Finite Group Databasebeta

← #153253

Subgroups — #153253

Order 512.

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

normal subgroups only

This group has 276 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#33607C211yesmaximal, characteristic
2#13638C211yesmaximal, characteristic, complemented
3C2×#1419C211yesmaximal, characteristic, complemented
4#13617C211yesmaximal, characteristic, complemented
5#13558C211yesmaximal, characteristic, complemented
6#12841C211yesmaximal, characteristic, complemented
7#7475C211yesmaximal, characteristic, complemented
8C22×D16C2211yescharacteristic
9C2×#504C2211yescharacteristic
10#3058—21no—
11C2×M64C2211yescharacteristic
12C2×M64C2211yescharacteristic
13#2218—22no—
14#2215—22no—
15C2×#368C2211yescharacteristic
16#2158C2211yescharacteristic
17#2158C2211yescharacteristic
18#2140—21no—
19#1814—21no—
20#1426—22no—
21#1419—22no—
22#1415—22no—
23#1401—22no—
24#1389—21no—
25C22×D8—21no—
26C2×#131D411yescharacteristic
27#535—41no—
28C2×D16—21no—
29C2×D16—21no—
30C2×D16—21no—
31C2×D16—21no—
32C2×D16—21no—
33C2×D16D412yes—
34#504—21no—
35#504—21no—
36#504—21no—
37#504—21no—
38C2×M32—21no—
39C2×M32D411yescharacteristic
40C22×C16—21no—
41C22×C16C2311yescharacteristic
42#469—42no—
43#466—24no—
44C2×#108—21no—
45#427—21no—
46#427—21no—
47#421—41no—
48#418—41no—
49M64—21no—
50M64—21no—
51M64—21no—
52M64—21no—
53C2×C32—21no—
54C2×C32D412yescomplemented
55#368—21no—
56#368—21no—
57#367—21no—
58#367D412yes—
59#366—21no—
60#366—21no—
61#357—24no—
62#344—41no—
63C2×#41D811yescharacteristic
64C22×D4—41no—
65C2×D8—22no—
66C2×D8—41no—
67C2×D8—41no—
68C2×D8—41no—
69C2×D8—41no—
70C2×D8—41no—
71#131—21no—
72#131—21no—
73#131—21no—
74#131—21no—
75C2×M16—21no—
76C2×M16—41no—
77C2×M16D811yescharacteristic
78C22×C8—21no—
79C22×C8C2×D411yescharacteristic
80#127—81no—
81#124—81no—
82#121—44no—
83#120—82no—
84#117—41no—
85#117—41no—
86D16—22no—
87D16—22no—
88D16—41no—
89D16—41no—
90M32—21no—
91M32—21no—
92M32—41no—
93M32—41no—
94C2×C16—21no—
95C2×C16—41no—
96C2×C16—41no—
97C2×C16C2×D411yescharacteristic
98C2×C16C2×D411yescharacteristic
99C2×C16C2×D411yescharacteristic
100C2×C16#4111yescharacteristic
101C2×C16#4111yescharacteristic
102C2×C16#4111yescharacteristic
103#108—41no—
104#108—41no—
105#107—22no—
106#107—41no—
107#106—41no—
108#106—41no—
109#104—82no—
110#102—44no—
111C32—22no—
112C32—41no—
113C24—81no—
114#41—21no—
115#41—21no—
116#41—21no—
117#41—21no—
118C2×Q8SD3211yescharacteristic
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—84no—
122C2×D4—81no—
123C2×D4—81no—
124C2×D4—81no—
125C2×D4—81no—
126C2×D4—81no—
127C2×D4SD3211yescharacteristic
128C22×C4—21no—
129C22×C4C2×D811yescharacteristic
130D8—42no—
131D8—42no—
132D8—81no—
133D8—81no—
134M16—21no—
135M16—21no—
136M16—41no—
137M16—41no—
138M16—81no—
139M16—81no—
140C2×C8—21no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—81no—
144C2×C8#12011yescharacteristic
145C2×C8#12111yescharacteristic
146C2×C8#12311yescharacteristic
147C2×C8#13511yescharacteristic
148C2×C8AGL(1,ℤ/8)11yescharacteristic
149C2×C8#13711yescharacteristic
150C4⋊C4—42no—
151C4⋊C4—81no—
152C22⋊C4—84no—
153C22⋊C4—81no—
154C22⋊C4—81no—
155C42—81no—
156C42—81no—
157C16—41no—
158C16#12111yescharacteristic
159C16#12111yescharacteristic
160C16#12311yescharacteristic
161C16#12411yescharacteristic
162C23—21no—
163C23—82no—
164C23—161no—
165C23—161no—
166C23—322no—
167C23#50911yescharacteristic
168Q8—21no—
169Q8—21no—
170D4—21no—
171D4—21no—
172D4—41no—
173D4—41no—
174D4—82no—
175D4—82no—
176D4—161no—
177D4—161no—
178D4—324no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4—161no—
184C2×C4#44711yescharacteristic
185C2×C4#46611yescharacteristic
186C2×C4#48011yescharacteristic
187C2×C4C2×SD3211yescharacteristic
188C2×C4#50811yescharacteristic
189C2×C4#50911yescharacteristic
190C8—41no—
191C8—161no—
192C8#39411yescharacteristic
193C8#45211yescharacteristic
194C8#46811yescharacteristic
195C8#48211yescharacteristic
196C22—41no—
197C22—42no—
198C22—41no—
199C22—164no—
200C22—162no—
201C22—322no—
202C22—322no—
203C22#218711yescharacteristic
204C22#221811yescharacteristic
205C22#223411yescharacteristic
206C4—41no—
207C4—162no—
208C4—321no—
209C4#203311yescharacteristic
210C4#218611yescharacteristic
211C4#221711yescharacteristic
212C4#223311yescharacteristic
213C2—21no—
214C2—21no—
215C2—21no—
216C2—81no—
217C2—322no—
218C2#1338111yescharacteristic
subgroupG/H
center, socleC2#13381
commutator, Frattini, cosocleC22×C16C23
prime radical#33607C2
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13638C2semidirect
3. C2×#1419C2semidirect
4. #13617C2semidirect
5. #13558C2semidirect
6. #12841C2semidirect
7. #7475C2semidirect
54. C2×C32D4semidirect

Maximal subgroups

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

subgroupindex
#336072
#136382
C2×#14192
#136172
#135582
#128412
#74752