Finite Group Databasebeta

← #146137

Subgroups — #146137

Order 512.

173 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 173 conjugacy classes of subgroups — too many to draw. Only its 54 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#12029C211yesmaximal, characteristic
2#11643C211yesmaximal, characteristic
3#8679C211yesmaximal, characteristic
4#8637C211yesmaximal, characteristic
5#7295C211yesmaximal, characteristic
6#7221C211yesmaximal, characteristic
7#7221C211yesmaximal, characteristic
8#2960C2211yescharacteristic
9#2163C2211yescharacteristic
10#2163C2211yescharacteristic
11#1835C2211yescharacteristic
12#1566C2211yescharacteristic
13#1566C2211yescharacteristic
14#1564C2211yescharacteristic
15#1561—21no—
16#1518—21no—
17#1368—22no—
18#1338—22no—
19#1309—21no—
20#1286—21no—
21#533D411yescharacteristic
22#446—21no—
23#446—21no—
24C8×Q8—21no—
25C8×Q8—21no—
26#431—21no—
27#424—21no—
28#423—21no—
29#423—21no—
30#423D411yescharacteristic
31#405—21no—
32C2×C4×C8C2311yescharacteristic
33#382—21no—
34C4⋊C16—21no—
35C4⋊C16—21no—
36C22⋊C16—21no—
37C22⋊C16—21no—
38#346—21no—
39#346D412yes—
40#340—21no—
41#340—21no—
42#335—21no—
43#334—21no—
44#334—21no—
45#334—21no—
46#334—21no—
47#334—21no—
48#332—41no—
49#332—41no—
50#328—42no—
51C2×M16—21no—
52C2×M16—41no—
53C2×M16—41no—
54C22×C8#4111yescharacteristic
55C22×C8#4111yescharacteristic
56C4⋊Q8SD1611yescharacteristic
57C4⋊Q8SD1611yescharacteristic
58#125—21no—
59C22⋊Q8—21no—
60C22⋊Q8—21no—
61C4×Q8—21no—
62C4×Q8—21no—
63#117—21no—
64C2×C4⋊C4—41no—
65C2×C42C2×D411yescharacteristic
66C2×C16—22no—
67C2×C16—41no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70C4⋊C8—41no—
71C4⋊C8—41no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—41no—
77C22⋊C8—41no—
78#97—41no—
79C4×C8—41no—
80C4×C8—41no—
81C4×C8C2×D411yescharacteristic
82C4×C8#4111yescharacteristic
83C4×C8#4111yescharacteristic
84C4×C8#4111yescharacteristic
85#95—41no—
86#95—41no—
87C2×Q8—21no—
88C2×Q8—21no—
89C22×C4—81no—
90C22×C4#12311yescharacteristic
91C22×C4#13511yescharacteristic
92C22×C4AGL(1,ℤ/8)11yescharacteristic
93M16—81no—
94M16—161no—
95M16—161no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—81no—
100C2×C8—81no—
101C2×C8—81no—
102C2×C8#12311yescharacteristic
103C2×C8#12411yescharacteristic
104C2×C8#12611yescharacteristic
105C2×C8#12611yescharacteristic
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—81no—
113C4⋊C4—81no—
114C22⋊C4—41no—
115C42—41no—
116C42#12111yescharacteristic
117C42#12311yescharacteristic
118C42C2×SD1611yescharacteristic
119C42#13711yescharacteristic
120C16—42no—
121C16—81no—
122C23#48211yescharacteristic
123Q8—81no—
124Q8—81no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—161no—
132C2×C4—161no—
133C2×C4—161no—
134C2×C4#39711yescharacteristic
135C2×C4#46511yescharacteristic
136C2×C4#46811yescharacteristic
137C2×C4#48111yescharacteristic
138C2×C4#48311yescharacteristic
139C2×C4#48511yescharacteristic
140C8—21no—
141C8—21no—
142C8—21no—
143C8—21no—
144C8—81no—
145C8—161no—
146C8—161no—
147C8—161no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22#207711yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—81no—
158C4—81no—
159C4—321no—
160C4#208011yescharacteristic
161C4#208211yescharacteristic
162C2—41no—
163C2#1064311yescharacteristic
164C2#1350211yescharacteristic
165C2#1350411yescharacteristic
subgroupG/H
center, socleC22#2077
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#482
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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
#120292
#116432
#86792
#86372
#72952
#72212