Finite Group Databasebeta

← #27604

Subgroups — #27604

Order 256.

526 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 129 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#2280C211yesmaximal, characteristic
2C2×#380C211yesmaximal, characteristic
3#2077C212yesmaximal, complemented
4#2075C212yesmaximal, complemented
5#1854C212yesmaximal, complemented
6#1853C211yesmaximal, characteristic, complemented
7#1852C211yesmaximal, characteristic, complemented
8#1825C212yesmaximal, complemented
9#1824C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1779C211yesmaximal, characteristic
12C24×C4C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4—21no—
18C2×#107C2212yescomplemented
19C2×#106C2212yescomplemented
20C2×C22⋊C8C2211yescharacteristic
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2212yes—
23C2×C22⋊C8C2212yes—
24#381—21no—
25#380—21no—
26#380—21no—
27#380C2211yescharacteristic
28#380C2212yes—
29#380C2211yescharacteristic
30#380C2211yescharacteristic
31#380C2211yescharacteristic
32#380C2212yes—
33#379—21no—
34C4×C22⋊C4—21no—
35C4×C22⋊C4—21no—
36C2×#95—21no—
37#340C2212yescomplemented
38#340C2212yescomplemented
39#340C2212yescomplemented
40#340C2212yescomplemented
41#340C2212yescomplemented
42#340C2212yescomplemented
43C25D411yescharacteristic
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—24no—
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—42no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C22×C8C2312yes—
57C22×C8C2312yes—
58#117—24no—
59#117—42no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—44no—
80C2×C42—21no—
81C2×C42—21no—
82#107—42no—
83#106—42no—
84C22⋊C8—22no—
85C22⋊C8—22no—
86C22⋊C8—24no—
87C22⋊C8—42no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—41no—
100C24—41no—
101C24C2×D411yescharacteristic
102C24C2×D411yescharacteristic
103C24C2×D411yescharacteristic
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—48no—
129C22×C4—48no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4#4112yes—
140C22×C4#4112yes—
141C22×C4#4114yes—
142C22×C4C2411yescharacteristic
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—44no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—44no—
166C22⋊C4—42no—
167C42—42no—
168C42—42no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—28no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—44no—
187C23—44no—
188C23—44no—
189C23—44no—
190C23—44no—
191C23—44no—
192C23#12312yes—
193C23#12312yes—
194C23#13512yes—
195C23AGL(1,ℤ/8)12yes—
196C23#13712yes—
197C23C22×D411yescharacteristic
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4#12312yes—
235C2×C4#12312yes—
236C2×C4C2×#4111yescharacteristic
237C2×C4C2×#4111yescharacteristic
238C2×C4C2×#4112yes—
239C2×C4H(5,𝔽2)12yes—
240C8—42no—
241C8—42no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—416no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22#48514yes—
264C22C2×#12311yescharacteristic
265C22C2×#12311yescharacteristic
266C22#53411yescharacteristic
267C22C2×#13511yescharacteristic
268C22C2×AGL(1,ℤ/8)11yescharacteristic
269C22C2×#13711yescharacteristic
270C22#57511yescharacteristic
271C4—22no—
272C4—22no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—82no—
277C4—81no—
278C4—82no—
279C4—81no—
280C4#55112yes—
281C4#55112yes—
282C2—24no—
283C2—42no—
284C2—42no—
285C2#248311yescharacteristic
286C2C2×#48511yescharacteristic
287C2#309111yescharacteristic
288C2#309211yescharacteristic
289C2#309411yescharacteristic
290C2#318811yescharacteristic
291C2#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
3. #2077C2semidirect
4. #2075C2semidirect
5. #1854C2semidirect
6. #1853C2semidirect
7. #1852C2semidirect
8. #1825C2semidirect
9. #1824C2semidirect
10. #1824C2semidirect
18. C2×#107C22semidirect
19. C2×#106C22semidirect
37. #340C22semidirect
38. #340C22semidirect
39. #340C22semidirect
40. #340C22semidirect
41. #340C22semidirect
42. #340C22semidirect

Maximal subgroups

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

subgroupindex
#22802
C2×#3802
#20772
#20752
#18542
#18532
#18522
#18252
#18242
#17792