Finite Group Databasebeta

← #20134

Subgroups — #20134

Order 256.

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

normal subgroups only

This group has 295 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#3354C211yesmaximal, characteristic
3#3340C211yesmaximal, characteristic, complemented
4#3331C211yesmaximal, characteristic, complemented
5#1716C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1691C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1672C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1634C211yesmaximal, characteristic, complemented
13#1632C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#547C2211yescharacteristic
20#498C2211yescharacteristic
21#498C2211yescharacteristic
22#498C2211yescharacteristic
23#494C2211yescharacteristic
24#488—21no—
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#483—21no—
29#482—21no—
30#482—21no—
31#481—21no—
32#479—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#474C2211yescharacteristic
38#474C2211yescharacteristic
39#470—21no—
40#469—21no—
41#465—21no—
42#463—21no—
43#462C2211yescharacteristic
44#461C2211yescharacteristic
45#452—21no—
46C8×Q8C2211yescharacteristic
47C4×SD16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54#427—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—41no—
70C2×SD16—21no—
71C2×SD16D411yescharacteristic
72C2×D8D411yescharacteristic
73C2×M16—21no—
74C2×M16—21no—
75C22×C8—21no—
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80#126—41no—
81#125—21no—
82#124—21no—
83#124—41no—
84#123—41no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89#121—21no—
90#121—21no—
91#121—21no—
92C4×Q8—21no—
93C4×Q8C2311yescharacteristic
94C4×D4—21no—
95C4×D4C2311yescharacteristic
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98#117—21no—
99#117—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107D411yescharacteristic
107#106—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—41no—
115#106D411yescharacteristic
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#103D411yescharacteristic
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
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×D4—41no—
152C2×D4—41no—
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C2×D4C2×D411yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161Q16—82no—
162Q16—81no—
163SD16—41no—
164SD16—42no—
165D8—41no—
166M16—42no—
167M16—81no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—42no—
174C2×C8C2×D411yescharacteristic
175C2×C8C2×D411yescharacteristic
176C2×C8C2×D411yescharacteristic
177C2×C8C2×D411yescharacteristic
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⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—21no—
205C42—41no—
206C42—41no—
207C42C2411yescharacteristic
208C23—21no—
209C23—21no—
210C23—21no—
211Q8—21no—
212Q8—21no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—81no—
218Q8—81no—
219D4—21no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—82no—
226D4—161no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—82no—
243C2×C4—161no—
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—
262C22—41no—
263C22—41no—
264C22—41no—
265C22D4211yescharacteristic
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4#57511yescharacteristic
277C4#57511yescharacteristic
278C4#57711yescharacteristic
279C4#57811yescharacteristic
280C2—81no—
281C2—81no—
282C2—81no—
283C2#326711yescharacteristic
284C2#326911yescharacteristic
285C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3354C2
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
3. #3340C2semidirect
4. #3331C2semidirect
5. #1716C2semidirect
6. #1703C2semidirect
7. #1691C2semidirect
8. #1683C2semidirect
9. #1672C2semidirect
10. #1654C2semidirect
11. #1652C2semidirect
12. #1634C2semidirect
13. #1632C2semidirect
14. #1588C2semidirect
15. #1586C2semidirect

Maximal subgroups

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

subgroupindex
#33602
#33542
#33402
#33312
#17162
#17032
#16912
#16832
#16722
#16542
#16522
#16342
#16322
#15882
#15862