Finite Group Databasebeta

← #24528

Subgroups — #24528

Order 256.

532 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 143 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×#448C211yesmaximal, characteristic
2C22×#103C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic, complemented
5#2072C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1893C211yesmaximal, characteristic
8#1880C212yesmaximal, complemented
9#1850C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1803C212yesmaximal, complemented
12C23×Q8C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×#123—21no—
17C2×#123—21no—
18C2×#121C2211yescharacteristic
19C2×C4×Q8C2212yes—
20C22×C4⋊C4C2211yescharacteristic
21#448—24no—
22C2×C4⋊C8C2212yes—
23C2×#103—24no—
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2211yescharacteristic
28C2×#102C2212yes—
29C2×C22⋊C8C2211yescharacteristic
30#399—22no—
31#391—22no—
32#391C2212yes—
33#390C2212yes—
34#384C2212yes—
35#380—21no—
36#380C2211yescharacteristic
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—42no—
47C22×Q8D414yes—
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C2×SD16—22no—
56C2×SD16—22no—
57C2×SD16—24no—
58C2×SD16—42no—
59C22×C8—22no—
60C22×C8—42no—
61C22×C8C2312yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64#123—44no—
65#123—44no—
66#121—22no—
67#121—22no—
68C4×Q8—24no—
69C4×Q8—42no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4D411yescharacteristic
84C2×C22⋊C4D411yescharacteristic
85C2×C42—22no—
86C2×C42C2312yes—
87C4⋊C8—42no—
88#103—24no—
89#103—24no—
90#103—22no—
91#103—22no—
92#102—42no—
93C22⋊C8—22no—
94#95—22no—
95#95—22no—
96#95—21no—
97#95D412yes—
98C24—21no—
99C24C2×D411yescharacteristic
100C2×Q8—24no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—28no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—44no—
108C2×Q8—44no—
109C2×Q8—48no—
110C2×Q8C2×D412yes—
111C2×Q8C2×D412yes—
112C2×Q8#4114yes—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—44no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2×D412yes—
136C22×C4#4112yes—
137C22×C4C2411yescharacteristic
138SD16—44no—
139SD16—44no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—44no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C22⋊C4—22no—
159C22⋊C4—44no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—44no—
163C22⋊C4SD1614yescomplemented
164C42—22no—
165C42—22no—
166C42—44no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—42no—
171C23—41no—
172C23—41no—
173C23—42no—
174C23#12112yes—
175C23C2×SD1612yes—
176C23AGL(1,ℤ/8)12yes—
177C23C22×D411yescharacteristic
178Q8—22no—
179Q8—24no—
180Q8—22no—
181Q8—22no—
182Q8—48no—
183Q8—42no—
184Q8#12314yescomplemented
185D4—41no—
186D4—41no—
187D4—42no—
188D4—84no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—48no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—84no—
213C2×C4#12112yes—
214C2×C4C2×SD1614yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C2×#4112yes—
217C2×C4C2×#4111yescharacteristic
218C2×C4H(5,𝔽2)12yes—
219C8—41no—
220C8—42no—
221C8—41no—
222C22—22no—
223C22—24no—
224C22—22no—
225C22—24no—
226C22—81no—
227C22—81no—
228C22—82no—
229C22—81no—
230C22—82no—
231C22#46014yes—
232C22C2×#12111yescharacteristic
233C22#54612yes—
234C22C22×SD1611yescharacteristic
235C22C2×AGL(1,ℤ/8)11yescharacteristic
236C22#57812yes—
237C4—22no—
238C4—24no—
239C4—24no—
240C4—42no—
241C4—42no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4C2×#12311yescharacteristic
246C4#53411yescharacteristic
247C4#55112yes—
248C2—24no—
249C2—161no—
250C2#241211yescharacteristic
251C2C2×#46011yescharacteristic
252C2#304411yescharacteristic
253C2#329812yes—
254C2#330812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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. C22×#103C2semidirect
4. Q8×C22⋊C4C2semidirect
5. #2072C2semidirect
6. #2055C2semidirect
8. #1880C2semidirect
9. #1850C2semidirect
10. #1823C2semidirect
11. #1803C2semidirect
163. C22⋊C4SD16semidirect
184. Q8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4482
C22×#1032
#25012
Q8×C22⋊C42
#20722
#20552
#18932
#18802
#18502
#18232
#18032