Finite Group Databasebeta

← #508558

Subgroups — #508558

Order 512.

423 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 132 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#33211C211yesmaximal, characteristic
2#20827C211yesmaximal, characteristic
3#13541C212yesmaximal, complemented
4#13510C212yesmaximal, complemented
5#13501C214yesmaximal
6#13280C212yesmaximal, complemented
7#13184C212yesmaximal, complemented
8#12785C211yesmaximal, characteristic
9C2×#499C2211yescharacteristic
10#2920—21no—
11C2×C4×M16C2211yescharacteristic
12#2851—21no—
13C2×#365C2214yes—
14#2150C2212yes—
15C2×#349C2212yes—
16#1991—21no—
17#1944C2212yes—
18#1943—21no—
19#1849C2211yescharacteristic
20#1849C2211yescharacteristic
21#1846C2211yescharacteristic
22#1838—21no—
23#1811—21no—
24#1811C2212yes—
25#1809—21no—
26#1809C2212yes—
27C4×#108—21no—
28C4×#108C2212yes—
29#1387C2214yes—
30#1387C2214yes—
31#1383C2214yes—
32#1377C2212yes—
33C22×M16D411yescharacteristic
34C22×M16C2311yescharacteristic
35C2×#125—21no—
36C22×C42D411yescharacteristic
37C2×M32C2314yes—
38#499—24no—
39#432—42no—
40#431—42no—
41C2×#108—21no—
42C2×#108—21no—
43C2×#108C2312yes—
44C2×#108C2312yes—
45C2×#107—21no—
46C2×#107C2312yes—
47C2×#106—21no—
48C2×#106C2312yes—
49#423—21no—
50#423—21no—
51#407—22no—
52C4×M16—24no—
53C4×M16—22no—
54C2×#97D411yescharacteristic
55C2×#97C2311yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#389—22no—
59#387—42no—
60#385—42no—
61#382—21no—
62#380—22no—
63#365—44no—
64C4⋊C16—24no—
65#349—42no—
66#341—22no—
67#341—22no—
68#340—22no—
69#340—22no—
70C23×C4SD1612yes—
71C23×C4C2×D411yescharacteristic
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—21no—
76C2×M16—22no—
77C2×M16—21no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2411yescharacteristic
85#125—42no—
86#125—41no—
87#125—41no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42SD1612yes—
97C2×C42C2×D411yescharacteristic
98C2×C42C2×D411yescharacteristic
99M32—24no—
100M32—24no—
101C2×C16—24no—
102#108—22no—
103#108—22no—
104#108—24no—
105#108—42no—
106#108—42no—
107#107—24no—
108#107—42no—
109#106—22no—
110#106—22no—
111#106—42no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C22⋊C8—44no—
115#97—22no—
116#97—22no—
117#97—42no—
118C4×C8—22no—
119C4×C8—42no—
120C4×C8#4114yes—
121#95—42no—
122#95—42no—
123#95—42no—
124C24C2×SD1611yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—82no—
136C22×C4C2×SD1611yescharacteristic
137C22×C4C2×SD1612yes—
138C22×C4C2×SD1612yes—
139C22×C4C22×D411yescharacteristic
140M16—44no—
141M16—42no—
142M16—42no—
143M16—42no—
144M16—42no—
145M16—42no—
146M16—42no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8#12312yes—
157C2×C8#13712yes—
158C2×C8C2×#4112yes—
159C2×C8H(5,𝔽2)11yescharacteristic
160C2×C82−1+411yescharacteristic
161C2×C82−1+412yes—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—84no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C22⋊C4—84no—
170C42—21no—
171C42—21no—
172C42—22no—
173C42—22no—
174C42#12312yes—
175C42AGL(1,ℤ/8)11yescharacteristic
176C42AGL(1,ℤ/8)11yescharacteristic
177C16—44no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—42no—
182C23C22×SD1611yescharacteristic
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#48112yes—
202C2×C4#48112yes—
203C2×C4C2×#12311yescharacteristic
204C2×C4#53611yescharacteristic
205C2×C4#53711yescharacteristic
206C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C4C2×#13711yescharacteristic
208C2×C4#57511yescharacteristic
209C8—22no—
210C8—22no—
211C8—41no—
212C8—82no—
213C8—82no—
214C8#54112yes—
215C8#54212yes—
216C22—22no—
217C22—42no—
218C22—42no—
219C22—41no—
220C22—81no—
221C22C2×#48111yescharacteristic
222C22#322911yescharacteristic
223C22#324611yescharacteristic
224C4—22no—
225C4—22no—
226C4—42no—
227C4—41no—
228C4—162no—
229C4—162no—
230C4#262511yescharacteristic
231C4#309511yescharacteristic
232C4#323511yescharacteristic
233C4#325411yescharacteristic
234C2—21no—
235C2—21no—
236C2—81no—
237C2#2993711yescharacteristic
238C2#3365612yes—
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13541C2semidirect
4. #13510C2semidirect
6. #13280C2semidirect
7. #13184C2semidirect

Maximal subgroups

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

subgroupindex
#332112
#208272
#135412
#135102
#135012
#132802
#131842
#127852