Finite Group Databasebeta

← #19307

Subgroups — #19307

Order 256.

328 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3185C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic
3#3077C211yesmaximal, characteristic
4D4×M16C211yesmaximal, characteristic
5#1703C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1691C211yesmaximal, characteristic, complemented
8#1689C211yesmaximal, characteristic, complemented
9#1683C211yesmaximal, characteristic, complemented
10#1679C211yesmaximal, characteristic, complemented
11#1671C211yesmaximal, characteristic, complemented
12#1669C211yesmaximal, characteristic, complemented
13#1569C211yesmaximal, characteristic, complemented
14#1561C211yesmaximal, characteristic, complemented
15#1559C211yesmaximal, characteristic, complemented
16C22×M16—21no—
17#533C2211yescharacteristic
18#533C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#498C2211yescharacteristic, complemented
21#494C2211yescharacteristic, complemented
22#487C2211yescharacteristic, complemented
23#487C2211yescharacteristic, complemented
24#484—21no—
25#478—21no—
26#478—21no—
27#476C2211yescharacteristic
28#476C2211yescharacteristic
29#474C2211yescharacteristic
30#474C2211yescharacteristic
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#470—21no—
34#470—21no—
35#461C2211yescharacteristic
36#461C2211yescharacteristic
37#450—21no—
38#441—21no—
39#441—21no—
40#440—21no—
41#440—21no—
42#436C2211yescharacteristic
43#436C2211yescharacteristic
44#435—21no—
45#435—21no—
46C8×D4C2211yescharacteristic
47C8×D4C2211yescharacteristic
48#432C2211yescharacteristic
49#428—21no—
50#423C2211yescharacteristic
51#423C2211yescharacteristic
52#418—21no—
53#407—21no—
54C4×M16C2211yescharacteristic
55#335C2211yescharacteristic, complemented
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#334C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#332C2211yescharacteristic, complemented
62#332C2211yescharacteristic, complemented
63#332C2211yescharacteristic, complemented
64#332C2211yescharacteristic, complemented
65#332C2211yescharacteristic, complemented
66#332C2211yescharacteristic, complemented
67C22×D4D411yescharacteristic
68C23×C4—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×M16—21no—
74C2×M16—21no—
75C2×M16—21no—
76C2×M16—21no—
77C2×M16—44no—
78C2×M16D411yescharacteristic
79C2×M16D411yescharacteristic
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#125—21no—
87#125—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4D411yescharacteristic
99C4×D4D411yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103#117—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C22⋊C4—21no—
106C2×C42C2311yescharacteristic
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
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—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141#97D411yescharacteristic
142#97D411yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—41no—
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4#4111yescharacteristic
165Q16—42no—
166Q16—42no—
167SD16—42no—
168SD16—42no—
169M16—22no—
170M16—41no—
171M16—42no—
172M16—42no—
173M16—42no—
174M16—42no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C4⋊C4—21no—
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—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—41no—
209C42—41no—
210C42C2×D411yescharacteristic
211C42C2×D411yescharacteristic
212C42#4111yescharacteristic
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—42no—
217C23—42no—
218C23—81no—
219C23#12111yescharacteristic
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—42no—
231D4#13712yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4#12111yescharacteristic
251C2×C4#12111yescharacteristic
252C2×C4#12111yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C8—21no—
257C8—21no—
258C8—21no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8AGL(1,ℤ/8)12yes—
266C22—21no—
267C22—21no—
268C22—22no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—84no—
276C22C2×#12111yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—41no—
282C4—41no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4C2×AGL(1,ℤ/8)11yescharacteristic
288C4C2×#13711yescharacteristic
289C4#57511yescharacteristic
290C4#57511yescharacteristic
291C2—41no—
292C2—42no—
293C2—81no—
294C2#303611yescharacteristic
295C2#305611yescharacteristic
296C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3185C2semidirect
5. #1703C2semidirect
6. #1699C2semidirect
7. #1691C2semidirect
8. #1689C2semidirect
9. #1683C2semidirect
10. #1679C2semidirect
11. #1671C2semidirect
12. #1669C2semidirect
13. #1569C2semidirect
14. #1561C2semidirect
15. #1559C2semidirect
20. #498C22semidirect
21. #494C22semidirect
22. #487C22semidirect
23. #487C22semidirect
55. #335C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #334C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #332C22semidirect
62. #332C22semidirect
63. #332C22semidirect
64. #332C22semidirect
65. #332C22semidirect
66. #332C22semidirect

Maximal subgroups

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

subgroupindex
#31852
#31102
#30772
D4×M162
#17032
#16992
#16912
#16892
#16832
#16792
#16712
#16692
#15692
#15612
#15592