Finite Group Databasebeta

← #20144

Subgroups — #20144

Order 256.

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

normal subgroups only

This group has 303 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#3393C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic
4#3302C211yesmaximal, characteristic, complemented
5#1723C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1689C211yesmaximal, characteristic, complemented
8#1680C211yesmaximal, characteristic, complemented
9#1673C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1650C211yesmaximal, characteristic, complemented
13#1647C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#556C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#495—21no—
24#494C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#484—21no—
29#484—21no—
30#482—21no—
31#475C2211yescharacteristic
32#474C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#468—21no—
36#468—21no—
37#467—21no—
38#464—21no—
39#464—21no—
40#462C2211yescharacteristic
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#449—21no—
46C8×Q8C2211yescharacteristic
47C4×Q16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54#417—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×D8D411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#126—41no—
85#126—41no—
86#124—21no—
87#124—21no—
88#123—41no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94#121—21no—
95#121—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—41no—
112#107D411yescharacteristic
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#103—21no—
129#103—21no—
130#103D411yescharacteristic
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—42no—
146#41—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8C2×D411yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4C2×D411yescharacteristic
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163Q16—42no—
164Q16—81no—
165Q16—81no—
166SD16—41no—
167SD16—81no—
168SD16—81no—
169SD16—81no—
170D8—41no—
171M16—42no—
172M16—81no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—81no—
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—21no—
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—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—21no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216Q8—21no—
217Q8—21no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—82no—
233D4—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C8—22no—
255C8—21no—
256C8—21no—
257C8—21no—
258C8—21no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22D4211yescharacteristic
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4#57511yescharacteristic
284C4#57511yescharacteristic
285C4#57611yescharacteristic
286C4#57711yescharacteristic
287C2—81no—
288C2—81no—
289C2—81no—
290C2#326711yescharacteristic
291C2#326811yescharacteristic
292C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3330C2
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. #3393C2semidirect
2. #3353C2semidirect
4. #3302C2semidirect
5. #1723C2semidirect
6. #1699C2semidirect
7. #1689C2semidirect
8. #1680C2semidirect
9. #1673C2semidirect
10. #1654C2semidirect
11. #1652C2semidirect
12. #1650C2semidirect
13. #1647C2semidirect
14. #1588C2semidirect
15. #1586C2semidirect

Maximal subgroups

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

subgroupindex
#33932
#33532
#33302
#33022
#17232
#16992
#16892
#16802
#16732
#16542
#16522
#16502
#16472
#15882
#15862