Finite Group Databasebeta

← #126509

Subgroups — #126509

Order 512.

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

normal subgroups only

This group has 259 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#13541C212yesmaximal
3#10662C211yesmaximal, characteristic
4#7133C211yesmaximal, characteristic
5#7126C212yesmaximal
6#2546—21no—
7#2087—21no—
8#2057—22no—
9#1985C2211yescharacteristic
10#1944C2212yes—
11#1919—22no—
12#1849C2211yescharacteristic
13#1571—21no—
14#1487—21no—
15#1387—22no—
16#1383—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#340—22no—
38#340—21no—
39#340—22no—
40#336—22no—
41#335—42no—
42Q8⋊C8—42no—
43C22×Q8—41no—
44C22×Q8SD1611yescharacteristic
45C2×M16—41no—
46C22×C8—21no—
47C22×C8C2×D412yes—
48C4⋊Q8—81no—
49C4⋊Q8—81no—
50C4⋊Q8—84no—
51C4×Q8—41no—
52C4×Q8—41no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4SD1611yescharacteristic
60C2×C42—21no—
61C2×C42—41no—
62C2×C42C2×D411yescharacteristic
63M32—42no—
64M32—42no—
65C2×C16—42no—
66#108—44no—
67#107—42no—
68#107—81no—
69#106—22no—
70#106Q1614yes—
71C4⋊C8—41no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C22⋊C8—41no—
76#97—41no—
77C4×C8—22no—
78#95—41no—
79#95—42no—
80#95—42no—
81C2×Q8—42no—
82C2×Q8—41no—
83C2×Q8—82no—
84C2×Q8—82no—
85C2×Q8—82no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4#12011yescharacteristic
91C22×C4C2×SD1611yescharacteristic
92C22×C4AGL(1,ℤ/8)11yescharacteristic
93M16—82no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—42no—
97C2×C8—41no—
98C2×C8—42no—
99C2×C8—81no—
100C2×C8C2×Q1612yes—
101C2×C8#13712yes—
102C4⋊C4—24no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—81no—
108C4⋊C4—84no—
109C4⋊C4—84no—
110C4⋊C4—82no—
111C42—21no—
112C42—21no—
113C42—41no—
114C42—41no—
115C42—82no—
116C16—82no—
117C23#45011yescharacteristic
118Q8—42no—
119Q8—81no—
120Q8—164no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—44no—
131C2×C4—82no—
132C2×C4—82no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4#45112yes—
137C2×C4#45311yescharacteristic
138C2×C4#45412yes—
139C2×C4#45811yescharacteristic
140C8—22no—
141C8—22no—
142C8—82no—
143C8—81no—
144C8—161no—
145C22—21no—
146C22—21no—
147C22#161311yescharacteristic
148C22#163712yes—
149C4—42no—
150C4—41no—
151C4—41no—
152C4—84no—
153C4—81no—
154C4—162no—
155C4#161712yes—
156C4#163311yescharacteristic
157C4#165711yescharacteristic
158C2—22no—
159C2#782911yescharacteristic
160C2#1229811yescharacteristic
161C2#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
#135412
#106622
#71332
#71262