Finite Group Databasebeta

← #152007

Subgroups — #152007

Order 512.

618 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 144 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#62624C211yesmaximal, characteristic
2#13558C212yesmaximal, complemented
3C2×#1401C212yesmaximal, complemented
4#13556C211yesmaximal, characteristic, complemented
5#13554C211yesmaximal, characteristic, complemented
6#3573C411yescharacteristic
7#3407—24no—
8#3407—22no—
9#3407C412yes—
10#3407C411yescharacteristic
11C2×#504—24no—
12C2×#504—22no—
13C2×#504C2212yes—
14C2×#504C2211yescharacteristic
15C2×M64C2211yescharacteristic, complemented
16C2×M64C2212yescomplemented
17C22×C32C2211yescharacteristic, complemented
18#1401—24no—
19#1401—24no—
20#1400—22no—
21C22⋊C32—22no—
22#585C811yescharacteristic
23#568—24no—
24#568—22no—
25#568C811yescharacteristic
26#568C812yes—
27C2×#131—24no—
28C2×#131—22no—
29C2×#131C2×C412yes—
30C2×#131C2×C411yescharacteristic
31#504—24no—
32#504—28no—
33#504—24no—
34#504—24no—
35#504—24no—
36#504—24no—
37#504—22no—
38#504—22no—
39#504—22no—
40#504—22no—
41#504D414yes—
42#504D414yes—
43C2×M32—28no—
44C2×M32—24no—
45C2×M32—22no—
46C2×M32—24no—
47C2×M32—22no—
48C2×M32C2×C412yes—
49C2×M32C2×C411yescharacteristic
50C2×M32D412yes—
51C22×C16—24no—
52C22×C16—22no—
53C22×C16D412yes—
54C22×C16C2311yescharacteristic
55M64—22no—
56M64—22no—
57M64—22no—
58C2×C32—21no—
59C2×C32—21no—
60C2×C32—22no—
61C2×C32—22no—
622−1+4—22no—
632−1+4C1612yes—
64H(5,𝔽2)—22no—
65H(5,𝔽2)—22no—
66H(5,𝔽2)C1612yes—
67C2×#41—24no—
68C2×#41—22no—
69C2×#41C2×C812yes—
70C2×#41C2×C811yescharacteristic
71#131—24no—
72#131—28no—
73#131—24no—
74#131—24no—
75#131—24no—
76#131—24no—
77#131—22no—
78#131—22no—
79#131—22no—
80#131—22no—
81#131C22⋊C414yes—
82#131M1614yes—
83C2×M16—28no—
84C2×M16—24no—
85C2×M16—22no—
86C2×M16—24no—
87C2×M16—22no—
88C2×M16C2×C812yes—
89C2×M16C2×C811yescharacteristic
90C2×M16M1612yes—
91C22×C8—24no—
92C22×C8—22no—
93C22×C8C22⋊C412yes—
94C22×C8C22×C411yescharacteristic
95M32—24no—
96M32—28no—
97M32—22no—
98M32—22no—
99M32—24no—
100M32—24no—
101M32—24no—
102M32C22⋊C414yes—
103C2×C16—24no—
104C2×C16—28no—
105C2×C16—22no—
106C2×C16—24no—
107C2×C16—22no—
108C2×C16—24no—
109C2×C16—22no—
110C2×C16C22⋊C412yes—
111C2×C16C2×D412yes—
112C2×C16C2×D411yescharacteristic
113C2×C16C2×D411yescharacteristic
114C2×C16C2×D412yes—
115C32—41no—
116C32—42no—
117C32—41no—
118#41—24no—
119#41—28no—
120#41—24no—
121#41—24no—
122#41—24no—
123#41—24no—
124#41—22no—
125#41—22no—
126#41—22no—
127#41—22no—
128#41C22⋊C814yes—
129#41C2×C1614yes—
130C2×Q8—22no—
131C2×Q8—24no—
132C2×Q8C2×C1611yescharacteristic
133C2×Q8M3212yes—
134C2×D4—28no—
135C2×D4—22no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4C2×C1611yescharacteristic
140C2×D4M3212yes—
141C2×D4M3212yes—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4C22⋊C812yes—
145C22×C4C22×C811yescharacteristic
146M16—24no—
147M16—28no—
148M16—22no—
149M16—22no—
150M16—24no—
151M16—24no—
152M16—24no—
153M16C22⋊C814yes—
154C2×C8—24no—
155C2×C8—28no—
156C2×C8—22no—
157C2×C8—24no—
158C2×C8—22no—
159C2×C8—24no—
160C2×C8—22no—
161C2×C8C22⋊C812yes—
162C2×C8C2×C22⋊C412yes—
163C2×C8C2×C22⋊C411yescharacteristic
164C2×C8C2×M1612yes—
165C2×C8C2×M1611yescharacteristic
166C16—24no—
167C16—22no—
168C16#12011yescharacteristic
169C16#12012yes—
170C16#12011yescharacteristic
171C23—24no—
172C23—22no—
173C23C22⋊C1612yes—
174C23C2×M3211yescharacteristic
175Q8—24no—
176Q8—22no—
177Q8—22no—
178Q8C22⋊C1614yes—
179D4—28no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—24no—
184D4—22no—
185D4—22no—
186D4C22⋊C1614yes—
187C2×C4—24no—
188C2×C4—28no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4C22⋊C1612yes—
195C2×C4C2×C22⋊C812yes—
196C2×C4C2×C22⋊C811yescharacteristic
197C2×C4C22×C1611yescharacteristic
198C2×C4C2×M3212yes—
199C8—24no—
200C8—22no—
201C8#37911yescharacteristic
202C8#40712yes—
203C8#40711yescharacteristic
204C22—22no—
205C22—24no—
206C22—41no—
207C22—42no—
208C22—82no—
209C22—84no—
210C22C2×C22⋊C1611yescharacteristic
211C22#211212yes—
212C4—24no—
213C4—22no—
214C4#177911yescharacteristic
215C4C2×C22⋊C1612yes—
216C4#211211yescharacteristic
217C2—21no—
218C2—22no—
219C2—42no—
220C2—44no—
221C2#1267911yescharacteristic
subgroupG/H
centerC16#120
commutatorC2×C4C22×C16
Frattini, cosocleC22×C16C23
socleC2#12679
prime radical#585C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C16 ◁ C22×C16 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13558C2semidirect
3. C2×#1401C2semidirect
4. #13556C2semidirect
5. #13554C2semidirect
15. C2×M64C22semidirect
16. C2×M64C22semidirect
17. C22×C32C22semidirect

Maximal subgroups

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

subgroupindex
#626242
#135582
C2×#14012
#135562
#135542