Finite Group Databasebeta

← #18166

Subgroups — #18166

Order 256.

749 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 749 conjugacy classes of subgroups — too many to draw. Only its 117 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#3496C211yesmaximal, characteristic, complemented
2#3006C212yesmaximal, complemented
3#2388C211yesmaximal, characteristic, complemented
4#1604C212yesmaximal, complemented
5#1601C212yesmaximal, complemented
6#1598C212yesmaximal, complemented
7#1595C212yesmaximal, complemented
8#1507C212yesmaximal, complemented
9#1465C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11C22×#41C2211yescharacteristic, complemented
12C2×AGL(1,ℤ/8)—22no—
13#547—22no—
14#546—24no—
15D42—22no—
16#537C2212yes—
17#534—21no—
18#534C2212yes—
19#532—21no—
20C2×#121—21no—
21C2×#121C2212yes—
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24#449—24no—
25#448C2214yescomplemented
26#447C2214yescomplemented
27#417—22no—
28#407C2212yescomplemented
29#395C2211yescharacteristic, complemented
30#394—21no—
31#394C2211yescharacteristic, complemented
32#393C2211yescharacteristic, complemented
33#392—21no—
34#392C2211yescharacteristic, complemented
35C4×C22⋊C4—21no—
36C4×C22⋊C4C2211yescharacteristic, complemented
37#327C2214yescomplemented
38#327C2214yescomplemented
39#324C2212yescomplemented
40#323C2212yescomplemented
41H(5,𝔽2)—44no—
42H(5,𝔽2)—44no—
43C2×#41—24no—
44C2×#41—22no—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—44no—
48C22×Q8C2311yescharacteristic
49C22×D4—24no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—21no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4D411yescharacteristic
56C22×D4D412yescomplemented
57C22×D4C2311yescharacteristic
58C23×C4—21no—
59C23×C4D411yescharacteristic
60C23×C4C2311yescharacteristic
61AGL(1,ℤ/8)—48no—
62C2×SD16—24no—
63C2×D8—24no—
64C2×M16—22no—
65C4⋊Q8—21no—
66#127—21no—
67#124—22no—
68#123—24no—
69#123—24no—
70C22⋊Q8—24no—
71#121—22no—
72#121—22no—
73#121—24no—
74#121—22no—
75#121—22no—
76#121—22no—
77#121—22no—
78#121D414yescomplemented
79#121C2314yes—
80#120—24no—
81#120—24no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—44no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D412yescomplemented
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4C2311yescharacteristic
100C2×C42—21no—
101C2×C42D411yescharacteristic
102#103—24no—
103#102—24no—
104C22⋊C8—22no—
105C22⋊C8C2314yes—
106#95—21no—
107#95—21no—
108#95C2311yescharacteristic
109#95C2311yescharacteristic
110C24—22no—
111C24—21no—
112C24—22no—
113C24C2×D411yescharacteristic
114C24C2×D411yescharacteristic
115#41—28no—
116#41—44no—
117#41—44no—
118#41—44no—
119#41—416no—
120#41—44no—
121#41—44no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—42no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—24no—
130C2×D4—24no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—44no—
139C2×D4—42no—
140C2×D4—416no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—44no—
149C2×D4—82no—
150C2×D4C2×D414yes—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D412yes—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2411yescharacteristic
172SD16—48no—
173D8—48no—
174M16—44no—
175C2×C8—24no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4C2×D414yes—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—22no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C42—21no—
194C42—21no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—24no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—44no—
210C23—44no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23#12012yes—
219C23AGL(1,ℤ/8)12yescomplemented
220C23C22×D411yescharacteristic
221Q8—24no—
222Q8—42no—
223Q8—44no—
224D4—28no—
225D4—24no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—48no—
230D4—48no—
231D4—48no—
232D4—48no—
233D4—44no—
234D4—44no—
235D4—44no—
236D4—82no—
237D4—82no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—48no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4#12011yescharacteristic
266C2×C4#12011yescharacteristic
267C2×C4AGL(1,ℤ/8)12yescomplemented
268C2×C4C22×D412yes—
269C8—44no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—44no—
282C22—44no—
283C22—42no—
284C22—44no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—164no—
293C22—164no—
294C22C2×#12011yescharacteristic
295C22C2×AGL(1,ℤ/8)11yescharacteristic
296C22C2×AGL(1,ℤ/8)11yescharacteristic
297C4—22no—
298C4—22no—
299C4—21no—
300C4—42no—
301C4—44no—
302C4—41no—
303C4—42no—
304C4—41no—
305C4—81no—
306C4B(4,𝔽2)12yes—
307C2—21no—
308C2—21no—
309C2—42no—
310C2—42no—
311C2—82no—
312C2—84no—
313C2#300311yescharacteristic
314C2#301811yescharacteristic
315C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3496C2semidirect
2. #3006C2semidirect
3. #2388C2semidirect
4. #1604C2semidirect
5. #1601C2semidirect
6. #1598C2semidirect
7. #1595C2semidirect
8. #1507C2semidirect
9. #1465C2semidirect
11. C22×#41C22semidirect
25. #448C22semidirect
26. #447C22semidirect
28. #407C22semidirect
29. #395C22semidirect
31. #394C22semidirect
32. #393C22semidirect
34. #392C22semidirect
36. C4×C22⋊C4C22semidirect
37. #327C22semidirect
38. #327C22semidirect
39. #324C22semidirect
40. #323C22semidirect
56. C22×D4D4semidirect
78. #121D4semidirect
89. C2×C4⋊C4D4semidirect
219. C23AGL(1,ℤ/8)semidirect
267. C2×C4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#34962
#30062
#23882
#16042
#16012
#15982
#15952
#15072
#14652