Finite Group Databasebeta

← #19627

Subgroups — #19627

Order 256.

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

normal subgroups only

This group has 392 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#3209C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic
3#3167C211yesmaximal, characteristic, complemented
4#3141C211yesmaximal, characteristic, complemented
5#3138C211yesmaximal, characteristic, complemented
6C2×#486C211yesmaximal, characteristic, complemented
7#1720C212yesmaximal, complemented
8#1717C212yesmaximal, complemented
9#1711C212yesmaximal, complemented
10#1709C212yesmaximal, complemented
11C8×M16C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16#498C2211yescharacteristic, complemented
17#494C2212yescomplemented
18#493C2212yes—
19#492C2211yescharacteristic, complemented
20#490—22no—
21#489—22no—
22#487C2212yescomplemented
23#487C2211yescharacteristic, complemented
24#487C2212yescomplemented
25#486—22no—
26#486C2212yes—
27#486C2212yes—
28#486C2212yes—
29#486C2212yes—
30#486C2212yes—
31#486C2212yescomplemented
32#486C2211yescharacteristic, complemented
33#465—22no—
34#462—22no—
35#459—22no—
36#451—22no—
37#447—22no—
38C4×SD16—22no—
39#431C2211yescharacteristic
40#418—22no—
41#417—22no—
42C2×#102—22no—
43C4×M16C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#322C2212yescomplemented
46C82C2212yescomplemented
47C2×#41—21no—
48C22×Q8—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52C2×Q16—24no—
53C2×SD16—22no—
54C2×D8—24no—
55C2×M16D412yes—
56C22×C8D412yes—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8C2312yes—
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64#127—21no—
65#127—21no—
66#127—21no—
67#127C2312yes—
68#127C2311yescharacteristic
69#125—21no—
70#124—21no—
71C22⋊Q8—21no—
72C22⋊Q8—21no—
73#121—21no—
74#121—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×D4—21no—
78C4×D4—21no—
79#117—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C42C2311yescharacteristic
83#106—22no—
84C4⋊C8D412yes—
85#103—24no—
86#103—22no—
87#103—24no—
88#102—24no—
89#102—24no—
90#102—24no—
91#102—24no—
92#102—24no—
93#102—22no—
94#102—24no—
95C22⋊C8D412yes—
96#97D412yes—
97C4×C8D412yes—
98C4×C8C2311yescharacteristic
99C4×C8C2312yes—
100C4×C8C2312yes—
101C4×C8C2311yescharacteristic
102C24—41no—
103#41—44no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×Q8—41no—
109C2×Q8—82no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—44no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—44no—
122C2×D4—82no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128Q16—84no—
129SD16—44no—
130D8—84no—
131M16—24no—
132C2×C8—42no—
133C2×C8C2×D412yes—
134C2×C8C2×D412yes—
135C2×C8C2×D414yes—
136C2×C8C2×D414yes—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C22⋊C4—41no—
150C22⋊C4—41no—
151C22⋊C4—41no—
152C42—41no—
153C42C2×D411yescharacteristic
154C42C2×D411yescharacteristic
155C42C2×D411yescharacteristic
156C42C2411yescharacteristic
157C23—41no—
158C23—42no—
159C23—82no—
160C23—82no—
161C23—82no—
162C23#12711yescharacteristic
163Q8—42no—
164Q8—42no—
165Q8—42no—
166Q8—44no—
167Q8—82no—
168D4—44no—
169D4—42no—
170D4—42no—
171D4—44no—
172D4—42no—
173D4—44no—
174D4—82no—
175D4—82no—
176D4—82no—
177D4—82no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4#12711yescharacteristic
187C2×C4#12711yescharacteristic
188C2×C4#12711yescharacteristic
189C2×C4#13514yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C22×D411yescharacteristic
192C2×C4C22×D411yescharacteristic
193C8—24no—
194C8—22no—
195C8—22no—
196C8—24no—
197C22—21no—
198C22—21no—
199C22—81no—
200C22—82no—
201C22—82no—
202C22—81no—
203C22—82no—
204C22—81no—
205C22—164no—
206C22#49512yes—
207C22C2×#12711yescharacteristic
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4—82no—
213C4—81no—
214C4#49512yes—
215C4C2×#13512yes—
216C4#57512yes—
217C4#57612yes—
218C2—22no—
219C2—161no—
220C2—162no—
221C2C2×#49511yescharacteristic
222C2#315311yescharacteristic
223C2#315511yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
prime radical#3205C2
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. #3209C2semidirect
3. #3167C2semidirect
4. #3141C2semidirect
5. #3138C2semidirect
6. C2×#486C2semidirect
7. #1720C2semidirect
8. #1717C2semidirect
9. #1711C2semidirect
10. #1709C2semidirect
11. C8×M16C2semidirect
16. #498C22semidirect
17. #494C22semidirect
19. #492C22semidirect
22. #487C22semidirect
23. #487C22semidirect
24. #487C22semidirect
31. #486C22semidirect
32. #486C22semidirect
45. #322C22semidirect
46. C82C22semidirect

Maximal subgroups

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

subgroupindex
#32092
#32052
#31672
#31412
#31382
C2×#4862
#17202
#17172
#17112
#17092
C8×M162