Finite Group Databasebeta

← #19631

Subgroups — #19631

Order 256.

461 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 139 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#3226C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic, complemented
3#3169C211yesmaximal, characteristic, complemented
4#3152C211yesmaximal, characteristic, complemented
5#3151C211yesmaximal, characteristic, complemented
6C2×#486C211yesmaximal, characteristic, complemented
7#1722C212yesmaximal, complemented
8#1721C212yesmaximal, complemented
9#1709C212yesmaximal, complemented
10#1706C212yesmaximal, complemented
11C8×M16C211yesmaximal, characteristic, complemented
12C2×#137—22no—
13C2×AGL(1,ℤ/8)—22no—
14#532C2211yescharacteristic
15#532C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18#500C2211yescharacteristic, complemented
19#500C2212yescomplemented
20#497—22no—
21#496—22no—
22#494C2212yescomplemented
23#493C2212yes—
24#493C2212yes—
25#493C2211yescharacteristic, complemented
26#492C2212yes—
27#492C2212yescomplemented
28#486—22no—
29#486C2212yes—
30#486C2212yes—
31#486C2212yes—
32#486C2211yescharacteristic, complemented
33#486C2211yescharacteristic, complemented
34#486C2212yescomplemented
35#480—22no—
36#474—22no—
37#466—22no—
38#459—22no—
39#447—22no—
40C4×D8—22no—
41#431C2211yescharacteristic
42C2×#102—22no—
43C4×M16C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#322C2212yescomplemented
46C82C2212yescomplemented
47C2×#41—21no—
48C2×#41—21no—
49C22×Q8—21no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—21no—
53#137—48no—
54AGL(1,ℤ/8)—48no—
55C2×Q16—24no—
56C2×SD16—24no—
57C2×SD16—24no—
58C2×D8—24no—
59C2×D8—22no—
60C2×D8—24no—
61C2×M16D412yes—
62C22×C8D412yes—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8C2312yes—
67C4⋊Q8C2311yescharacteristic
68C4⋊Q8C2311yescharacteristic
69#127—21no—
70#127—21no—
71#127—21no—
72#127C2312yes—
73#127C2311yescharacteristic
74#127C2311yescharacteristic
75#124—21no—
76#124—21no—
77#121—21no—
78#121—21no—
79#121—21no—
80#121—21no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C42C2311yescharacteristic
88#107—22no—
89#107—24no—
90C4⋊C8D412yes—
91#102—24no—
92#102—24no—
93#102—22no—
94#102—22no—
95#102—24no—
96#102—24no—
97C22⋊C8D412yes—
98#97D412yes—
99C4×C8D412yes—
100C4×C8C2311yescharacteristic
101C4×C8C2312yes—
102C4×C8C2312yes—
103C4×C8C2311yescharacteristic
104C24—41no—
105#41—44no—
106#41—44no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—22no—
110C2×Q8—41no—
111C2×Q8—44no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—44no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—44no—
128C2×D4—82no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135Q16—48no—
136SD16—48no—
137SD16—48no—
138D8—44no—
139D8—48no—
140D8—84no—
141M16D818yescomplemented
142C2×C8—42no—
143C2×C8C2×D412yes—
144C2×C8C2×D412yes—
145C2×C8C2×D414yes—
146C2×C8C2×D414yes—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C22⋊C4—41no—
156C22⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C42C2×D411yescharacteristic
160C42C2×D411yescharacteristic
161C42C2×D411yescharacteristic
162C42C2411yescharacteristic
163C23—41no—
164C23—41no—
165C23—42no—
166C23—82no—
167C23—82no—
168C23—82no—
169C23#12711yescharacteristic
170Q8—42no—
171Q8—44no—
172Q8—42no—
173Q8—82no—
174D4—42no—
175D4—44no—
176D4—42no—
177D4—42no—
178D4—42no—
179D4—44no—
180D4—42no—
181D4—44no—
182D4—42no—
183D4—82no—
184D4—82no—
185D4—82no—
186D4—82no—
187D4—82no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4#12711yescharacteristic
197C2×C4#12711yescharacteristic
198C2×C4#12711yescharacteristic
199C2×C4C2×D814yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C22×D411yescharacteristic
202C2×C4C22×D411yescharacteristic
203C8—22no—
204C8—22no—
205C8—24no—
206C8C2×D818yes—
207C22—21no—
208C22—21no—
209C22—81no—
210C22—81no—
211C22—82no—
212C22—82no—
213C22—81no—
214C22—81no—
215C22—82no—
216C22—81no—
217C22—81no—
218C22—164no—
219C22#49312yes—
220C22C2×#12711yescharacteristic
221C4—21no—
222C4—21no—
223C4—81no—
224C4—81no—
225C4—82no—
226C4#49312yes—
227C4C22×D812yes—
228C4#57512yes—
229C4#57712yes—
230C2—22no—
231C2—161no—
232C2—161no—
233C2—162no—
234C2C2×#49311yescharacteristic
235C2#315311yescharacteristic
236C2#315411yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
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. #3226C2semidirect
2. #3205C2semidirect
3. #3169C2semidirect
4. #3152C2semidirect
5. #3151C2semidirect
6. C2×#486C2semidirect
7. #1722C2semidirect
8. #1721C2semidirect
9. #1709C2semidirect
10. #1706C2semidirect
11. C8×M16C2semidirect
18. #500C22semidirect
19. #500C22semidirect
22. #494C22semidirect
25. #493C22semidirect
27. #492C22semidirect
32. #486C22semidirect
33. #486C22semidirect
34. #486C22semidirect
45. #322C22semidirect
46. C82C22semidirect
141. M16D8semidirect

Maximal subgroups

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

subgroupindex
#32262
#32052
#31692
#31522
#31512
C2×#4862
#17222
#17212
#17092
#17062
C8×M162