Finite Group Databasebeta

← #19859

Subgroups — #19859

Order 256.

328 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3354C211yesmaximal, characteristic, complemented
2#3334C211yesmaximal, characteristic, complemented
3#3303C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic, complemented
5#1723C211yesmaximal, characteristic, complemented
6#1693C211yesmaximal, characteristic, complemented
7#1683C211yesmaximal, characteristic, complemented
8#1671C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1647C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1646C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15#1580C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#555C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#482—21no—
29#481—21no—
30#480—21no—
31#480—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#474C2211yescharacteristic
37#469—21no—
38#467—21no—
39#466—21no—
40#465—21no—
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#452—21no—
46#449—21no—
47C4×D8C2211yescharacteristic
48C4×D8C2211yescharacteristic
49#436C2211yescharacteristic
50C8×D4C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54C2×#107—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
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—
70#135—44no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×D8D411yescharacteristic
77C2×D8D411yescharacteristic
78C2×M16—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#124—41no—
87#123—41no—
88#123—41no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#107—41no—
113#107D411yescharacteristic
114#107D411yescharacteristic
115#106—21no—
116#106—21no—
117#106—21no—
118#106—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#103—41no—
129#102—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—81no—
149#41—81no—
150#41—81no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170Q16—42no—
171Q16—81no—
172SD16—42no—
173SD16—42no—
174SD16—81no—
175D8—22no—
176D8—22no—
177M16—42no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—81no—
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C4⋊C4—21no—
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⋊C4—41no—
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—82no—
235D4—82no—
236D4—82no—
237D4—161no—
238D4AGL(1,ℤ/8)12yes—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
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—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8#13512yescomplemented
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22D4211yescharacteristic
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4C2×#13511yescharacteristic
293C4C2×AGL(1,ℤ/8)11yescharacteristic
294C4#57611yescharacteristic
295C4#57811yescharacteristic
296C2—42no—
297C2—81no—
298C2—81no—
299C2—81no—
300C2#326411yescharacteristic
301C2#328311yescharacteristic
302C2#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. #3354C2semidirect
2. #3334C2semidirect
3. #3303C2semidirect
4. #3283C2semidirect
5. #1723C2semidirect
6. #1693C2semidirect
7. #1683C2semidirect
8. #1671C2semidirect
9. #1669C2semidirect
10. #1647C2semidirect
11. #1647C2semidirect
12. #1646C2semidirect
13. #1635C2semidirect
14. #1581C2semidirect
15. #1580C2semidirect
269. C8#135semidirect

Maximal subgroups

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

subgroupindex
#33542
#33342
#33032
#32832
#17232
#16932
#16832
#16712
#16692
#16472
#16462
#16352
#15812
#15802