Finite Group Databasebeta

← #24038

Subgroups — #24038

Order 256.

752 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 143 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×#546C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C22×#102C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1892C211yesmaximal, characteristic, complemented
8#1869C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1788C212yesmaximal, complemented
11#1786C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C23×C8C2211yescharacteristic
15#546—24no—
16#546—24no—
17C2×#124C2211yescharacteristic
18C2×#123—21no—
19C2×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#120—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C22⋊C4—21no—
28#483—24no—
29C2×#106C2212yescomplemented
30C2×#103C2212yescomplemented
31C2×#103C2212yescomplemented
32C2×#102—24no—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2212yescomplemented
39#398—21no—
40#397—21no—
41#389C2212yescomplemented
42#381—21no—
43#380C2211yescharacteristic
44#380C2212yescomplemented
45#340C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—24no—
51C2×#41—42no—
52C22×Q8C2311yescharacteristic
53C22×D4—24no—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—22no—
58C22×D4—42no—
59C22×D4D411yescharacteristic
60C22×D4D414yes—
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C22×C8—22no—
69C22×C8—42no—
70C22×C8C2312yes—
71C22×C8C2312yes—
72#124—22no—
73#124—22no—
74#123—28no—
75#123—44no—
76C22⋊Q8—22no—
77C22⋊Q8D414yes—
78#121—28no—
79#121—22no—
80#121—22no—
81#120—28no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—28no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42C2311yescharacteristic
106#106—24no—
107#103—24no—
108#103—42no—
109#102—24no—
110#102—24no—
111#102—24no—
112#102—42no—
113C22⋊C8—22no—
114C22⋊C8—42no—
115#95—21no—
116#95—22no—
117#95—22no—
118C24—21no—
119C24—21no—
120C24—24no—
121C24—44no—
122C24—42no—
123C24—42no—
124C24—82no—
125C24C2×D411yescharacteristic
126C24C2×D411yescharacteristic
127#41—48no—
128#41—44no—
129#41—44no—
130C2×Q8—21no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8C2×D412yes—
134C2×D4—28no—
135C2×D4—24no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—44no—
147C2×D4—48no—
148C2×D4—44no—
149C2×D4—44no—
150C2×D4—82no—
151C2×D4C2×D412yes—
152C2×D4C2×D412yes—
153C2×D4#4114yes—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—82no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2411yescharacteristic
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—44no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—44no—
193C4⋊C4—42no—
194C4⋊C4C2×D412yes—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—44no—
203C22⋊C4—42no—
204C42—21no—
205C42—21no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—22no—
211C23—21no—
212C23—22no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—48no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—48no—
225C23—84no—
226C23—88no—
227C23—88no—
228C23#12012yes—
229C23#13512yes—
230C23AGL(1,ℤ/8)12yes—
231C23C22×D411yescharacteristic
232Q8—24no—
233Q8—41no—
234Q8—41no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—24no—
239D4—48no—
240D4—48no—
241D4—44no—
242D4—42no—
243D4—84no—
244D4#12314yescomplemented
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—48no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4#12012yes—
273C2×C4C22×D411yescharacteristic
274C2×C4C22×D411yescharacteristic
275C2×C4C2×#4112yes—
276C2×C4H(5,𝔽2)12yes—
277C8—42no—
278C8—42no—
279C22—22no—
280C22—22no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—42no—
285C22—42no—
286C22—44no—
287C22—44no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—816no—
297C22—82no—
298C22—82no—
299C22#44914yes—
300C22C2×#12011yescharacteristic
301C22#54612yes—
302C22C2×#13511yescharacteristic
303C22C2×AGL(1,ℤ/8)11yescharacteristic
304C22#57712yes—
305C4—22no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—42no—
311C4—82no—
312C4—82no—
313C4C2×#12311yescharacteristic
314C4#53411yescharacteristic
315C4#54612yes—
316C2—24no—
317C2—44no—
318C2—81no—
319C2—82no—
320C2#240511yescharacteristic
321C2C2×#44911yescharacteristic
322C2#300911yescharacteristic
323C2#329612yes—
324C2#330612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#483C2semidirect
3. C22×#102C2semidirect
4. #2486C2semidirect
5. #2042C2semidirect
6. #2034C2semidirect
7. #1892C2semidirect
8. #1869C2semidirect
9. #1864C2semidirect
10. #1788C2semidirect
11. #1786C2semidirect
29. C2×#106C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
38. C2×C22⋊C8C22semidirect
41. #389C22semidirect
44. #380C22semidirect
45. #340C22semidirect
46. #336C22semidirect
244. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4832
C22×#1022
#24862
#20422
#20342
#18922
#18692
#18642
#17882
#17862