Finite Group Databasebeta

← #145990

Subgroups — #145990

Order 512.

251 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 64 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#12036C211yesmaximal, characteristic, complemented
2#12018C211yesmaximal, characteristic, complemented
3#11961C211yesmaximal, characteristic
4#11931C211yesmaximal, characteristic
5#9802C211yesmaximal, characteristic
6#9634C211yesmaximal, characteristic, complemented
7#7245C211yesmaximal, characteristic, complemented
8C2×#493C2211yescharacteristic
9#2162C2211yescharacteristic
10#2150C2211yescharacteristic
11#1965—21no—
12#1574—21no—
13#1567C2211yescharacteristic
14#1567C2211yescharacteristic
15#1564C2211yescharacteristic
16#1545—21no—
17C8×M16C2211yescharacteristic
18#1372—21no—
19#1371—21no—
20#1370—21no—
21#1367—21no—
22#1361—21no—
23#1361—21no—
24#1361—21no—
25#1361—21no—
26#1299—21no—
27C22×D8—21no—
28C22×D8Q812yes—
29C2×#127—21no—
30C2×M32—21no—
31#493—21no—
32#493—21no—
33#493—21no—
34#493—21no—
35#431—21no—
36#431D411yescharacteristic
37#423—21no—
38#423—21no—
39C2×#100—42no—
40C4×M16D411yescharacteristic
41C2×C4×C8C2311yescharacteristic
42C4⋊C16—21no—
43C4⋊C16—21no—
44C4⋊C16D411yescharacteristic
45C4⋊C16D411yescharacteristic
46C22⋊C16—21no—
47C4×C16—21no—
48#335—21no—
49#335—21no—
50#335—21no—
51#335—21no—
52#335—41no—
53#335D411yescharacteristic
54#335D411yescharacteristic
55#334—21no—
56#334—41no—
57#331—41no—
58#331—41no—
59#322—21no—
60C82—21no—
61C22×D4—22no—
62C22×D4—41no—
63C22×D4—41no—
64C2×D8—22no—
65C2×D8—22no—
66C2×D8—41no—
67C2×D8—41no—
68C2×D8—81no—
69C2×D8—81no—
70C2×D8—82no—
71C2×M16—21no—
72C2×M16—41no—
73C2×M16—41no—
74C22×C8C2×Q811yescharacteristic
75C22×C8#4111yescharacteristic
76#127—41no—
77#127—41no—
78#127—41no—
79#127—41no—
80C2×C42C2×D411yescharacteristic
81M32—81no—
82C2×C16—21no—
83C2×C16—21no—
84C2×C16—41no—
85C4⋊C8—21no—
86C4⋊C8—21no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8D811yescharacteristic
92C4⋊C8D811yescharacteristic
93#100—84no—
94C22⋊C8—21no—
95C22⋊C8—41no—
96#97—21no—
97C4×C8—21no—
98C4×C8—21no—
99C4×C8—41no—
100C4×C8C2×D411yescharacteristic
101C4×C8C2×D411yescharacteristic
102C4×C8#4111yescharacteristic
103C4×C8#4111yescharacteristic
104C24—42no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—82no—
108C2×D4—82no—
109C2×D4—82no—
110C2×D4—82no—
111C2×D4—81no—
112C2×D4—81no—
113C2×D4—81no—
114C2×D4—81no—
115C2×D4—161no—
116C2×D4—161no—
117C22×C4C22⋊Q811yescharacteristic
118C22×C4#13511yescharacteristic
119C22×C4AGL(1,ℤ/8)11yescharacteristic
120D8—82no—
121D8—82no—
122D8—82no—
123D8—81no—
124D8—81no—
125M16—22no—
126M16—41no—
127M16—82no—
128M16—161no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—81no—
138C2×C8—81no—
139C2×C8#12111yescharacteristic
140C2×C8C22⋊Q811yescharacteristic
141C2×C8C22⋊Q811yescharacteristic
142C2×C8#12611yescharacteristic
143C42#12111yescharacteristic
144C42#12111yescharacteristic
145C42C2×D811yescharacteristic
146C42#13711yescharacteristic
147C16—41no—
148C16—41no—
149C16—81no—
150C23—84no—
151C23—82no—
152C23—162no—
153C23—162no—
154C23#47811yescharacteristic
155D4—82no—
156D4—82no—
157D4—162no—
158D4—162no—
159D4—162no—
160D4—161no—
161D4—161no—
162D4—161no—
163D4—161no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4#39611yescharacteristic
171C2×C4#45911yescharacteristic
172C2×C4#46411yescharacteristic
173C2×C4#46611yescharacteristic
174C2×C4#46811yescharacteristic
175C2×C4#47411yescharacteristic
176C8—21no—
177C8—21no—
178C8—21no—
179C8—42no—
180C8—41no—
181C8—41no—
182C8—81no—
183C8—81no—
184C8—161no—
185C8—161no—
186C8#47111yescharacteristic
187C8#47211yescharacteristic
188C22—21no—
189C22—21no—
190C22—21no—
191C22—162no—
192C22—162no—
193C22—162no—
194C22—324no—
195C22#206711yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4#206311yescharacteristic
200C4#206911yescharacteristic
201C4#221211yescharacteristic
202C4#221311yescharacteristic
203C4#222111yescharacteristic
204C4#222311yescharacteristic
205C2—41no—
206C2—322no—
207C2#1060311yescharacteristic
208C2#1347011yescharacteristic
209C2#1347811yescharacteristic
subgroupG/H
center, socleC22#2067
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #12036C2semidirect
2. #12018C2semidirect
6. #9634C2semidirect
7. #7245C2semidirect

Maximal subgroups

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

subgroupindex
#120362
#120182
#119612
#119312
#98022
#96342
#72452