Finite Group Databasebeta

← #20257

Subgroups — #20257

Order 256.

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

normal subgroups only

This group has 304 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#3368C211yesmaximal, characteristic, complemented
3#3332C211yesmaximal, characteristic
4#3302C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1700C211yesmaximal, characteristic, complemented
8#1689C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1653C211yesmaximal, characteristic, complemented
12#1641C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1582C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic
17#556C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#498C2211yescharacteristic
21#497—21no—
22#494C2211yescharacteristic
23#490—21no—
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#481—21no—
30#478—21no—
31#475C2211yescharacteristic
32#474C2211yescharacteristic
33#474C2211yescharacteristic
34#469—21no—
35#467—21no—
36#464—21no—
37#462—21no—
38#462C2211yescharacteristic
39#462C2211yescharacteristic
40#461C2211yescharacteristic
41#460—21no—
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#449—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48#441—21no—
49#440—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#417—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×M16—21no—
79C22×C8—21no—
80C4⋊Q8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#126—41no—
87#125—21no—
88#125—41no—
89#124—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94#121—21no—
95#121—41no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#106—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#103—21no—
124#103D411yescharacteristic
125#103D411yescharacteristic
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—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146#41—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—42no—
166Q16—81no—
167Q16—81no—
168Q16—81no—
169SD16—41no—
170SD16—41no—
171SD16—42no—
172SD16—81no—
173D8—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—81no—
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—21no—
209C42—21no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—41no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
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—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—82no—
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—
274C22D4211yescharacteristic
275C4—21no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4#57511yescharacteristic
286C4#57511yescharacteristic
287C4#57611yescharacteristic
288C4#57711yescharacteristic
289C2—81no—
290C2—81no—
291C2—161no—
292C2#326811yescharacteristic
293C2#327511yescharacteristic
294C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3332C2
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. #3368C2semidirect
4. #3302C2semidirect
5. #1720C2semidirect
6. #1702C2semidirect
7. #1700C2semidirect
8. #1689C2semidirect
9. #1681C2semidirect
10. #1654C2semidirect
11. #1653C2semidirect
12. #1641C2semidirect
13. #1628C2semidirect
14. #1592C2semidirect
15. #1582C2semidirect

Maximal subgroups

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

subgroupindex
#33982
#33682
#33322
#33022
#17202
#17022
#17002
#16892
#16812
#16542
#16532
#16412
#16282
#15922
#15822