Finite Group Databasebeta

← #28571

Subgroups — #28571

Order 256.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#465C212yesmaximal
2#2857C211yesmaximal, characteristic, complemented
3C4×#121C211yesmaximal, characteristic
4#2084C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#1927C212yesmaximal, complemented
9#1892C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11C2×#336C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C23×C8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×#121C2211yescharacteristic
17C2×C4×D4—21no—
18C2×C4×D4C2211yescharacteristic
19C22×C42C2211yescharacteristic
20#465—28no—
21C2×#106C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26#391C2212yes—
27#390C2211yescharacteristic
28#389—21no—
29#389—21no—
30#389C2211yescharacteristic
31#388—21no—
32#386—21no—
33#384C2211yescharacteristic
34#380—21no—
35#380—21no—
36#380C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43#336C2212yes—
44C22×Q8C2312yes—
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—42no—
54C22×C8—24no—
55C22×C8—24no—
56C22×C8—42no—
57C22×C8C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—42no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62#121—22no—
63#121D414yes—
64C4×D4—42no—
65C4×D4—44no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—21no—
83C2×C42—42no—
84C2×C42D411yescharacteristic
85C2×C42D411yescharacteristic
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88#106—24no—
89#103—24no—
90#103—24no—
91#103—42no—
92#102—24no—
93#95—21no—
94#95—21no—
95#95—21no—
96#95—22no—
97#95—22no—
98C24—21no—
99C24C2×D411yescharacteristic
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4C2×D412yes—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4SD1614yescomplemented
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4#4112yes—
136C22×C4C2411yescharacteristic
137SD16—48no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—48no—
144C2×C8#4114yes—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4C2×D412yes—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C22⋊C4—44no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C42—22no—
164C42—22no—
165C42—41no—
166C42—41no—
167C42—42no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—22no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23#12112yes—
180C23C2×SD1612yes—
181C23#13712yes—
182C23C22×D411yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—42no—
186Q8—42no—
187D4—24no—
188D4—48no—
189D4—42no—
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—24no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—84no—
221C2×C4#12112yes—
222C2×C4C2×SD1612yes—
223C2×C4C2×SD1612yes—
224C2×C4#13512yes—
225C2×C4C22×D411yescharacteristic
226C2×C4C2×#4112yes—
227C2×C4C2×#4111yescharacteristic
228C2×C4H(5,𝔽2)12yes—
229C8—24no—
230C8—42no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22#46114yes—
245C22C2×#12111yescharacteristic
246C22#53211yescharacteristic
247C22#53411yescharacteristic
248C22C22×SD1611yescharacteristic
249C22C2×#13511yescharacteristic
250C22C2×#13711yescharacteristic
251C22#57511yescharacteristic
252C4—22no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—42no—
258C4—81no—
259C4—81no—
260C4—82no—
261C4—82no—
262C4#54612yes—
263C4#55312yes—
264C2—24no—
265C2—82no—
266C2#254311yescharacteristic
267C2C2×#46111yescharacteristic
268C2#303511yescharacteristic
269C2#316711yescharacteristic
270C2#318111yescharacteristic
271C2#318711yescharacteristic
272C2#319011yescharacteristic
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. #2857C2semidirect
4. #2084C2semidirect
5. #2071C2semidirect
6. #2043C2semidirect
7. #2013C2semidirect
8. #1927C2semidirect
9. #1892C2semidirect
10. #1877C2semidirect
11. C2×#336C2semidirect
128. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
#28572
C4×#1212
#20842
#20712
#20432
#20132
#19272
#18922
#18772
C2×#3362