Finite Group Databasebeta

← #24571

Subgroups — #24571

Order 256.

455 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#483C211yesmaximal, characteristic, complemented
2#3007C211yesmaximal, characteristic
3C2×#417C211yesmaximal, characteristic
4#2880C211yesmaximal, characteristic
5#2079C212yesmaximal, complemented
6#2052C212yesmaximal, complemented
7#1908C212yesmaximal, complemented
8#1901C211yesmaximal, characteristic, complemented
9#1855C211yesmaximal, characteristic, complemented
10#1821C211yesmaximal, characteristic, complemented
11#1815C212yesmaximal, complemented
12C22×#41C2211yescharacteristic
13C2×#137C2211yescharacteristic
14C2×#137C2211yescharacteristic
15C2×#131C2212yes—
16C22×M16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#483—21no—
20#483—21no—
21#483—22no—
22#452—21no—
23#452—21no—
24#451—21no—
25#450—21no—
26C2×#108C2212yescomplemented
27C2×#106C2212yescomplemented
28#417—21no—
29#417—21no—
30#417C2212yes—
31#417C2211yescharacteristic
32#417C2212yes—
33#417C2211yescharacteristic
34C2×#103—21no—
35C2×#103C2212yescomplemented
36C2×#102—21no—
37C2×#101C2212yescomplemented
38#408—22no—
39#408C2212yes—
40#407—21no—
41#407C2211yescharacteristic
42#407C2211yescharacteristic
43C2×C22⋊C8—21no—
44C2×C22⋊C8C2211yescharacteristic
45#343C2212yescomplemented
46#343C2212yescomplemented
47#343C2212yescomplemented
48#341C2212yescomplemented
49C2×#41—21no—
50C2×#41—21no—
51C2×#41—24no—
52C2×#41—42no—
53C2×#41D411yescharacteristic
54C2×#41D411yescharacteristic
55C2×#41C2311yescharacteristic
56C2×#41C2311yescharacteristic
57C22×Q8D411yescharacteristic
58C22×Q8C2311yescharacteristic
59C22×D4—21no—
60C22×D4D411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64#137—24no—
65#137—42no—
66C2×Q16—21no—
67C2×Q16—21no—
68C2×SD16—21no—
69C2×SD16—21no—
70#131—24no—
71#131—42no—
72C2×M16—21no—
73C2×M16—22no—
74C2×M16—42no—
75C2×M16C2312yes—
76C2×M16C2312yes—
77C2×M16C2311yescharacteristic
78C2×M16C2311yescharacteristic
79C22×C8—21no—
80C22×C8—22no—
81C22×C8C2312yes—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95#108—42no—
96#106—22no—
97#106—22no—
98#103—22no—
99#103—22no—
100#103—21no—
101#103—21no—
102#103—22no—
103#103—22no—
104#102—22no—
105#102—22no—
106#102—21no—
107#102—21no—
108#101—42no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C22⋊C8—21no—
112C22⋊C8—21no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8D411yescharacteristic
116C22⋊C8D411yescharacteristic
117C22⋊C8D412yes—
118C24—21no—
119C24#4111yescharacteristic
120#41—22no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125#41—44no—
126#41—41no—
127#41—41no—
128#41—44no—
129#41#4114yes—
130C2×Q8—21no—
131C2×Q8—22no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—41no—
137C2×Q8C2×D411yescharacteristic
138C2×Q8C2×D411yescharacteristic
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—41no—
146C2×D4—44no—
147C2×D4C2×D411yescharacteristic
148C2×D4C2×D411yescharacteristic
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—82no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4#4111yescharacteristic
168C22×C4C2411yescharacteristic
169Q16—42no—
170Q16—42no—
171SD16—42no—
172SD16—42no—
173M16—22no—
174M16—22no—
175M16—22no—
176M16—22no—
177M16—22no—
178M16—22no—
179M16—44no—
180M16—42no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—22no—
187C2×C8—44no—
188C2×C8—42no—
189C2×C8C2×D412yes—
190C2×C8C2×D412yes—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—21no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23—81no—
209C23#12111yescharacteristic
210C23#12111yescharacteristic
211C23C2×#4111yescharacteristic
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—42no—
216Q8—41no—
217Q8—41no—
218Q8—42no—
219Q8#12312yes—
220D4—24no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—42no—
225D4—42no—
226D4—42no—
227D4—42no—
228D4#12312yes—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4#12111yescharacteristic
252C2×C4#12111yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C2×#4112yes—
256C2×C4H(5,𝔽2)12yes—
257C8—22no—
258C8—22no—
259C8—41no—
260C8—42no—
261C8—41no—
262C8—42no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—22no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—84no—
278C22#47112yes—
279C22C2×#12111yescharacteristic
280C22#55112yes—
281C4—21no—
282C4—21no—
283C4—22no—
284C4—41no—
285C4—81no—
286C4—82no—
287C4—81no—
288C4C2×#12311yescharacteristic
289C4#53411yescharacteristic
290C4#54612yes—
291C2—22no—
292C2—22no—
293C2—41no—
294C2—42no—
295C2—81no—
296C2#241211yescharacteristic
297C2C2×#47111yescharacteristic
298C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#483C2semidirect
5. #2079C2semidirect
6. #2052C2semidirect
7. #1908C2semidirect
8. #1901C2semidirect
9. #1855C2semidirect
10. #1821C2semidirect
11. #1815C2semidirect
26. C2×#108C22semidirect
27. C2×#106C22semidirect
35. C2×#103C22semidirect
37. C2×#101C22semidirect
45. #343C22semidirect
46. #343C22semidirect
47. #343C22semidirect
48. #341C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
#30072
C2×#4172
#28802
#20792
#20522
#19082
#19012
#18552
#18212
#18152