Finite Group Databasebeta

← #314620

Subgroups — #314620

Order 512.

465 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 118 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#21923C211yesmaximal, characteristic
2#21846C211yesmaximal, characteristic
3#10669C212yesmaximal, complemented
4#10628C212yesmaximal, complemented
5#9580C212yesmaximal, complemented
6#9579C212yesmaximal, complemented
7#8863C212yesmaximal
8#8733C211yesmaximal, characteristic
9#8410C212yesmaximal
10#3458C2211yescharacteristic
11#2921C2211yescharacteristic
12#2298C2211yescharacteristic
13#1995C2212yes—
14#1994C2212yes—
15#1986—22no—
16#1923—22no—
17#1916—22no—
18#1843C2211yescharacteristic
19#1832C2211yescharacteristic
20#1826—22no—
21#1798C2212yes—
22C4×C4⋊C8C2212yes—
23#1571C2212yes—
24#1570C2212yes—
25#1542C2212yes—
26#1542C2212yes—
27#1481C2212yes—
28#1481C2212yes—
29#1458C2212yes—
30#1301C2212yes—
31#1301C2212yes—
32#1276C2211yescharacteristic
33#1276C2211yescharacteristic
34#1274C2212yes—
35#544—22no—
36#544—22no—
37#544—22no—
38#543—22no—
39#533—21no—
40#533—41no—
41#533—41no—
42#533C2312yes—
43C2×#117—21no—
44C2×#117C2311yescharacteristic
45#433—41no—
46#433—41no—
47#431C2312yes—
48#431C2312yes—
49#423C2312yes—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53#407—21no—
54C2×C22⋊C8—22no—
55C4×M16—21no—
56C2×C4×C8—22no—
57#388—22no—
58#387—22no—
59C4×C4⋊C4D412yes—
60C4×C22⋊C4D412yes—
61#376C2311yescharacteristic
62C43C2311yescharacteristic
63#339—24no—
64#339—24no—
65#336—22no—
66#336—22no—
67#336—22no—
68#335—42no—
69#334—42no—
70#333—22no—
71#333—22no—
72#332—22no—
73#332—22no—
74Q8⋊C8—42no—
75Q8⋊C8—42no—
76#324—22no—
77#323—22no—
78C23×C4C2×D411yescharacteristic
79C2×M16—22no—
80C22×C8—22no—
81C22×C8—22no—
82C4⋊Q8—22no—
83C4⋊Q8—42no—
84C4⋊Q8—42no—
85C4⋊Q8#4112yes—
86C4⋊Q8#4112yes—
87#125—21no—
88#125—21no—
89#125—22no—
90#125—42no—
91#125—42no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—42no—
95C22⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—42no—
102#117—22no—
103#117—21no—
104#117—21no—
105#117—21no—
106#117—21no—
107#117—22no—
108#117—41no—
109#117—41no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4#4112yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4#4112yes—
116C2×C42—22no—
117C2×C42—41no—
118C2×C42C2×D412yes—
119C2×C42C2×D412yes—
120C2×C42C2×D411yescharacteristic
121C2×C42C2411yescharacteristic
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—82no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—42no—
134C22⋊C8—82no—
135#97—41no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—41no—
139C4×C8—82no—
140#95—22no—
141#95—22no—
142#95—22no—
143C24#12311yescharacteristic
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—42no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—42no—
154C22×C4#12311yescharacteristic
155C22×C4#12312yes—
156C22×C4C22×D411yescharacteristic
157C22×C4C2×#4112yes—
158M16—42no—
159M16—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—44no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—42no—
188C42—21no—
189C42—21no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—44no—
195C42—41no—
196C42—41no—
197C42—42no—
198C42C2×#4112yes—
199C42H(5,𝔽2)11yescharacteristic
200C42H(5,𝔽2)11yescharacteristic
201C23—21no—
202C23—22no—
203C23—42no—
204C23C2×#12311yescharacteristic
205Q8—82no—
206Q8—82no—
207Q8—162no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—162no—
225C2×C4#52511yescharacteristic
226C2×C4#53511yescharacteristic
227C2×C4#55112yes—
228C2×C4#55112yes—
229C2×C4#57712yes—
230C2×C4#57812yes—
231C8—82no—
232C8—82no—
233C8—82no—
234C8—82no—
235C8—82no—
236C22—22no—
237C22—41no—
238C22—42no—
239C22—82no—
240C22#248211yescharacteristic
241C22#309712yes—
242C4—22no—
243C4—22no—
244C4—44no—
245C4—41no—
246C4—42no—
247C4—82no—
248C4—82no—
249C4—82no—
250C4#312611yescharacteristic
251C4#312711yescharacteristic
252C4#322512yes—
253C2—22no—
254C2—81no—
255C2#2759211yescharacteristic
256C2#2759812yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2482
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10669C2semidirect
4. #10628C2semidirect
5. #9580C2semidirect
6. #9579C2semidirect

Maximal subgroups

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

subgroupindex
#219232
#218462
#106692
#106282
#95802
#95792
#88632
#87332
#84102