Finite Group Databasebeta

← #146002

Subgroups — #146002

Order 512.

189 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 60 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#12031C211yesmaximal, characteristic
2#12017C211yesmaximal, characteristic
3#11969C211yesmaximal, characteristic
4#11935C211yesmaximal, characteristic
5#9788C211yesmaximal, characteristic
6#9633C211yesmaximal, characteristic
7#7244C211yesmaximal, characteristic
8C2×#499C2211yescharacteristic
9#2163C2211yescharacteristic
10#2150C2211yescharacteristic
11#1574—21no—
12#1568C2211yescharacteristic
13#1566C2211yescharacteristic
14#1564C2211yescharacteristic
15#1528—21no—
16#1470—21no—
17C8×M16C2211yescharacteristic
18#1370—21no—
19#1368—22no—
20#1367—21no—
21#1363—21no—
22#1363—21no—
23#1363—22no—
24#1299—21no—
25C2×#125—21no—
26C2×M32—21no—
27#499—21no—
28#499—21no—
29#499—21no—
30#499—21no—
31#432—21no—
32#431—21no—
33#431D411yescharacteristic
34C2×#107Q811yescharacteristic
35C2×#107Q811yescharacteristic
36C2×#106—21no—
37#423—21no—
38C4×M16D411yescharacteristic
39C2×C4×C8C2311yescharacteristic
40C4⋊C16—21no—
41C4⋊C16—21no—
42C4⋊C16D411yescharacteristic
43C4⋊C16D411yescharacteristic
44C22⋊C16—21no—
45#346—21no—
46#335—21no—
47#335—21no—
48#335—41no—
49#334—21no—
50#334—21no—
51#334—21no—
52#334—41no—
53#334D412yes—
54#330—41no—
55#330—41no—
56#324—42no—
57#322—21no—
58C82—21no—
59C2×M16—21no—
60C2×M16—41no—
61C22×C8C2×Q811yescharacteristic
62C22×C8#4111yescharacteristic
63#125—41no—
64#125—41no—
65#125—41no—
66#125—41no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C42C2×D411yescharacteristic
72M32—81no—
73C2×C16—21no—
74C2×C16—21no—
75C2×C16—41no—
76#107—21no—
77#107—21no—
78#107—21no—
79#107—21no—
80#106—41no—
81#106—41no—
82C4⋊C8—22no—
83C4⋊C8—21no—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C22⋊C8—21no—
88C22⋊C8—41no—
89C22⋊C8—41no—
90#97—41no—
91#97SD1611yescharacteristic
92#97SD1611yescharacteristic
93C4×C8—21no—
94C4×C8—21no—
95C4×C8—41no—
96C4×C8C2×D411yescharacteristic
97C4×C8C2×D411yescharacteristic
98C4×C8#4111yescharacteristic
99C4×C8#4111yescharacteristic
100C22×C4—41no—
101C22×C4—41no—
102C22×C4C22⋊Q811yescharacteristic
103C22×C4#13511yescharacteristic
104C22×C4AGL(1,ℤ/8)11yescharacteristic
105M16—42no—
106M16—161no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—41no—
112C2×C8—42no—
113C2×C8—41no—
114C2×C8—81no—
115C2×C8—81no—
116C2×C8#12111yescharacteristic
117C2×C8C22⋊Q811yescharacteristic
118C2×C8C22⋊Q811yescharacteristic
119C2×C8#12611yescharacteristic
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—81no—
125C4⋊C4—81no—
126C4⋊C4—81no—
127C4⋊C4—81no—
128C42#12111yescharacteristic
129C42#12111yescharacteristic
130C42C2×SD1611yescharacteristic
131C42AGL(1,ℤ/8)11yescharacteristic
132C16—41no—
133C16—41no—
134C16—81no—
135C23#47811yescharacteristic
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—82no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—161no—
144C2×C4#39611yescharacteristic
145C2×C4#46011yescharacteristic
146C2×C4#46311yescharacteristic
147C2×C4#46511yescharacteristic
148C2×C4#46911yescharacteristic
149C2×C4#47611yescharacteristic
150C8—22no—
151C8—21no—
152C8—21no—
153C8—21no—
154C8—41no—
155C8—82no—
156C8—81no—
157C8—161no—
158C8—161no—
159C8#47111yescharacteristic
160C8#47311yescharacteristic
161C22—21no—
162C22—21no—
163C22—21no—
164C22#206511yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—161no—
171C4—161no—
172C4#206411yescharacteristic
173C4#207011yescharacteristic
174C2—41no—
175C2#1060811yescharacteristic
176C2#1346611yescharacteristic
177C2#1346911yescharacteristic
subgroupG/H
center, socleC22#2065
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#478
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ 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
#120312
#120172
#119692
#119352
#97882
#96332
#72442