Finite Group Databasebeta

← #32761

Subgroups — #32761

Order 256.

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

normal subgroups only

This group has 419 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#2857C211yesmaximal, characteristic, complemented
2#2850C211yesmaximal, characteristic
3#2074C212yesmaximal, complemented
4#2072C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2016C211yesmaximal, characteristic, complemented
7#2015C212yesmaximal, complemented
8#2013C211yesmaximal, characteristic, complemented
9#2001C212yesmaximal, complemented
10#2000C212yesmaximal
11#1817C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2212yes—
18C2×#121C2211yescharacteristic
19C22×C42C2211yescharacteristic
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25#391C2211yescharacteristic
26#391C2212yes—
27#390C2211yescharacteristic
28#389—21no—
29#389—21no—
30#388—21no—
31#388—21no—
32#386—21no—
33#386—21no—
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36#380—21no—
37#380—21no—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×Q8C2312yes—
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C2×Q16—42no—
51C2×Q16—42no—
52C2×Q16—42no—
53C2×SD16—42no—
54C2×SD16—42no—
55C2×SD16—42no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—42no—
59#124—42no—
60C22⋊Q8—42no—
61C22⋊Q8—42no—
62C22⋊Q8—42no—
63#121—42no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C42—42no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85#103—42no—
86#103—42no—
87#103—42no—
88#102—42no—
89C22⋊C8—42no—
90#95—22no—
91#95—22no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24—21no—
97C24C2×D411yescharacteristic
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—42no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136Q16—48no—
137SD16—48no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—44no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—22no—
162C42—22no—
163C42—41no—
164C42—41no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23#13712yes—
179C23C22×D411yescharacteristic
180Q8—42no—
181Q8—42no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—84no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
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#13512yes—
222C2×C4#13712yes—
223C2×C4H(5,𝔽2)12yes—
224C2×C4H(5,𝔽2)12yes—
225C2×C4H(5,𝔽2)11yescharacteristic
226C2×C4H(5,𝔽2)11yescharacteristic
227C8—42no—
228C8—42no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22#53411yescharacteristic
244C22#53511yescharacteristic
245C22#53511yescharacteristic
246C22C2×#13511yescharacteristic
247C22C2×#13711yescharacteristic
248C22C2×#13711yescharacteristic
249C22#57511yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—81no—
256C4—82no—
257C4—81no—
258C4—81no—
259C4—82no—
260C4#56012yes—
261C4#56012yes—
262C2—42no—
263C2—161no—
264C2#281511yescharacteristic
265C2#319011yescharacteristic
266C2#319611yescharacteristic
267C2#321011yescharacteristic
268C2#321211yescharacteristic
269C2#321511yescharacteristic
270C2#321811yescharacteristic
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
1. #2857C2semidirect
3. #2074C2semidirect
4. #2072C2semidirect
5. #2071C2semidirect
6. #2016C2semidirect
7. #2015C2semidirect
8. #2013C2semidirect
9. #2001C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#28502
#20742
#20722
#20712
#20162
#20152
#20132
#20012
#20002
#18172