Finite Group Databasebeta

← #144563

Subgroups — #144563

Order 512.

307 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 72 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#33291C211yesmaximal, characteristic, complemented
2#11856C212yesmaximal
3#9844C211yesmaximal, characteristic
4#7217C212yesmaximal, complemented
5#7148C211yesmaximal, characteristic, complemented
6#3169C2211yescharacteristic
7#2859—21no—
8#2162C2212yes—
9#1991—24no—
10#1962—21no—
11#1834—21no—
12#1834C2212yes—
13C42×C8C2211yescharacteristic
14#1567C2211yescharacteristic
15#1492—21no—
16#1347—24no—
17#532—21no—
18C2×#125—22no—
19#500Q811yescharacteristic
20#493Q811yescharacteristic
21#486—21no—
22#466—22no—
23C4×D8—22no—
24#423—21no—
25C2×#104—42no—
26C2×C4×C8—22no—
27C2×C4×C8D411yescharacteristic
28C2×C4×C8C2311yescharacteristic
29#389—42no—
30#389—42no—
31#384—22no—
32#382—22no—
33C43D411yescharacteristic
34C4⋊C16—22no—
35C22⋊C16—22no—
36C4×C16—22no—
37#340—24no—
38#340—22no—
39#340D414yes—
40#335—22no—
41D4⋊C8—42no—
42C2×#41—41no—
43C2×D8—22no—
44C2×M16—41no—
45C22×C8—21no—
46C22×C8C2×D412yes—
47C4⋊Q8—21no—
48#127—21no—
49#125—48no—
50#124—41no—
51#121—41no—
52#121—41no—
53C4×D4—41no—
54C4×D4—41no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—44no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—42no—
59C2×C42—21no—
60C2×C42—21no—
61C2×C42D812yes—
62C2×C42C2×D411yescharacteristic
63C2×C16—42no—
64#107—22no—
65C4⋊C8—41no—
66C4⋊C8—41no—
67#104—84no—
68#102—42no—
69C4×C8—21no—
70C4×C8—44no—
71C4×C8SD1612yes—
72C4×C8C2×Q811yescharacteristic
73C4×C8#4111yescharacteristic
74C4×C8#4112yes—
75#95—44no—
76#41—82no—
77#41—82no—
78C2×Q8—41no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—81no—
83C22×C4—22no—
84C22×C4—44no—
85C22×C4—81no—
86C22×C4#12011yescharacteristic
87C22×C4C2×D811yescharacteristic
88C22×C4AGL(1,ℤ/8)11yescharacteristic
89D8—82no—
90M16—82no—
91C2×C8—22no—
92C2×C8—42no—
93C2×C8—42no—
94C2×C8—81no—
95C2×C8C22⋊Q812yes—
96C2×C8#12412yes—
97C4⋊C4—48no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—84no—
101C4⋊C4—88no—
102C4⋊C4—82no—
103C4⋊C4—82no—
104C22⋊C4—81no—
105C22⋊C4—81no—
106C42—24no—
107C42—22no—
108C42—22no—
109C42—42no—
110C42C22⋊Q811yescharacteristic
111C42#12411yescharacteristic
112C42C2×SD1611yescharacteristic
113C42#13711yescharacteristic
114C16—82no—
115C23—81no—
116C23#44711yescharacteristic
117Q8—82no—
118D4—82no—
119D4—82no—
120D4—82no—
121D4—162no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—84no—
128C2×C4—84no—
129C2×C4—84no—
130C2×C4—81no—
131C2×C4—161no—
132C2×C4—161no—
133C2×C4#39311yescharacteristic
134C2×C4#44811yescharacteristic
135C2×C4#47411yescharacteristic
136C2×C4#47611yescharacteristic
137C2×C4#48611yescharacteristic
138C2×C4#48911yescharacteristic
139C2×C4#50812yes—
140C2×C4#50912yes—
141C8—22no—
142C8—22no—
143C8—42no—
144C8—161no—
145C22—21no—
146C22—21no—
147C22—161no—
148C22—161no—
149C22—161no—
150C22#202311yescharacteristic
151C22#218612yes—
152C4—21no—
153C4—21no—
154C4—44no—
155C4—164no—
156C4—161no—
157C4#203112yes—
158C4#222812yes—
159C4#223112yes—
160C4#224211yescharacteristic
161C4#224411yescharacteristic
162C2—22no—
163C2—321no—
164C2#1036211yescharacteristic
165C2#1336511yescharacteristic
166C2#1337211yescharacteristic
subgroupG/H
center, socleC22#2023
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33291C2semidirect
4. #7217C2semidirect
5. #7148C2semidirect

Maximal subgroups

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

subgroupindex
#332912
#118562
#98442
#72172
#71482