Finite Group Databasebeta

← #19678

Subgroups — #19678

Order 256.

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

normal subgroups only

This group has 414 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#3226C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic, complemented
3#3167C211yesmaximal, characteristic, complemented
4#3151C211yesmaximal, characteristic
5#3139C211yesmaximal, characteristic, complemented
6C2×#486C211yesmaximal, characteristic, complemented
7#1719C212yesmaximal, complemented
8#1717C212yesmaximal, complemented
9#1717C212yesmaximal, complemented
10#1714C212yesmaximal, complemented
11#1451C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)—22no—
13#533C2211yescharacteristic
14#532C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17#500C2212yescomplemented
18#498C2211yescharacteristic, complemented
19#496—22no—
20#493C2212yes—
21#492C2212yes—
22#492C2212yes—
23#492C2211yescharacteristic, complemented
24#491—22no—
25#487C2212yescomplemented
26#487C2212yescomplemented
27#487C2211yescharacteristic, complemented
28#486—22no—
29#486C2212yes—
30#486C2212yes—
31#486C2212yes—
32#486C2211yescharacteristic, complemented
33#486C2212yescomplemented
34#480—22no—
35#474—22no—
36#465—22no—
37#460—22no—
38#450—22no—
39C4×SD16—22no—
40#431C2211yescharacteristic
41#419—22no—
42C2×#102—22no—
43C4×M16C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#322C2212yescomplemented
46#322C2212yescomplemented
47C2×#41—21no—
48C22×Q8—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52AGL(1,ℤ/8)—48no—
53C2×SD16—24no—
54C2×SD16—22no—
55C2×SD16—24no—
56C2×D8—24no—
57C2×M16D412yes—
58C22×C8D412yes—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8C2312yes—
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66#127—21no—
67#127—21no—
68#127—21no—
69#127C2312yes—
70#127C2311yescharacteristic
71#125—21no—
72#124—21no—
73C22⋊Q8—21no—
74C22⋊Q8—21no—
75#121—21no—
76#121—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×D4—21no—
80C4×D4—21no—
81#117—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C42C2311yescharacteristic
85#107—24no—
86#106—22no—
87C4⋊C8D412yes—
88#103—24no—
89#103—24no—
90#103—22no—
91#102—24no—
92#102—24no—
93#102—22no—
94#102—24no—
95#102—24no—
96C22⋊C8D412yes—
97#97D412yes—
98C4×C8D412yes—
99C4×C8C2311yescharacteristic
100C4×C8C2311yescharacteristic
101C4×C8C2312yes—
102C4×C8C2312yes—
103C24—41no—
104#41—44no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—41no—
110C2×Q8—44no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—44no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—44no—
123C2×D4—82no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129SD16—44no—
130SD16—48no—
131SD16—84no—
132D8—48no—
133M16—24no—
134C2×C8—42no—
135C2×C8C2×D412yes—
136C2×C8C2×D412yes—
137C2×C8C2×D414yes—
138C2×C8C2×D414yes—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C22⋊C4—41no—
154C42—41no—
155C42C2×D411yescharacteristic
156C42C2×D411yescharacteristic
157C42C2×D411yescharacteristic
158C42C2411yescharacteristic
159C23—41no—
160C23—42no—
161C23—82no—
162C23—82no—
163C23—82no—
164C23#12711yescharacteristic
165Q8—42no—
166Q8—44no—
167Q8—42no—
168Q8—42no—
169Q8—82no—
170D4—42no—
171D4—44no—
172D4—42no—
173D4—44no—
174D4—44no—
175D4—42no—
176D4—82no—
177D4—82no—
178D4—82no—
179D4—82no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4#12711yescharacteristic
189C2×C4#12711yescharacteristic
190C2×C4#12711yescharacteristic
191C2×C4AGL(1,ℤ/8)14yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C22×D411yescharacteristic
194C2×C4C22×D411yescharacteristic
195C8—24no—
196C8—22no—
197C8—22no—
198C8—24no—
199C22—21no—
200C22—21no—
201C22—81no—
202C22—82no—
203C22—82no—
204C22—81no—
205C22—82no—
206C22—81no—
207C22—164no—
208C22#49612yes—
209C22C2×#12711yescharacteristic
210C4—21no—
211C4—21no—
212C4—81no—
213C4—82no—
214C4—81no—
215C4—81no—
216C4#49612yes—
217C4C2×AGL(1,ℤ/8)12yes—
218C4#57512yes—
219C4#57712yes—
220C2—22no—
221C2—161no—
222C2—162no—
223C2C2×#49611yescharacteristic
224C2#315311yescharacteristic
225C2#315411yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
prime radical#3151C2
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. #3208C2semidirect
3. #3167C2semidirect
5. #3139C2semidirect
6. C2×#486C2semidirect
7. #1719C2semidirect
8. #1717C2semidirect
9. #1717C2semidirect
10. #1714C2semidirect
11. #1451C2semidirect
17. #500C22semidirect
18. #498C22semidirect
23. #492C22semidirect
25. #487C22semidirect
26. #487C22semidirect
27. #487C22semidirect
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
#32262
#32082
#31672
#31512
#31392
C2×#4862
#17192
#17172
#17142
#14512