Finite Group Databasebeta

← #153251

Subgroups — #153251

Order 512.

261 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#2222C211yesmaximal, characteristic
2C2×#1425C211yesmaximal, characteristic, complemented
3C2×#1420C211yesmaximal, characteristic, complemented
4C2×#1416C211yesmaximal, characteristic, complemented
5#13554C211yesmaximal, characteristic
6#12842C211yesmaximal, characteristic
7#7478C211yesmaximal, characteristic, complemented
8C22×Q32C2211yescharacteristic
9C2×#504C2211yescharacteristic
10C2×#473—21no—
11C2×M64C2211yescharacteristic
12C22×C32C2211yescharacteristic
13#2222—22no—
14#2222—22no—
15C2×#368C2211yescharacteristic
16C2×#368C2211yescharacteristic
17C2×#366C2211yescharacteristic
18C2×#362—21no—
19#1816—21no—
20#1425—22no—
21#1420—22no—
22#1416—22no—
23C22⋊C32—22no—
24#1387—21no—
25C2×#137—41no—
26C22×Q16—21no—
27C2×#131D411yescharacteristic
28C2×Q32—21no—
29C2×Q32—21no—
30C2×Q32—21no—
31C2×Q32—21no—
32C2×Q32—21no—
33C2×Q32D412yes—
34#504—21no—
35#504—21no—
36#504—21no—
37#504—21no—
38C2×M32—21no—
39C2×M32D411yescharacteristic
40C22×C16—21no—
41C22×C16C2311yescharacteristic
42#473—24no—
43#473—42no—
44C2×#108—21no—
45C2×#108—21no—
46C2×#107—21no—
47#417—41no—
48C2×#101—41no—
49M64—21no—
50M64—21no—
51C2×C32—21no—
52C2×C32—21no—
53C2×C32—21no—
54C2×C32D412yes—
55#368—21no—
56#368—21no—
57#368—21no—
58#368D412yes—
59#366—21no—
60#366—21no—
61#362—24no—
62#343—41no—
63C2×#41D811yescharacteristic
64C22×Q8—41no—
65#137—44no—
66#137—82no—
67C2×Q16—22no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×Q16—81no—
74C2×SD16—81no—
75#131—21no—
76#131—21no—
77#131—21no—
78#131—21no—
79C2×M16—21no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16D811yescharacteristic
83C22×C8—21no—
84C22×C8C2×D411yescharacteristic
85C2×C4⋊C4—41no—
86Q32—22no—
87Q32—22no—
88Q32—41no—
89Q32—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—22no—
104#108—41no—
105#108—41no—
106#108—41no—
107#107—41no—
108#107—41no—
109#103—81no—
110#102—81no—
111#101—44no—
112C32—22no—
113C32—41no—
114#41—21no—
115#41—21no—
116#41—21no—
117#41—21no—
118C2×Q8—42no—
119C2×Q8—81no—
120C2×Q8—81no—
121C2×Q8—81no—
122C2×Q8—81no—
123C2×Q8—81no—
124C2×Q8D1611yescharacteristic
125C2×D4—21no—
126C2×D4D1611yescharacteristic
127C22×C4—21no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4C2×D811yescharacteristic
131Q16—42no—
132Q16—42no—
133Q16—84no—
134Q16—81no—
135Q16—81no—
136SD16—84no—
137M16—21no—
138M16—21no—
139M16—41no—
140M16—41no—
141M16—42no—
142M16—81no—
143M16—81no—
144M16—81no—
145C2×C8—21no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8#12011yescharacteristic
151C2×C8#12111yescharacteristic
152C2×C8#12311yescharacteristic
153C2×C8#13511yescharacteristic
154C2×C8AGL(1,ℤ/8)11yescharacteristic
155C2×C8#13711yescharacteristic
156C4⋊C4—81no—
157C4⋊C4—81no—
158C16—41no—
159C16#12111yescharacteristic
160C16#12111yescharacteristic
161C16#12311yescharacteristic
162C16#12411yescharacteristic
163C23—21no—
164C23#50811yescharacteristic
165Q8—21no—
166Q8—21no—
167Q8—82no—
168Q8—82no—
169Q8—161no—
170Q8—161no—
171D4—21no—
172D4—21no—
173D4—41no—
174D4—41no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—82no—
179C2×C4—161no—
180C2×C4—161no—
181C2×C4—161no—
182C2×C4—321no—
183C2×C4#44711yescharacteristic
184C2×C4#46611yescharacteristic
185C2×C4#48011yescharacteristic
186C2×C4C2×D1611yescharacteristic
187C2×C4#50911yescharacteristic
188C2×C4#51011yescharacteristic
189C8—41no—
190C8—82no—
191C8—161no—
192C8#39411yescharacteristic
193C8#45211yescharacteristic
194C8#46811yescharacteristic
195C8#48211yescharacteristic
196C22—21no—
197C22—21no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22#218911yescharacteristic
202C22#221511yescharacteristic
203C22#223411yescharacteristic
204C4—41no—
205C4—162no—
206C4—321no—
207C4#203311yescharacteristic
208C4#218411yescharacteristic
209C4#222011yescharacteristic
210C4#223211yescharacteristic
211C2—21no—
212C2—21no—
213C2—81no—
214C2#1338011yescharacteristic
215C2#1370111yescharacteristic
216C2#1370211yescharacteristic
subgroupG/H
center, socleC22#2215
commutator, Frattini, cosocleC22×C16C23
prime radicalC2×#41D8
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. C2×#1425C2semidirect
3. C2×#1420C2semidirect
4. C2×#1416C2semidirect
7. #7478C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22222
C2×#14252
C2×#14202
C2×#14162
#135542
#128422
#74782