Finite Group Databasebeta

← #126478

Subgroups — #126478

Order 512.

277 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 52 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#32469C211yesmaximal, characteristic
2#13541C212yesmaximal, complemented
3#10661C211yesmaximal, characteristic
4#7126C212yesmaximal, complemented
5#7105C211yesmaximal, characteristic
6#2486—21no—
7#2087—21no—
8#2077—22no—
9#1955C2211yescharacteristic
10#1944C2212yes—
11#1852—22no—
12#1849C2211yescharacteristic
13#1570—21no—
14#1486—21no—
15#1387—22no—
16#1383—22no—
17#1377C2212yes—
18#1275C2211yescharacteristic
19C2×#123—41no—
20C2×C4×D4D411yescharacteristic
21C22×C4⋊C4—21no—
22C2×M32—22no—
23#431—21no—
24C2×#108—22no—
25C2×#107—21no—
26C2×#107D412yes—
27C2×#106D412yes—
28#423—21no—
29C2×C4⋊C8D411yescharacteristic
30C2×C22⋊C8—22no—
31C2×#97C2311yescharacteristic
32#398—41no—
33#397—41no—
34#389—22no—
35#381—41no—
36#380—21no—
37#380—22no—
38#340—21no—
39#340—22no—
40#340—22no—
41#334—42no—
42D4⋊C8—42no—
43C22×D4SD1611yescharacteristic
44C23×C4—21no—
45C23×C4—41no—
46C2×M16—41no—
47C22×C8—21no—
48C22×C8C2×D412yes—
49#123—84no—
50C4×D4—41no—
51C4×D4—41no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—81no—
57C2×C4⋊C4—81no—
58C2×C4⋊C4SD1611yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—41no—
61C2×C22⋊C4—42no—
62C2×C42C2×D411yescharacteristic
63M32—42no—
64M32—42no—
65C2×C16—42no—
66#108—44no—
67#107—42no—
68#107—81no—
69#106—22no—
70#106—22no—
71C4⋊C8—41no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C22⋊C8—41no—
76C22⋊C8—42no—
77C22⋊C8—42no—
78#97—41no—
79C4×C8—22no—
80#95—42no—
81#95—41no—
82#95—42no—
83C24—21no—
84C2×D4—41no—
85C2×D4—42no—
86C22×C4—21no—
87C22×C4—42no—
88C22×C4—41no—
89C22×C4—42no—
90C22×C4—41no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—82no—
94C22×C4—82no—
95C22×C4—82no—
96C22×C4#12011yescharacteristic
97C22×C4C2×SD1611yescharacteristic
98C22×C4AGL(1,ℤ/8)11yescharacteristic
99M16—82no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—41no—
105C2×C8—81no—
106C2×C8#13512yes—
107C2×C8#13712yes—
108C4⋊C4—41no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—82no—
114C4⋊C4—84no—
115C22⋊C4—81no—
116C22⋊C4—84no—
117C22⋊C4—82no—
118C42—21no—
119C42—21no—
120C16—82no—
121C23—22no—
122C23—41no—
123C23—42no—
124C23—41no—
125C23—42no—
126C23#45011yescharacteristic
127D4—42no—
128D4—81no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—82no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—82no—
141C2×C4—82no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4—164no—
145C2×C4#45212yes—
146C2×C4#45311yescharacteristic
147C2×C4#45412yes—
148C2×C4#45811yescharacteristic
149C8—22no—
150C8—22no—
151C8—82no—
152C8—81no—
153C8—161no—
154C22—21no—
155C22—21no—
156C22—44no—
157C22—42no—
158C22—82no—
159C22—81no—
160C22—81no—
161C22#161311yescharacteristic
162C22#163512yes—
163C4—41no—
164C4—41no—
165C4—81no—
166C4—162no—
167C4—162no—
168C4#161512yes—
169C4#163011yescharacteristic
170C4#165711yescharacteristic
171C2—22no—
172C2—82no—
173C2#782411yescharacteristic
174C2#1229811yescharacteristic
175C2#1230211yescharacteristic
subgroupG/H
center, socleC22#1613
commutator, Frattini, cosocleC2×#97C23
prime radicalC22×D4SD16
LCSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13541C2semidirect
4. #7126C2semidirect

Maximal subgroups

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

subgroupindex
#324692
#135412
#106612
#71262
#71052