Finite Group Databasebeta

← #23944

Subgroups — #23944

Order 256.

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

normal subgroups only

This group has 603 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×#541C211yesmaximal, characteristic, complemented
2C2×#452C212yesmaximal
3#2311C211yesmaximal, characteristic
4#2026C211yesmaximal, characteristic, complemented
5#2025C211yesmaximal, characteristic, complemented
6#2016C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#1880C212yesmaximal, complemented
9#1878C212yesmaximal, complemented
10#1872C212yesmaximal, complemented
11#1818C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15#541—28no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—22no—
18C2×#125—22no—
19C2×#125C2211yescharacteristic
20C2×C22⋊Q8C2212yes—
21C2×C4×Q8C2212yes—
22C2×#117C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#452—28no—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×C22⋊C8C2212yes—
30#391C2211yescharacteristic
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#386—22no—
35#380—22no—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—24no—
44C2×#41—42no—
45C22×Q8C2312yes—
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C2×Q16—24no—
53C2×Q16—24no—
54C2×Q16—42no—
55C2×SD16—24no—
56C2×SD16—24no—
57C2×SD16—42no—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—24no—
61#126—216no—
62#125—216no—
63#125—24no—
64C22⋊Q8—24no—
65C22⋊Q8D418yes—
66C4×Q8—24no—
67C4×Q8—24no—
68#117—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4D412yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4D412yes—
83C2×C42—22no—
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86#103—24no—
87#103—42no—
88#103—42no—
89#102—24no—
90C22⋊C8—24no—
91#95—22no—
92#95—22no—
93C24—21no—
94C24C2×D411yescharacteristic
95#41—48no—
96#41—48no—
97C2×Q8—22no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×D414yes—
106C2×Q8#4112yes—
107C2×D4—24no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4#4112yes—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—82no—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D414yes—
131C22×C4C2411yescharacteristic
132Q16—48no—
133SD16—48no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—28no—
141C4⋊C4—28no—
142C4⋊C4—28no—
143C4⋊C4—28no—
144C4⋊C4—24no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4C2×D414yes—
150C22⋊C4—28no—
151C22⋊C4—24no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C42—28no—
155C42—22no—
156C42—22no—
157C23—21no—
158C23—21no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—42no—
165C23—42no—
166C23#12012yes—
167C23C22×D411yescharacteristic
168Q8—24no—
169Q8—24no—
170Q8—24no—
171Q8—42no—
172Q8—42no—
173Q8—42no—
174D4—24no—
175D4—48no—
176D4—42no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—48no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—84no—
202C2×C4#12012yes—
203C2×C4C22×D412yes—
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4H(5,𝔽2)12yes—
207C8—42no—
208C8—42no—
209C22—22no—
210C22—24no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—44no—
215C22—42no—
216C22—81no—
217C22—81no—
218C22—82no—
219C22C2×#12011yescharacteristic
220C22#52511yescharacteristic
221C22#53511yescharacteristic
222C22#57712yes—
223C22#57812yes—
224C4—22no—
225C4—42no—
226C4—44no—
227C4—42no—
228C4—42no—
229C4—44no—
230C4—81no—
231C4—81no—
232C4#54612yes—
233C4#54612yes—
234C2—42no—
235C2—82no—
236C2#240411yescharacteristic
237C2#301312yes—
238C2#312012yes—
239C2#322211yescharacteristic
240C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#541C2semidirect
4. #2026C2semidirect
5. #2025C2semidirect
6. #2016C2semidirect
7. #2013C2semidirect
8. #1880C2semidirect
9. #1878C2semidirect
10. #1872C2semidirect
11. #1818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4522
#23112
#20262
#20252
#20162
#20132
#18802
#18782
#18722
#18182