Finite Group Databasebeta

← #18762

Subgroups — #18762

Order 256.

482 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 121 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×#556C211yesmaximal, characteristic, complemented
2#3227C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic
4#3037C211yesmaximal, characteristic, complemented
5#1652C214yesmaximal, complemented
6#1628C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C2×#137—21no—
11#556—22no—
12#556—22no—
13#556—22no—
14#556C2214yes—
15#532C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—22no—
18C2×#124—21no—
19C2×C22⋊Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117—21no—
24#498C2212yescomplemented
25#486C2212yescomplemented
26#481—22no—
27#480—21no—
28#476—22no—
29#474C2212yes—
30#464—22no—
31#462C2212yescomplemented
32#460C2212yescomplemented
33#459C2212yes—
34#447—21no—
35C4×D8—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#332C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—21no—
45C22×Q8D411yescharacteristic
46C22×D4D411yescharacteristic
47C23×C4—21no—
48#137—44no—
49C2×Q16—22no—
50C2×SD16—22no—
51C2×D8—22no—
52C2×M16—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8D412yes—
56C4⋊Q8C2311yescharacteristic
57C4⋊Q8C2312yes—
58#127C2311yescharacteristic
59#126—28no—
60#126—28no—
61#124—24no—
62#124—24no—
63#124—21no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66#121—21no—
67#121—21no—
68C4×Q8—24no—
69C4×Q8—24no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8D412yes—
73C4×Q8C2312yes—
74C4×D4—21no—
75C4×D4—21no—
76C4×D4—21no—
77C4×D4—21no—
78C4×D4D412yes—
79C4×D4C2312yes—
80#117—24no—
81#117—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D412yes—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94#107—22no—
95#106—24no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—22no—
101#102—22no—
102#102—22no—
103#102—22no—
104#102—24no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C4×C8—21no—
108C4×C8C2312yes—
109C24—21no—
110#41—42no—
111#41—42no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—44no—
119C2×Q8C2×D412yes—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—41no—
126C2×D4C2×D412yes—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139Q16—44no—
140SD16—44no—
141D8—44no—
142M16—42no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—28no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D414yes—
158C4⋊C4C2×D412yes—
159C22⋊C4—28no—
160C22⋊C4—28no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C42—28no—
166C42—22no—
167C42—22no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2411yescharacteristic
172C23—22no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—41no—
177C23#12011yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—44no—
181Q8—44no—
182Q8—42no—
183Q8—41no—
184Q8—41no—
185D4—22no—
186D4—22no—
187D4—42no—
188D4—42no—
189D4—41no—
190D4—41no—
191D4—42no—
192D4—82no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—41no—
207C2×C4—82no—
208C2×C4#12011yescharacteristic
209C2×C4#12011yescharacteristic
210C2×C4#12011yescharacteristic
211C2×C4#13512yes—
212C2×C4AGL(1,ℤ/8)12yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C22×D411yescharacteristic
216C8—42no—
217C8—42no—
218C8—42no—
219C22—21no—
220C22—21no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—84no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22#44912yes—
229C22C2×#12011yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—42no—
235C4—44no—
236C4—42no—
237C4—81no—
238C4#44912yes—
239C4C2×#13511yescharacteristic
240C4C2×AGL(1,ℤ/8)11yescharacteristic
241C4#57512yes—
242C2—22no—
243C2—82no—
244C2—161no—
245C2C2×#44911yescharacteristic
246C2#300511yescharacteristic
247C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3100C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#556C2semidirect
2. #3227C2semidirect
4. #3037C2semidirect
5. #1652C2semidirect
6. #1628C2semidirect
7. #1549C2semidirect
8. #1486C2semidirect
9. #1479C2semidirect
24. #498C22semidirect
25. #486C22semidirect
31. #462C22semidirect
32. #460C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
#32272
#31002
#30372
#16522
#16282
#15492
#14862
#14792