Finite Group Databasebeta

← #144572

Subgroups — #144572

Order 512.

296 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 76 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#33258C211yesmaximal, characteristic
2#11857C212yesmaximal
3#9845C211yesmaximal, characteristic
4#7217C212yesmaximal
5#7148C211yesmaximal, characteristic
6#3170C2211yescharacteristic
7#2859—21no—
8#2162C2212yes—
9#1994—21no—
10#1942—24no—
11#1845—21no—
12#1834C2212yes—
13C42×C8C2211yescharacteristic
14#1567C2211yescharacteristic
15#1493—21no—
16#1348—24no—
17#533—21no—
18C2×#125—41no—
19#500Q811yescharacteristic
20#494Q811yescharacteristic
21#487—21no—
22#467—22no—
23C4×Q16—22no—
24C2×#107—24no—
25#423—21no—
26C2×C4×C8—22no—
27C2×C4×C8D411yescharacteristic
28C2×C4×C8C2311yescharacteristic
29#389—24no—
30#389—42no—
31#384—22no—
32#382—41no—
33C43D411yescharacteristic
34C4⋊C16—22no—
35C22⋊C16—22no—
36C4×C16—22no—
37#340—22no—
38#340D414yes—
39#339—41no—
40#339—41no—
41#335—22no—
42Q8⋊C8—42no—
43C2×Q16—22no—
44C2×M16—41no—
45C22×C8—21no—
46C22×C8C2×D412yes—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49#125—41no—
50#125—82no—
51#125—82no—
52C22⋊Q8—41no—
53C22⋊Q8—41no—
54C4×Q8—41no—
55C4×Q8—41no—
56#117—41no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—44no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C42—21no—
62C2×C42—21no—
63C2×C42D811yescharacteristic
64C2×C42D811yescharacteristic
65C2×C42C2×D411yescharacteristic
66C2×C16—42no—
67#107—22no—
68#107—48no—
69C4⋊C8—41no—
70C4⋊C8—41no—
71#103—42no—
72C4×C8—21no—
73C4×C8—44no—
74C4×C8SD1612yes—
75C4×C8C2×Q811yescharacteristic
76C4×C8#4111yescharacteristic
77C4×C8#4112yes—
78#95—44no—
79C2×Q8—41no—
80C2×Q8—41no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—44no—
84C22×C4#12011yescharacteristic
85C22×C4C2×D811yescharacteristic
86C22×C4AGL(1,ℤ/8)11yescharacteristic
87Q16—82no—
88M16—81no—
89M16—81no—
90C2×C8—24no—
91C2×C8—22no—
92C2×C8—42no—
93C2×C8—81no—
94C2×C8C22⋊Q812yes—
95C2×C8#12412yes—
96C4⋊C4—48no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—84no—
102C4⋊C4—88no—
103C4⋊C4—81no—
104C4⋊C4—81no—
105C4⋊C4—82no—
106C4⋊C4—82no—
107C22⋊C4—81no—
108C42—21no—
109C42—21no—
110C42—42no—
111C42—42no—
112C42—81no—
113C42SD3212yes—
114C42Q3212yes—
115C42C22⋊Q811yescharacteristic
116C42#12411yescharacteristic
117C42C2×SD1611yescharacteristic
118C42#13711yescharacteristic
119C16—82no—
120C23#44711yescharacteristic
121Q8—82no—
122Q8—82no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—84no—
132C2×C4—84no—
133C2×C4—84no—
134C2×C4—81no—
135C2×C4—81no—
136C2×C4#39311yescharacteristic
137C2×C4#44811yescharacteristic
138C2×C4#47411yescharacteristic
139C2×C4#47611yescharacteristic
140C2×C4#48611yescharacteristic
141C2×C4#48911yescharacteristic
142C2×C4C2×SD3211yescharacteristic
143C2×C4C2×Q3211yescharacteristic
144C2×C4#50911yescharacteristic
145C2×C4#51011yescharacteristic
146C8—24no—
147C8—22no—
148C8—22no—
149C8—161no—
150C22—21no—
151C22—21no—
152C22#202311yescharacteristic
153C22#218711yescharacteristic
154C22#218811yescharacteristic
155C4—21no—
156C4—21no—
157C4—42no—
158C4—42no—
159C4—164no—
160C4—161no—
161C4—161no—
162C4#203112yes—
163C4#222511yescharacteristic
164C4#222711yescharacteristic
165C4#223112yes—
166C4#224111yescharacteristic
167C4#224311yescharacteristic
168C2—21no—
169C2—21no—
170C2#1036411yescharacteristic
171C2#1336411yescharacteristic
172C2#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 ◁ C42×C8 ◁ 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
#332582
#118572
#98452
#72172
#71482