Finite Group Databasebeta

← #22898

Subgroups — #22898

Order 256.

498 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 165 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#3018C211yesmaximal, characteristic
2#2956C212yesmaximal, complemented
3#2901C212yesmaximal, complemented
4#1963C211yesmaximal, characteristic, complemented
5#1932C212yesmaximal, complemented
6#1906C212yesmaximal, complemented
7#1884C211yesmaximal, characteristic, complemented
8#1861C212yesmaximal, complemented
9#1759C212yesmaximal, complemented
10#585C2211yescharacteristic
11#575C2212yes—
12#568C2212yes—
13#568C2211yescharacteristic, complemented
14#532C2211yescharacteristic
15C4×#41C2212yes—
16#456C412yescomplemented
17#455C412yescomplemented
18#454C412yescomplemented
19#453C412yescomplemented
20#444—24no—
21#443—24no—
22#430C2212yescomplemented
23#423C2211yescharacteristic, complemented
24#421—24no—
25#421C2212yes—
26#421C2212yes—
27C2×#104—24no—
28C2×#104C2212yes—
29C2×#104C2212yes—
30C2×#104C2212yes—
31C2×#104C2212yescomplemented
32#413C2211yescharacteristic
33#410C2211yescharacteristic, complemented
34C4×M16C2212yescomplemented
35#376C2211yescharacteristic, complemented
36#344C2212yescomplemented
37#339C2212yescomplemented
38#339C2212yescomplemented
392−1+4—22no—
402−1+4C2×C412yes—
41H(5,𝔽2)—22no—
42H(5,𝔽2)—41no—
43H(5,𝔽2)C2×C412yes—
44C2×#41—24no—
45C2×#41—41no—
46C2×#41C2311yescharacteristic
47C2×#41C2311yescharacteristic
48C2×#41C2312yes—
49#137—22no—
50#137C2×C414yescomplemented
51AGL(1,ℤ/8)—22no—
52AGL(1,ℤ/8)C2×C414yescomplemented
53#135—24no—
54#131—24no—
55#131—24no—
56#131—42no—
57C2×M16—24no—
58C2×M16—41no—
59C2×M16C2312yes—
60C2×M16C2312yes—
61C2×M16C2312yes—
62C4⋊Q8C2×C411yescharacteristic, complemented
63#127C2×C411yescharacteristic, complemented
64#124C2×C411yescharacteristic, complemented
65#124C2×C411yescharacteristic, complemented
66#121—41no—
67C4×Q8—22no—
68C4×D4—22no—
69C4×D4—41no—
70C4×D4—42no—
71#117—22no—
72#117C2312yes—
73C2×C42—22no—
74C2×C42C2311yescharacteristic
75C2×C42C2312yes—
76#108—24no—
77C4⋊C8—41no—
78#104—24no—
79#104—24no—
80#104—22no—
81#104—22no—
82#104—22no—
83#104—22no—
84#104C2×C414yescomplemented
85#104C2×C414yescomplemented
86#104D414yescomplemented
87#104D414yescomplemented
88#101C2×C412yescomplemented
89#100C2×C412yescomplemented
90#99—41no—
91#97—22no—
92C4×C8—22no—
93#95—41no—
94#41—22no—
95#41—22no—
96#41—24no—
97#41—24no—
98#41—22no—
99#41—22no—
100#41—22no—
101#41—22no—
102#41—42no—
103#41—42no—
104#41—44no—
105#41—42no—
106#41C22×C414yes—
107#41C2×D414yes—
108C2×Q8—24no—
109C2×Q8—41no—
110C2×Q8C22×C411yescharacteristic
111C2×Q8C22×C411yescharacteristic
112C2×Q8#4112yes—
113C2×D4—22no—
114C2×D4—24no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—41no—
118C2×D4—42no—
119C2×D4—41no—
120C2×D4—44no—
121C2×D4C22×C411yescharacteristic
122C2×D4C22×C411yescharacteristic
123C2×D4#4112yes—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2411yescharacteristic
132Q16—24no—
133SD16—24no—
134SD16—24no—
135D8—24no—
136M16—22no—
137M16—22no—
138M16—22no—
139M16—22no—
140M16—42no—
141M16—44no—
142M16C22×C414yes—
143M16C2×D414yes—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—42no—
148C2×C8—44no—
149C4⋊C4—22no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C22⋊C4—22no—
153C22⋊C4—41no—
154C22⋊C4—41no—
155C22⋊C4—42no—
156C42—22no—
157C42—21no—
158C42—22no—
159C42—42no—
160C42C22×C411yescharacteristic
161C42C22×C411yescharacteristic
162C42C2×D412yes—
163C42C2×D412yes—
164C23—22no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23H(5,𝔽2)11yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—22no—
172Q8—42no—
173Q8C4×D414yes—
174D4—24no—
175D4—24no—
176D4—22no—
177D4—22no—
178D4—22no—
179D4—44no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—82no—
184D4C4×D414yes—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—84no—
204C2×C4C23×C411yescharacteristic
205C2×C4C22×D412yes—
206C2×C4C2×#4112yes—
207C2×C4H(5,𝔽2)11yescharacteristic
208C8—24no—
209C8—24no—
210C8—42no—
211C22—24no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—82no—
216C22—81no—
217C22—81no—
218C22—162no—
219C22#51811yescharacteristic
220C22#54612yes—
221C4—24no—
222C4—42no—
223C4—41no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—82no—
229C4C2×C4×D412yes—
230C4D4211yescharacteristic
231C4#55111yescharacteristic
232C2—22no—
233C2—21no—
234C2—44no—
235C2—81no—
236C2—81no—
237C2#235811yescharacteristic
subgroupG/H
centerC4D42
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC2#2358
prime radical#3018C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2956C2semidirect
3. #2901C2semidirect
4. #1963C2semidirect
5. #1932C2semidirect
6. #1906C2semidirect
7. #1884C2semidirect
8. #1861C2semidirect
9. #1759C2semidirect
13. #568C22semidirect
16. #456C4semidirect
17. #455C4semidirect
18. #454C4semidirect
19. #453C4semidirect
22. #430C22semidirect
23. #423C22semidirect
31. C2×#104C22semidirect
33. #410C22semidirect
34. C4×M16C22semidirect
35. #376C22semidirect
36. #344C22semidirect
37. #339C22semidirect
38. #339C22semidirect
50. #137C2×C4semidirect
52. AGL(1,ℤ/8)C2×C4semidirect
62. C4⋊Q8C2×C4semidirect
63. #127C2×C4semidirect
64. #124C2×C4semidirect
65. #124C2×C4semidirect
84. #104C2×C4semidirect
85. #104C2×C4semidirect
86. #104D4semidirect
87. #104D4semidirect
88. #101C2×C4semidirect
89. #100C2×C4semidirect

Maximal subgroups

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

subgroupindex
#30182
#29562
#29012
#19632
#19322
#19062
#18842
#18612
#17592