Finite Group Databasebeta

← #19825

Subgroups — #19825

Order 256.

321 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3394C211yesmaximal, characteristic, complemented
2#3361C211yesmaximal, characteristic, complemented
3#3324C211yesmaximal, characteristic, complemented
4#3295C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1682C211yesmaximal, characteristic, complemented
8#1678C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1644C211yesmaximal, characteristic, complemented
11#1638C211yesmaximal, characteristic, complemented
12#1620C211yesmaximal, characteristic, complemented
13#1619C211yesmaximal, characteristic
14#1583C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic, complemented
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19D42C2211yescharacteristic
20#495—21no—
21#493C2211yescharacteristic
22#492C2211yescharacteristic, complemented
23#488—21no—
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic, complemented
28#482—21no—
29#481—21no—
30#478—21no—
31#477C2211yescharacteristic, complemented
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#468—21no—
35#465—21no—
36#464—21no—
37#463—21no—
38#463—21no—
39#462C2211yescharacteristic, complemented
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#450—21no—
45#450—21no—
46C8×Q8C2211yescharacteristic, complemented
47C8×Q8C2211yescharacteristic, complemented
48C4×SD16—21no—
49C4×SD16C2211yescharacteristic
50C4×SD16C2211yescharacteristic
51C4×D8—21no—
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#418—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic, complemented
67C22×D4—21no—
68C22×D4—21no—
69C2×Q16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16D411yescharacteristic
75C2×SD16D411yescharacteristic
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—41no—
79C2×M16—21no—
80C22×C8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#127C2311yescharacteristic
85#126—41no—
86#125—21no—
87#124—21no—
88#124—21no—
89#124—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96#120—41no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#106—21no—
110#106D411yescharacteristic
111#106D411yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—41no—
121#103D411yescharacteristic
122#103D411yescharacteristic
123#102—21no—
124#102—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C24—41no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8C2×D411yescharacteristic
148C2×Q8C2×D411yescharacteristic
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—81no—
160C2×D4—81no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167Q16—81no—
168SD16—22no—
169SD16—41no—
170SD16—42no—
171SD16—81no—
172SD16—81no—
173SD16—81no—
174D8—42no—
175D8—81no—
176D8—82no—
177M16—81no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—81no—
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218C23—82no—
219C23—82no—
220C23—161no—
221Q8—21no—
222Q8—21no—
223Q8—21no—
224Q8—21no—
225Q8—41no—
226Q8—41no—
227Q8—81no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—81no—
240D4—81no—
241D4—82no—
242D4—82no—
243D4#13512yes—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—161no—
257C2×C4—161no—
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C8—21no—
262C8—21no—
263C8—21no—
264C8—21no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—164no—
283C22D4211yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4C2×#13511yescharacteristic
294C4#57511yescharacteristic
295C4#57611yescharacteristic
296C4#57711yescharacteristic
297C2—42no—
298C2—81no—
299C2—161no—
300C2—161no—
301C2#326811yescharacteristic
302C2#328211yescharacteristic
303C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#1619C2
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. #3394C2semidirect
2. #3361C2semidirect
3. #3324C2semidirect
4. #3295C2semidirect
5. #1717C2semidirect
6. #1697C2semidirect
7. #1682C2semidirect
8. #1678C2semidirect
9. #1665C2semidirect
10. #1644C2semidirect
11. #1638C2semidirect
12. #1620C2semidirect
14. #1583C2semidirect
15. SD16⋊C8C2semidirect
16. #559C22semidirect
22. #492C22semidirect
24. #487C22semidirect
27. #486C22semidirect
31. #477C22semidirect
39. #462C22semidirect
46. C8×Q8C22semidirect
47. C8×Q8C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
61. #331C22semidirect
62. Q8⋊C8C22semidirect
63. Q8⋊C8C22semidirect
66. #322C22semidirect

Maximal subgroups

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

subgroupindex
#33942
#33612
#33242
#32952
#17172
#16972
#16822
#16782
#16652
#16442
#16382
#16202
#16192
#15832
SD16⋊C82