Finite Group Databasebeta

← #24556

Subgroups — #24556

Order 256.

578 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 137 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×#484C211yesmaximal, characteristic, complemented
2C2×#407C211yesmaximal, characteristic
3#2848C211yesmaximal, characteristic
4C2×#380C211yesmaximal, characteristic
5#2075C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1894C212yesmaximal, complemented
8#1872C211yesmaximal, characteristic, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1809C212yesmaximal, complemented
12C24×C4C2211yescharacteristic
13C22×M16C2212yes—
14C2×C22⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—21no—
20#484—24no—
21C2×#107C2212yescomplemented
22C2×#106C2212yescomplemented
23C2×#103C2212yescomplemented
24C2×#103C2212yescomplemented
25#407—24no—
26C2×C22⋊C8C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#385—21no—
30#385—21no—
31#385—21no—
32#380—21no—
33#380—21no—
34#380C2211yescharacteristic
35#380C2212yes—
36#380C2211yescharacteristic
37#380C2212yes—
38#380C2211yescharacteristic
39#379—21no—
40C2×#95—21no—
41#340C2212yescomplemented
42#340C2212yescomplemented
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25D411yescharacteristic
46C22×Q8C2311yescharacteristic
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—24no—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—42no—
53C23×C4D411yescharacteristic
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C2×M16—24no—
60C2×M16—24no—
61C2×M16—42no—
62C22×C8C2312yes—
63C22×C8C2312yes—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C22⋊Q8—24no—
67C22⋊Q8—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—44no—
90#107—24no—
91#106—42no—
92#103—24no—
93#103—42no—
94C22⋊C8—22no—
95C22⋊C8—24no—
96C22⋊C8D414yes—
97#95—22no—
98#95—22no—
99#95—21no—
100#95—21no—
101#95—21no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—41no—
108C24—41no—
109C24C2×D411yescharacteristic
110C24C2×D411yescharacteristic
111C24C2×D411yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—48no—
141C22×C4—48no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D412yes—
151C22×C4C2×D411yescharacteristic
152C22×C4#4112yes—
153C22×C4#4114yes—
154C22×C4C2411yescharacteristic
155M16—48no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8C2×D414yes—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—44no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—44no—
182C22⋊C4—42no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—28no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23—44no—
206C23#12112yes—
207C23#12312yes—
208C23AGL(1,ℤ/8)12yes—
209C23#13712yes—
210C23#13712yes—
211C23C22×D411yescharacteristic
212Q8—42no—
213Q8—42no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—84no—
249C2×C4—84no—
250C2×C4#12112yes—
251C2×C4#12312yes—
252C2×C4C22×D411yescharacteristic
253C2×C4C2×#4112yes—
254C2×C4C2×#4111yescharacteristic
255C2×C4H(5,𝔽2)12yes—
256C8—24no—
257C8—42no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—416no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22#47014yes—
280C22C2×#12111yescharacteristic
281C22C2×#12311yescharacteristic
282C22#53411yescharacteristic
283C22C2×AGL(1,ℤ/8)11yescharacteristic
284C22C2×#13711yescharacteristic
285C22C2×#13711yescharacteristic
286C22#57511yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4—82no—
293C4—81no—
294C4—81no—
295C4—82no—
296C4#54612yes—
297C4#55112yes—
298C2—24no—
299C2—42no—
300C2—42no—
301C2#241211yescharacteristic
302C2C2×#47011yescharacteristic
303C2#305511yescharacteristic
304C2#309211yescharacteristic
305C2#309511yescharacteristic
306C2#319311yescharacteristic
307C2#319611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
5. #2075C2semidirect
6. #2035C2semidirect
7. #1894C2semidirect
8. #1872C2semidirect
9. #1853C2semidirect
10. #1824C2semidirect
11. #1809C2semidirect
21. C2×#107C22semidirect
22. C2×#106C22semidirect
23. C2×#103C22semidirect
24. C2×#103C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4072
#28482
C2×#3802
#20752
#20352
#18942
#18722
#18532
#18242
#18092