Finite Group Databasebeta

← #20080

Subgroups — #20080

Order 256.

332 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3360C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3295C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic
5#1719C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1673C211yesmaximal, characteristic, complemented
9#1666C211yesmaximal, characteristic, complemented
10#1643C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#556C2211yescharacteristic, complemented
18#554C2211yescharacteristic
19#547C2211yescharacteristic
20D42C2211yescharacteristic
21#500C2211yescharacteristic, complemented
22#492C2211yescharacteristic
23#488—21no—
24#487C2211yescharacteristic, complemented
25#487C2211yescharacteristic, complemented
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#484—21no—
29#482—21no—
30#481—21no—
31#479—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic, complemented
34#475C2211yescharacteristic, complemented
35#474C2211yescharacteristic
36#468—21no—
37#468—21no—
38#465—21no—
39#463—21no—
40#461C2211yescharacteristic
41#461C2211yescharacteristic
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#450—21no—
45#450—21no—
46#449—21no—
47C8×Q8C2211yescharacteristic, complemented
48C4×SD16—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic, complemented
53C8×D4C2211yescharacteristic
54#428—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70AGL(1,ℤ/8)—44no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C2×M16—21no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127C2311yescharacteristic
86#126—41no—
87#125—21no—
88#124—21no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96#121—21no—
97#120—41no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107D411yescharacteristic
111#106—21no—
112#106—21no—
113#106—21no—
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—41no—
126#103D411yescharacteristic
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—41no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170SD16—41no—
171SD16—42no—
172SD16—42no—
173SD16—42no—
174SD16—81no—
175SD16—81no—
176D8—22no—
177D8—42no—
178M16—42no—
179M16—42no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—81no—
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—82no—
221C23—82no—
222C23—161no—
223Q8—21no—
224Q8—21no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—81no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—81no—
240D4—82no—
241D4—82no—
242D4—82no—
243D4AGL(1,ℤ/8)12yes—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—82no—
259C2×C4—161no—
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C8—21no—
264C8—21no—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8—41no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—164no—
285C22D4211yescharacteristic
286C4—21no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4C2×AGL(1,ℤ/8)11yescharacteristic
296C4#57511yescharacteristic
297C4#57711yescharacteristic
298C4#57711yescharacteristic
299C2—42no—
300C2—81no—
301C2—81no—
302C2—161no—
303C2#326511yescharacteristic
304C2#326711yescharacteristic
305C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3272C2
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
2. #3353C2semidirect
3. #3295C2semidirect
5. #1719C2semidirect
6. #1703C2semidirect
7. #1697C2semidirect
8. #1673C2semidirect
9. #1666C2semidirect
10. #1643C2semidirect
11. #1639C2semidirect
12. #1635C2semidirect
13. #1628C2semidirect
14. #1588C2semidirect
15. #1586C2semidirect
17. #556C22semidirect
21. #500C22semidirect
24. #487C22semidirect
25. #487C22semidirect
33. #476C22semidirect
34. #475C22semidirect
47. C8×Q8C22semidirect
52. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect

Maximal subgroups

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

subgroupindex
#33602
#33532
#32952
#32722
#17192
#17032
#16972
#16732
#16662
#16432
#16392
#16352
#16282
#15882
#15862