Finite Group Databasebeta

← #20140

Subgroups — #20140

Order 256.

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

normal subgroups only

This group has 291 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#3353C211yesmaximal, characteristic
4#3302C211yesmaximal, characteristic
5#1723C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1674C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1652C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1648C211yesmaximal, characteristic, complemented
13#1647C211yesmaximal, characteristic
14#1588C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic
17#556C2211yescharacteristic
18#556C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#497—21no—
24#494C2211yescharacteristic
25#488—21no—
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#485—21no—
29#484—21no—
30#482—21no—
31#479—21no—
32#475C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#468—21no—
37#467—21no—
38#464—21no—
39#464—21no—
40#462—21no—
41#462C2211yescharacteristic
42#462C2211yescharacteristic
43#462C2211yescharacteristic
44#461C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#441—21no—
49C4×Q16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16D411yescharacteristic
75C2×SD16D411yescharacteristic
76C2×M16—21no—
77C22×C8—21no—
78C4⋊Q8—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#126—41no—
84#126—41no—
85#125—21no—
86#125—41no—
87#124—21no—
88#124—21no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—41no—
111#106—21no—
112#106—41no—
113#106D411yescharacteristic
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103D411yescharacteristic
131#103D411yescharacteristic
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
152C2×D4—41no—
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158Q16—42no—
159Q16—42no—
160Q16—81no—
161Q16—81no—
162Q16—81no—
163SD16—41no—
164SD16—41no—
165SD16—81no—
166M16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—81no—
173C2×C8C2×D411yescharacteristic
174C2×C8C2×D411yescharacteristic
175C2×C8C2×D411yescharacteristic
176C2×C8C2×D411yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—21no—
205C42—21no—
206C42—41no—
207C42—41no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—21no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—82no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—82no—
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C2×C4H(5,𝔽2)11yescharacteristic
247C8—22no—
248C8—21no—
249C8—21no—
250C8—21no—
251C8—21no—
252C8—41no—
253C8—41no—
254C8—41no—
255C8—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22D4211yescharacteristic
263C4—21no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4#57511yescharacteristic
275C4#57511yescharacteristic
276C4#57611yescharacteristic
277C4#57811yescharacteristic
278C2—81no—
279C2—81no—
280C2#326811yescharacteristic
281C2#326911yescharacteristic
282C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#547C22
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. #1723C2semidirect
6. #1700C2semidirect
7. #1699C2semidirect
8. #1674C2semidirect
9. #1669C2semidirect
10. #1652C2semidirect
11. #1652C2semidirect
12. #1648C2semidirect
14. #1588C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33982
#33642
#33532
#33022
#17232
#17002
#16992
#16742
#16692
#16522
#16482
#16472
#15882
#15852