidirect
4. #3324C2semidirect
5. #1719C2semidirect
6. #1703C2semidirect
7. #1701C2semidirect
8. #1695C2semidirect
9. #1678C2semidirect
10. #1646C2semidirect
11. #1638C2semidirect
12. #1629C2semidirect
13. #1622C2semidirect
14. #1594C2semidirect
15. #1583C2semidirect
18. #554C22semidirect
20. #500C22semidirect
23. #487C22semidirect
24. #487C22semidirect
32. #477C22semidirect
41. #461C22semidirect
46. C8×Q8C22semidirect
52. #436C22semidirect
56. #334C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #333C22semidirect
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
#33602
#33572
#33252
#33242
#17192
#17032
#17012
#16952
#16782
#16462
#16382
#16292
#16222
#15942
#15832