Finite Group Databasebeta

← #20258

Subgroups — #20258

Order 256.

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

normal subgroups only

This group has 329 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#3397C211yesmaximal, characteristic, complemented
2#3325C211yesmaximal, characteristic, complemented
3#3324C211yesmaximal, characteristic, complemented
4#3305C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1638C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1619C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#550C2211yescharacteristic, complemented
19D42C2211yescharacteristic
20#496—21no—
21#493C2211yescharacteristic
22#492C2211yescharacteristic, complemented
23#487C2211yescharacteristic, complemented
24#486C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic, complemented
27#482—21no—
28#482—21no—
29#482—21no—
30#477C2211yescharacteristic, complemented
31#476C2211yescharacteristic
32#474C2211yescharacteristic
33#469—21no—
34#468—21no—
35#466—21no—
36#463—21no—
37#461C2211yescharacteristic, complemented
38#460—21no—
39#460C2211yescharacteristic
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#450—21no—
44#449—21no—
45#447—21no—
46#446C2211yescharacteristic, complemented
47#440—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic, complemented
53#418—21no—
54#418—21no—
55#416—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, 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×D4—21no—
69C22×D4—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16D411yescharacteristic
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—41no—
79C2×D8D411yescharacteristic
80C2×M16—21no—
81C2×M16—21no—
82C22×C8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127C2311yescharacteristic
87#127C2311yescharacteristic
88#126—41no—
89#124—21no—
90#124—41no—
91C22⋊Q8—21no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—41no—
98#120—41no—
99C4×Q8—21no—
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106#117—21no—
107#117—21no—
108#107—21no—
109#107—21no—
110#107D411yescharacteristic
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—41no—
121#103D411yescharacteristic
122#102—21no—
123#102—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—21no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C24—41no—
145#41—42no—
146#41—81no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
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—
172SD16—41no—
173SD16—42no—
174SD16—81no—
175SD16—81no—
176SD16—81no—
177D8—41no—
178D8—81no—
179D8—81no—
180D8—81no—
181D8—82no—
182M16—81no—
183M16—81no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—81no—
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—41no—
215C42—41no—
216C42C2411yescharacteristic
217C23—21no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—82no—
223C23—82no—
224C23—161no—
225Q8—21no—
226Q8—21no—
227Q8—41no—
228Q8—41no—
229Q8—81no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—81no—
245D4—81no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—161no—
250D4—161no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C8—21no—
269C8—21no—
270C8—21no—
271C8—21no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—164no—
293C22D4211yescharacteristic
294C4—21no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4#57511yescharacteristic
303C4#57511yescharacteristic
304C4#57611yescharacteristic
305C4#57611yescharacteristic
306C2—81no—
307C2—81no—
308C2—81no—
309C2—161no—
310C2—161no—
311C2#326811yescharacteristic
312C2#327311yescharacteristic
313C2#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. #3397C2semidirect
2. #3325C2semidirect
3. #3324C2semidirect
4. #3305C2semidirect
5. #1717C2semidirect
6. #1698C2semidirect
7. #1697C2semidirect
8. #1685C2semidirect
9. #1681C2semidirect
10. #1646C2semidirect
11. #1638C2semidirect
12. #1621C2semidirect
13. #1619C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect
18. #550C22semidirect
22. #492C22semidirect
23. #487C22semidirect
26. #486C22semidirect
30. #477C22semidirect
37. #461C22semidirect
46. #446C22semidirect
52. C8×D4C22semidirect
56. #335C22semidirect
57. #335C22semidirect
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
#33972
#33252
#33242
#33052
#17172
#16982
#16972
#16852
#16812
#16462
#16382
#16212
#16192
#15922
#15812