Finite Group Databasebeta

← #20105

Subgroups — #20105

Order 256.

310 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3398C211yesmaximal, characteristic, complemented
2#3381C211yesmaximal, characteristic, complemented
3#3324C211yesmaximal, characteristic
4#3295C211yesmaximal, characteristic
5#1720C211yesmaximal, characteristic, complemented
6#1698C211yesmaximal, characteristic, complemented
7#1691C211yesmaximal, characteristic, complemented
8#1682C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1644C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic
12#1638C211yesmaximal, characteristic, complemented
13#1626C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic, complemented
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19D42C2211yescharacteristic
20#499—21no—
21#498C2211yescharacteristic, complemented
22#497—21no—
23#494C2211yescharacteristic, complemented
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#482—21no—
29#481—21no—
30#479—21no—
31#478—21no—
32#477C2211yescharacteristic, complemented
33#476C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#468—21no—
37#465—21no—
38#464—21no—
39#462—21no—
40#462C2211yescharacteristic, complemented
41#461C2211yescharacteristic
42#460C2211yescharacteristic
43#460C2211yescharacteristic
44#450—21no—
45#450—21no—
46#446C2211yescharacteristic, complemented
47C8×Q8C2211yescharacteristic, complemented
48#441—21no—
49C4×SD16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#418—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic, complemented
67C22×D4—21no—
68C22×D4—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×SD16D411yescharacteristic
77C2×M16—21no—
78C22×C8—21no—
79C4⋊Q8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#126—41no—
85#125—21no—
86#125—21no—
87#125—41no—
88#124—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—41no—
92#121—21no—
93#121—21no—
94#120—41no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103#117—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106D411yescharacteristic
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—41no—
123#103D411yescharacteristic
124#103D411yescharacteristic
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—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C24—41no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—81no—
158C2×D4—81no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164Q16—42no—
165Q16—81no—
166Q16—81no—
167SD16—22no—
168SD16—41no—
169SD16—81no—
170SD16—81no—
171SD16—81no—
172M16—81no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—81no—
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
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⋊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—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42C2411yescharacteristic
212C23—21no—
213C23—21no—
214C23—41no—
215C23—82no—
216C23—82no—
217C23—161no—
218Q8—21no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236D4—82no—
237D4#13712yes—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—161no—
252C2×C4C22×D411yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4H(5,𝔽2)11yescharacteristic
255C8—21no—
256C8—21no—
257C8—21no—
258C8—21no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—164no—
274C22D4211yescharacteristic
275C4—21no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4C2×#13711yescharacteristic
286C4#57511yescharacteristic
287C4#57611yescharacteristic
288C4#57711yescharacteristic
289C2—42no—
290C2—81no—
291C2—161no—
292C2#326611yescharacteristic
293C2#326811yescharacteristic
294C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalD42C22
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. #3398C2semidirect
2. #3381C2semidirect
5. #1720C2semidirect
6. #1698C2semidirect
7. #1691C2semidirect
8. #1682C2semidirect
9. #1669C2semidirect
10. #1644C2semidirect
12. #1638C2semidirect
13. #1626C2semidirect
14. #1588C2semidirect
15. #1585C2semidirect
16. #557C22semidirect
21. #498C22semidirect
23. #494C22semidirect
24. #487C22semidirect
32. #477C22semidirect
40. #462C22semidirect
46. #446C22semidirect
47. C8×Q8C22semidirect
56. #335C22semidirect
57. #335C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
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
#33982
#33812
#33242
#32952
#17202
#16982
#16912
#16822
#16692
#16442
#16392
#16382
#16262
#15882
#15852