Finite Group Databasebeta

← #20267

Subgroups — #20267

Order 256.

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

normal subgroups only

This group has 308 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#3398C211yesmaximal, characteristic, complemented
2#3364C211yesmaximal, characteristic, complemented
3#3324C211yesmaximal, characteristic
4#3324C211yesmaximal, characteristic
5#1720C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1689C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1652C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1639C211yesmaximal, characteristic
13#1626C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic, complemented
17#556C2211yescharacteristic
18#556C2211yescharacteristic
19D42C2211yescharacteristic
20#498C2211yescharacteristic, complemented
21#497—21no—
22#494C2211yescharacteristic, complemented
23#488—21no—
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#482—21no—
29#482—21no—
30#479—21no—
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#474C2211yescharacteristic
34#468—21no—
35#468—21no—
36#464—21no—
37#464—21no—
38#462—21no—
39#462C2211yescharacteristic, complemented
40#462C2211yescharacteristic, complemented
41#461C2211yescharacteristic
42#460C2211yescharacteristic
43#460C2211yescharacteristic
44#450—21no—
45#450—21no—
46#446C2211yescharacteristic, complemented
47C8×Q8C2211yescharacteristic, complemented
48#441—21no—
49C4×Q16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#418—21no—
55#418—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#332C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic, complemented
67C22×D4—21no—
68C22×D4—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×SD16D411yescharacteristic
78C2×M16—21no—
79C2×M16—21no—
80C4⋊Q8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127C2311yescharacteristic
85#126—41no—
86#126—41no—
87#125—21no—
88#125—41no—
89#124—21no—
90#124—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94#121—21no—
95#120—41no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105#117—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#106—41no—
110#106D411yescharacteristic
111#106D411yescharacteristic
112C4⋊C8—21no—
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—21no—
123#103D411yescharacteristic
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—21no—
134#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—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×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—81no—
158C2×D4—81no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163Q16—42no—
164Q16—42no—
165Q16—81no—
166Q16—81no—
167SD16—41no—
168SD16—41no—
169SD16—81no—
170SD16—81no—
171SD16—81no—
172M16—81no—
173M16—81no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C4⋊C4—21no—
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⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—41no—
216C23—82no—
217C23—82no—
218C23—161no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—21no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—81no—
238D4—82no—
239D4—82no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
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—81no—
272C22—81no—
273C22—81no—
274C22—164no—
275C22D4211yescharacteristic
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4#57511yescharacteristic
287C4#57511yescharacteristic
288C4#57611yescharacteristic
289C4#57711yescharacteristic
290C2—81no—
291C2—81no—
292C2—161no—
293C2#326811yescharacteristic
294C2#327511yescharacteristic
295C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalD42C22
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. #3398C2semidirect
2. #3364C2semidirect
5. #1720C2semidirect
6. #1702C2semidirect
7. #1699C2semidirect
8. #1689C2semidirect
9. #1682C2semidirect
10. #1652C2semidirect
11. #1652C2semidirect
13. #1626C2semidirect
14. #1592C2semidirect
15. #1583C2semidirect
16. #557C22semidirect
20. #498C22semidirect
22. #494C22semidirect
24. #487C22semidirect
39. #462C22semidirect
40. #462C22semidirect
46. #446C22semidirect
47. C8×Q8C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
60. #332C22semidirect
62. Q8⋊C8C22semidirect
63. Q8⋊C8C22semidirect
66. #322C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#33982
#33642
#33242
#17202
#17022
#16992
#16892
#16822
#16522
#16392
#16262
#15922
#15832