Finite Group Databasebeta

← #24423

Subgroups — #24423

Order 256.

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

normal subgroups only

This group has 642 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
2#3069C212yesmaximal, complemented
3#3022C211yesmaximal, characteristic, complemented
4#3011C211yesmaximal, characteristic, complemented
5#2889C211yesmaximal, characteristic, complemented
6#2049C212yesmaximal, complemented
7#2020C212yesmaximal, complemented
8#1915C211yesmaximal, characteristic, complemented
9#1881C212yesmaximal, complemented
10#1858C211yesmaximal, characteristic, complemented
11#1791C211yesmaximal, 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×#137C2212yes—
21C2×#137C2211yescharacteristic, complemented
22C22×Q16C2211yescharacteristic, complemented
23#568C2211yescharacteristic
24#568C2212yescomplemented
25#536C2211yescharacteristic
26#473—24no—
27#471—24no—
28#456—22no—
29#455—22no—
30#452—22no—
31#451—22no—
32#430C2212yescomplemented
33#421C2212yes—
34#419C2211yescharacteristic, complemented
35#416C2211yescharacteristic, complemented
36#413—22no—
37#413C2212yes—
38#413C2211yescharacteristic
39#413C2211yescharacteristic
40C2×#101—22no—
41C2×#101C2212yescomplemented
42C2×#101C2212yescomplemented
43#408C2211yescharacteristic
44#408C2212yes—
45#344C2212yescomplemented
46#341C2212yescomplemented
472−1+4—24no—
482−1+4—22no—
492−1+4—21no—
502−1+4—42no—
51H(5,𝔽2)—22no—
52H(5,𝔽2)—22no—
53H(5,𝔽2)—21no—
54C2×#41—21no—
55C2×#41—22no—
56C2×#41—22no—
57C2×#41—42no—
58C2×#41D412yes—
59C2×#41C2312yes—
60C2×#41C2311yescharacteristic
61C2×#41C2311yescharacteristic
62C22×Q8—21no—
63C22×Q8D412yes—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66#137—28no—
67#137—24no—
68#137—22no—
69#137—24no—
70#137—24no—
71#137—24no—
72#137—24no—
73#137—42no—
74#137D414yes—
75AGL(1,ℤ/8)—24no—
76AGL(1,ℤ/8)—22no—
77AGL(1,ℤ/8)D414yes—
78#135—28no—
79#135—24no—
80C2×Q16—24no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×Q16—22no—
84C2×Q16—21no—
85C2×Q16—42no—
86C2×SD16—24no—
87C2×SD16—22no—
88C2×SD16—22no—
89C2×SD16—21no—
90#131—24no—
91#131—24no—
92#131—42no—
93C2×M16—22no—
94C2×M16—24no—
95C2×M16C2312yes—
96C2×M16C2311yescharacteristic
97C2×M16C2312yes—
98C2×M16C2312yes—
99C22×C8C2311yescharacteristic
100C4⋊Q8—21no—
101#124—21no—
102#123—22no—
103C22⋊Q8—22no—
104#117C2311yescharacteristic
105#108—24no—
106#104—24no—
107#103—21no—
108#103—22no—
109#102—21no—
110#101—21no—
111#101—22no—
112#101—24no—
113#101—24no—
114#101D412yes—
115#100—21no—
116#100—22no—
117#100D412yes—
118C22⋊C8—22no—
119C22⋊C8—42no—
120#41—22no—
121#41—22no—
122#41—22no—
123#41—24no—
124#41—24no—
125#41—22no—
126#41—22no—
127#41—22no—
128#41—22no—
129#41—42no—
130#41—44no—
131#41—44no—
132#41—42no—
133#41—48no—
134#41—42no—
135#41—42no—
136#41—42no—
137#41C2×D414yes—
138C2×Q8—22no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×Q8—24no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—42no—
151C2×Q8C2×D412yes—
152C2×Q8C2×D412yes—
153C2×Q8C2×D411yescharacteristic
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—42no—
162C2×D4C2×D412yes—
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C2×D4#4112yes—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4C2×D412yes—
174C22×C4C2411yescharacteristic
175Q16—28no—
176Q16—24no—
177Q16—44no—
178Q16—44no—
179Q16—42no—
180SD16—28no—
181SD16—24no—
182SD16—24no—
183SD16—44no—
184SD16—42no—
185D8—24no—
186M16—24no—
187M16—22no—
188M16—22no—
189M16—22no—
190M16—22no—
191M16—22no—
192M16—44no—
193M16C2×D414yes—
194M16C2×D414yes—
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—44no—
203C4⋊C4—21no—
204C4⋊C4—42no—
205C22⋊C4—21no—
206C22⋊C4—42no—
207C42—21no—
208C23—21no—
209C23—22no—
210C23—42no—
211C23#12112yes—
212C23C2×#4111yescharacteristic
213Q8—21no—
214Q8—22no—
215Q8—22no—
216Q8—22no—
217Q8—22no—
218Q8—22no—
219Q8—44no—
220Q8—42no—
221Q8—42no—
222Q8—41no—
223Q8—41no—
224Q8—42no—
225Q8#12012yes—
226D4—24no—
227D4—24no—
228D4—22no—
229D4—21no—
230D4—22no—
231D4—22no—
232D4—42no—
233D4—44no—
234D4—44no—
235D4—42no—
236D4—42no—
237D4#12012yes—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4#12112yes—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D412yes—
257C2×C4C22×D411yescharacteristic
258C2×C4H(5,𝔽2)12yes—
259C8—24no—
260C8—24no—
261C8—41no—
262C8—41no—
263C8—42no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—42no—
268C22—42no—
269C22—81no—
270C22—82no—
271C22—162no—
272C22C2×#12111yescharacteristic
273C22#54612yes—
274C4—22no—
275C4—22no—
276C4—42no—
277C4—42no—
278C4—81no—
279C4—81no—
280C4C2×#12011yescharacteristic
281C4#53411yescharacteristic
282C4D4212yes—
283C2—21no—
284C2—22no—
285C2—42no—
286C2—42no—
287C2—82no—
288C2#241011yescharacteristic
subgroupG/H
center, socleC2#2410
commutator, Frattini, cosocleC22×C4C24
prime radical#3586C2
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
2. #3069C2semidirect
3. #3022C2semidirect
4. #3011C2semidirect
5. #2889C2semidirect
6. #2049C2semidirect
7. #2020C2semidirect
8. #1915C2semidirect
9. #1881C2semidirect
10. #1858C2semidirect
11. #1791C2semidirect
21. C2×#137C22semidirect
22. C22×Q16C22semidirect
24. #568C22semidirect
32. #430C22semidirect
34. #419C22semidirect
35. #416C22semidirect
41. C2×#101C22semidirect
42. C2×#101C22semidirect
45. #344C22semidirect
46. #341C22semidirect

Maximal subgroups

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

subgroupindex
#35862
#30692
#30222
#30112
#28892
#20492
#20202
#19152
#18812
#18582
#17912