Finite Group Databasebeta

← #126461

Subgroups — #126461

Order 512.

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

normal subgroups only

This group has 267 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#32693C211yesmaximal, characteristic
2#13510C212yesmaximal
3#10663C211yesmaximal, characteristic
4#7133C211yesmaximal, characteristic
5#7126C212yesmaximal
6#2546—21no—
7C2×#365—22no—
8#2087—21no—
9#2057—22no—
10#1985C2211yescharacteristic
11#1944C2212yes—
12#1919—22no—
13#1849C2211yescharacteristic
14#1570—21no—
15#1487—21no—
16#1387—22no—
17#1377C2212yes—
18#1275C2211yescharacteristic
19C2×C4⋊Q8—21no—
20C2×C4⋊Q8—41no—
21C2×C4×Q8D411yescharacteristic
22C2×M32—22no—
23#431—21no—
24C2×#108—22no—
25C2×#107—21no—
26C2×#107D412yes—
27C2×#106D412yes—
28#423—21no—
29C2×C4⋊C8D411yescharacteristic
30C2×#97C2311yescharacteristic
31#398—41no—
32#398—41no—
33#391—41no—
34#389—22no—
35#389—22no—
36C4×C4⋊C4—21no—
37#365—44no—
38#340—22no—
39#340—21no—
40#340—22no—
41#336—22no—
42#334—42no—
43Q8⋊C8—42no—
44C22×Q8—41no—
45C22×Q8SD1611yescharacteristic
46C2×M16—41no—
47C22×C8—21no—
48C22×C8C2×D412yes—
49C4⋊Q8—81no—
50C4⋊Q8—81no—
51C4⋊Q8—84no—
52C4×Q8—41no—
53C4×Q8—41no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4SD1611yescharacteristic
61C2×C42—21no—
62C2×C42—41no—
63C2×C42C2×D411yescharacteristic
64M32—24no—
65M32—42no—
66C2×C16—42no—
67#108—42no—
68#108—42no—
69#107—42no—
70#107—81no—
71#106—22no—
72#106SD1614yes—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—41no—
77C22⋊C8—41no—
78#97—41no—
79C4×C8—22no—
80#95—41no—
81#95—42no—
82#95—42no—
83C2×Q8—42no—
84C2×Q8—41no—
85C2×Q8—82no—
86C2×Q8—82no—
87C2×Q8—82no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4#12011yescharacteristic
93C22×C4C2×SD1611yescharacteristic
94C22×C4AGL(1,ℤ/8)11yescharacteristic
95M16—82no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—42no—
99C2×C8—41no—
100C2×C8—42no—
101C2×C8—81no—
102C2×C8C2×SD1612yes—
103C2×C8#13712yes—
104C4⋊C4—24no—
105C4⋊C4—41no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—81no—
110C4⋊C4—84no—
111C4⋊C4—84no—
112C4⋊C4—82no—
113C42—21no—
114C42—21no—
115C42—41no—
116C42—41no—
117C42—82no—
118C16—44no—
119C23#45011yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—164no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—44no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4#44812yes—
139C2×C4#45311yescharacteristic
140C2×C4#45612yes—
141C2×C4#45811yescharacteristic
142C8—22no—
143C8—22no—
144C8—82no—
145C8—81no—
146C8—161no—
147C22—21no—
148C22—21no—
149C22#161311yescharacteristic
150C22#165112yes—
151C4—42no—
152C4—41no—
153C4—41no—
154C4—84no—
155C4—81no—
156C4—162no—
157C4#161112yes—
158C4#162311yescharacteristic
159C4#166011yescharacteristic
160C2—22no—
161C2#781711yescharacteristic
162C2#1229811yescharacteristic
163C2#1230211yescharacteristic
subgroupG/H
center, socleC22#1613
commutator, Frattini, cosocleC2×#97C23
prime radicalC23#450
LCSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#326932
#135102
#106632
#71332
#71262