Finite Group Databasebeta

← #153262

Subgroups — #153262

Order 512.

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

normal subgroups only

This group has 258 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
1C2×#2222C211yesmaximal, characteristic
2#13638C211yesmaximal, characteristic, complemented
3#13618C211yesmaximal, characteristic, complemented
4#13618C211yesmaximal, characteristic, complemented
5#13558C211yesmaximal, characteristic
6#12839C211yesmaximal, characteristic
7#7480C211yesmaximal, characteristic, complemented
8C22×Q32C2211yescharacteristic
9C2×#504C2211yescharacteristic
10C2×#473—21no—
11C2×M64C2211yescharacteristic
12C2×M64C2211yescharacteristic
13#2222—22no—
14#2222—22no—
15C2×#368C2211yescharacteristic
16#2158C2211yescharacteristic
17#2158C2211yescharacteristic
18C2×#362—21no—
19#1812—21no—
20#1426—22no—
21#1416—22no—
22#1416—22no—
23#1401—22no—
24#1384—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—
45#427—21no—
46#427—21no—
47C2×#101—41no—
48#410—41no—
49M64—21no—
50M64—21no—
51M64—21no—
52M64—21no—
53C2×C32—21no—
54C2×C32D412yes—
55#368—21no—
56#368D412yes—
57#367—21no—
58#367—21no—
59#366—21no—
60#366—21no—
61#362—24no—
62H(3,ℤ/4)—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×M16D811yescharacteristic
82C22×C8—21no—
83C22×C8C2×D411yescharacteristic
84#117—41no—
85#117—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#107—41no—
106#107—41no—
107#106—41no—
108#106—41no—
109#101—44no—
110#99—82no—
111C32—22no—
112C32—41no—
113#41—21no—
114#41—21no—
115#41—21no—
116#41—21no—
117C2×Q8—42no—
118C2×Q8—81no—
119C2×Q8—81no—
120C2×Q8—81no—
121C2×Q8—81no—
122C2×Q8—81no—
123C2×Q8SD3211yescharacteristic
124C2×D4—21no—
125C2×D4SD3211yescharacteristic
126C22×C4—21no—
127C22×C4—81no—
128C22×C4C2×D811yescharacteristic
129Q16—42no—
130Q16—42no—
131Q16—84no—
132Q16—81no—
133Q16—81no—
134SD16—84no—
135M16—21no—
136M16—21no—
137M16—41no—
138M16—41no—
139M16—42no—
140M16—81no—
141C2×C8—21no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—81no—
145C2×C8#12011yescharacteristic
146C2×C8#12111yescharacteristic
147C2×C8#12311yescharacteristic
148C2×C8#13511yescharacteristic
149C2×C8AGL(1,ℤ/8)11yescharacteristic
150C2×C8#13711yescharacteristic
151C4⋊C4—81no—
152C4⋊C4—81no—
153C22⋊C4—81no—
154C22⋊C4—81no—
155C42—81no—
156C42—81no—
157C16—41no—
158C16#12111yescharacteristic
159C16#12111yescharacteristic
160C16#12311yescharacteristic
161C16#12411yescharacteristic
162C23—21no—
163C23#50911yescharacteristic
164Q8—21no—
165Q8—21no—
166Q8—82no—
167Q8—82no—
168Q8—161no—
169Q8—161no—
170D4—21no—
171D4—21no—
172D4—41no—
173D4—41no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4—161no—
179C2×C4—161no—
180C2×C4—161no—
181C2×C4—161no—
182C2×C4#44711yescharacteristic
183C2×C4#46611yescharacteristic
184C2×C4#48011yescharacteristic
185C2×C4C2×SD3211yescharacteristic
186C2×C4#50811yescharacteristic
187C2×C4#50911yescharacteristic
188C8—41no—
189C8—82no—
190C8#39411yescharacteristic
191C8#45211yescharacteristic
192C8#46811yescharacteristic
193C8#48211yescharacteristic
194C22—41no—
195C22—41no—
196C22—81no—
197C22#218611yescharacteristic
198C22#221811yescharacteristic
199C22#223311yescharacteristic
200C4—41no—
201C4—162no—
202C4—321no—
203C4—321no—
204C4#203311yescharacteristic
205C4#218711yescharacteristic
206C4#221711yescharacteristic
207C4#223411yescharacteristic
208C2—21no—
209C2—21no—
210C2—21no—
211C2—81no—
212C2#1338111yescharacteristic
subgroupG/H
center, socleC2#13381
commutator, Frattini, cosocleC22×C16C23
prime radicalC2×#41D8
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. #13618C2semidirect
4. #13618C2semidirect
7. #7480C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22222
#136382
#136182
#135582
#128392
#74802