Finite Group Databasebeta

← #19735

Subgroups — #19735

Order 256.

390 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3228C211yesmaximal, characteristic, complemented
2#3210C211yesmaximal, characteristic, complemented
3#3208C211yesmaximal, characteristic, complemented
4#3207C211yesmaximal, characteristic, complemented
5#3205C211yesmaximal, characteristic, complemented
6#3205C211yesmaximal, characteristic, complemented
7#1721C211yesmaximal, characteristic, complemented
8#1721C211yesmaximal, characteristic, complemented
9#1720C211yesmaximal, characteristic, complemented
10#1719C211yesmaximal, characteristic, complemented
11#1717C211yesmaximal, characteristic, complemented
12#1715C211yesmaximal, characteristic, complemented
13#1709C211yesmaximal, characteristic, complemented
14#1706C211yesmaximal, characteristic, complemented
15#1454C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#532C2211yescharacteristic
19C2×#127C2211yescharacteristic
20#500C2211yescharacteristic, complemented
21#498C2211yescharacteristic, complemented
22#494C2211yescharacteristic, complemented
23#493C2211yescharacteristic
24#493C2211yescharacteristic
25#493C2211yescharacteristic
26#493C2211yescharacteristic
27#493C2211yescharacteristic, complemented
28#492C2211yescharacteristic
29#492C2211yescharacteristic
30#492C2211yescharacteristic, complemented
31#492C2211yescharacteristic, complemented
32#487C2211yescharacteristic, complemented
33#487C2211yescharacteristic, complemented
34#486C2211yescharacteristic
35#486C2211yescharacteristic
36#486C2211yescharacteristic
37#486C2211yescharacteristic
38#486C2211yescharacteristic
39#486C2211yescharacteristic
40#486C2211yescharacteristic, complemented
41#486C2211yescharacteristic, complemented
42#486C2211yescharacteristic, complemented
43#486C2211yescharacteristic, complemented
44#482—22no—
45#478—22no—
46#464—22no—
47#463—22no—
48#460—22no—
49#459—22no—
50#459—22no—
51#452—22no—
52#450—22no—
53#449—22no—
54#447—22no—
55#447—22no—
56#431C2211yescharacteristic
57#431C2211yescharacteristic
58#431C2211yescharacteristic
59#322C2211yescharacteristic, complemented
60#322C2211yescharacteristic, complemented
61#322C2211yescharacteristic, complemented
62C82C2211yescharacteristic, complemented
63C2×#41—21no—
64C2×#41—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C2×Q16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#127—21no—
83#127—21no—
84#127—21no—
85#127C2311yescharacteristic
86#127C2311yescharacteristic
87#127C2311yescharacteristic
88#127C2311yescharacteristic
89#125—21no—
90#124—21no—
91#124—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104#117—21no—
105C2×C42C2311yescharacteristic
106#107—22no—
107#106—22no—
108C4⋊C8D412yes—
109C4⋊C8D412yes—
110C4⋊C8D412yes—
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—
125C22⋊C8D412yes—
126C4×C8C2311yescharacteristic
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129C4×C8C2311yescharacteristic
130C4×C8C2311yescharacteristic
131C4×C8C2311yescharacteristic
132C24—41no—
133#41—82no—
134#41—82no—
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—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—82no—
157C2×D4—82no—
158C2×D4—82no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164Q16—82no—
165SD16—82no—
166SD16—82no—
167SD16—82no—
168SD16—82no—
169D8—82no—
170D8—82no—
171D8—82no—
172D8—82no—
173D8—82no—
174C2×C8C2×D412yes—
175C2×C8C2×D412yes—
176C2×C8C2×D412yes—
177C2×C8C2×D412yes—
178C2×C8C2×D412yes—
179C2×C8C2×D412yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C42—41no—
196C42C2×D411yescharacteristic
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42C2411yescharacteristic
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—82no—
205C23—82no—
206C23—82no—
207C23#12711yescharacteristic
208Q8—42no—
209Q8—42no—
210Q8—42no—
211Q8—42no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—42no—
216D4—42no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—42no—
223D4—42no—
224D4—82no—
225D4—82no—
226D4—82no—
227D4—82no—
228D4—82no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4#12711yescharacteristic
239C2×C4#12711yescharacteristic
240C2×C4#12711yescharacteristic
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D411yescharacteristic
243C2×C4C22×D411yescharacteristic
244C8—22no—
245C8—22no—
246C8—22no—
247C8—22no—
248C8—22no—
249C8—22no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—164no—
266C22C2×#12711yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4#57612yes—
275C4#57612yes—
276C4#57712yes—
277C2—41no—
278C2—161no—
279C2—161no—
280C2—161no—
281C2—161no—
282C2#315411yescharacteristic
283C2#315511yescharacteristic
284C2#315511yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
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. #3228C2semidirect
2. #3210C2semidirect
3. #3208C2semidirect
4. #3207C2semidirect
5. #3205C2semidirect
6. #3205C2semidirect
7. #1721C2semidirect
8. #1721C2semidirect
9. #1720C2semidirect
10. #1719C2semidirect
11. #1717C2semidirect
12. #1715C2semidirect
13. #1709C2semidirect
14. #1706C2semidirect
15. #1454C2semidirect
20. #500C22semidirect
21. #498C22semidirect
22. #494C22semidirect
27. #493C22semidirect
30. #492C22semidirect
31. #492C22semidirect
32. #487C22semidirect
33. #487C22semidirect
40. #486C22semidirect
41. #486C22semidirect
42. #486C22semidirect
43. #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
#32282
#32102
#32082
#32072
#32052
#17212
#17202
#17192
#17172
#17152
#17092
#17062
#14542