Finite Group Databasebeta

← #20068

Subgroups — #20068

Order 256.

322 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3367C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3298C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic
5#1723C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1690C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic, complemented
9#1673C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1650C211yesmaximal, characteristic, complemented
13#1647C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#556C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#489—21no—
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#484—21no—
29#483—21no—
30#482—21no—
31#477—21no—
32#475C2211yescharacteristic
33#475C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#468—21no—
37#467—21no—
38#465—21no—
39#463—21no—
40#461C2211yescharacteristic
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#452—21no—
46#448—21no—
47#446C2211yescharacteristic
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55C2×#103—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C22×Q8—21no—
70#135—44no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16D411yescharacteristic
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#126—41no—
88#124—21no—
89#123—41no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—41no—
96#121—21no—
97#121—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—41no—
112#107D411yescharacteristic
113#106—21no—
114#106—21no—
115#106—41no—
116#106D411yescharacteristic
117C4⋊C8—21no—
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#103—21no—
131#103—21no—
132#103D411yescharacteristic
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×Q8—21no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—81no—
156C2×Q8C2×D411yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168Q16—42no—
169Q16—81no—
170SD16—41no—
171SD16—42no—
172SD16—42no—
173SD16—42no—
174SD16—81no—
175D8—22no—
176M16—42no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—42no—
183C2×C8—81no—
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
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—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219Q8—21no—
220Q8—21no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—82no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—82no—
236D4—82no—
237D4AGL(1,ℤ/8)12yes—
238C2×C4—21no—
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—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C8—22no—
260C8—21no—
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22D4211yescharacteristic
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4C2×AGL(1,ℤ/8)11yescharacteristic
289C4#57511yescharacteristic
290C4#57711yescharacteristic
291C4#57711yescharacteristic
292C2—42no—
293C2—81no—
294C2—81no—
295C2#326511yescharacteristic
296C2#326711yescharacteristic
297C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3283C2
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. #3367C2semidirect
2. #3353C2semidirect
3. #3298C2semidirect
5. #1723C2semidirect
6. #1699C2semidirect
7. #1690C2semidirect
8. #1679C2semidirect
9. #1673C2semidirect
10. #1654C2semidirect
11. #1651C2semidirect
12. #1650C2semidirect
13. #1647C2semidirect
14. #1586C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33672
#33532
#32982
#32832
#17232
#16992
#16902
#16792
#16732
#16542
#16512
#16502
#16472
#15862
#15852