Finite Group Databasebeta

← #20259

Subgroups — #20259

Order 256.

342 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic, complemented
2#3340C211yesmaximal, characteristic, complemented
3#3325C211yesmaximal, characteristic, complemented
4#3274C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1691C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1667C211yesmaximal, characteristic, complemented
10#1639C211yesmaximal, characteristic, complemented
11#1631C211yesmaximal, characteristic, complemented
12#1628C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#556C2211yescharacteristic, complemented
18#555C2211yescharacteristic
19#550C2211yescharacteristic
20D42C2211yescharacteristic
21#498C2211yescharacteristic, complemented
22#496—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic, complemented
25#492C2211yescharacteristic, complemented
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#482—21no—
30#482—21no—
31#477C2211yescharacteristic, complemented
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#469—21no—
35#469—21no—
36#465—21no—
37#463—21no—
38#461C2211yescharacteristic
39#460—21no—
40#460C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic, complemented
44#449—21no—
45#447—21no—
46#447—21no—
47#446C2211yescharacteristic, complemented
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic, complemented
54#427—21no—
55#418—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
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—41no—
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8D411yescharacteristic
83C2×M16—21no—
84C2×M16—21no—
85C22×C8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127—21no—
89#127C2311yescharacteristic
90#127C2311yescharacteristic
91#126—41no—
92#124—21no—
93#124—41no—
94C22⋊Q8—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—41no—
101#120—41no—
102C4×Q8—21no—
103C4×Q8C2311yescharacteristic
104C4×D4—21no—
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108#117—21no—
109#117—21no—
110#117—21no—
111#107—21no—
112#107—21no—
113#107D411yescharacteristic
114#106—21no—
115#106—21no—
116#106—21no—
117#106D411yescharacteristic
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#103D411yescharacteristic
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—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—41no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C2×D4C2×D411yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173SD16—22no—
174SD16—42no—
175SD16—42no—
176SD16—81no—
177SD16—81no—
178SD16—82no—
179D8—41no—
180D8—42no—
181D8—42no—
182D8—81no—
183D8—81no—
184D8—81no—
185M16—42no—
186M16—81no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—41no—
218C42—41no—
219C42C2411yescharacteristic
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23—82no—
226C23—82no—
227C23—161no—
228Q8—21no—
229Q8—21no—
230Q8—41no—
231Q8—41no—
232Q8—81no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—81no—
248D4—81no—
249D4—82no—
250D4—82no—
251D4—82no—
252D4—161no—
253D4—161no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C8—21no—
272C8—21no—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C8—41no—
279C8AGL(1,ℤ/8)12yes—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—164no—
296C22D4211yescharacteristic
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4C2×AGL(1,ℤ/8)11yescharacteristic
306C4#57511yescharacteristic
307C4#57611yescharacteristic
308C4#57711yescharacteristic
309C2—81no—
310C2—81no—
311C2—81no—
312C2—161no—
313C2—161no—
314C2#326811yescharacteristic
315C2#327211yescharacteristic
316C2#328811yescharacteristic
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. #3397C2semidirect
2. #3340C2semidirect
3. #3325C2semidirect
4. #3274C2semidirect
5. #1715C2semidirect
6. #1698C2semidirect
7. #1691C2semidirect
8. #1685C2semidirect
9. #1667C2semidirect
10. #1639C2semidirect
11. #1631C2semidirect
12. #1628C2semidirect
13. #1624C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect
17. #556C22semidirect
21. #498C22semidirect
24. #492C22semidirect
25. #492C22semidirect
31. #477C22semidirect
43. #459C22semidirect
47. #446C22semidirect
53. C8×D4C22semidirect
56. #335C22semidirect
57. #335C22semidirect
58. #333C22semidirect
59. #332C22semidirect
61. #331C22semidirect
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
#33972
#33402
#33252
#32742
#17152
#16982
#16912
#16852
#16672
#16392
#16312
#16282
#16242
#15922
#15812