Finite Group Databasebeta

← #27895

Subgroups — #27895

Order 256.

406 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 119 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#2850C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic, complemented
3#2036C212yesmaximal, complemented
4#2034C212yesmaximal
5#1959C211yesmaximal, characteristic, complemented
6#1959C211yesmaximal, characteristic, complemented
7#1927C212yesmaximal, complemented
8#1925C212yesmaximal, complemented
9#1847C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#117—21no—
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2212yes—
24C2×#97C2211yescharacteristic
25C2×#97C2211yescharacteristic
26#391C2211yescharacteristic
27#390C2211yescharacteristic
28#388—21no—
29#388—21no—
30#386—21no—
31#386—21no—
32#384C2211yescharacteristic
33#384C2211yescharacteristic
34#382—21no—
35#382C2211yescharacteristic
36#380—21no—
37#380C2212yes—
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51#124—22no—
52#124—22no—
53C22⋊Q8—42no—
54#121—42no—
55#117—44no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—42no—
80#106—42no—
81#103—24no—
82#103—42no—
83#102—24no—
84#102—42no—
85C22⋊C8—24no—
86#97—22no—
87#97—22no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24—21no—
95C24C2×D411yescharacteristic
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×D4—22no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8#4114yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C22⋊C4—44no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—21no—
155C42—21no—
156C42—44no—
157C42#4112yes—
158C42#4112yes—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23#13712yes—
173C23C22×D411yescharacteristic
174Q8—42no—
175Q8—42no—
176D4—42no—
177D4—42no—
178D4—84no—
179C2×C4—28no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4#12312yes—
211C2×C4#12312yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4C2×#4111yescharacteristic
214C2×C4C2×#4112yes—
215C2×C4C2×#4111yescharacteristic
216C2×C4H(5,𝔽2)12yes—
217C8—24no—
218C8—42no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22C2×#12311yescharacteristic
234C22#53211yescharacteristic
235C22#53511yescharacteristic
236C22C2×AGL(1,ℤ/8)11yescharacteristic
237C22C2×#13711yescharacteristic
238C22#57511yescharacteristic
239C22#57511yescharacteristic
240C4—22no—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—41no—
245C4—82no—
246C4—81no—
247C4—82no—
248C4—81no—
249C4—81no—
250C4#55312yes—
251C4#55512yes—
252C2—42no—
253C2—161no—
254C2#250011yescharacteristic
255C2#309211yescharacteristic
256C2#309311yescharacteristic
257C2#317111yescharacteristic
258C2#318111yescharacteristic
259C2#321411yescharacteristic
260C2#321511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2285C2semidirect
3. #2036C2semidirect
5. #1959C2semidirect
6. #1959C2semidirect
7. #1927C2semidirect
8. #1925C2semidirect
9. #1847C2semidirect
10. #1836C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
#28502
#22852
#20362
#20342
#19592
#19272
#19252
#18472
#18362
#17492