Finite Group Databasebeta

← #19778

Subgroups — #19778

Order 256.

303 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#3360C211yesmaximal, characteristic, complemented
2#3334C211yesmaximal, characteristic
3#3331C211yesmaximal, characteristic, complemented
4#3302C211yesmaximal, characteristic, complemented
5#1718C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1683C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1674C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1644C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#498C2211yescharacteristic
22#488—21no—
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#483—21no—
29#480—21no—
30#480—21no—
31#479—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#474C2211yescharacteristic
37#470—21no—
38#467—21no—
39#466—21no—
40#463—21no—
41#462C2211yescharacteristic
42#462C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#452—21no—
46C8×Q8C2211yescharacteristic
47C4×SD16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C2×#107—21no—
54#417—21no—
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
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×SD16—21no—
72C2×SD16D411yescharacteristic
73C2×D8D411yescharacteristic
74C2×M16—21no—
75C22×C8—21no—
76C22×C8—21no—
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#126—41no—
82#125—21no—
83#124—21no—
84#123—41no—
85#123—41no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—41no—
91#121—21no—
92#121—21no—
93C4×Q8—21no—
94C4×Q8C2311yescharacteristic
95C4×D4—21no—
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99#117—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107D411yescharacteristic
109#106—21no—
110#106—21no—
111#106—41no—
112#106D411yescharacteristic
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#103—21no—
125#103—21no—
126#103—21no—
127#103—41no—
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—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—
163C22×C4—41no—
164Q16—81no—
165Q16—81no—
166SD16—22no—
167SD16—42no—
168D8—41no—
169M16—81no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—42no—
176C2×C8—81no—
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C4⋊C4—21no—
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—41no—
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—41no—
220Q8—41no—
221Q8—81no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—82no—
228D4—82no—
229D4#13512yes—
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—21no—
252C8—21no—
253C8—21no—
254C8—21no—
255C8—41no—
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×#13511yescharacteristic
281C4#57511yescharacteristic
282C4#57611yescharacteristic
283C4#57811yescharacteristic
284C2—42no—
285C2—81no—
286C2—81no—
287C2#326911yescharacteristic
288C2#328011yescharacteristic
289C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3334C2
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. #3331C2semidirect
4. #3302C2semidirect
5. #1718C2semidirect
6. #1699C2semidirect
7. #1683C2semidirect
8. #1683C2semidirect
9. #1674C2semidirect
10. #1654C2semidirect
11. #1652C2semidirect
12. #1644C2semidirect
13. #1635C2semidirect
14. #1581C2semidirect
15. SD16⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#33602
#33342
#33312
#33022
#17182
#16992
#16832
#16742
#16542
#16522
#16442
#16352
#15812
SD16⋊C82