Finite Group Databasebeta

← #25804

Subgroups — #25804

Order 256.

727 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 143 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
2#3156C211yesmaximal, characteristic, complemented
3C2×#495C211yesmaximal, characteristic, complemented
4C2×#473C211yesmaximal, characteristic, complemented
5C2×#455C212yesmaximal, complemented
6#3009C212yesmaximal, complemented
7#2019C212yesmaximal, complemented
8#2003C212yesmaximal, complemented
9#1978C211yesmaximal, characteristic, complemented
10#1974C211yesmaximal, characteristic, complemented
11#1764C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)C2212yes—
13#576—24no—
14#576—24no—
15C2×#137C2212yes—
16C2×#137C2212yescomplemented
17C2×AGL(1,ℤ/8)—22no—
18C2×AGL(1,ℤ/8)C2212yes—
19C2×#135—21no—
20C2×#135C2211yescharacteristic
21C22×Q16C2211yescharacteristic, complemented
22C22×SD16C2212yescomplemented
23C22×D8—21no—
24C22×D8C2211yescharacteristic
25C2×#131C2211yescharacteristic
26#536C2212yescomplemented
27C2×#124C2212yescomplemented
28#497—22no—
29#495—24no—
30#495—21no—
31#494—21no—
32#473—24no—
33#455—28no—
34#452—24no—
35#450—24no—
36C2×#108C2211yescharacteristic, complemented
37C2×#104C2211yescharacteristic
38C2×#104C2212yescomplemented
39#418C2212yescomplemented
40#416C2211yescharacteristic
41C2×#101C2212yescomplemented
42C2×#100C2212yescomplemented
43#408C2212yescomplemented
44#405C2211yescharacteristic, complemented
45C2×C4×C8C2211yescharacteristic, complemented
46#341C2211yescharacteristic, complemented
47#339C2211yescharacteristic, complemented
48H(5,𝔽2)—24no—
49H(5,𝔽2)—24no—
50H(5,𝔽2)—24no—
51H(5,𝔽2)—42no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41C2312yes—
55C2×#41C2311yescharacteristic
56C22×Q8C2312yes—
57C22×D4—22no—
58C22×D4—22no—
59C22×D4—42no—
60C22×D4C2312yes—
61#137—24no—
62#137—24no—
63#137—24no—
64#137—42no—
65AGL(1,ℤ/8)—216no—
66AGL(1,ℤ/8)—24no—
67AGL(1,ℤ/8)—24no—
68#135—28no—
69#135—22no—
70#135—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—21no—
75C2×Q16—22no—
76C2×Q16—21no—
77C2×Q16D411yescharacteristic
78C2×SD16—22no—
79C2×SD16—24no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—24no—
83C2×SD16—42no—
84C2×SD16D412yes—
85C2×D8—28no—
86C2×D8—22no—
87C2×D8—21no—
88C2×D8—21no—
89C2×D8—22no—
90C2×D8D411yescharacteristic
91C2×D8D414yes—
92#131—22no—
93#131D414yes—
94C2×M16—21no—
95C2×M16C2312yes—
96C2×M16C2311yescharacteristic
97C2×M16C2311yescharacteristic
98C22×C8—21no—
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C4⋊Q8—22no—
102#124—24no—
103#124—22no—
104#124—24no—
105#123—42no—
106C22⋊Q8—24no—
107#117C2311yescharacteristic
108C2×C22⋊C4—42no—
109C2×C42C2311yescharacteristic
110#108—22no—
111#104—24no—
112#104D414yes—
113#103D412yes—
114#102—24no—
115#102D412yes—
116#101—24no—
117#100—24no—
118C22⋊C8—24no—
119#97—21no—
120C4×C8—22no—
121C4×C8—21no—
122C24—22no—
123C24—42no—
124#41—21no—
125#41—21no—
126#41—28no—
127#41—22no—
128#41—22no—
129#41—22no—
130#41—22no—
131#41—44no—
132#41—44no—
133#41—44no—
134#41C2×D412yes—
135#41C2×D412yes—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8C2×D412yes—
143C2×Q8C2×D411yescharacteristic
144C2×D4—21no—
145C2×D4—28no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—44no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—44no—
156C2×D4—48no—
157C2×D4—48no—
158C2×D4—44no—
159C2×D4C2×D412yes—
160C2×D4C2×D414yes—
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4C2411yescharacteristic
168Q16—22no—
169Q16—44no—
170Q16—44no—
171Q16—42no—
172Q16—42no—
173SD16—28no—
174SD16—24no—
175SD16—48no—
176SD16—44no—
177SD16—44no—
178D8—28no—
179D8—22no—
180D8—24no—
181D8—24no—
182M16—21no—
183M16—21no—
184M16—24no—
185M16—22no—
186M16—22no—
187M16C2×D412yes—
188M16C2×D412yes—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8C2×D412yes—
196C2×C8C2×D411yescharacteristic
197C2×C8C2×D412yes—
198C2×C8C2×D411yescharacteristic
199C2×C8C2×D412yes—
200C4⋊C4—44no—
201C4⋊C4C2×D412yes—
202C22⋊C4—21no—
203C22⋊C4—48no—
204C22⋊C4—44no—
205C42—21no—
206C42—21no—
207C42C2×D412yes—
208C23—21no—
209C23—22no—
210C23—24no—
211C23—48no—
212C23—42no—
213C23—42no—
214C23—84no—
215C23—164no—
216C23H(5,𝔽2)11yescharacteristic
217Q8—21no—
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—42no—
222Q8—42no—
223Q8—42no—
224Q8#12712yes—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—24no—
229D4—21no—
230D4—24no—
231D4—22no—
232D4—48no—
233D4—48no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—42no—
238D4#12712yes—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—48no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4C22×D411yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C8—22no—
259C8—22no—
260C8—22no—
261C8—42no—
262C8#12012yes—
263C8#12012yes—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—44no—
269C22—44no—
270C22—84no—
271C22—84no—
272C22—162no—
273C22—168no—
274C22#53511yescharacteristic
275C22D4211yescharacteristic
276C22D4211yescharacteristic
277C4—22no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4C2×#12011yescharacteristic
283C4C2×#12711yescharacteristic
284C4D4211yescharacteristic
285C4D4211yescharacteristic
286C2—21no—
287C2—21no—
288C2—42no—
289C2—82no—
290C2—84no—
291C2#243311yescharacteristic
292C2#329012yes—
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#576C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3156C2semidirect
3. C2×#495C2semidirect
4. C2×#473C2semidirect
5. C2×#455C2semidirect
6. #3009C2semidirect
7. #2019C2semidirect
8. #2003C2semidirect
9. #1978C2semidirect
10. #1974C2semidirect
11. #1764C2semidirect
16. C2×#137C22semidirect
21. C22×Q16C22semidirect
22. C22×SD16C22semidirect
26. #536C22semidirect
27. C2×#124C22semidirect
36. C2×#108C22semidirect
38. C2×#104C22semidirect
39. #418C22semidirect
41. C2×#101C22semidirect
42. C2×#100C22semidirect
43. #408C22semidirect
44. #405C22semidirect
45. C2×C4×C8C22semidirect
46. #341C22semidirect
47. #339C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5762
#31562
C2×#4952
C2×#4732
C2×#4552
#30092
#20192
#20032
#19782
#19742
#17642