Finite Group Databasebeta

← #30040

Subgroups — #30040

Order 256.

475 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 113 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#2861C211yesmaximal, characteristic, complemented
2C4×#127C211yesmaximal, characteristic, complemented
3#2072C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic, complemented
5#2025C212yesmaximal, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2012C212yesmaximal, complemented
8#2011C211yesmaximal, characteristic
9#1879C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1839C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×D4—21no—
18C2×C4×D4—21no—
19C2×C4×D4C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24#391C2211yescharacteristic
25#391C2211yescharacteristic
26#390—21no—
27#390—21no—
28#390C2212yes—
29#390C2211yescharacteristic
30#390C2211yescharacteristic
31#388—21no—
32#388—21no—
33#388—21no—
34#388—21no—
35#384C2211yescharacteristic
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C22×D4C2312yes—
48C23×C4—22no—
49C2×SD16—24no—
50C2×SD16—42no—
51C2×SD16—42no—
52C2×D8—24no—
53C2×D8—42no—
54C2×D8—42no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57#127—24no—
58#125—42no—
59#124—42no—
60C4×D4—42no—
61C4×D4—42no—
62C4×D4—44no—
63C4×D4—44no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
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×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87#103—42no—
88#102—42no—
89#102—42no—
90#102—42no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95C24—22no—
96C24—21no—
97C2×Q8—22no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4#4114yes—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131SD16—48no—
132D8—48no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—44no—
149C22⋊C4—44no—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42—22no—
163C42—41no—
164C23—24no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23C22×D411yescharacteristic
179Q8—42no—
180Q8—42no—
181D4—24no—
182D4—24no—
183D4—48no—
184D4—48no—
185D4—42no—
186D4—42no—
187D4—42no—
188D4—42no—
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—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—88no—
217C2×C4#13512yes—
218C2×C4AGL(1,ℤ/8)12yes—
219C2×C4AGL(1,ℤ/8)12yes—
220C2×C4C2×#4112yes—
221C2×C4H(5,𝔽2)12yes—
222C2×C4H(5,𝔽2)11yescharacteristic
223C2×C4H(5,𝔽2)11yescharacteristic
224C8—42no—
225C8—42no—
226C22—44no—
227C22—44no—
228C22—44no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#52511yescharacteristic
241C22#53511yescharacteristic
242C22#53511yescharacteristic
243C22C2×#13511yescharacteristic
244C22C2×AGL(1,ℤ/8)11yescharacteristic
245C22C2×AGL(1,ℤ/8)11yescharacteristic
246C22#57511yescharacteristic
247C4—22no—
248C4—22no—
249C4—22no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4—82no—
257C4#53812yes—
258C4#53812yes—
259C2—84no—
260C2—161no—
261C2#263111yescharacteristic
262C2#310511yescharacteristic
263C2#311411yescharacteristic
264C2#320711yescharacteristic
265C2#321211yescharacteristic
266C2#321411yescharacteristic
267C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2861C2semidirect
2. C4×#127C2semidirect
3. #2072C2semidirect
4. #2071C2semidirect
5. #2025C2semidirect
6. #2014C2semidirect
7. #2012C2semidirect
9. #1879C2semidirect
10. #1877C2semidirect
11. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28612
C4×#1272
#20722
#20712
#20252
#20142
#20122
#20112
#18792
#18772
#18392