Finite Group Databasebeta

← #32776

Subgroups — #32776

Order 256.

422 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 109 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#2856C211yesmaximal, characteristic
2#2851C211yesmaximal, characteristic, complemented
3#2092C212yesmaximal, complemented
4#2084C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2034C212yesmaximal, complemented
9#2033C212yesmaximal, complemented
10#2025C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12C2×#127C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#121—21no—
15C2×#121—21no—
16C2×#121C2212yes—
17C22×C42C2211yescharacteristic
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25#390—21no—
26#390—21no—
27#390C2212yes—
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#387—21no—
31#387—21no—
32#385—21no—
33#385—21no—
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36#380C2211yescharacteristic
37#380C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×D4—22no—
44C22×D4—21no—
45C22×D4—21no—
46C22×D4C2312yes—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52#127—42no—
53#125—42no—
54#121—42no—
55#121—42no—
56#121—44no—
57#121—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—42no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—42no—
80#106—42no—
81#102—42no—
82#102—42no—
83#102—42no—
84#102—42no—
85C22⋊C8—42no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92C24—22no—
93C24C2×D411yescharacteristic
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—42no—
108C2×D4—42no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2411yescharacteristic
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—44no—
151C22⋊C4—44no—
152C22⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C42—22no—
156C42—22no—
157C42—41no—
158C42—41no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23AGL(1,ℤ/8)12yes—
173C23C22×D411yescharacteristic
174D4—42no—
175D4—42no—
176D4—42no—
177D4—42no—
178D4—84no—
179D4—84no—
180D4—88no—
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—21no—
193C2×C4—21no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4#13512yes—
214C2×C4AGL(1,ℤ/8)12yes—
215C2×C4H(5,𝔽2)12yes—
216C2×C4H(5,𝔽2)11yescharacteristic
217C2×C4H(5,𝔽2)11yescharacteristic
218C2×C42−1+412yes—
219C8—42no—
220C8—42no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22#53411yescharacteristic
236C22#53511yescharacteristic
237C22#53611yescharacteristic
238C22C2×#13511yescharacteristic
239C22C2×AGL(1,ℤ/8)11yescharacteristic
240C22C2×AGL(1,ℤ/8)11yescharacteristic
241C22#57511yescharacteristic
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—81no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4—82no—
252C4#56212yes—
253C4#56212yes—
254C2—42no—
255C2—162no—
256C2#281611yescharacteristic
257C2#318811yescharacteristic
258C2#319411yescharacteristic
259C2#321211yescharacteristic
260C2#321711yescharacteristic
261C2#322811yescharacteristic
262C2#323411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2856C2
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. #2851C2semidirect
3. #2092C2semidirect
4. #2084C2semidirect
5. #2083C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2034C2semidirect
9. #2033C2semidirect
10. #2025C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28562
#28512
#20922
#20842
#20832
#20412
#20342
#20332
#20252
#18172