Finite Group Databasebeta

← #24410

Subgroups — #24410

Order 256.

661 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 133 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#3586C211yesmaximal, characteristic, complemented
2#3043C212yesmaximal, complemented
3#3027C211yesmaximal, characteristic, complemented
4#3010C211yesmaximal, characteristic, complemented
5#2884C211yesmaximal, characteristic, complemented
6#2053C212yesmaximal, complemented
7#2008C212yesmaximal, complemented
8#1910C211yesmaximal, characteristic, complemented
9#1905C212yesmaximal, complemented
10#1868C211yesmaximal, characteristic, complemented
11#1794C211yesmaximal, characteristic, complemented
12#585C2211yescharacteristic
13C2×2−1+4C2211yescharacteristic
14#578—24no—
15#577—24no—
16#575—22no—
17#575C2212yes—
18C2×#137—22no—
19C2×#137C2212yes—
20C2×AGL(1,ℤ/8)C2211yescharacteristic, complemented
21C22×SD16C2211yescharacteristic, complemented
22#568C2211yescharacteristic
23#535C2211yescharacteristic
24#535C2212yes—
25#519C2212yescomplemented
26#464—24no—
27#460—24no—
28B(4,𝔽2)—24no—
29#449—22no—
30#448—22no—
31#424C2212yescomplemented
32#421C2212yes—
33#419C2211yescharacteristic, complemented
34#418C2212yescomplemented
35#416C2212yes—
36#416C2211yescharacteristic, complemented
37#410—24no—
38#410C2211yescharacteristic
39#410C2212yes—
40#410C2211yescharacteristic
41#410C2212yescomplemented
42#408C2211yescharacteristic
43#344C2212yescomplemented
44H(3,ℤ/4)C2212yescomplemented
452−1+4—24no—
462−1+4—22no—
472−1+4—21no—
482−1+4—42no—
49H(5,𝔽2)—22no—
50H(5,𝔽2)—22no—
51H(5,𝔽2)—21no—
52C2×#41—22no—
53C2×#41—21no—
54C2×#41—22no—
55C2×#41—42no—
56C2×#41D412yescomplemented
57C2×#41C2312yes—
58C2×#41C2311yescharacteristic
59C2×#41C2311yescharacteristic
60C22×Q8—21no—
61C22×Q8D412yescomplemented
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64#137—28no—
65#137—24no—
66#137—22no—
67#137—24no—
68#137—24no—
69#137—22no—
70AGL(1,ℤ/8)—24no—
71AGL(1,ℤ/8)—22no—
72AGL(1,ℤ/8)—22no—
73AGL(1,ℤ/8)—42no—
74#135—28no—
75#135—24no—
76C2×Q16—24no—
77C2×Q16D414yescomplemented
78C2×SD16—24no—
79C2×SD16—22no—
80C2×SD16—24no—
81C2×SD16—21no—
82C2×SD16D414yescomplemented
83C2×D8—21no—
84#131—24no—
85C2×M16—22no—
86C2×M16C2311yescharacteristic
87C2×M16C2312yes—
88C22×C8C2311yescharacteristic
89#127—22no—
90#127—21no—
91#124—22no—
92#124—21no—
93#121—22no—
94#121—42no—
95#120—22no—
96#120—24no—
97C4×Q8—24no—
98C4×Q8—42no—
99#117—24no—
100#117C2312yes—
101#117C2311yescharacteristic
102#117C2312yes—
103C4⋊C8—24no—
104#104—42no—
105#103—22no—
106#103—21no—
107#103—22no—
108#102—22no—
109#102—21no—
110#102—24no—
111#99—22no—
112#99—24no—
113#99—24no—
114#99D414yes—
115C22⋊C8—22no—
116C24—21no—
117#41—22no—
118#41—22no—
119#41—24no—
120#41—24no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127#41—42no—
128#41—48no—
129#41—44no—
130#41—42no—
131#41—44no—
132#41—42no—
133#41—42no—
134#41—42no—
135C2×Q8—22no—
136C2×Q8—21no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—24no—
140C2×Q8—42no—
141C2×Q8—44no—
142C2×Q8—42no—
143C2×Q8C2×D412yes—
144C2×Q8C2×D411yescharacteristic
145C2×Q8C2×D412yes—
146C2×Q8C2×D414yes—
147C2×Q8#4111yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—22no—
156C2×D4—44no—
157C2×D4—42no—
158C2×D4—44no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4C2×D412yes—
162C2×D4C2×D412yes—
163C2×D4C2×D411yescharacteristic
164C2×D4#4111yescharacteristic
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4C2×D412yes—
172C22×C4C2411yescharacteristic
173Q16—24no—
174Q16—28no—
175SD16—24no—
176SD16—24no—
177SD16—28no—
178SD16—42no—
179SD16—44no—
180D8—24no—
181D8—42no—
182M16—24no—
183M16—22no—
184M16—22no—
185M16—22no—
186C2×C8—24no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8C2×D414yes—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—44no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C22⋊C4—44no—
198C22⋊C4—42no—
199C22⋊C4C2×D414yes—
200C42—22no—
201C42—21no—
202C42—22no—
203C42—44no—
204C23—21no—
205C23—22no—
206C23—42no—
207C23—41no—
208C23—42no—
209C23—161no—
210C23#12112yes—
211C23C22×D411yescharacteristic
212Q8—22no—
213Q8—24no—
214Q8—22no—
215Q8—21no—
216Q8—22no—
217Q8—21no—
218Q8—22no—
219Q8—44no—
220Q8—42no—
221Q8#12014yes—
222D4—24no—
223D4—21no—
224D4—22no—
225D4—21no—
226D4—22no—
227D4—22no—
228D4—44no—
229D4—42no—
230D4—44no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—41no—
235D4—41no—
236D4—42no—
237D4—164no—
238D4—162no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4#12112yes—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D412yes—
258C2×C4C2×#4111yescharacteristic
259C2×C4H(5,𝔽2)12yes—
260C8—24no—
261C8—41no—
262C8—41no—
263C22—22no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—42no—
268C22—82no—
269C22—81no—
270C22—84no—
271C22—82no—
272C22—81no—
273C22—162no—
274C22C2×#12111yescharacteristic
275C22D4212yes—
276C4—24no—
277C4—22no—
278C4—44no—
279C4—42no—
280C4—41no—
281C4—82no—
282C4—81no—
283C4C2×#12011yescharacteristic
284C4#53411yescharacteristic
285C4#54612yes—
286C2—22no—
287C2—21no—
288C2—42no—
289C2—81no—
290C2—82no—
291C2—161no—
292C2#241011yescharacteristic
subgroupG/H
center, socleC2#2410
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3586C2semidirect
2. #3043C2semidirect
3. #3027C2semidirect
4. #3010C2semidirect
5. #2884C2semidirect
6. #2053C2semidirect
7. #2008C2semidirect
8. #1910C2semidirect
9. #1905C2semidirect
10. #1868C2semidirect
11. #1794C2semidirect
20. C2×AGL(1,ℤ/8)C22semidirect
21. C22×SD16C22semidirect
25. #519C22semidirect
31. #424C22semidirect
33. #419C22semidirect
34. #418C22semidirect
36. #416C22semidirect
41. #410C22semidirect
43. #344C22semidirect
44. H(3,ℤ/4)C22semidirect
56. C2×#41D4semidirect
61. C22×Q8D4semidirect
77. C2×Q16D4semidirect
82. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
#35862
#30432
#30272
#30102
#28842
#20532
#20082
#19102
#19052
#18682
#17942