Finite Group Databasebeta

← #32944

Subgroups — #32944

Order 256.

660 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 147 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×#568C211yesmaximal, characteristic, complemented
2C2×#471C212yesmaximal, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4#3007C211yesmaximal, characteristic, complemented
5#3006C211yesmaximal, characteristic
6#1908C212yesmaximal, complemented
7#1907C212yesmaximal, complemented
8#1901C211yesmaximal, characteristic, complemented
9#1900C211yesmaximal, characteristic, complemented
10#1816C212yesmaximal, complemented
11#1811C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C2×#137C2212yes—
14C2×AGL(1,ℤ/8)C2212yes—
15C22×SD16C2211yescharacteristic, complemented
16#568—28no—
17C2×#131—22no—
18C2×#131C2212yes—
19C2×#131C2212yes—
20C22×M16—22no—
21C22×M16C2211yescharacteristic
22C22×M16C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25#471—28no—
26#468—22no—
27#468—22no—
28#452—22no—
29#451—21no—
30#450—21no—
31#449—22no—
32#448—21no—
33#447—21no—
34C2×#108C2212yescomplemented
35C2×#107C2211yescharacteristic, complemented
36#417C2212yes—
37#417C2212yes—
38C2×#103C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×#101C2212yescomplemented
41C2×#100C2212yescomplemented
42#407C2211yescharacteristic
43#407C2211yescharacteristic
44#343C2212yescomplemented
45#343C2212yescomplemented
46#341C2212yescomplemented
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—21no—
50C2×#41—21no—
51C2×#41D412yes—
52C2×#41C2312yes—
53C22×Q8D411yescharacteristic
54C22×Q8C2311yescharacteristic
55C22×D4—21no—
56C22×D4—21no—
57C22×D4D411yescharacteristic
58C22×D4C2311yescharacteristic
59C23×C4—21no—
60C23×C4C2311yescharacteristic
61#137—24no—
62#137—24no—
63AGL(1,ℤ/8)—24no—
64AGL(1,ℤ/8)—24no—
65C2×Q16—22no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—24no—
70C2×D8—22no—
71#131—216no—
72#131—24no—
73#131—24no—
74#131—24no—
75#131D418yescomplemented
76C2×M16—216no—
77C2×M16—21no—
78C2×M16—24no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—24no—
82C2×M16D412yes—
83C2×M16D412yes—
84C2×M16C2312yes—
85C2×M16C2312yes—
86C2×M16C2312yes—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105#108—24no—
106#107—22no—
107#103—22no—
108#103—22no—
109#103—22no—
110#102—22no—
111#102—22no—
112#102—22no—
113#101—24no—
114#100—24no—
115C22⋊C8—21no—
116C22⋊C8—21no—
117C22⋊C8—21no—
118C22⋊C8—21no—
119C24—21no—
120C24—21no—
121C24#4111yescharacteristic
122#41—24no—
123#41—28no—
124#41—24no—
125#41—24no—
126#41—24no—
127#41—22no—
128#41—22no—
129#41C2×D414yes—
130#41#4114yes—
131C2×Q8—24no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—22no—
137C2×Q8C2×D412yes—
138C2×D4—28no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—24no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—82no—
150C2×D4C2×D412yes—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4#4111yescharacteristic
166C22×C4C2411yescharacteristic
167Q16—44no—
168SD16—44no—
169SD16—44no—
170SD16—44no—
171D8—44no—
172M16—24no—
173M16—28no—
174M16—22no—
175M16—24no—
176M16—28no—
177M16—24no—
178M16—22no—
179M16—22no—
180M16—22no—
181M16—22no—
182M16C2×D414yes—
183C2×C8—24no—
184C2×C8—28no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—28no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8C2×D412yes—
191C2×C8C2×D412yes—
192C2×C8C2×D414yes—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23#12111yescharacteristic
215C23#12111yescharacteristic
216C23C2×#4111yescharacteristic
217Q8—24no—
218Q8—22no—
219Q8—41no—
220Q8—42no—
221Q8—41no—
222Q8#12114yes—
223D4—28no—
224D4—24no—
225D4—24no—
226D4—24no—
227D4—22no—
228D4—41no—
229D4—42no—
230D4—41no—
231D4—82no—
232D4—82no—
233D4#12114yes—
234C2×C4—24no—
235C2×C4—28no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4#12111yescharacteristic
253C2×C4#12111yescharacteristic
254C2×C4C22×D412yes—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4C2×#4112yes—
258C8—24no—
259C8—22no—
260C8—42no—
261C8—42no—
262C8#12014yes—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—24no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—82no—
275C22—84no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—162no—
280C22C2×#12111yescharacteristic
281C22#52512yes—
282C4—21no—
283C4—24no—
284C4—21no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4C2×#12011yescharacteristic
289C4C2×#12112yes—
290C4#52511yescharacteristic
291C2—22no—
292C2—41no—
293C2—41no—
294C2—44no—
295C2—161no—
296C2#284711yescharacteristic
297C2#306811yescharacteristic
298C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#3006C2
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
1. C2×#568C2semidirect
2. C2×#471C2semidirect
3. C2×#468C2semidirect
4. #3007C2semidirect
6. #1908C2semidirect
7. #1907C2semidirect
8. #1901C2semidirect
9. #1900C2semidirect
10. #1816C2semidirect
11. #1811C2semidirect
15. C22×SD16C22semidirect
34. C2×#108C22semidirect
35. C2×#107C22semidirect
38. C2×#103C22semidirect
39. C2×#102C22semidirect
40. C2×#101C22semidirect
41. C2×#100C22semidirect
44. #343C22semidirect
45. #343C22semidirect
46. #341C22semidirect
75. #131D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5682
C2×#4712
C2×#4682
#30072
#30062
#19082
#19072
#19012
#19002
#18162
#18112