Finite Group Databasebeta

← #19697

Subgroups — #19697

Order 256.

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

normal subgroups only

This group has 410 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#3229C211yesmaximal, characteristic, complemented
2#3226C211yesmaximal, characteristic, complemented
3#3208C211yesmaximal, characteristic, complemented
4#3205C211yesmaximal, characteristic
5C2×#492C211yesmaximal, characteristic, complemented
6#3131C211yesmaximal, characteristic, complemented
7#1719C212yesmaximal, complemented
8#1717C212yesmaximal, complemented
9#1715C212yesmaximal, complemented
10#1714C212yesmaximal, complemented
11#1452C211yesmaximal, characteristic, complemented
12C22×SD16—22no—
13#533C2211yescharacteristic
14#532C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17#500C2212yescomplemented
18#498C2212yescomplemented
19#493C2212yes—
20#492—22no—
21#492C2212yes—
22#492C2212yes—
23#492C2212yes—
24#488—22no—
25#487C2211yescharacteristic, complemented
26#487C2211yescharacteristic, complemented
27#487C2212yescomplemented
28#486C2212yes—
29#486C2212yes—
30#486C2211yescharacteristic, complemented
31#486C2211yescharacteristic, complemented
32#486C2212yescomplemented
33#483—22no—
34#480—22no—
35#475—22no—
36#474—22no—
37#460—22no—
38#459—22no—
39#450—22no—
40#447—22no—
41#431C2211yescharacteristic
42#431C2211yescharacteristic
43#416—22no—
44C2×C4×C8C2211yescharacteristic
45#322C2212yescomplemented
46#322C2212yescomplemented
47C2×#41—21no—
48C22×Q8—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—24no—
55C2×SD16—48no—
56C2×D8—24no—
57C22×C8D412yes—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8C2312yes—
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65#127—21no—
66#127—21no—
67#127—21no—
68#127C2312yes—
69#127C2311yescharacteristic
70#125—21no—
71#124—21no—
72C22⋊Q8—21no—
73C22⋊Q8—21no—
74#121—21no—
75#121—21no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×D4—21no—
79C4×D4—21no—
80#117—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C42C2311yescharacteristic
84#107—24no—
85#106—24no—
86C4⋊C8D412yes—
87C4⋊C8D412yes—
88#103—22no—
89#103—22no—
90#103—24no—
91#102—22no—
92#102—22no—
93#102—24no—
94#102—24no—
95#102—24no—
96C22⋊C8D412yes—
97C22⋊C8D412yes—
98C4×C8D412yes—
99C4×C8C2311yescharacteristic
100C4×C8C2311yescharacteristic
101C4×C8C2312yes—
102C4×C8C2312yes—
103C24—41no—
104#41—44no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—41no—
110C2×Q8—44no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—44no—
122C2×D4—82no—
123C2×D4—82no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129SD16—48no—
130SD16—48no—
131SD16—84no—
132D8—84no—
133C2×C8—24no—
134C2×C8C2×D412yes—
135C2×C8C2×D412yes—
136C2×C8C2×D414yes—
137C2×C8C2×D414yes—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C22⋊C4—41no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C42—41no—
154C42C2×D411yescharacteristic
155C42C2×D411yescharacteristic
156C42C2×D411yescharacteristic
157C42C2411yescharacteristic
158C23—41no—
159C23—42no—
160C23—82no—
161C23—82no—
162C23—82no—
163C23#12711yescharacteristic
164Q8—44no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168Q8—82no—
169D4—44no—
170D4—42no—
171D4—42no—
172D4—44no—
173D4—44no—
174D4—42no—
175D4—82no—
176D4—82no—
177D4—82no—
178D4—82no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4#12711yescharacteristic
188C2×C4#12711yescharacteristic
189C2×C4#12711yescharacteristic
190C2×C4AGL(1,ℤ/8)14yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C22×D411yescharacteristic
193C2×C4C22×D411yescharacteristic
194C8—22no—
195C8—22no—
196C8—24no—
197C8—24no—
198C22—21no—
199C22—21no—
200C22—81no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22—164no—
207C22#49612yes—
208C22C2×#12711yescharacteristic
209C4—21no—
210C4—21no—
211C4—81no—
212C4—82no—
213C4—81no—
214C4—81no—
215C4#49612yes—
216C4C2×AGL(1,ℤ/8)12yes—
217C4#57712yes—
218C4#57712yes—
219C2—22no—
220C2—161no—
221C2—162no—
222C2C2×#49611yescharacteristic
223C2#315411yescharacteristic
224C2#315411yescharacteristic
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. #3229C2semidirect
2. #3226C2semidirect
3. #3208C2semidirect
5. C2×#492C2semidirect
6. #3131C2semidirect
7. #1719C2semidirect
8. #1717C2semidirect
9. #1715C2semidirect
10. #1714C2semidirect
11. #1452C2semidirect
17. #500C22semidirect
18. #498C22semidirect
25. #487C22semidirect
26. #487C22semidirect
27. #487C22semidirect
30. #486C22semidirect
31. #486C22semidirect
32. #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
#32292
#32262
#32082
#32052
C2×#4922
#31312
#17192
#17172
#17152
#17142
#14522