Finite Group Databasebeta

← #25799

Subgroups — #25799

Order 256.

729 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 729 conjugacy classes of subgroups — too many to draw. Only its 141 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#3584C211yesmaximal, characteristic, complemented
2#3155C212yesmaximal, complemented
3#3058C211yesmaximal, characteristic, complemented
4#3027C212yesmaximal, complemented
5#3021C212yesmaximal, complemented
6#2008C214yesmaximal, complemented
7#1976C212yesmaximal, complemented
8#1757C211yesmaximal, characteristic, complemented
9#585C2212yes—
10#578—22no—
11#577—24no—
12#576—22no—
13#575—22no—
14#575C2212yes—
15C2×AGL(1,ℤ/8)C2214yescomplemented
16C2×#135—22no—
17C2×#135C2212yes—
18C22×D8C2211yescharacteristic, complemented
19C2×#131C2211yescharacteristic
20#535C2212yes—
21#535C2214yescomplemented
22#496—24no—
23#495—22no—
24#493—22no—
25#469—22no—
26#466—22no—
27B(4,𝔽2)—28no—
28#454—24no—
29#453—24no—
30#427C2211yescharacteristic, complemented
31#421C2214yescomplemented
32#418C2212yescomplemented
33#413C2211yescharacteristic
34#410C2211yescharacteristic
35#410C2214yescomplemented
36#405C2212yescomplemented
37#344C2212yescomplemented
382−1+4—24no—
392−1+4—22no—
40H(5,𝔽2)—24no—
41H(5,𝔽2)—24no—
42H(5,𝔽2)—22no—
43C2×#41—24no—
44C2×#41—44no—
45C2×#41C2314yes—
46C2×#41C2311yescharacteristic
47C22×D4C2312yes—
48#137—28no—
49#137—24no—
50AGL(1,ℤ/8)—28no—
51AGL(1,ℤ/8)—24no—
52AGL(1,ℤ/8)—28no—
53AGL(1,ℤ/8)—44no—
54#135—24no—
55#135—24no—
56#135—22no—
57#135—24no—
58#135—22no—
59#135—22no—
60#135—22no—
61C2×Q16—22no—
62C2×Q16D412yescomplemented
63C2×SD16—24no—
64C2×SD16—24no—
65C2×SD16D414yescomplemented
66C2×D8—22no—
67C2×D8—24no—
68C2×D8—21no—
69C2×D8—21no—
70C2×D8—22no—
71C2×D8—22no—
72C2×D8D412yescomplemented
73#131—22no—
74#131—22no—
75C2×M16D412yescomplemented
76C2×M16C2312yes—
77C2×M16C2311yescharacteristic
78C22×C8D412yescomplemented
79C22×C8C2311yescharacteristic
80#127—22no—
81#127—24no—
82#124—22no—
83#124—24no—
84#121—24no—
85#121—44no—
86#120—24no—
87#120—28no—
88#117C2312yes—
89#117C2312yes—
90#107—21no—
91#106—21no—
92#104—28no—
93#102—24no—
94#101D412yescomplemented
95#100D412yescomplemented
96#99—28no—
97#99D414yes—
98#97—22no—
99C4×C8—22no—
100C24—22no—
101#41—22no—
102#41—22no—
103#41—28no—
104#41—24no—
105#41—24no—
106#41—24no—
107#41—24no—
108#41—48no—
109#41—44no—
110#41—44no—
111#41—44no—
112#41—44no—
113C2×Q8—24no—
114C2×Q8—44no—
115C2×Q8C2×D414yes—
116C2×Q8C2×D411yescharacteristic
117C2×D4—24no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—24no—
123C2×D4—48no—
124C2×D4—44no—
125C2×D4—48no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4C2×D412yes—
129C2×D4C2×D414yes—
130C2×D4C2×D411yescharacteristic
131C22×C4—24no—
132C22×C4—44no—
133C22×C4C2×D412yes—
134C22×C4C2411yescharacteristic
135Q16—24no—
136Q16—22no—
137Q16—22no—
138SD16—28no—
139SD16—24no—
140SD16—24no—
141SD16—48no—
142D8—24no—
143D8—22no—
144D8—22no—
145D8—48no—
146D8—41no—
147D8—41no—
148D8—42no—
149M16—22no—
150M16—22no—
151M16—24no—
152M16C2×D414yes—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8C2×D412yes—
157C2×C8C2×D412yes—
158C2×C8C2×D412yes—
159C2×C8C2×D411yescharacteristic
160C2×C8C2×D411yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—48no—
165C22⋊C4—44no—
166C22⋊C4C2×D414yes—
167C42—22no—
168C42—22no—
169C23—24no—
170C23—44no—
171C23—42no—
172C23—44no—
173C23—162no—
174C23#12712yes—
175C23C22×D411yescharacteristic
176Q8—22no—
177Q8—24no—
178Q8—24no—
179Q8—44no—
180D4—24no—
181D4—24no—
182D4—22no—
183D4—24no—
184D4—44no—
185D4—48no—
186D4—44no—
187D4—44no—
188D4—42no—
189D4—42no—
190D4—44no—
191D4—168no—
192D4—164no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—48no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4#12712yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C22×D412yes—
205C2×C4C22×D411yescharacteristic
206C2×C4H(5,𝔽2)12yes—
207C8—22no—
208C8—24no—
209C8#12011yescharacteristic
210C8#12011yescharacteristic
211C8#12012yes—
212C22—22no—
213C22—41no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—88no—
218C22—84no—
219C22—82no—
220C22—164no—
221C22C2×#12711yescharacteristic
222C22D4212yes—
223C4—22no—
224C4—44no—
225C4—44no—
226C4—82no—
227C4C2×#12011yescharacteristic
228C4#53511yescharacteristic
229C4D4212yes—
230C2—22no—
231C2—21no—
232C2—42no—
233C2—84no—
234C2—162no—
235C2#243311yescharacteristic
subgroupG/H
center, socleC2#2433
commutator, Frattini, cosocleC22×C4C24
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. #3584C2semidirect
2. #3155C2semidirect
3. #3058C2semidirect
4. #3027C2semidirect
5. #3021C2semidirect
6. #2008C2semidirect
7. #1976C2semidirect
8. #1757C2semidirect
15. C2×AGL(1,ℤ/8)C22semidirect
18. C22×D8C22semidirect
21. #535C22semidirect
30. #427C22semidirect
31. #421C22semidirect
32. #418C22semidirect
35. #410C22semidirect
36. #405C22semidirect
37. #344C22semidirect
62. C2×Q16D4semidirect
65. C2×SD16D4semidirect
72. C2×D8D4semidirect
75. C2×M16D4semidirect
78. C22×C8D4semidirect
94. #101D4semidirect
95. #100D4semidirect

Maximal subgroups

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

subgroupindex
#35842
#31552
#30582
#30272
#30212
#20082
#19762
#17572