Finite Group Databasebeta

← #146207

Subgroups — #146207

Order 512.

226 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 64 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#12030C212yesmaximal
2#11655C212yesmaximal
3#9853C211yesmaximal, characteristic
4#8809C211yesmaximal, characteristic
5#7143C211yesmaximal, characteristic
6#3170C2211yescharacteristic
7#2163C2212yes—
8#1994—21no—
9#1842—21no—
10#1744C2211yescharacteristic
11#1566C2211yescharacteristic
12C2×#334—21no—
13C2×#334C2212yes—
14#1493—21no—
15#1368—24no—
16#1334—24no—
17#533—21no—
18#500Q811yescharacteristic
19#494Q811yescharacteristic
20#487—21no—
21#467—22no—
22C4×Q16—22no—
23#423—21no—
24C2×C4⋊C8—22no—
25C2×C4⋊C8—42no—
26C2×C4×C8D411yescharacteristic
27C2×C4×C8C2311yescharacteristic
28C43D411yescharacteristic
29C4⋊C16—22no—
30C22⋊C16—22no—
31#346D414yes—
32#339—42no—
33#336—22no—
34#334—22no—
35#334—22no—
36#334—22no—
37#334—42no—
38Q8⋊C8—42no—
39C2×Q16—22no—
40C2×M16—41no—
41C22×C8—21no—
42C22×C8—42no—
43C22×C8#4112yes—
44C4⋊Q8—21no—
45C4⋊Q8—21no—
46#125—41no—
47C22⋊Q8—41no—
48C22⋊Q8—41no—
49C4×Q8—41no—
50C4×Q8—41no—
51#117—41no—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42SD1612yes—
55C2×C42C2×D411yescharacteristic
56C2×C16—24no—
57#107—22no—
58C4⋊C8—42no—
59C4⋊C8—41no—
60C4⋊C8—41no—
61C4⋊C8—42no—
62C4⋊C8—84no—
63#103—42no—
64C4×C8—21no—
65C4×C8SD1612yes—
66C4×C8C2×D412yes—
67C4×C8C2×Q811yescharacteristic
68C4×C8#4111yescharacteristic
69C2×Q8—41no—
70C2×Q8—41no—
71C22×C4—22no—
72C22×C4#12311yescharacteristic
73C22×C4C2×SD1611yescharacteristic
74C22×C4AGL(1,ℤ/8)11yescharacteristic
75Q16—44no—
76M16—82no—
77C2×C8—22no—
78C2×C8—42no—
79C2×C8—42no—
80C2×C8—84no—
81C2×C8—81no—
82C2×C8—82no—
83C2×C8C22⋊Q812yes—
84C2×C8#12312yes—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—81no—
90C4⋊C4—81no—
91C22⋊C4—81no—
92C42—24no—
93C42—42no—
94C42—42no—
95C42—81no—
96C42C22⋊Q811yescharacteristic
97C42#12711yescharacteristic
98C42C2×SD1611yescharacteristic
99C42AGL(1,ℤ/8)11yescharacteristic
100C16—44no—
101C23#48111yescharacteristic
102Q8—82no—
103Q8—82no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—24no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—81no—
111C2×C4—81no—
112C2×C4#39911yescharacteristic
113C2×C4#47611yescharacteristic
114C2×C4#47611yescharacteristic
115C2×C4#48111yescharacteristic
116C2×C4#49211yescharacteristic
117C2×C4#49611yescharacteristic
118C8—22no—
119C8—22no—
120C8—42no—
121C8—162no—
122C8—161no—
123C22—21no—
124C22—21no—
125C22#209411yescharacteristic
126C22#223912yes—
127C4—21no—
128C4—21no—
129C4—21no—
130C4—21no—
131C4—44no—
132C4—161no—
133C4—161no—
134C4#209812yes—
135C4#223012yes—
136C2—22no—
137C2#1066611yescharacteristic
138C2#1353311yescharacteristic
139C2#1353611yescharacteristic
subgroupG/H
center, socleC22#2094
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#481
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#120302
#116552
#98532
#88092
#71432