Finite Group Databasebeta

← #18934

Subgroups — #18934

Order 256.

510 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3549C211yesmaximal, characteristic, complemented
2#3245C211yesmaximal, characteristic, complemented
3#3214C211yesmaximal, characteristic, complemented
4#3207C211yesmaximal, characteristic, complemented
5#1646C212yesmaximal, complemented
6#1639C212yesmaximal, complemented
7#1624C212yesmaximal, complemented
8#1619C212yesmaximal, complemented
9#1551C211yesmaximal, characteristic, complemented
10#1496C211yesmaximal, characteristic, complemented
11#1480C211yesmaximal, characteristic, complemented
12H(3,𝔽4)—22no—
13#555—21no—
14#555—21no—
15#555C2212yes—
16D42—21no—
17D42—21no—
18D42C2212yes—
19#535—22no—
20#535—22no—
21#535—22no—
22#532C2211yescharacteristic
23#532C2211yescharacteristic
24#532C2211yescharacteristic
25#532C2211yescharacteristic, complemented
26#496—21no—
27#493C2211yescharacteristic
28#492C2211yescharacteristic, complemented
29#489—21no—
30#487C2211yescharacteristic, complemented
31#486C2211yescharacteristic
32#474—21no—
33#474C2212yescomplemented
34#466—22no—
35#463—22no—
36#461C2212yescomplemented
37#459—21no—
38#459C2212yescomplemented
39#459C2212yescomplemented
40#449—22no—
41#449—22no—
42#432C2211yescharacteristic, complemented
43#431C2211yescharacteristic, complemented
44C2×C4⋊C8C2211yescharacteristic, complemented
45#333C2212yescomplemented
46#331C2212yes—
47D4⋊C8C2212yescomplemented
48D4⋊C8C2212yescomplemented
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51C2×#41—21no—
52C2×#41—21no—
53C2×#41—21no—
54C2×#41D412yes—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—22no—
58C2×SD16—22no—
59C2×SD16—22no—
60C2×D8—22no—
61C2×D8—22no—
62C2×D8—22no—
63C22×C8—21no—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D411yescharacteristic
66C4⋊Q8C2311yescharacteristic
67C4⋊Q8C2311yescharacteristic
68#127—24no—
69#127D411yescharacteristic
70#127D411yescharacteristic
71#127C2311yescharacteristic
72#127C2311yescharacteristic
73#124—24no—
74#124—24no—
75#124—24no—
76#124—21no—
77#124—21no—
78#124—21no—
79#124D412yes—
80#121—22no—
81#121—22no—
82#121—22no—
83#121—22no—
84#121—21no—
85#121—21no—
86#121—21no—
87#121—21no—
88#121—22no—
89#121—22no—
90#121—21no—
91#121—21no—
92#121D412yes—
93#121D412yes—
94#120—24no—
95#120—24no—
96#120—24no—
97#120—24no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4C2312yes—
105C4×D4C2312yes—
106#117—22no—
107#117—22no—
108#117—22no—
109C2×C42C2311yescharacteristic
110#107—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8C2312yes—
114C4⋊C8C2312yes—
115#103—22no—
116#103—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C24—24no—
127#41—81no—
128#41—82no—
129#41—81no—
130#41—82no—
131#41—82no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8C2×D412yes—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
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—82no—
153C2×D4—82no—
154C2×D4—82no—
155C2×D4—82no—
156C2×D4—84no—
157C2×D4C2×D412yes—
158C2×D4C2×D412yes—
159C2×D4C2×D412yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167SD16—82no—
168SD16—82no—
169D8—82no—
170D8—82no—
171D8—82no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—81no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4C2×D412yes—
184C4⋊C4C2×D412yes—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C42—24no—
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42C2×D411yescharacteristic
200C42C2411yescharacteristic
201C23—22no—
202C23—22no—
203C23—22no—
204C23—41no—
205C23—84no—
206C23—84no—
207C23—84no—
208C23#12011yescharacteristic
209Q8—42no—
210Q8—42no—
211Q8—81no—
212Q8—81no—
213D4—22no—
214D4—22no—
215D4—22no—
216D4—22no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—41no—
223D4—41no—
224D4—42no—
225D4—41no—
226D4—41no—
227D4—81no—
228D4—81no—
229D4—82no—
230D4—82no—
231D4—82no—
232D4—84no—
233D4—84no—
234D4—84no—
235D4—82no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4#12011yescharacteristic
251C2×C4#12011yescharacteristic
252C2×C4#12011yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C8—41no—
257C8—41no—
258C8—42no—
259C8—42no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—168no—
276C22C2×#12011yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4#57612yes—
286C4#57611yescharacteristic
287C4#57711yescharacteristic
288C2—41no—
289C2—82no—
290C2—82no—
291C2—82no—
292C2—161no—
293C2#300811yescharacteristic
294C2#301211yescharacteristic
295C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
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. #3549C2semidirect
2. #3245C2semidirect
3. #3214C2semidirect
4. #3207C2semidirect
5. #1646C2semidirect
6. #1639C2semidirect
7. #1624C2semidirect
8. #1619C2semidirect
9. #1551C2semidirect
10. #1496C2semidirect
11. #1480C2semidirect
25. #532C22semidirect
28. #492C22semidirect
30. #487C22semidirect
33. #474C22semidirect
36. #461C22semidirect
38. #459C22semidirect
39. #459C22semidirect
42. #432C22semidirect
43. #431C22semidirect
44. C2×C4⋊C8C22semidirect
45. #333C22semidirect
47. D4⋊C8C22semidirect
48. D4⋊C8C22semidirect
49. D4⋊C8C22semidirect
50. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#32452
#32142
#32072
#16462
#16392
#16242
#16192
#15512
#14962
#14802