Finite Group Databasebeta

← #146138

Subgroups — #146138

Order 512.

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

normal subgroups only

This group has 187 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#12028C211yesmaximal, characteristic
2#11644C211yesmaximal, characteristic
3#8657C211yesmaximal, characteristic
4#8625C211yesmaximal, characteristic
5#7294C211yesmaximal, characteristic
6#7222C211yesmaximal, characteristic
7#7222C211yesmaximal, characteristic
8C2×#446C2211yescharacteristic
9#2163C2211yescharacteristic
10#2163C2211yescharacteristic
11#1834C2211yescharacteristic
12#1803—21no—
13#1568C2211yescharacteristic
14#1568C2211yescharacteristic
15C2×#334C2211yescharacteristic
16#1467—21no—
17#1368—22no—
18#1338—22no—
19#1296—21no—
20#1290—21no—
21C2×C4×Q8D411yescharacteristic
22#446—21no—
23#446—21no—
24#446—21no—
25#446—21no—
26#432—21no—
27#432—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8D411yescharacteristic
31C2×#103—41no—
32C2×#97—21no—
33C2×C4×C8C2311yescharacteristic
34#384—21no—
35C4⋊C16—21no—
36C4⋊C16—21no—
37C22⋊C16—21no—
38C22⋊C16—21no—
39#346—21no—
40#346D412yes—
41#340—21no—
42#340—21no—
43#340—41no—
44#335—21no—
45#335—21no—
46#334—21no—
47#334—21no—
48#334—21no—
49#334—21no—
50#324—42no—
51C22×Q8SD1611yescharacteristic
52C22×C8—21no—
53C22×C8—41no—
54C22×C8#4111yescharacteristic
55C22×C8#4111yescharacteristic
56C4×Q8—21no—
57C4×Q8—21no—
58C4×Q8—21no—
59C4×Q8—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4SD1611yescharacteristic
65C2×C42—21no—
66C2×C42C2×D411yescharacteristic
67C2×C16—22no—
68C2×C16—41no—
69C4⋊C8—21no—
70C4⋊C8—21no—
71C4⋊C8—41no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—41no—
77#103—82no—
78C22⋊C8—41no—
79C22⋊C8—41no—
80#97—41no—
81#97—41no—
82#97—41no—
83#97—41no—
84C4×C8C2×D411yescharacteristic
85C4×C8#4111yescharacteristic
86C4×C8#4111yescharacteristic
87C4×C8#4111yescharacteristic
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—41no—
91C2×Q8—41no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—81no—
95C22×C4#12311yescharacteristic
96C22×C4C2×SD1611yescharacteristic
97C22×C4AGL(1,ℤ/8)11yescharacteristic
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—81no—
102C2×C8—81no—
103C2×C8—81no—
104C2×C8—81no—
105C2×C8—81no—
106C2×C8—81no—
107C2×C8#12311yescharacteristic
108C2×C8#12411yescharacteristic
109C2×C8#12611yescharacteristic
110C2×C8#12611yescharacteristic
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—81no—
116C4⋊C4—81no—
117C4⋊C4—81no—
118C4⋊C4—81no—
119C4⋊C4—161no—
120C42—41no—
121C42—41no—
122C42#12111yescharacteristic
123C42#12311yescharacteristic
124C42#13511yescharacteristic
125C42#13711yescharacteristic
126C16—42no—
127C16—81no—
128C23#48111yescharacteristic
129Q8—42no—
130Q8—81no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—81no—
141C2×C4—161no—
142C2×C4—161no—
143C2×C4—161no—
144C2×C4#39711yescharacteristic
145C2×C4#46511yescharacteristic
146C2×C4#46811yescharacteristic
147C2×C4#48211yescharacteristic
148C2×C4#48311yescharacteristic
149C2×C4#48511yescharacteristic
150C8—21no—
151C8—21no—
152C8—21no—
153C8—21no—
154C8—81no—
155C8—161no—
156C8—161no—
157C8—161no—
158C22—21no—
159C22—21no—
160C22#207711yescharacteristic
161C22#223811yescharacteristic
162C22#223911yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—81no—
169C4—81no—
170C4—321no—
171C4#208011yescharacteristic
172C4#208211yescharacteristic
173C2—21no—
174C2—21no—
175C2#1064411yescharacteristic
176C2#1350111yescharacteristic
177C2#1350411yescharacteristic
subgroupG/H
center, socleC22#2077
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#481
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ 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
#120282
#116442
#86572
#86252
#72942
#72222