Finite Group Databasebeta

← #20042

Subgroups — #20042

Order 256.

304 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3384C211yesmaximal, characteristic, complemented
2#3335C211yesmaximal, characteristic, complemented
3#3329C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic
5#1716C211yesmaximal, characteristic, complemented
6#1704C211yesmaximal, characteristic, complemented
7#1691C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1671C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1634C211yesmaximal, characteristic, complemented
13#1632C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#554C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20#501—21no—
21#498C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#485—21no—
29#483—21no—
30#481—21no—
31#478—21no—
32#477C2211yescharacteristic
33#476—21no—
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#475C2211yescharacteristic
38#474C2211yescharacteristic
39#468—21no—
40#465—21no—
41#465—21no—
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#449—21no—
45#448—21no—
46#446C2211yescharacteristic
47#442—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C2×#106—21no—
54#419—21no—
55#416—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—
68C22×Q8—21no—
69C2×Q16—41no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16D411yescharacteristic
73C2×D8D411yescharacteristic
74C2×M16—21no—
75C22×C8—21no—
76C22×C8—21no—
77C4⋊Q8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#125—21no—
83#124—41no—
84#123—41no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90#121—21no—
91#121—21no—
92#121—21no—
93C4×Q8C2311yescharacteristic
94C4×D4—21no—
95C4×D4—21no—
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99#117—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107D411yescharacteristic
106#106—21no—
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#103—21no—
129#103D411yescharacteristic
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—
140#97—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×Q8—41no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8C2×D411yescharacteristic
154C2×D4—41no—
155C2×D4—41no—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D411yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—82no—
166SD16—41no—
167SD16—81no—
168SD16—81no—
169D8—22no—
170M16—81no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—42no—
177C2×C8—81no—
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C4⋊C4—21no—
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⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212C23—21no—
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—81no—
220Q8—81no—
221Q8—82no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—82no—
228D4—161no—
229D4#13712yes—
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—41no—
244C2×C4—41no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—161no—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4H(5,𝔽2)11yescharacteristic
251C8—22no—
252C8—21no—
253C8—21no—
254C8—21no—
255C8—21no—
256C8—41no—
257C8—41no—
258C8—41no—
259C8—41no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22D4211yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4C2×#13711yescharacteristic
281C4#57511yescharacteristic
282C4#57711yescharacteristic
283C4#57711yescharacteristic
284C2—42no—
285C2—81no—
286C2—81no—
287C2#326611yescharacteristic
288C2#326711yescharacteristic
289C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3303C2
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. #3384C2semidirect
2. #3335C2semidirect
3. #3329C2semidirect
5. #1716C2semidirect
6. #1704C2semidirect
7. #1691C2semidirect
8. #1683C2semidirect
9. #1671C2semidirect
10. #1654C2semidirect
11. #1651C2semidirect
12. #1634C2semidirect
13. #1632C2semidirect
14. #1586C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33842
#33352
#33292
#33032
#17162
#17042
#16912
#16832
#16712
#16542
#16512
#16342
#16322
#15862
#15852