Finite Group Databasebeta

← #32950

Subgroups — #32950

Order 256.

641 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 641 conjugacy classes of subgroups — too many to draw. Only its 145 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#3573C211yesmaximal, characteristic, complemented
2C2×#471C212yesmaximal, complemented
3#3057C211yesmaximal, characteristic, complemented
4#3022C211yesmaximal, characteristic, complemented
5#3021C211yesmaximal, characteristic
6#1915C211yesmaximal, characteristic, complemented
7#1914C211yesmaximal, characteristic, complemented
8#1903C212yesmaximal, complemented
9#1902C212yesmaximal, complemented
10#1814C212yesmaximal, complemented
11#1813C211yesmaximal, characteristic, complemented
12#585C2211yescharacteristic
13C2×#137C2212yes—
14C2×AGL(1,ℤ/8)C2212yes—
15C22×SD16C2211yescharacteristic, complemented
16#568—24no—
17#568—22no—
18#568C2212yes—
19#568C2211yescharacteristic
20C2×#131—24no—
21C2×#131—22no—
22C2×#131C2212yes—
23C2×#131C2211yescharacteristic
24#536C2211yescharacteristic
25#535C2211yescharacteristic
26#471—24no—
27#471—24no—
28#468—22no—
29#465—22no—
30#456—22no—
31#455—22no—
32#454—22no—
33#453—22no—
34C2×#108C2212yescomplemented
35#427C2211yescharacteristic, complemented
36#421C2212yes—
37#421C2212yes—
38#419C2211yescharacteristic, complemented
39#418C2211yescharacteristic, complemented
40#413C2211yescharacteristic
41#413C2211yescharacteristic
42C2×#101C2212yescomplemented
43C2×#100C2212yescomplemented
44#344C2212yescomplemented
45#344C2212yescomplemented
46#338C2212yescomplemented
472−1+4—22no—
482−1+4D412yes—
49H(5,𝔽2)—22no—
50H(5,𝔽2)—22no—
51H(5,𝔽2)D412yes—
52C2×#41—24no—
53C2×#41—22no—
54C2×#41C2312yes—
55C2×#41C2311yescharacteristic
56C22×Q8C2311yescharacteristic
57C22×D4C2311yescharacteristic
58#137—24no—
59#137—24no—
60AGL(1,ℤ/8)—24no—
61AGL(1,ℤ/8)—24no—
62C2×Q16—22no—
63C2×SD16—22no—
64C2×SD16—22no—
65C2×SD16—21no—
66C2×SD16—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×D8—22no—
70#131—24no—
71#131—28no—
72#131—24no—
73#131—24no—
74#131—24no—
75#131—22no—
76#131—22no—
77#131—24no—
78#131—22no—
79#131—22no—
80#131D414yescomplemented
81#131D414yescomplemented
82C2×M16—28no—
83C2×M16—24no—
84C2×M16—22no—
85C2×M16—24no—
86C2×M16—22no—
87C2×M16D412yes—
88C2×M16C2312yes—
89C2×M16C2312yes—
90C2×M16C2312yes—
91C2×M16C2311yescharacteristic
92C22×C8—24no—
93C22×C8—22no—
94C22×C8D412yes—
95C22×C8C2311yescharacteristic
96C4⋊Q8—21no—
97#127—21no—
98#124—21no—
99#124—21no—
100#123—22no—
101C22⋊Q8—22no—
102#121—22no—
103#120—22no—
104#117C2311yescharacteristic
105#117C2311yescharacteristic
106#108—22no—
107#108—22no—
108#107—21no—
109#106—21no—
110#104—24no—
111#104—24no—
112#103—22no—
113#102—22no—
114#101—21no—
115#101—21no—
116#101—24no—
117#100—21no—
118#100—21no—
119#100—24no—
120C24—21no—
121#41—24no—
122#41—28no—
123#41—24no—
124#41—24no—
125#41—24no—
126#41—24no—
127#41—22no—
128#41—22no—
129#41—22no—
130#41—22no—
131#41C2×D414yes—
132#41C2×D414yes—
133C2×Q8—22no—
134C2×Q8—24no—
135C2×Q8—22no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8C2×D411yescharacteristic
140C2×Q8#4112yes—
141C2×D4—28no—
142C2×D4—22no—
143C2×D4—24no—
144C2×D4—22no—
145C2×D4—24no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—22no—
150C2×D4—42no—
151C2×D4C2×D411yescharacteristic
152C2×D4#4112yes—
153C2×D4#4112yes—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
159Q16—44no—
160SD16—44no—
161SD16—44no—
162SD16—41no—
163SD16—41no—
164SD16—42no—
165D8—44no—
166M16—24no—
167M16—28no—
168M16—22no—
169M16—22no—
170M16—24no—
171M16—24no—
172M16—24no—
173M16—22no—
174M16—22no—
175M16—22no—
176M16—22no—
177M16C2×D414yes—
178C2×C8—24no—
179C2×C8—28no—
180C2×C8—22no—
181C2×C8—24no—
182C2×C8—22no—
183C2×C8—24no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8C2×D412yes—
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D412yes—
191C2×C8C2×D412yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C42—21no—
200C42—21no—
201C23—24no—
202C23—22no—
203C23—41no—
204C23—42no—
205C23—161no—
206C23#12112yes—
207C23C2×#4111yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—41no—
212Q8—42no—
213Q8—41no—
214Q8#12114yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—42no—
224D4—41no—
225D4—162no—
226D4#12114yes—
227C2×C4—24no—
228C2×C4—28no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4#12112yes—
239C2×C4C22×D412yes—
240C2×C4C22×D411yescharacteristic
241C2×C4C22×D411yescharacteristic
242C2×C4C2×#4112yes—
243C8—24no—
244C8—22no—
245C8—42no—
246C8—42no—
247C8#12011yescharacteristic
248C8#12012yes—
249C8#12011yescharacteristic
250C22—22no—
251C22—24no—
252C22—41no—
253C22—42no—
254C22—82no—
255C22—82no—
256C22—84no—
257C22—81no—
258C22—162no—
259C22C2×#12111yescharacteristic
260C22#52512yes—
261C4—24no—
262C4—22no—
263C4—81no—
264C4—81no—
265C4—81no—
266C4C2×#12011yescharacteristic
267C4C2×#12112yes—
268C4#52511yescharacteristic
269C2—21no—
270C2—22no—
271C2—42no—
272C2—44no—
273C2—161no—
274C2#284711yescharacteristic
subgroupG/H
center, socleC2#2847
commutator, Frattini, cosocleC22×C4C24
prime radical#3021C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3573C2semidirect
2. C2×#471C2semidirect
3. #3057C2semidirect
4. #3022C2semidirect
6. #1915C2semidirect
7. #1914C2semidirect
8. #1903C2semidirect
9. #1902C2semidirect
10. #1814C2semidirect
11. #1813C2semidirect
15. C22×SD16C22semidirect
34. C2×#108C22semidirect
35. #427C22semidirect
38. #419C22semidirect
39. #418C22semidirect
42. C2×#101C22semidirect
43. C2×#100C22semidirect
44. #344C22semidirect
45. #344C22semidirect
46. #338C22semidirect
80. #131D4semidirect
81. #131D4semidirect

Maximal subgroups

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

subgroupindex
#35732
C2×#4712
#30572
#30222
#30212
#19152
#19142
#19032
#19022
#18142
#18132