Finite Group Databasebeta

← #20138

Subgroups — #20138

Order 256.

326 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 99 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#3399C212yesmaximal, complemented
2#3305C212yesmaximal, complemented
3#1721C211yesmaximal, characteristic, complemented
4#1698C212yesmaximal, complemented
5#1667C212yesmaximal, complemented
6#1630C211yesmaximal, characteristic, complemented
7#1624C211yesmaximal, characteristic, complemented
8#1621C212yesmaximal, complemented
9#1585C212yesmaximal, complemented
10#559C2211yescharacteristic
11#555C2211yescharacteristic
12#550C2212yes—
13#496—22no—
14#493C2212yes—
15#493C2211yescharacteristic
16#492C2211yescharacteristic
17#486C2212yes—
18#482—22no—
19#477C2212yes—
20#466—22no—
21#463—22no—
22#460C2212yes—
23#459—22no—
24#459C2212yes—
25#459C2212yes—
26#449—22no—
27#446C2212yes—
28#442—22no—
29C4×SD16C2212yes—
30C8×D4C2212yes—
31#416—22no—
32#335C2212yes—
33#333C2211yescharacteristic
34#331C2212yes—
35#331C2211yescharacteristic
36Q8⋊C8C2212yes—
37D4⋊C8C2212yes—
38#322C2211yescharacteristic
39C2×#41—22no—
40C2×SD16—22no—
41C2×SD16—41no—
42C2×SD16D412yes—
43C2×D8—22no—
44C2×D8—22no—
45C2×D8—22no—
46C2×D8—22no—
47C2×D8—41no—
48C22×C8—22no—
49C4⋊Q8C2311yescharacteristic
50#127—22no—
51#127C2312yes—
52#127C2311yescharacteristic
53#124—22no—
54#124—41no—
55#123—41no—
56#121—22no—
57#121—22no—
58#121—42no—
59C4×Q8C2312yes—
60C4×D4—22no—
61C4×D4—22no—
62C4×D4C2312yes—
63#117—22no—
64#107—22no—
65#106D412yes—
66C4⋊C8—22no—
67C4⋊C8C2312yes—
68C4⋊C8C2312yes—
69#103—22no—
70#103D412yes—
71#102—22no—
72#102—22no—
73#102—22no—
74#102—22no—
75#102—22no—
76#102D412yes—
77C22⋊C8—22no—
78#97—22no—
79C4×C8C2311yescharacteristic
80C4×C8C2312yes—
81#41—44no—
82#41—82no—
83C2×Q8—21no—
84C2×Q8C2×D412yes—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—21no—
90C2×D4—42no—
91C2×D4—42no—
92C2×D4—41no—
93C2×D4C2×D412yes—
94C22×C4—22no—
95C22×C4—42no—
96C22×C4—41no—
97SD16—42no—
98SD16—82no—
99SD16—82no—
100D8—44no—
101D8—82no—
102D8—82no—
103D8—82no—
104D8—82no—
105C2×C8—21no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—82no—
109C2×C8C2×D412yes—
110C2×C8C2×D412yes—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4C2×D412yes—
114C4⋊C4C2×D412yes—
115C22⋊C4—22no—
116C22⋊C4—42no—
117C22⋊C4—42no—
118C22⋊C4—41no—
119C22⋊C4—42no—
120C42—22no—
121C42—41no—
122C42C2411yescharacteristic
123C23—22no—
124C23—41no—
125C23—42no—
126Q8—22no—
127Q8—22no—
128Q8—81no—
129D4—22no—
130D4—22no—
131D4—42no—
132D4—42no—
133D4—42no—
134D4—42no—
135D4—42no—
136D4—42no—
137D4—82no—
138D4—81no—
139D4—84no—
140D4—162no—
141D4—161no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—84no—
149C2×C4—161no—
150C2×C4C22×D412yes—
151C2×C4H(5,𝔽2)11yescharacteristic
152C8—22no—
153C8—22no—
154C8—22no—
155C8—42no—
156C8—42no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—82no—
161C22—82no—
162C22—82no—
163C22—81no—
164C22—82no—
165C22D4211yescharacteristic
166C4—21no—
167C4—42no—
168C4—42no—
169C4—42no—
170C4—81no—
171C4#57512yes—
172C4#57612yes—
173C2—82no—
174C2—162no—
175C2—161no—
176C2#326812yes—
177C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3399C2semidirect
2. #3305C2semidirect
3. #1721C2semidirect
4. #1698C2semidirect
5. #1667C2semidirect
6. #1630C2semidirect
7. #1624C2semidirect
8. #1621C2semidirect
9. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33052
#17212
#16982
#16672
#16302
#16242
#16212
#15852