Finite Group Databasebeta

← #19673

Subgroups — #19673

Order 256.

431 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 123 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#3206C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic, complemented
3#3169C211yesmaximal, characteristic, complemented
4#3151C211yesmaximal, characteristic, complemented
5#3141C211yesmaximal, characteristic, complemented
6C2×#486C211yesmaximal, characteristic, complemented
7#1721C212yesmaximal, complemented
8#1719C212yesmaximal, complemented
9#1717C212yesmaximal, complemented
10#1717C212yesmaximal, complemented
11#1451C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)—22no—
13#532C2211yescharacteristic
14#532C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17#500C2211yescharacteristic, complemented
18#496—22no—
19#493C2212yes—
20#493C2212yes—
21#493C2211yescharacteristic, complemented
22#492C2212yes—
23#492C2212yescomplemented
24#489—22no—
25#487C2212yescomplemented
26#487C2212yescomplemented
27#486—22no—
28#486C2212yes—
29#486C2212yes—
30#486C2212yes—
31#486C2212yescomplemented
32#486C2211yescharacteristic, complemented
33#486C2211yescharacteristic, complemented
34#466—22no—
35#461—22no—
36#459—22no—
37#448—22no—
38#447—22no—
39C4×D8—22no—
40#431C2211yescharacteristic
41#417—22no—
42C2×#102—22no—
43C4×M16C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#322C2212yescomplemented
46#322C2212yescomplemented
47C2×#41—21no—
48C2×#41—21no—
49C22×Q8—21no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—21no—
53AGL(1,ℤ/8)—48no—
54C2×SD16—24no—
55C2×SD16—24no—
56C2×D8—24no—
57C2×D8—22no—
58C2×D8—24no—
59C2×M16D412yes—
60C22×C8D412yes—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8C2312yes—
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#127—21no—
68#127—21no—
69#127—21no—
70#127C2312yes—
71#127C2311yescharacteristic
72#127C2311yescharacteristic
73#124—21no—
74#124—21no—
75#121—21no—
76#121—21no—
77#121—21no—
78#121—21no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C42C2311yescharacteristic
86#107—22no—
87C4⋊C8D412yes—
88#103—24no—
89#103—24no—
90#102—24no—
91#102—24no—
92#102—24no—
93#102—22no—
94#102—22no—
95#102—24no—
96C22⋊C8D412yes—
97#97D412yes—
98C4×C8D412yes—
99C4×C8C2311yescharacteristic
100C4×C8C2311yescharacteristic
101C4×C8C2312yes—
102C4×C8C2312yes—
103C24—41no—
104#41—44no—
105#41—44no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—21no—
109C2×Q8—41no—
110C2×Q8—82no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—22no—
120C2×D4—44no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—44no—
127C2×D4—82no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134SD16—48no—
135SD16—84no—
136D8—44no—
137D8—48no—
138D8—84no—
139M16—24no—
140C2×C8—42no—
141C2×C8C2×D412yes—
142C2×C8C2×D412yes—
143C2×C8C2×D414yes—
144C2×C8C2×D414yes—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C22⋊C4—41no—
154C22⋊C4—41no—
155C22⋊C4—41no—
156C22⋊C4—41no—
157C42C2×D411yescharacteristic
158C42C2×D411yescharacteristic
159C42C2×D411yescharacteristic
160C42C2411yescharacteristic
161C23—41no—
162C23—41no—
163C23—42no—
164C23—82no—
165C23—82no—
166C23—82no—
167C23#12711yescharacteristic
168Q8—42no—
169Q8—42no—
170Q8—44no—
171Q8—82no—
172D4—42no—
173D4—44no—
174D4—42no—
175D4—42no—
176D4—42no—
177D4—44no—
178D4—42no—
179D4—44no—
180D4—42no—
181D4—82no—
182D4—82no—
183D4—82no—
184D4—82no—
185D4—82no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#12711yescharacteristic
195C2×C4#12711yescharacteristic
196C2×C4#12711yescharacteristic
197C2×C4AGL(1,ℤ/8)14yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C8—24no—
202C8—22no—
203C8—22no—
204C8—24no—
205C22—21no—
206C22—21no—
207C22—81no—
208C22—81no—
209C22—82no—
210C22—82no—
211C22—81no—
212C22—81no—
213C22—82no—
214C22—81no—
215C22—81no—
216C22—164no—
217C22#49612yes—
218C22C2×#12711yescharacteristic
219C4—21no—
220C4—21no—
221C4—81no—
222C4—81no—
223C4—82no—
224C4#49612yes—
225C4C2×AGL(1,ℤ/8)12yes—
226C4#57512yes—
227C4#57612yes—
228C2—22no—
229C2—161no—
230C2—161no—
231C2—162no—
232C2C2×#49611yescharacteristic
233C2#315311yescharacteristic
234C2#315511yescharacteristic
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. #3206C2semidirect
2. #3205C2semidirect
3. #3169C2semidirect
4. #3151C2semidirect
5. #3141C2semidirect
6. C2×#486C2semidirect
7. #1721C2semidirect
8. #1719C2semidirect
9. #1717C2semidirect
10. #1717C2semidirect
11. #1451C2semidirect
17. #500C22semidirect
21. #493C22semidirect
23. #492C22semidirect
25. #487C22semidirect
26. #487C22semidirect
31. #486C22semidirect
32. #486C22semidirect
33. #486C22semidirect
45. #322C22semidirect
46. #322C22semidirect

Maximal subgroups

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

subgroupindex
#32062
#32052
#31692
#31512
#31412
C2×#4862
#17212
#17192
#17172
#14512