Finite Group Databasebeta

← #19460

Subgroups — #19460

Order 256.

321 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#3210C211yesmaximal, characteristic, complemented
2#3100C211yesmaximal, characteristic
3#3077C211yesmaximal, characteristic, complemented
4#2974C211yesmaximal, characteristic, complemented
5#1703C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1689C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1683C211yesmaximal, characteristic, complemented
10#1677C211yesmaximal, characteristic, complemented
11#1669C211yesmaximal, characteristic, complemented
12#1665C211yesmaximal, characteristic, complemented
13#1574C211yesmaximal, characteristic, complemented
14#1557C211yesmaximal, characteristic, complemented
15#1554C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#494C2211yescharacteristic, complemented
20#492C2211yescharacteristic, complemented
21#487C2211yescharacteristic, complemented
22#486C2211yescharacteristic, complemented
23#480—21no—
24#478—21no—
25#478—21no—
26#476C2211yescharacteristic
27#476C2211yescharacteristic
28#474C2211yescharacteristic
29#474C2211yescharacteristic
30#474C2211yescharacteristic
31#474C2211yescharacteristic
32#464—21no—
33#464—21no—
34#459C2211yescharacteristic
35#459C2211yescharacteristic
36#452—21no—
37#452—21no—
38#447—21no—
39C4×D8—21no—
40C4×D8—21no—
41#436C2211yescharacteristic
42#436C2211yescharacteristic
43#435—21no—
44#435—21no—
45C8×D4C2211yescharacteristic
46C8×D4C2211yescharacteristic
47#432C2211yescharacteristic
48#431C2211yescharacteristic
49#431C2211yescharacteristic
50#428—21no—
51#423C2211yescharacteristic
52#418—21no—
53#407—21no—
54C2×C22⋊C8—21no—
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#331C2211yescharacteristic, complemented
66#331C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4D411yescharacteristic
69C23×C4—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×D8—21no—
75C2×D8—21no—
76C2×M16—21no—
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127C2311yescharacteristic
85#125—21no—
86#124—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89#121—21no—
90#121—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4D411yescharacteristic
98C4×D4D411yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C22⋊C4—21no—
104C2×C42C2311yescharacteristic
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#106—21no—
110#106—21no—
111C4⋊C8—21no—
112C4⋊C8D411yescharacteristic
113C4⋊C8D411yescharacteristic
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#102—21no—
123#102—21no—
124#102—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132C22⋊C8—21no—
133C22⋊C8—21no—
134C22⋊C8—21no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8D411yescharacteristic
139C22⋊C8D411yescharacteristic
140#97—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C24—21no—
145#41—81no—
146#41—81no—
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—41no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4#4111yescharacteristic
170Q16—81no—
171Q16—81no—
172SD16—81no—
173SD16—81no—
174D8—42no—
175D8—42no—
176M16—42no—
177M16—81no—
178M16—81no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—81no—
186C2×C8—81no—
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
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⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—41no—
210C42C2×D411yescharacteristic
211C42C2×D411yescharacteristic
212C42#4111yescharacteristic
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—42no—
217C23—42no—
218C23—41no—
219C23—81no—
220C23#12111yescharacteristic
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
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—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
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—21no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
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×#13711yescharacteristic
295C4#57511yescharacteristic
296C4#57611yescharacteristic
297C4#57611yescharacteristic
298C2—41no—
299C2—42no—
300C2—81no—
301C2—161no—
302C2#304111yescharacteristic
303C2#305611yescharacteristic
304C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3100C2
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. #3210C2semidirect
3. #3077C2semidirect
4. #2974C2semidirect
5. #1703C2semidirect
6. #1697C2semidirect
7. #1689C2semidirect
8. #1685C2semidirect
9. #1683C2semidirect
10. #1677C2semidirect
11. #1669C2semidirect
12. #1665C2semidirect
13. #1574C2semidirect
14. #1557C2semidirect
15. #1554C2semidirect
19. #494C22semidirect
20. #492C22semidirect
21. #487C22semidirect
22. #486C22semidirect
55. #335C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #334C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #332C22semidirect
62. #332C22semidirect
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
#32102
#31002
#30772
#29742
#17032
#16972
#16892
#16852
#16832
#16772
#16692
#16652
#15742
#15572
#15542