Finite Group Databasebeta

← #33539

Subgroups — #33539

Order 256.

565 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 125 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×#576C211yesmaximal, characteristic, complemented
2#3414C211yesmaximal, characteristic, complemented
3C2×#509C211yesmaximal, characteristic, complemented
4#3067C211yesmaximal, characteristic, complemented
5#2194C214yesmaximal, complemented
6#2192C214yesmaximal, complemented
7C2×#361C211yesmaximal, characteristic, complemented
8#2143C211yesmaximal, characteristic, complemented
9#2117C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11#576—22no—
12#576—22no—
13#576—22no—
14#576C2214yes—
15#575—21no—
16C2×AGL(1,ℤ/8)—22no—
17C2×AGL(1,ℤ/8)—21no—
18C2×#135—21no—
19C2×#135C2211yescharacteristic
20C2×#135C2211yescharacteristic, complemented
21C22×D8—21no—
22C22×D8C2211yescharacteristic
23C2×#131C2211yescharacteristic
24#510—21no—
25#510C2212yescomplemented
26#509—22no—
27#509—22no—
28#509—21no—
29#509C2212yescomplemented
30#508—22no—
31C2×SD32C2212yescomplemented
32C2×D16C2212yescomplemented
33C2×M32C2211yescharacteristic, complemented
34C2×M32C2211yescharacteristic, complemented
35#473C2212yescomplemented
36#472C2212yescomplemented
37#471—22no—
38C2×#108C2211yescharacteristic, complemented
39#413—21no—
40#361C2214yescomplemented
41#359C2214yescomplemented
42#350C2214yescomplemented
43H(5,𝔽2)—24no—
44H(5,𝔽2)—24no—
45H(5,𝔽2)—42no—
46H(5,𝔽2)—42no—
47C2×#41—22no—
48C2×#41—21no—
49C2×#41—21no—
50C2×#41—41no—
51C2×#41D411yescharacteristic
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—42no—
55C22×D4—41no—
56#137—22no—
57#137—41no—
58AGL(1,ℤ/8)—24no—
59AGL(1,ℤ/8)—28no—
60AGL(1,ℤ/8)—28no—
61AGL(1,ℤ/8)—22no—
62AGL(1,ℤ/8)—24no—
63AGL(1,ℤ/8)—41no—
64#135—24no—
65#135—24no—
66#135—21no—
67#135—21no—
68#135—21no—
69#135—21no—
70#135—21no—
71#135—21no—
72#135—42no—
73#135D412yes—
74#135C2312yes—
75C2×Q16D411yescharacteristic
76C2×Q16C2311yescharacteristic, complemented
77C2×SD16—22no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×D8—22no—
81C2×D8—24no—
82C2×D8—24no—
83C2×D8—21no—
84C2×D8—21no—
85C2×D8—21no—
86C2×D8D411yescharacteristic
87C2×D8D412yes—
88C2×D8D412yes—
89C2×D8C2312yes—
90C2×D8C2311yescharacteristic, complemented
91#131—21no—
92#131—21no—
93#131D412yes—
94#131C2312yes—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16D411yescharacteristic
98C22×C8—21no—
99C22×C8C2311yescharacteristic
100Q32—22no—
101SD32—22no—
102SD32—24no—
103SD32—22no—
104D16—24no—
105D16—22no—
106M32—21no—
107M32—21no—
108M32—21no—
109M32C2312yescomplemented
110C2×C16—21no—
111C2×C16C2312yescomplemented
112#108—21no—
113#108C2312yescomplemented
114#101—22no—
115#100—22no—
116C24—42no—
117C24—41no—
118#41—21no—
119#41—21no—
120#41—28no—
121#41—22no—
122#41—22no—
123#41—44no—
124#41—44no—
125#41—44no—
126#41—41no—
127#41—41no—
128#41—41no—
129#41—41no—
130#41—41no—
131#41—41no—
132#41D812yes—
133#41C2×D412yes—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8D811yescharacteristic
137C2×D4—21no—
138C2×D4—28no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—42no—
145C2×D4—44no—
146C2×D4—48no—
147C2×D4—48no—
148C2×D4—44no—
149C2×D4—42no—
150C2×D4—41no—
151C2×D4—44no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4D811yescharacteristic
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4C2×D411yescharacteristic
162Q16—21no—
163Q16—21no—
164Q16—21no—
165Q16—42no—
166Q16C2×D412yes—
167SD16—28no—
168SD16—22no—
169SD16—22no—
170SD16—42no—
171SD16—42no—
172SD16—41no—
173SD16—41no—
174D8—28no—
175D8—21no—
176D8—22no—
177D8—22no—
178D8—22no—
179D8—21no—
180D8—21no—
181D8—42no—
182D8C2×D412yes—
183D8C2×D414yes—
184M16—21no—
185M16—22no—
186M16—22no—
187M16—22no—
188M16—41no—
189M16C2×D412yes—
190C2×C8—21no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—41no—
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C2×C8C2×D411yescharacteristic
198C2×C8C2×D411yescharacteristic
199C2×C8C2411yescharacteristic
200C16—22no—
201C16—22no—
202C23—21no—
203C23—48no—
204C23—42no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23—41no—
209C23—162no—
210C23—164no—
211C23AGL(1,ℤ/8)11yescharacteristic
212Q8—21no—
213Q8—22no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8C2×D812yes—
218D4—22no—
219D4—22no—
220D4—22no—
221D4—24no—
222D4—21no—
223D4—48no—
224D4—48no—
225D4—42no—
226D4—42no—
227D4—41no—
228D4—42no—
229D4—41no—
230D4—41no—
231D4—42no—
232D4—42no—
233D4—81no—
234D4—81no—
235D4C2×D812yes—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—48no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4#12011yescharacteristic
247C2×C4#12011yescharacteristic
248C2×C4C2×D811yescharacteristic
249C2×C4C2×D811yescharacteristic
250C2×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C4C22×D411yescharacteristic
252C8—22no—
253C8—42no—
254C8#12011yescharacteristic
255C8#12011yescharacteristic
256C8C22×D411yescharacteristic
257C8C22×D411yescharacteristic
258C22—22no—
259C22—41no—
260C22—42no—
261C22—44no—
262C22—42no—
263C22—84no—
264C22—82no—
265C22—81no—
266C22—162no—
267C22—162no—
268C22—161no—
269C22—168no—
270C22#44711yescharacteristic
271C22#44711yescharacteristic
272C22C2×AGL(1,ℤ/8)11yescharacteristic
273C4—22no—
274C4—42no—
275C4—81no—
276C4#44711yescharacteristic
277C4#44711yescharacteristic
278C4C2×#12011yescharacteristic
279C4C22×D811yescharacteristic
280C2—21no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2—84no—
285C2—82no—
286C2—161no—
287C2C2×#44711yescharacteristic
subgroupG/H
center, socleC2C2×#447
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#576C2semidirect
2. #3414C2semidirect
3. C2×#509C2semidirect
4. #3067C2semidirect
5. #2194C2semidirect
6. #2192C2semidirect
7. C2×#361C2semidirect
8. #2143C2semidirect
9. #2117C2semidirect
20. C2×#135C22semidirect
25. #510C22semidirect
29. #509C22semidirect
31. C2×SD32C22semidirect
32. C2×D16C22semidirect
33. C2×M32C22semidirect
34. C2×M32C22semidirect
35. #473C22semidirect
36. #472C22semidirect
38. C2×#108C22semidirect
40. #361C22semidirect
41. #359C22semidirect
42. #350C22semidirect
76. C2×Q16C23semidirect
90. C2×D8C23semidirect
109. M32C23semidirect
111. C2×C16C23semidirect
113. #108C23semidirect

Maximal subgroups

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

subgroupindex
C2×#5762
#34142
C2×#5092
#30672
#21942
#21922
C2×#3612
#21432
#21172