Finite Group Databasebeta

← #20130

Subgroups — #20130

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#3354C211yesmaximal, characteristic, complemented
2#3341C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic, complemented
5#1718C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1695C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic, complemented
9#1671C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1646C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#555C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#498C2211yescharacteristic
22#487C2211yescharacteristic
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#485—21no—
27#484—21no—
28#482—21no—
29#482—21no—
30#482—21no—
31#481—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#474C2211yescharacteristic
37#474C2211yescharacteristic
38#469—21no—
39#468—21no—
40#466—21no—
41#465—21no—
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#449—21no—
45#449—21no—
46C4×D8C2211yescharacteristic
47C4×D8C2211yescharacteristic
48#436C2211yescharacteristic
49#436C2211yescharacteristic
50C8×D4C2211yescharacteristic
51C8×D4C2211yescharacteristic
52#428—21no—
53#427—21no—
54#417—21no—
55#416—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×SD16—21no—
70C2×SD16—21no—
71C2×D8D411yescharacteristic
72C2×D8D411yescharacteristic
73C2×M16—21no—
74C2×M16—21no—
75C22×C8—21no—
76C22×C8—21no—
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#124—41no—
82#124—41no—
83#123—41no—
84#123—41no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93C4×D4C2311yescharacteristic
94C4×D4C2311yescharacteristic
95C4×D4C2311yescharacteristic
96C4×D4C2311yescharacteristic
97#117—21no—
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#107—21no—
107#107D411yescharacteristic
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—41no—
126#103—41no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102D411yescharacteristic
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊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×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D411yescharacteristic
159C2×D4C2×D411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166SD16—81no—
167SD16—81no—
168D8—41no—
169D8—41no—
170M16—42no—
171M16—81no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—42no—
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—41no—
192C4⋊C4—41no—
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—41no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—81no—
219Q8—81no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—82no—
229D4—82no—
230D4—161no—
231D4—161no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
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—82no—
249C2×C4—82no—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4H(5,𝔽2)11yescharacteristic
253C8—21no—
254C8—21no—
255C8—21no—
256C8—21no—
257C8—41no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22D4211yescharacteristic
275C4—21no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4#57511yescharacteristic
285C4#57511yescharacteristic
286C4#57711yescharacteristic
287C4#57711yescharacteristic
288C2—81no—
289C2—81no—
290C2—81no—
291C2—81no—
292C2#326711yescharacteristic
293C2#326711yescharacteristic
294C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
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. #3354C2semidirect
2. #3341C2semidirect
3. #3330C2semidirect
4. #3303C2semidirect
5. #1718C2semidirect
6. #1703C2semidirect
7. #1695C2semidirect
8. #1679C2semidirect
9. #1671C2semidirect
10. #1646C2semidirect
11. #1646C2semidirect
12. #1635C2semidirect
13. #1635C2semidirect
14. #1586C2semidirect
15. #1586C2semidirect

Maximal subgroups

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

subgroupindex
#33542
#33412
#33302
#33032
#17182
#17032
#16952
#16792
#16712
#16462
#16352
#15862