Finite Group Databasebeta

← #144586

Subgroups — #144586

Order 512.

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

normal subgroups only

This group has 282 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#23836C211yesmaximal, characteristic
2#11745C212yesmaximal
3#9853C211yesmaximal, characteristic
4#7302C212yesmaximal
5#7143C211yesmaximal, characteristic
6#3170C2211yescharacteristic
7C4×#125—21no—
8#2163C2212yes—
9#2085—22no—
10#2059—22no—
11#1994—21no—
12C4×#107—21no—
13C4×#106C2212yes—
14#1744C2211yescharacteristic
15#1566C2211yescharacteristic
16#1493—21no—
17#1348—22no—
18#1348—22no—
19#533—21no—
20C2×#125—21no—
21#500Q811yescharacteristic
22#494Q811yescharacteristic
23#487—21no—
24#467—22no—
25C4×Q16—22no—
26C2×#107—22no—
27C2×#106—22no—
28#423—21no—
29C2×C4×C8D411yescharacteristic
30C2×C4×C8C2311yescharacteristic
31#389—22no—
32#387—42no—
33#382—21no—
34C4×C4⋊C4—22no—
35C4×C4⋊C4—42no—
36C43D411yescharacteristic
37C4⋊C16—22no—
38C22⋊C16—22no—
39#346—22no—
40#340—21no—
41#340—21no—
42#340D412yes—
43#340D412yes—
44#339—42no—
45#336—22no—
46#334—22no—
47Q8⋊C8—42no—
48C2×Q16—22no—
49C2×M16—41no—
50C22×C8—21no—
51C22×C8C2×D412yes—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54#125—44no—
55#125—41no—
56C22⋊Q8—41no—
57C22⋊Q8—41no—
58C4×Q8—41no—
59C4×Q8—41no—
60#117—41no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42—42no—
68C2×C42D812yes—
69C2×C42C2×D411yescharacteristic
70C2×C16—42no—
71#107—22no—
72#107—44no—
73#106—44no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76#103—42no—
77C4×C8—21no—
78C4×C8SD1612yes—
79C4×C8C2×Q811yescharacteristic
80C4×C8#4111yescharacteristic
81C4×C8#4112yes—
82#95—42no—
83#95—42no—
84#95—42no—
85C2×Q8—41no—
86C2×Q8—41no—
87C22×C4—22no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4#12011yescharacteristic
91C22×C4C2×D811yescharacteristic
92C22×C4AGL(1,ℤ/8)11yescharacteristic
93Q16—82no—
94M16—82no—
95C2×C8—22no—
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—81no—
99C2×C8C22⋊Q812yes—
100C2×C8#12412yes—
101C4⋊C4—24no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4—48no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—84no—
111C4⋊C4—81no—
112C4⋊C4—81no—
113C22⋊C4—81no—
114C42—42no—
115C42—42no—
116C42—42no—
117C42—84no—
118C42—81no—
119C42C22⋊Q811yescharacteristic
120C42#12411yescharacteristic
121C42C2×SD1611yescharacteristic
122C42#13711yescharacteristic
123C16—82no—
124C23#44711yescharacteristic
125Q8—82no—
126Q8—82no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—44no—
133C2×C4—42no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4—82no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4#39311yescharacteristic
142C2×C4#44811yescharacteristic
143C2×C4#47411yescharacteristic
144C2×C4#47611yescharacteristic
145C2×C4#48611yescharacteristic
146C2×C4#48911yescharacteristic
147C8—22no—
148C8—22no—
149C8—42no—
150C8—161no—
151C22—21no—
152C22—21no—
153C22#202311yescharacteristic
154C22#219112yes—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—82no—
160C4—84no—
161C4—162no—
162C4—161no—
163C4—161no—
164C4#203112yes—
165C4#223112yes—
166C2—22no—
167C2#1037011yescharacteristic
168C2#1336811yescharacteristic
169C2#1337211yescharacteristic
subgroupG/H
center, socleC22#2023
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
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
#238362
#117452
#98532
#73022
#71432