Finite Group Databasebeta

← #24544

Subgroups — #24544

Order 256.

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

normal subgroups only

This group has 553 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×#449C211yesmaximal, characteristic
2C2×#417C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic, complemented
4#2339C211yesmaximal, characteristic, complemented
5#2072C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1895C211yesmaximal, characteristic, complemented
8#1879C212yesmaximal, complemented
9#1851C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1802C212yesmaximal, complemented
12C22×#41C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14C22×D8C2211yescharacteristic
15C22×M16C2211yescharacteristic
16C2×#123—22no—
17C2×#121C2211yescharacteristic
18C2×C4×D4C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20#449—24no—
21C2×C4⋊C8C2212yes—
22#417—24no—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×C22⋊C8C2211yescharacteristic
29#399—21no—
30#399—21no—
31#391C2212yes—
32#390C2212yes—
33#386—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×D4—21no—
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—22no—
55C23×C4C2311yescharacteristic
56C2×SD16—22no—
57C2×SD16—42no—
58C2×D8—22no—
59C2×D8—24no—
60C2×M16—22no—
61C2×M16—42no—
62C22×C8C2312yes—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65#123—48no—
66#121—22no—
67#121—22no—
68C4×D4—24no—
69C4×D4—42no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4D412yes—
84C2×C42C2312yes—
85C4⋊C8—42no—
86#103—24no—
87#103—22no—
88#102—24no—
89#102—22no—
90#102—42no—
91C22⋊C8—22no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95D411yescharacteristic
96#95D411yescharacteristic
97C24—21no—
98C24—21no—
99C24C2×D411yescharacteristic
100#41—28no—
101#41—44no—
102#41—48no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8C2×D412yes—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—42no—
117C2×D4C2×D412yes—
118C2×D4#4114yes—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—82no—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2×D412yes—
136C22×C4#4112yes—
137C22×C4C2411yescharacteristic
138SD16—44no—
139D8—44no—
140M16—44no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C22⋊C4—24no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C22⋊C4—42no—
161C22⋊C4—44no—
162C22⋊C4—81no—
163C42—22no—
164C42—22no—
165C23—22no—
166C23—24no—
167C23—22no—
168C23—22no—
169C23—21no—
170C23—21no—
171C23—42no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—42no—
176C23—42no—
177C23—41no—
178C23—41no—
179C23#12112yes—
180C23C22×D411yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184D4—22no—
185D4—24no—
186D4—48no—
187D4—41no—
188D4—41no—
189D4—42no—
190D4—84no—
191D4#12314yescomplemented
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—48no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—88no—
213C2×C4—84no—
214C2×C4#12112yes—
215C2×C4AGL(1,ℤ/8)14yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4112yes—
218C2×C4C2×#4111yescharacteristic
219C2×C4H(5,𝔽2)12yes—
220C8—41no—
221C8—42no—
222C8—41no—
223C22—22no—
224C22—24no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—82no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—82no—
238C22C2×#12111yescharacteristic
239C22#54612yes—
240C22C2×AGL(1,ℤ/8)12yes—
241C22#57711yescharacteristic
242C22#57811yescharacteristic
243C4—22no—
244C4—24no—
245C4—42no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4C2×#12311yescharacteristic
251C4#53411yescharacteristic
252C4#55112yes—
253C2—42no—
254C2—44no—
255C2—161no—
256C2#241211yescharacteristic
257C2#303811yescharacteristic
258C2#304611yescharacteristic
259C2#330612yes—
260C2#333012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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
2. C2×#417C2semidirect
3. #2501C2semidirect
4. #2339C2semidirect
5. #2072C2semidirect
6. #2055C2semidirect
7. #1895C2semidirect
8. #1879C2semidirect
9. #1851C2semidirect
10. #1824C2semidirect
11. #1802C2semidirect
191. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
C2×#4172
#25012
#23392
#20722
#20552
#18952
#18792
#18512
#18242
#18022