Finite Group Databasebeta

← #24019

Subgroups — #24019

Order 256.

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

normal subgroups only

This group has 643 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
1#3486C211yesmaximal, characteristic
2#3092C212yesmaximal, complemented
3C2×#391C211yesmaximal, characteristic, complemented
4#2038C214yesmaximal, complemented
5#1896C212yesmaximal, complemented
6#1857C212yesmaximal, complemented
7#1793C212yesmaximal, complemented
8#1784C211yesmaximal, characteristic, complemented
9C2×2−1+4C2211yescharacteristic
10C23×Q8C2211yescharacteristic
11D4×Q8—22no—
12D4×Q8—22no—
13#548—22no—
14#548—22no—
15#536—22no—
16#536C2212yes—
17#525—22no—
18#525C2212yes—
19#519C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#484—22no—
23#483—22no—
24#481—22no—
25#480—22no—
26#428C2214yes—
27C2×#104C2214yes—
28#417C2214yes—
29C2×#101C2212yes—
30C2×#101C2212yes—
31#407C2212yes—
32#391—22no—
33#391C2212yes—
34#391C2212yes—
35C2×#95—22no—
36#339C2214yes—
372−1+4—24no—
382−1+4—42no—
39C2×#41—22no—
40C2×#41—22no—
41C2×#41C2312yes—
42C22×Q8—22no—
43C22×Q8—24no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—42no—
47C22×Q8D412yes—
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C23×C4—21no—
51C23×C4—21no—
52C23×C4D412yes—
53C23×C4C2311yescharacteristic
54C2×M16C2314yes—
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57#124—24no—
58#124—22no—
59#123—24no—
60#123—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—24no—
64C22⋊Q8—24no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8D412yes—
68C22⋊Q8D412yes—
69#121—22no—
70#121—22no—
71#121D412yes—
72#121D412yes—
73#120—22no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×D4—24no—
77C4×D4—24no—
78C4×D4—22no—
79C4×D4—22no—
80#117—22no—
81#117C2312yes—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2312yes—
86C2×C42—21no—
87C2×C42—42no—
88C2×C42C2312yes—
89#107—24no—
90#106—24no—
91#104—44no—
92#103—24no—
93#102—24no—
94#101—24no—
95#101—42no—
96C22⋊C8—22no—
97C22⋊C8—22no—
98#95—24no—
99#95—24no—
100C24C2×D411yescharacteristic
101#41—24no—
102#41—24no—
103#41—44no—
104#41—48no—
105#41—44no—
106C2×Q8—22no—
107C2×Q8—24no—
108C2×Q8—24no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—42no—
115C2×Q8—48no—
116C2×Q8—48no—
117C2×Q8—48no—
118C2×Q8C2×D412yes—
119C2×Q8#4114yes—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4C2×D412yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4C2×D412yes—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D412yes—
145C22×C4C2×D412yes—
146C22×C4C2411yescharacteristic
147M16—24no—
148M16—24no—
149C2×C8—24no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4—42no—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C42—24no—
168C42—22no—
169C42—24no—
170C42—42no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23#12011yescharacteristic
177C23#12011yescharacteristic
178C23C22×D411yescharacteristic
179Q8—28no—
180Q8—24no—
181Q8—48no—
182Q8—44no—
183Q8—44no—
184D4—24no—
185D4—44no—
186D4—42no—
187D4—42no—
188D4—44no—
189D4—44no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—48no—
206C2×C4—48no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#12012yes—
215C2×C4#12314yes—
216C2×C4C22×D412yes—
217C2×C4C2×#4112yes—
218C2×C4H(5,𝔽2)12yes—
219C8—44no—
220C22—24no—
221C22—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22#45412yes—
232C22#45612yes—
233C22C2×#12011yescharacteristic
234C22C2×#12311yescharacteristic
235C22#53411yescharacteristic
236C4—22no—
237C4—24no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—82no—
244C4#54614yes—
245C2—21no—
246C2—21no—
247C2—22no—
248C2—22no—
249C2—82no—
250C2#240511yescharacteristic
251C2C2×#45411yescharacteristic
252C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3486C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3092C2semidirect
3. C2×#391C2semidirect
4. #2038C2semidirect
5. #1896C2semidirect
6. #1857C2semidirect
7. #1793C2semidirect
8. #1784C2semidirect

Maximal subgroups

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

subgroupindex
#34862
#30922
C2×#3912
#20382
#18962
#18572
#17932
#17842