Finite Group Databasebeta

← #20056

Subgroups — #20056

Order 256.

334 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#3334C211yesmaximal, characteristic, complemented
2#3330C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic, complemented
5#1723C211yesmaximal, characteristic, complemented
6#1693C211yesmaximal, characteristic, complemented
7#1689C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic, complemented
9#1679C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1654C211yesmaximal, characteristic, complemented
12#1647C211yesmaximal, characteristic, complemented
13#1647C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#547C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#484—21no—
28#484—21no—
29#480—21no—
30#480—21no—
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#468—21no—
36#468—21no—
37#467—21no—
38#466—21no—
39#461C2211yescharacteristic
40#461C2211yescharacteristic
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#452—21no—
44#452—21no—
45#449—21no—
46#449—21no—
47C4×D8C2211yescharacteristic
48C4×D8C2211yescharacteristic
49#436C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C2×#107—21no—
54#417—21no—
55#417—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C2×#41—21no—
71#135—44no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×D8D411yescharacteristic
79C2×D8D411yescharacteristic
80C2×M16—21no—
81C2×M16—21no—
82C22×C8—21no—
83C22×C8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#123—41no—
89#123—41no—
90#123—41no—
91#123—41no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#107—21no—
113#107—21no—
114#107—41no—
115#107D411yescharacteristic
116#107D411yescharacteristic
117#106—41no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—42no—
148#41—42no—
149#41—81no—
150#41—81no—
151#41—81no—
152#41—81no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173Q16—42no—
174Q16—81no—
175SD16—42no—
176SD16—42no—
177SD16—81no—
178SD16—81no—
179D8—22no—
180D8—41no—
181M16—81no—
182M16—81no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—42no—
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—21no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223Q8—41no—
224Q8—41no—
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—21no—
238D4—82no—
239D4—82no—
240D4—82no—
241D4—82no—
242D4AGL(1,ℤ/8)12yes—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
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—41no—
258C2×C4—41no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C8—22no—
267C8—21no—
268C8—21no—
269C8—21no—
270C8—21no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—41no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22D4211yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4C2×AGL(1,ℤ/8)11yescharacteristic
298C4#57511yescharacteristic
299C4#57611yescharacteristic
300C4#57711yescharacteristic
301C2—42no—
302C2—81no—
303C2—81no—
304C2—81no—
305C2#326411yescharacteristic
306C2#326711yescharacteristic
307C2#329311yescharacteristic
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. #3334C2semidirect
2. #3330C2semidirect
3. #3330C2semidirect
4. #3283C2semidirect
5. #1723C2semidirect
6. #1693C2semidirect
7. #1689C2semidirect
8. #1679C2semidirect
9. #1679C2semidirect
10. #1654C2semidirect
11. #1654C2semidirect
12. #1647C2semidirect
13. #1647C2semidirect
14. #1586C2semidirect
15. #1586C2semidirect

Maximal subgroups

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

subgroupindex
#33342
#33302
#32832
#17232
#16932
#16892
#16792
#16542
#16472
#15862