Finite Group Databasebeta

← #19945

Subgroups — #19945

Order 256.

327 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#3397C211yesmaximal, characteristic, complemented
2#3354C211yesmaximal, characteristic, complemented
3#3340C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic, complemented
5#1705C211yesmaximal, characteristic, complemented
6#1692C211yesmaximal, characteristic, complemented
7#1685C211yesmaximal, characteristic, complemented
8#1671C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15C8×D8C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#556C2211yescharacteristic
18#555C2211yescharacteristic
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21#498C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#496—21no—
25#492C2211yescharacteristic
26#492C2211yescharacteristic
27#492C2211yescharacteristic
28#485—21no—
29#482—21no—
30#482—21no—
31#481—21no—
32#477C2211yescharacteristic
33#476C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#469—21no—
37#465—21no—
38#465—21no—
39#463—21no—
40#462C2211yescharacteristic
41#461C2211yescharacteristic
42#460—21no—
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#449—21no—
47#446C2211yescharacteristic
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55#427—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—41no—
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8—21no—
80C2×D8D411yescharacteristic
81C2×M16—21no—
82C22×C8—21no—
83C22×C8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127—21no—
88#127C2311yescharacteristic
89#126—41no—
90#124—21no—
91#124—41no—
92#123—41no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—41no—
100C4×Q8—21no—
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106#117—21no—
107#117—21no—
108#117—21no—
109C2×C4⋊C4—21no—
110#107—21no—
111#107—21no—
112#107—21no—
113#107D411yescharacteristic
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118#106—21no—
119#106—41no—
120#106D411yescharacteristic
121C4⋊C8—21no—
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125C4⋊C8C2311yescharacteristic
126#103—21no—
127#103—21no—
128#103—21no—
129#103D411yescharacteristic
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
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—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168Q16—42no—
169SD16—22no—
170SD16—42no—
171SD16—42no—
172SD16—81no—
173SD16—81no—
174SD16—82no—
175SD16—81no—
176D8—22no—
177D8—42no—
178D8—81no—
179M16—42no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—41no—
215C42—41no—
216C42C2411yescharacteristic
217C23—21no—
218C23—21no—
219C23—21no—
220C23—41no—
221Q8—21no—
222Q8—21no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—81no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—81no—
239D4—82no—
240D4—82no—
241D4—161no—
242D4—161no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
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—82no—
258C2×C4—82no—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4H(5,𝔽2)11yescharacteristic
262C8—22no—
263C8—21no—
264C8—21no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8#13512yescomplemented
270C8AGL(1,ℤ/8)12yes—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22D4211yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4C2×#13511yescharacteristic
294C4C2×AGL(1,ℤ/8)11yescharacteristic
295C4#57711yescharacteristic
296C4#57711yescharacteristic
297C2—81no—
298C2—81no—
299C2—81no—
300C2—161no—
301C2#327211yescharacteristic
302C2#328411yescharacteristic
303C2#329111yescharacteristic
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. #3354C2semidirect
3. #3340C2semidirect
4. #3284C2semidirect
5. #1705C2semidirect
6. #1692C2semidirect
7. #1685C2semidirect
8. #1671C2semidirect
9. #1665C2semidirect
10. #1654C2semidirect
11. #1652C2semidirect
12. #1631C2semidirect
13. #1624C2semidirect
14. #1590C2semidirect
15. C8×D8C2semidirect
269. C8#135semidirect

Maximal subgroups

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

subgroupindex
#33972
#33542
#33402
#32842
#17052
#16922
#16852
#16712
#16652
#16542
#16522
#16312
#16242
#15902
C8×D82