Finite Group Databasebeta

← #330940

Subgroups — #330940

Order 512.

601 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 130 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#32915C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#10644C214yesmaximal, complemented
4#10447C214yesmaximal, complemented
5#9195C212yesmaximal
6#8862C212yesmaximal
7#8323C211yesmaximal, characteristic
8C2×#519C2211yescharacteristic
9C2×#462C2212yes—
10C2×#451—21no—
11#2905C2212yes—
12C22×#101—21no—
13#2621C2211yescharacteristic
14#1878—22no—
15#1873—21no—
16#1805C2212yes—
17#1801—22no—
18#1801C2214yes—
19#1798C2211yescharacteristic
20#1742—21no—
21#1570C2214yes—
22#1546C2214yes—
23#1479C2214yes—
24#1458C2212yes—
25#1301C2214yes—
26#1286C2214yes—
27C23×Q8D411yescharacteristic
28C22×Q16—22no—
29C22×M16—22no—
30C2×C4⋊Q8C2311yescharacteristic
31C2×C22⋊Q8—21no—
32#519—44no—
33C2×C4×Q8—22no—
34C2×C4×Q8D412yes—
35C2×C4×Q8C2312yes—
36C2×#117—22no—
37C2×#117D411yescharacteristic
38C2×#117C2311yescharacteristic
39#462—44no—
40#451—44no—
41#431C2314yes—
42#424—44no—
43#423—22no—
44#423C2314yes—
45C2×C4⋊C8C2312yes—
46C2×#103—22no—
47C2×#103—22no—
48C2×#101—24no—
49C2×#101—44no—
50C2×C22⋊C8—21no—
51#398—22no—
52#397—22no—
53#391—21no—
54#382C2311yescharacteristic
55#343—24no—
56#340—22no—
57#336—21no—
58#334—44no—
59#332—44no—
60Q8⋊C8—44no—
61#324—22no—
62#323—22no—
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—21no—
66C22×Q8—42no—
67C22×Q8—42no—
68C22×Q8SD1614yes—
69C22×Q8C2×D412yes—
70C23×C4—21no—
71C23×C4C2×D411yescharacteristic
72C2×Q16—44no—
73C2×Q16—44no—
74C2×Q16—44no—
75C2×M16—24no—
76C2×M16—42no—
77C2×M16—44no—
78C2×M16—44no—
79C22×C8—22no—
80C22×C8—42no—
81C4⋊Q8—24no—
82C22⋊Q8—42no—
83C22⋊Q8—42no—
84C4×Q8—24no—
85C4×Q8—24no—
86C4×Q8—42no—
87C4×Q8—42no—
88C4×Q8—48no—
89#117—24no—
90#117—48no—
91#117—42no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4C2×D412yes—
98C2×C4⋊C4#4112yes—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4#4112yes—
102C2×C42—22no—
103C2×C42—21no—
104C2×C42—22no—
105C2×C42C2×D411yescharacteristic
106C2×C42C2411yescharacteristic
107C4⋊C8—24no—
108C4⋊C8—24no—
109C4⋊C8—24no—
110C4⋊C8—24no—
111C4⋊C8—44no—
112#103—44no—
113#103—44no—
114#101—48no—
115C22⋊C8—24no—
116C22⋊C8—42no—
117C4×C8—24no—
118#95—22no—
119#95—42no—
120C24#12311yescharacteristic
121C2×Q8—22no—
122C2×Q8—24no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—48no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—48no—
131C2×Q8—42no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—42no—
135C2×Q8C2×SD1614yes—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4#12311yescharacteristic
146C22×C4#12312yes—
147C22×C4C2×SD1612yes—
148C22×C4AGL(1,ℤ/8)12yes—
149C22×C4C22×D411yescharacteristic
150C22×C4C2×#4112yes—
151Q16—88no—
152M16—44no—
153M16—44no—
154M16—84no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—84no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—82no—
170C22⋊C4—24no—
171C22⋊C4—42no—
172C22⋊C4—82no—
173C42—22no—
174C42—42no—
175C42—44no—
176C42—44no—
177C42H(5,𝔽2)14yes—
178C23—21no—
179C23—21no—
180C23—42no—
181C23#48112yes—
182C23C2×#12311yescharacteristic
183Q8—24no—
184Q8—24no—
185Q8—48no—
186Q8—42no—
187Q8—42no—
188Q8—82no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—48no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4#48112yes—
209C2×C4#53412yes—
210C2×C4#53814yes—
211C2×C4C22×SD1611yescharacteristic
212C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C4#57512yes—
214C8—84no—
215C8—82no—
216C8—84no—
217C22—22no—
218C22—22no—
219C22—42no—
220C22—42no—
221C22—81no—
222C22#262211yescharacteristic
223C22C2×#48111yescharacteristic
224C22#309311yescharacteristic
225C4—22no—
226C4—24no—
227C4—44no—
228C4—42no—
229C4—82no—
230C4—161no—
231C4—161no—
232C4#318712yes—
233C4#319412yes—
234C2—21no—
235C2—21no—
236C2—42no—
237C2#2989211yescharacteristic
238C2#2990512yes—
subgroupG/H
center, socleC22#2622
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10644C2semidirect
4. #10447C2semidirect

Maximal subgroups

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

subgroupindex
#329152
#206392
#106442
#104472
#91952
#88622
#83232