Finite Group Databasebeta

← #19637

Subgroups — #19637

Order 256.

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

normal subgroups only

This group has 375 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#3208C211yesmaximal, characteristic
3#3170C211yesmaximal, characteristic, complemented
4#3139C211yesmaximal, characteristic, complemented
5#3138C211yesmaximal, characteristic
6C2×#486C211yesmaximal, characteristic
7#1720C212yesmaximal, complemented
8#1719C212yesmaximal, complemented
9#1711C212yesmaximal, complemented
10#1708C212yesmaximal, complemented
11C8×M16C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13#533C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16#500C2211yescharacteristic, complemented
17#498C2212yescomplemented
18#494C2211yescharacteristic, complemented
19#492C2212yes—
20#491—22no—
21#490—22no—
22#487C2212yescomplemented
23#487C2212yescomplemented
24#487C2211yescharacteristic, complemented
25#487C2211yescharacteristic, complemented
26#487C2212yescomplemented
27#486—22no—
28#486C2212yes—
29#486C2212yes—
30#486C2212yes—
31#486C2212yes—
32#486C2212yes—
33#483—22no—
34#475—22no—
35#467—22no—
36#460—22no—
37#450—22no—
38C4×Q16—22no—
39#431C2211yescharacteristic
40#419—22no—
41#418—22no—
42C2×#102—22no—
43C4×M16C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#322C2212yescomplemented
46C82C2212yescomplemented
47C22×Q8—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51C2×Q16—22no—
52C2×SD16—24no—
53C2×M16D412yes—
54C22×C8D412yes—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8C2312yes—
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63#127—21no—
64#127—21no—
65#127—21no—
66#127C2312yes—
67#125—21no—
68#125—21no—
69C22⋊Q8—21no—
70C22⋊Q8—21no—
71C22⋊Q8—21no—
72C22⋊Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77#117—21no—
78#117—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C42C2311yescharacteristic
82#107—22no—
83#106—24no—
84C4⋊C8D412yes—
85#103—24no—
86#103—24no—
87#103—22no—
88#103—22no—
89#103—24no—
90#102—24no—
91#102—24no—
92#102—24no—
93#102—24no—
94#102—24no—
95C22⋊C8D412yes—
96#97D412yes—
97C4×C8D412yes—
98C4×C8C2311yescharacteristic
99C4×C8C2312yes—
100C4×C8C2312yes—
101C4×C8C2311yescharacteristic
102C24—41no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—41no—
109C2×Q8—44no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—44no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—44no—
118C2×D4—82no—
119C22×C4—41no—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123Q16—44no—
124SD16—84no—
125M16—24no—
126C2×C8—42no—
127C2×C8C2×D412yes—
128C2×C8C2×D412yes—
129C2×C8C2×D414yes—
130C2×C8C2×D414yes—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C22⋊C4—41no—
148C22⋊C4—41no—
149C42—41no—
150C42—41no—
151C42C2×D411yescharacteristic
152C42C2×D411yescharacteristic
153C42C2×D411yescharacteristic
154C42C2411yescharacteristic
155C23—42no—
156C23—82no—
157C23—82no—
158C23—82no—
159C23#12711yescharacteristic
160Q8—44no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—42no—
165Q8—82no—
166D4—44no—
167D4—44no—
168D4—44no—
169D4—82no—
170D4—82no—
171D4—82no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4#12711yescharacteristic
181C2×C4#12711yescharacteristic
182C2×C4#12711yescharacteristic
183C2×C4#13514yes—
184C2×C4C22×D411yescharacteristic
185C2×C4C22×D411yescharacteristic
186C2×C4C22×D411yescharacteristic
187C8—24no—
188C8—22no—
189C8—22no—
190C8—24no—
191C22—21no—
192C22—21no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22—164no—
197C22#49512yes—
198C22C2×#12711yescharacteristic
199C4—21no—
200C4—21no—
201C4—82no—
202C4—81no—
203C4—81no—
204C4—81no—
205C4—81no—
206C4#49512yes—
207C4C2×#13512yes—
208C4#57512yes—
209C4#57712yes—
210C2—22no—
211C2—162no—
212C2C2×#49511yescharacteristic
213C2#315311yescharacteristic
214C2#315411yescharacteristic
subgroupG/H
center, socleC22C2×#127
commutator, Frattini, cosocleC42C24
prime radicalC2×#127C22
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
3. #3170C2semidirect
4. #3139C2semidirect
7. #1720C2semidirect
8. #1719C2semidirect
9. #1711C2semidirect
10. #1708C2semidirect
11. C8×M16C2semidirect
16. #500C22semidirect
17. #498C22semidirect
18. #494C22semidirect
22. #487C22semidirect
23. #487C22semidirect
24. #487C22semidirect
25. #487C22semidirect
26. #487C22semidirect
45. #322C22semidirect
46. C82C22semidirect

Maximal subgroups

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

subgroupindex
#32292
#32082
#31702
#31392
#31382
C2×#4862
#17202
#17192
#17112
#17082
C8×M162