Finite Group Databasebeta

← #18139

Subgroups — #18139

Order 256.

481 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 113 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×#539C211yesmaximal, characteristic
2#3095C211yesmaximal, characteristic, complemented
3#3094C211yesmaximal, characteristic
4#2484C211yesmaximal, characteristic, complemented
5#1615C214yesmaximal, complemented
6#1609C214yesmaximal, complemented
7#1511C211yesmaximal, characteristic, complemented
8#1510C211yesmaximal, characteristic
9#1463C211yesmaximal, characteristic, complemented
10#539—22no—
11#539—22no—
12#539—22no—
13#539C2214yes—
14C2×#126—21no—
15C2×#125—21no—
16C2×#123—21no—
17C2×#123—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4—21no—
22C2×#117C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#485—22no—
25#484C2212yescomplemented
26#483C2212yescomplemented
27#482—22no—
28#481C2212yes—
29#480C2212yes—
30#428—21no—
31#428—21no—
32#419—21no—
33#418—21no—
34#407C2211yescharacteristic
35#407C2211yescharacteristic
36#400C2212yescomplemented
37#398C2212yescomplemented
38#387—22no—
39#385—21no—
40#381—21no—
41#381C2211yescharacteristic
42#328C2214yescomplemented
43#327C2214yes—
44#324C2214yescomplemented
45C22×Q8D411yescharacteristic
46C22×D4—21no—
47C22×D4D411yescharacteristic
48C23×C4—21no—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×M16—21no—
53C2×M16—21no—
54#126—24no—
55#126—24no—
56#125—24no—
57#125—24no—
58#123—24no—
59#123—24no—
60#123—24no—
61#123—24no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64C22⋊Q8—21no—
65C22⋊Q8—21no—
66C22⋊Q8D412yes—
67C22⋊Q8C2312yes—
68#121—21no—
69#121—21no—
70#121D412yes—
71#121C2312yes—
72C4×D4—24no—
73C4×D4—24no—
74#117—21no—
75#117—21no—
76#117—21no—
77#117—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4D412yes—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42—21no—
102#107—22no—
103#107—22no—
104#106—22no—
105#106—22no—
106#103—22no—
107#103—22no—
108#102—22no—
109#102—22no—
110C22⋊C8—21no—
111C22⋊C8—21no—
112C22⋊C8C2312yes—
113C22⋊C8C2312yes—
114#95—22no—
115#95—22no—
116#95—21no—
117#95—22no—
118#95C2312yes—
119C24—21no—
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—42no—
123C2×Q8C2×D412yes—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—42no—
128C2×D4—44no—
129C2×D4C2×D412yes—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—82no—
145C22×C4C2×D412yes—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D412yes—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2411yescharacteristic
153M16—42no—
154M16—42no—
155C2×C8—22no—
156C2×C8—22no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—44no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C42—24no—
186C42—41no—
187C42—41no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23#12011yescharacteristic
196C23#12011yescharacteristic
197C23C22×D411yescharacteristic
198Q8—42no—
199Q8—81no—
200D4—42no—
201D4—44no—
202D4—44no—
203D4—81no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—84no—
233C2×C4#12011yescharacteristic
234C2×C4#12011yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C8—42no—
238C8—42no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—84no—
251C22#45812yes—
252C22C2×#12011yescharacteristic
253C22#57512yes—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—42no—
260C4—42no—
261C4—42no—
262C4—42no—
263C4—82no—
264C4—81no—
265C4—81no—
266C2—22no—
267C2—22no—
268C2—41no—
269C2—82no—
270C2#300511yescharacteristic
271C2#302011yescharacteristic
272C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3095C2semidirect
4. #2484C2semidirect
5. #1615C2semidirect
6. #1609C2semidirect
7. #1511C2semidirect
9. #1463C2semidirect
25. #484C22semidirect
26. #483C22semidirect
36. #400C22semidirect
37. #398C22semidirect
42. #328C22semidirect
44. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5392
#30952
#30942
#24842
#16152
#16092
#15112
#15102
#14632