Finite Group Databasebeta

← #24543

Subgroups — #24543

Order 256.

490 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 127 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#2503C211yesmaximal, characteristic, complemented
4#2341C211yesmaximal, characteristic
5#2073C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1878C212yesmaximal, complemented
9#1851C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1805C212yesmaximal, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C22×M16C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19#452—24no—
20C2×C4⋊C8C2212yes—
21#417—24no—
22C2×#103C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28#400—21no—
29#400—21no—
30#391C2212yes—
31#390—22no—
32#390C2212yes—
33#382C2212yes—
34#381—22no—
35#380—21no—
36#380C2211yescharacteristic
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2211yescharacteristic
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×C4C2311yescharacteristic
54C2×Q16—22no—
55C2×Q16—24no—
56C2×SD16—22no—
57C2×SD16—42no—
58C2×M16—22no—
59C2×M16—42no—
60C22×C8C2312yes—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C4×Q8—24no—
66C4×Q8—42no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4D412yes—
80C2×C42—22no—
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—22no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8C2×D412yes—
108C2×Q8#4114yes—
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—44no—
114C2×D4C2×D412yes—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4C2×D412yes—
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D412yes—
130C22×C4#4112yes—
131C22×C4C2411yescharacteristic
132Q16—44no—
133SD16—44no—
134M16—44no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—82no—
155C22⋊C4—81no—
156C42—22no—
157C42—22no—
158C42—44no—
159C23—22no—
160C23—22no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—42no—
165C23—42no—
166C23—41no—
167C23—41no—
168C23#12112yes—
169C23C22×D411yescharacteristic
170Q8—22no—
171Q8—24no—
172Q8—41no—
173Q8—41no—
174Q8—42no—
175Q8#12314yes—
176D4—22no—
177D4—22no—
178D4—48no—
179D4—42no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—48no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—84no—
204C2×C4#12112yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C2×#4112yes—
207C2×C4C2×#4111yescharacteristic
208C2×C4H(5,𝔽2)12yes—
209C8—41no—
210C8—42no—
211C8—41no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—82no—
221C22—82no—
222C22C2×#12111yescharacteristic
223C22#54612yes—
224C22#57512yes—
225C22#57711yescharacteristic
226C22#57811yescharacteristic
227C4—22no—
228C4—24no—
229C4—42no—
230C4—42no—
231C4—81no—
232C4—82no—
233C4—81no—
234C4—82no—
235C4C2×#12311yescharacteristic
236C4#53411yescharacteristic
237C4#55112yes—
238C2—42no—
239C2—82no—
240C2#241211yescharacteristic
241C2#304011yescharacteristic
242C2#304611yescharacteristic
243C2#331712yes—
244C2#332712yes—
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. #2503C2semidirect
5. #2073C2semidirect
6. #2055C2semidirect
7. #1894C2semidirect
8. #1878C2semidirect
9. #1851C2semidirect
10. #1824C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×#4172
#25032
#23412
#20732
#20552
#18942
#18782
#18512
#18242
#18052