Finite Group Databasebeta

← #150146

Subgroups — #150146

Order 512.

230 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 66 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#33463C211yesmaximal, characteristic
2#13642C211yesmaximal, characteristic, complemented
3#12020C211yesmaximal, characteristic, complemented
4#7389C214yesmaximal, complemented
5#3414C411yescharacteristic
6C2×#443—21no—
7#2179—21no—
8#2179C2212yes—
9#2178—21no—
10#2178C412yes—
11#2161C411yescharacteristic
12#2143—21no—
13C4×M32C2211yescharacteristic
14#1566—21no—
15#1422—21no—
16#1400—21no—
17#1398C2212yescomplemented
18#1368—21no—
19#1368C2212yescomplemented
20C2×#135—21no—
21C2×#131—41no—
22#510—22no—
23#509—22no—
24#508—21no—
25#508C2×C412yes—
26C2×M32C2×C411yescharacteristic
27C2×M32D411yescharacteristic
28#443—22no—
29#443—22no—
30#443—41no—
31#443—41no—
32C2×#108—21no—
33#423—41no—
34C2×#104—41no—
35C2×C4×C8D411yescharacteristic
36M64—22no—
37#368—21no—
38#368C2×C412yes—
39#359—22no—
40#359—22no—
41#346D411yescharacteristic
42#346C2311yescharacteristic
43C4×C16C2×C411yescharacteristic
44C4×C16D411yescharacteristic
45#334—22no—
46#334—41no—
47C2×#41—41no—
48#135—22no—
49#135—22no—
50#135—41no—
51#135—41no—
52C2×Q16—21no—
53C2×SD16—41no—
54C2×D8—21no—
55#131—42no—
56#131—42no—
57#131—81no—
58#131—81no—
59C2×M16—41no—
60C2×M16—81no—
61C2×M16—81no—
62C22×C8—81no—
63C22×C8C22⋊C411yescharacteristic
64C22×C8SD1611yescharacteristic
65C2×C42D811yescharacteristic
66Q32—21no—
67Q32M1612yescomplemented
68SD32—22no—
69SD32—22no—
70D16—21no—
71D16M1612yescomplemented
72M32—22no—
73M32—41no—
74C2×C16C22⋊C411yescharacteristic
75C2×C16C22⋊C411yescharacteristic
76C2×C16C22×C411yescharacteristic
77C2×C16C2×D411yescharacteristic
78#108—22no—
79#108—41no—
80C4⋊C8—42no—
81C4⋊C8—81no—
82#104—42no—
83#104—42no—
84C4×C8C22⋊C411yescharacteristic
85C4×C8D811yescharacteristic
86C4×C8SD1611yescharacteristic
87C4×C8C2×D411yescharacteristic
88C32—42no—
89#41—42no—
90#41—42no—
91#41—81no—
92#41—81no—
93C2×Q8—41no—
94C2×D4—41no—
95C2×D4—81no—
96C22×C4—21no—
97C22×C4—81no—
98C22×C4#10211yescharacteristic
99Q16—22no—
100Q16—41no—
101SD16—42no—
102SD16—42no—
103D8—22no—
104D8—41no—
105M16—42no—
106M16—82no—
107M16—81no—
108M16—82no—
109M16—162no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—82no—
115C2×C8—81no—
116C2×C8—82no—
117C2×C8—82no—
118C2×C8#10211yescharacteristic
119C2×C8#10211yescharacteristic
120C2×C8C2×C22⋊C411yescharacteristic
121C2×C8C2×SD1611yescharacteristic
122C42—21no—
123C42#10211yescharacteristic
124C42C2×D811yescharacteristic
125C16—21no—
126C16—21no—
127C16—21no—
128C16C2×M1612yes—
129C23—81no—
130C23#36011yescharacteristic
131Q8—42no—
132Q8—81no—
133D4—42no—
134D4—81no—
135D4—82no—
136D4—82no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—82no—
143C2×C4—81no—
144C2×C4—82no—
145C2×C4#36011yescharacteristic
146C2×C4C2×#10211yescharacteristic
147C8—21no—
148C8—21no—
149C8—82no—
150C8—162no—
151C8#36111yescharacteristic
152C8#36111yescharacteristic
153C8#40711yescharacteristic
154C8#42111yescharacteristic
155C22—21no—
156C22—41no—
157C22—82no—
158C22—162no—
159C22C2×#36011yescharacteristic
160C4—22no—
161C4—21no—
162C4—41no—
163C4—82no—
164C4#148611yescharacteristic
165C4C2×#36111yescharacteristic
166C2—21no—
167C2—41no—
168C2—162no—
169C2#1166311yescharacteristic
subgroupG/H
center, socleC2#11663
commutatorC2×C16C22×C4
Frattini, cosocle#346C23
prime radical#3414C4
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G
LECSG ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #13642C2semidirect
3. #12020C2semidirect
4. #7389C2semidirect
17. #1398C22semidirect
19. #1368C22semidirect
67. Q32M16semidirect
71. D16M16semidirect

Maximal subgroups

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

subgroupindex
#334632
#136422
#120202
#73892