Finite Group Databasebeta

← #24548

Subgroups — #24548

Order 256.

512 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 131 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×#452C211yesmaximal, characteristic
2C2×#417C211yesmaximal, characteristic
3#2502C211yesmaximal, characteristic, complemented
4#2339C211yesmaximal, characteristic
5#2072C212yesmaximal, complemented
6#2056C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1877C212yesmaximal, complemented
9#1851C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1802C212yesmaximal, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C22×M16C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×D4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19#452—24no—
20C2×C4⋊C8C2212yes—
21#417—24no—
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#398—21no—
29#398—21no—
30#391C2212yes—
31#391C2212yes—
32#386—22no—
33#385—22no—
34#384C2212yes—
35#381—21no—
36#380C2211yescharacteristic
37C4×C22⋊C4C2211yescharacteristic
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C2×#41—22no—
44C2×#41—24no—
45C2×#41—42no—
46C2×#41D414yes—
47C22×Q8C2311yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4C2311yescharacteristic
55C2×Q16—22no—
56C2×Q16—42no—
57C2×SD16—22no—
58C2×SD16—24no—
59C2×M16—22no—
60C2×M16—42no—
61C22×C8C2312yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C4×D4—24no—
67C4×D4—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4D412yes—
81C2×C42C2312yes—
82C4⋊C8—42no—
83#103—24no—
84#103—22no—
85#103—42no—
86#102—24no—
87#102—22no—
88C22⋊C8—22no—
89#95—21no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95D411yescharacteristic
95#95D411yescharacteristic
96C24—21no—
97C24C2×D411yescharacteristic
98#41—28no—
99#41—44no—
100#41—48no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8C2×D412yes—
109C2×D4—24no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—44no—
113C2×D4C2×D412yes—
114C2×D4#4114yes—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4—44no—
127C22×C4—82no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D412yes—
133C22×C4C2×D412yes—
134C22×C4#4112yes—
135C22×C4C2411yescharacteristic
136Q16—44no—
137SD16—44no—
138M16—44no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C22⋊C4—24no—
157C22⋊C4—44no—
158C22⋊C4—42no—
159C22⋊C4—81no—
160C42—22no—
161C42—22no—
162C23—22no—
163C23—24no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—21no—
168C23—42no—
169C23—41no—
170C23—41no—
171C23#12112yes—
172C23C22×D411yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—42no—
176Q8—41no—
177Q8—41no—
178Q8—42no—
179D4—22no—
180D4—24no—
181D4—48no—
182D4#12314yes—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—48no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—88no—
207C2×C4—84no—
208C2×C4#12112yes—
209C2×C4#13714yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C2×#4112yes—
212C2×C4C2×#4111yescharacteristic
213C2×C4H(5,𝔽2)12yes—
214C8—41no—
215C8—42no—
216C8—41no—
217C22—22no—
218C22—24no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—44no—
223C22—42no—
224C22—42no—
225C22—44no—
226C22—42no—
227C22C2×#12111yescharacteristic
228C22#54612yes—
229C22C2×#13712yes—
230C22#57611yescharacteristic
231C22#57711yescharacteristic
232C4—22no—
233C4—24no—
234C4—42no—
235C4—42no—
236C4—81no—
237C4—81no—
238C4—82no—
239C4—82no—
240C4C2×#12311yescharacteristic
241C4#53411yescharacteristic
242C4#55112yes—
243C2—42no—
244C2—44no—
245C2#241211yescharacteristic
246C2#303911yescharacteristic
247C2#304511yescharacteristic
248C2#331412yes—
249C2#333112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2502C2semidirect
5. #2072C2semidirect
6. #2056C2semidirect
7. #1894C2semidirect
8. #1877C2semidirect
9. #1851C2semidirect
10. #1823C2semidirect
11. #1802C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×#4172
#25022
#23392
#20722
#20562
#18942
#18772
#18512
#18232
#18022