Finite Group Databasebeta

← #126496

Subgroups — #126496

Order 512.

268 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 56 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#31256C211yesmaximal, characteristic
2#13543C212yesmaximal
3#10686C211yesmaximal, characteristic
4#7119C212yesmaximal
5#7108C211yesmaximal, characteristic
6#3082—22no—
7#2064—22no—
8#2060—21no—
9#1990—21no—
10#1956C2211yescharacteristic
11#1945C2212yes—
12#1886—21no—
13#1844C2211yescharacteristic
14#1834—21no—
15#1386—22no—
16#1383—22no—
17#1376C2212yes—
18#1273C2211yescharacteristic
19#544—21no—
20C2×C4⋊Q8D411yescharacteristic
21#519—21no—
22C2×M32—22no—
23#479—42no—
24#479—42no—
25#477—42no—
26#475—42no—
27C2×#108D412yes—
28#428—22no—
29#427D412yes—
30C2×#106—22no—
31#424—22no—
32#419—22no—
33C2×#103—22no—
34C2×#101—21no—
35C2×C4×C8—21no—
36C2×C4×C8C2311yescharacteristic
37#384—21no—
38#384D411yescharacteristic
39#343—21no—
40#340—42no—
41#340—42no—
42#339—22no—
43C22×Q8—21no—
44C2×M16—22no—
45C22×C8—22no—
46C22×C8C2×D412yes—
47C4⋊Q8—21no—
48C4⋊Q8—41no—
49C4⋊Q8—42no—
50C4⋊Q8Q1612yes—
51#125—42no—
52C22⋊Q8—81no—
53C4×Q8—42no—
54C4×Q8—42no—
55#117—22no—
56#117—81no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C42C2×D411yescharacteristic
64M32—42no—
65M32—42no—
66C2×C16—42no—
67#108—42no—
68#107—42no—
69#107SD1612yes—
70#107SD1612yes—
71#106—22no—
72#106—42no—
73#106—44no—
74C4⋊C8—42no—
75C4⋊C8—42no—
76#103—42no—
77#103—42no—
78#103—42no—
79#103—42no—
80#101—81no—
81C4×C8—41no—
82C4×C8—41no—
83C4×C8—41no—
84C2×Q8—22no—
85C2×Q8—41no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—81no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—81no—
93C22×C4#12011yescharacteristic
94M16—42no—
95M16—42no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—42no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8C2×SD1612yes—
103C2×C8AGL(1,ℤ/8)12yes—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—81no—
109C4⋊C4—82no—
110C4⋊C4—82no—
111C4⋊C4—82no—
112C4⋊C4—81no—
113C4⋊C4—81no—
114C4⋊C4—81no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C22⋊C4—42no—
118C42—21no—
119C42—42no—
120C42—82no—
121C42C2×Q1611yescharacteristic
122C42#13711yescharacteristic
123C16—82no—
124C23#45311yescharacteristic
125Q8—44no—
126Q8—82no—
127Q8—321no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—161no—
139C2×C4—161no—
140C2×C4—161no—
141C2×C4—161no—
142C2×C4#45012yes—
143C2×C4#45111yescharacteristic
144C2×C4#45412yes—
145C2×C4#45811yescharacteristic
146C8—22no—
147C8—22no—
148C8—82no—
149C8—82no—
150C22—21no—
151C22—21no—
152C22#161411yescharacteristic
153C22#162911yescharacteristic
154C22#165711yescharacteristic
155C4—22no—
156C4—22no—
157C4—82no—
158C4—82no—
159C4—82no—
160C4—161no—
161C4—321no—
162C4#161612yes—
163C4#163712yes—
164C2—21no—
165C2—21no—
166C2#782611yescharacteristic
167C2#1236511yescharacteristic
168C2#1236711yescharacteristic
subgroupG/H
center, socleC22#1629
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#453
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#312562
#135432
#106862
#71192
#71082