Finite Group Databasebeta

← #19305

Subgroups — #19305

Order 256.

339 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3182C211yesmaximal, characteristic, complemented
2#3109C211yesmaximal, characteristic
3#3077C211yesmaximal, characteristic, complemented
4D4×M16C211yesmaximal, characteristic, complemented
5#1703C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1695C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1683C211yesmaximal, characteristic, complemented
10#1677C211yesmaximal, characteristic, complemented
11#1675C211yesmaximal, characteristic, complemented
12#1665C211yesmaximal, characteristic, complemented
13#1569C211yesmaximal, characteristic, complemented
14#1560C211yesmaximal, characteristic, complemented
15#1559C211yesmaximal, characteristic, complemented
16C22×M16—21no—
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#500C2211yescharacteristic, complemented
21#492C2211yescharacteristic, complemented
22#487C2211yescharacteristic, complemented
23#486C2211yescharacteristic, complemented
24#481—21no—
25#478—21no—
26#478—21no—
27#476C2211yescharacteristic
28#476C2211yescharacteristic
29#476C2211yescharacteristic
30#476C2211yescharacteristic
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#468—21no—
34#468—21no—
35#459C2211yescharacteristic
36#459C2211yescharacteristic
37#447—21no—
38#442—21no—
39#442—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#333C2211yescharacteristic, complemented
62#333C2211yescharacteristic, complemented
63#332C2211yescharacteristic, complemented
64#332C2211yescharacteristic, complemented
65#331C2211yescharacteristic, complemented
66#331C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4D411yescharacteristic
69C23×C4—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×D8—21no—
73C2×D8—21no—
74C2×M16—21no—
75C2×M16—21no—
76C2×M16—21no—
77C2×M16—21no—
78C2×M16—44no—
79C2×M16D411yescharacteristic
80C2×M16D411yescharacteristic
81C22×C8—21no—
82C22×C8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127C2311yescharacteristic
87#125—21no—
88#124—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91#121—21no—
92#121—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4D411yescharacteristic
100C4×D4D411yescharacteristic
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
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#102—21no—
126#102—21no—
127#102—21no—
128#102—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—
146#41—82no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D411yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4#4111yescharacteristic
170SD16—42no—
171SD16—42no—
172D8—42no—
173D8—42no—
174M16—22no—
175M16—41no—
176M16—42no—
177M16—42no—
178M16—42no—
179M16—42no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—41no—
211C42C2×D411yescharacteristic
212C42C2×D411yescharacteristic
213C42#4111yescharacteristic
214C42C2411yescharacteristic
215C23—21no—
216C23—21no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—81no—
221C23#12111yescharacteristic
222Q8—41no—
223Q8—41no—
224Q8—81no—
225Q8—81no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—42no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236D4#13712yes—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4#12111yescharacteristic
256C2×C4#12111yescharacteristic
257C2×C4#12111yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8AGL(1,ℤ/8)12yes—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—84no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22C2×#12111yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4C2×AGL(1,ℤ/8)11yescharacteristic
295C4C2×#13711yescharacteristic
296C4#57511yescharacteristic
297C4#57511yescharacteristic
298C2—41no—
299C2—42no—
300C2—81no—
301C2—161no—
302C2#303611yescharacteristic
303C2#305611yescharacteristic
304C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3109C2
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. #3182C2semidirect
3. #3077C2semidirect
4. D4×M16C2semidirect
5. #1703C2semidirect
6. #1697C2semidirect
7. #1695C2semidirect
8. #1685C2semidirect
9. #1683C2semidirect
10. #1677C2semidirect
11. #1675C2semidirect
12. #1665C2semidirect
13. #1569C2semidirect
14. #1560C2semidirect
15. #1559C2semidirect
20. #500C22semidirect
21. #492C22semidirect
22. #487C22semidirect
23. #486C22semidirect
55. #335C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #334C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #333C22semidirect
62. #333C22semidirect
63. #332C22semidirect
64. #332C22semidirect
65. #331C22semidirect
66. #331C22semidirect

Maximal subgroups

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

subgroupindex
#31822
#31092
#30772
D4×M162
#17032
#16972
#16952
#16852
#16832
#16772
#16752
#16652
#15692
#15602
#15592