Finite Group Databasebeta

← #19715

Subgroups — #19715

Order 256.

377 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3231C211yesmaximal, characteristic, complemented
2#3210C211yesmaximal, characteristic, complemented
3#3208C211yesmaximal, characteristic, complemented
4#3205C211yesmaximal, characteristic
5#3182C211yesmaximal, characteristic, complemented
6#3138C211yesmaximal, characteristic, complemented
7#1722C211yesmaximal, characteristic, complemented
8#1720C211yesmaximal, characteristic, complemented
9#1719C211yesmaximal, characteristic, complemented
10#1717C211yesmaximal, characteristic, complemented
11#1717C211yesmaximal, characteristic, complemented
12#1715C211yesmaximal, characteristic, complemented
13#1711C211yesmaximal, characteristic, complemented
14#1706C211yesmaximal, characteristic, complemented
15#1453C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×#127C2211yescharacteristic
20#500C2211yescharacteristic, complemented
21#500C2211yescharacteristic, complemented
22#498C2211yescharacteristic, complemented
23#494C2211yescharacteristic, complemented
24#493C2211yescharacteristic
25#493C2211yescharacteristic
26#492C2211yescharacteristic
27#492C2211yescharacteristic
28#492C2211yescharacteristic, complemented
29#492C2211yescharacteristic, complemented
30#490—22no—
31#487C2211yescharacteristic, complemented
32#487C2211yescharacteristic, complemented
33#487C2211yescharacteristic, complemented
34#487C2211yescharacteristic, complemented
35#486C2211yescharacteristic
36#486C2211yescharacteristic
37#486C2211yescharacteristic
38#486C2211yescharacteristic
39#486C2211yescharacteristic
40#486C2211yescharacteristic
41#486C2211yescharacteristic
42#486C2211yescharacteristic
43#486C2211yescharacteristic, complemented
44#486C2211yescharacteristic, complemented
45#485—22no—
46#479—22no—
47#468—22no—
48#464—22no—
49#460—22no—
50#459—22no—
51#452—22no—
52#450—22no—
53#447—22no—
54#440—22no—
55#431C2211yescharacteristic
56#431C2211yescharacteristic
57#418—22no—
58C4×M16C2211yescharacteristic
59#322C2211yescharacteristic, complemented
60#322C2211yescharacteristic, complemented
61#322C2211yescharacteristic, complemented
62C82C2211yescharacteristic, complemented
63C2×#41—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C2×Q16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×D8—22no—
73C2×D8—22no—
74C2×M16D412yes—
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80#127—21no—
81#127—21no—
82#127—21no—
83#127C2311yescharacteristic
84#127C2311yescharacteristic
85#127C2311yescharacteristic
86#125—21no—
87#125—21no—
88#124—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94#121—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×D4—21no—
100C4×D4—21no—
101#117—21no—
102#117—21no—
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#106—22no—
107C4⋊C8D412yes—
108C4⋊C8D412yes—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123C22⋊C8D412yes—
124C22⋊C8D412yes—
125#97D412yes—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C4×C8C2311yescharacteristic
130C4×C8C2311yescharacteristic
131C4×C8C2311yescharacteristic
132C24—41no—
133#41—82no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—44no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—82no—
154C2×D4—82no—
155C22×C4—41no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159Q16—82no—
160SD16—44no—
161SD16—82no—
162SD16—82no—
163SD16—82no—
164D8—82no—
165D8—82no—
166M16—42no—
167C2×C8C2×D412yes—
168C2×C8C2×D412yes—
169C2×C8C2×D412yes—
170C2×C8C2×D412yes—
171C2×C8C2×D412yes—
172C2×C8C2×D412yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C42—41no—
192C42—41no—
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2411yescharacteristic
197C23—41no—
198C23—41no—
199C23—41no—
200C23—82no—
201C23—82no—
202C23—82no—
203C23#12711yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—42no—
207Q8—42no—
208Q8—42no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—42no—
216D4—42no—
217D4—42no—
218D4—82no—
219D4—82no—
220D4—82no—
221D4—82no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—82no—
231C2×C4#12711yescharacteristic
232C2×C4#12711yescharacteristic
233C2×C4#12711yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C8—22no—
238C8—22no—
239C8—22no—
240C8—22no—
241C8—22no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—164no—
256C22C2×#12711yescharacteristic
257C4—21no—
258C4—21no—
259C4—21no—
260C4—81no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4#57512yes—
266C4#57612yes—
267C4#57712yes—
268C2—41no—
269C2—161no—
270C2—161no—
271C2—161no—
272C2#315311yescharacteristic
273C2#315411yescharacteristic
274C2#315511yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
prime radical#3205C2
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. #3231C2semidirect
2. #3210C2semidirect
3. #3208C2semidirect
5. #3182C2semidirect
6. #3138C2semidirect
7. #1722C2semidirect
8. #1720C2semidirect
9. #1719C2semidirect
10. #1717C2semidirect
11. #1717C2semidirect
12. #1715C2semidirect
13. #1711C2semidirect
14. #1706C2semidirect
15. #1453C2semidirect
20. #500C22semidirect
21. #500C22semidirect
22. #498C22semidirect
23. #494C22semidirect
28. #492C22semidirect
29. #492C22semidirect
31. #487C22semidirect
32. #487C22semidirect
33. #487C22semidirect
34. #487C22semidirect
43. #486C22semidirect
44. #486C22semidirect
59. #322C22semidirect
60. #322C22semidirect
61. #322C22semidirect
62. C82C22semidirect

Maximal subgroups

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

subgroupindex
#32312
#32102
#32082
#32052
#31822
#31382
#17222
#17202
#17192
#17172
#17152
#17112
#17062
#14532