Finite Group Databasebeta

← #20240

Subgroups — #20240

Order 256.

329 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3332C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic, complemented
5#1722C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1693C211yesmaximal, characteristic, complemented
8#1690C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1650C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1642C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1582C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#556C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#496—21no—
24#494C2211yescharacteristic
25#492C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#484—21no—
29#482—21no—
30#481—21no—
31#475C2211yescharacteristic
32#475C2211yescharacteristic
33#474C2211yescharacteristic
34#469—21no—
35#468—21no—
36#467—21no—
37#463—21no—
38#461C2211yescharacteristic
39#461C2211yescharacteristic
40#461C2211yescharacteristic
41#460—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#452—21no—
46#449—21no—
47#446C2211yescharacteristic
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55#417—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70#135—44no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16D411yescharacteristic
79C2×D8—21no—
80C2×D8—21no—
81C2×D8D411yescharacteristic
82C2×M16—21no—
83C2×M16—21no—
84C22×C8—21no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127—21no—
89#127C2311yescharacteristic
90#126—41no—
91#124—21no—
92#123—41no—
93#123—41no—
94C22⋊Q8—21no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—41no—
101C4×Q8—21no—
102C4×Q8C2311yescharacteristic
103C4×D4—21no—
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107#117—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110#107—21no—
111#107—21no—
112#107—21no—
113#107—41no—
114#107D411yescharacteristic
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#103—21no—
126#103—21no—
127#103—21no—
128#103D411yescharacteristic
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
145#41—42no—
146#41—42no—
147#41—42no—
148#41—81no—
149#41—81no—
150#41—81no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171Q16—42no—
172Q16—81no—
173SD16—41no—
174SD16—42no—
175SD16—42no—
176SD16—42no—
177SD16—81no—
178SD16—81no—
179D8—22no—
180D8—81no—
181D8—81no—
182M16—42no—
183M16—81no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—42no—
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—41no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—21no—
222C23—41no—
223Q8—21no—
224Q8—21no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—81no—
242D4—82no—
243D4—82no—
244D4—82no—
245D4—161no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C8—21no—
267C8—21no—
268C8—21no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8AGL(1,ℤ/8)12yes—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22D4211yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4C2×AGL(1,ℤ/8)11yescharacteristic
298C4#57511yescharacteristic
299C4#57611yescharacteristic
300C4#57711yescharacteristic
301C2—81no—
302C2—81no—
303C2—81no—
304C2—161no—
305C2#326711yescharacteristic
306C2#327411yescharacteristic
307C2#328911yescharacteristic
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. #3353C2semidirect
3. #3332C2semidirect
4. #3283C2semidirect
5. #1722C2semidirect
6. #1701C2semidirect
7. #1693C2semidirect
8. #1690C2semidirect
9. #1665C2semidirect
10. #1650C2semidirect
11. #1647C2semidirect
12. #1642C2semidirect
13. #1625C2semidirect
14. #1594C2semidirect
15. #1582C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33532
#33322
#32832
#17222
#17012
#16932
#16902
#16652
#16502
#16472
#16422
#16252
#15942
#15822