Finite Group Databasebeta

← #32943

Subgroups — #32943

Order 256.

658 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#131C211yesmaximal, characteristic
2C2×#473C212yesmaximal, complemented
3C2×#467C211yesmaximal, characteristic, complemented
4#3007C212yesmaximal
5#1908C214yesmaximal, complemented
6#1901C212yesmaximal, complemented
7#1816C212yesmaximal, complemented
8#1810C211yesmaximal, characteristic, complemented
9C22×#41C2211yescharacteristic
10C2×#137C2214yes—
11C22×Q16C2211yescharacteristic, complemented
12C2×#131—28no—
13C2×#131—21no—
14C2×#131—21no—
15C2×#131C2212yes—
16C2×#131C2212yes—
17C22×M16—21no—
18C22×M16C2211yescharacteristic
19C23×C8—21no—
20C23×C8C2211yescharacteristic
21C2×C22⋊Q8C2212yes—
22#473—28no—
23#467—21no—
24#467—21no—
25#467—21no—
26#467—21no—
27#452—24no—
28#451—22no—
29#450—22no—
30C2×#108C2212yescomplemented
31C2×#107C2211yescharacteristic, complemented
32#417C2214yes—
33C2×#103C2212yescomplemented
34C2×#101C2214yescomplemented
35#407C2212yes—
36#343C2214yescomplemented
37#341C2212yescomplemented
38C2×#41—24no—
39C2×#41—24no—
40C2×#41—21no—
41C2×#41—21no—
42C2×#41D412yes—
43C2×#41C2312yes—
44C22×Q8D411yescharacteristic
45C22×Q8C2312yes—
46C22×D4—21no—
47C22×D4D411yescharacteristic
48C23×C4—21no—
49C23×C4C2311yescharacteristic
50#137—28no—
51#137—28no—
52C2×Q16—24no—
53C2×Q16—21no—
54C2×Q16—21no—
55C2×Q16—24no—
56C2×SD16—24no—
57#131—28no—
58#131—28no—
59#131—24no—
60#131—24no—
61#131—24no—
62#131D418yes—
63C2×M16—28no—
64C2×M16—21no—
65C2×M16—24no—
66C2×M16—22no—
67C2×M16—22no—
68C2×M16C2314yes—
69C2×M16C2312yes—
70C22×C8—28no—
71C22×C8—21no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—24no—
75C22×C8D411yescharacteristic
76C22×C8D411yescharacteristic
77C22×C8D411yescharacteristic
78C22×C8D411yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4C2312yes—
87C2×C22⋊C4—22no—
88#108—24no—
89#107—21no—
90#107—21no—
91#103—24no—
92#103—22no—
93#103—22no—
94#102—24no—
95#101—28no—
96C22⋊C8—22no—
97C22⋊C8—22no—
98C24—21no—
99C24#4111yescharacteristic
100#41—24no—
101#41—28no—
102#41—24no—
103#41—24no—
104#41—24no—
105#41—22no—
106#41—22no—
107#41C2×D414yes—
108#41#4114yes—
109C2×Q8—24no—
110C2×Q8—21no—
111C2×Q8—24no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—24no—
115C2×Q8C2×D412yes—
116C2×D4—28no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—24no—
120C2×D4—21no—
121C2×D4C2×D412yes—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4#4111yescharacteristic
136C22×C4C2411yescharacteristic
137Q16—48no—
138Q16—42no—
139Q16—42no—
140SD16—48no—
141M16—24no—
142M16—28no—
143M16—22no—
144M16—28no—
145M16—24no—
146M16—24no—
147M16C2×D414yes—
148C2×C8—24no—
149C2×C8—28no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—28no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—24no—
158C2×C8C2×D411yescharacteristic
159C2×C8C2×D411yescharacteristic
160C2×C8C2×D411yescharacteristic
161C2×C8C2×D411yescharacteristic
162C2×C8C2×D414yes—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23#12111yescharacteristic
177C23#12111yescharacteristic
178C23C2×#4111yescharacteristic
179Q8—24no—
180Q8—22no—
181Q8—42no—
182Q8—44no—
183Q8—42no—
184Q8#12114yes—
185D4—28no—
186D4—24no—
187D4—24no—
188D4—24no—
189D4—22no—
190D4#12114yes—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—82no—
207C2×C4#12111yescharacteristic
208C2×C4#12111yescharacteristic
209C2×C4C22×D412yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C2×C4C2×#4112yes—
213C8—21no—
214C8—24no—
215C8—21no—
216C8—44no—
217C8#12012yes—
218C8#12012yes—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—48no—
229C22—42no—
230C22—42no—
231C22#47312yescomplemented
232C22C2×#12111yescharacteristic
233C22#52512yes—
234C4—21no—
235C4—24no—
236C4—21no—
237C4—82no—
238C4—82no—
239C4C2×#12011yescharacteristic
240C4C2×#12112yes—
241C4#52511yescharacteristic
242C2—22no—
243C2—22no—
244C2—41no—
245C2—44no—
246C2#284711yescharacteristic
247C2C2×#47311yescharacteristic
248C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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. C2×#473C2semidirect
3. C2×#467C2semidirect
5. #1908C2semidirect
6. #1901C2semidirect
7. #1816C2semidirect
8. #1810C2semidirect
11. C22×Q16C22semidirect
30. C2×#108C22semidirect
31. C2×#107C22semidirect
33. C2×#103C22semidirect
34. C2×#101C22semidirect
36. #343C22semidirect
37. #341C22semidirect
231. C22#473semidirect

Maximal subgroups

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

subgroupindex
C22×#1312
C2×#4732
C2×#4672
#30072
#19082
#19012
#18162
#18102