Finite Group Databasebeta

← #314643

Subgroups — #314643

Order 512.

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

normal subgroups only

This group has 455 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#21927C211yesmaximal, characteristic
2#21846C211yesmaximal, characteristic
3#10670C211yesmaximal, characteristic, complemented
4#10668C211yesmaximal, characteristic, complemented
5#10630C211yesmaximal, characteristic, complemented
6#10630C211yesmaximal, characteristic, complemented
7#9584C212yesmaximal, complemented
8#9583C212yesmaximal, complemented
9#8863C212yesmaximal
10#8732C211yesmaximal, characteristic
11#8410C212yesmaximal
12#3458C2211yescharacteristic
13#2921C2211yescharacteristic
14#2298C2211yescharacteristic
15#1995C2211yescharacteristic
16#1995C2211yescharacteristic
17#1994C2211yescharacteristic
18#1994C2211yescharacteristic
19#1986—22no—
20#1924—22no—
21#1917—22no—
22#1843C2211yescharacteristic
23#1832C2211yescharacteristic
24#1826—22no—
25#1798C2212yes—
26#1767C2212yes—
27#1571C2212yes—
28#1570C2212yes—
29#1542C2212yes—
30#1542C2212yes—
31#1481C2212yes—
32#1481C2212yes—
33#1458C2212yes—
34#1301C2212yes—
35#1301C2212yes—
36#1276C2212yes—
37#1274C2211yescharacteristic
38#1274C2211yescharacteristic
39#544—22no—
40#544—22no—
41#544—22no—
42#543—22no—
43#533—21no—
44#533—42no—
45#533C2311yescharacteristic
46#533C2311yescharacteristic
47C2×#117—21no—
48C2×#117C2311yescharacteristic
49#433—42no—
50#431C2312yes—
51#431C2312yes—
52#423C2311yescharacteristic
53#423C2311yescharacteristic
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57#407—21no—
58C2×C22⋊C8—22no—
59C4×M16—21no—
60C2×#97—22no—
61#388—22no—
62#387—22no—
63C4×C4⋊C4D412yes—
64C4×C22⋊C4D412yes—
65#376C2311yescharacteristic
66C43C2311yescharacteristic
67#339—22no—
68#339—22no—
69#339—22no—
70#339—22no—
71#336—22no—
72#336—22no—
73#336—22no—
74#335—42no—
75#334—42no—
76#333—22no—
77#333—22no—
78#332—22no—
79#332—22no—
80Q8⋊C8—42no—
81Q8⋊C8—42no—
82#324—22no—
83#323—22no—
84C23×C4C2×D411yescharacteristic
85C2×M16—21no—
86C2×M16—21no—
87C22×C8—22no—
88C22×C8—22no—
89C4⋊Q8—22no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8#4112yes—
93C4⋊Q8#4112yes—
94#125—22no—
95#125—21no—
96#125—21no—
97#125—42no—
98#125—42no—
99C22⋊Q8—22no—
100C22⋊Q8—22no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104C22⋊Q8—42no—
105C22⋊Q8—42no—
106C4×Q8—22no—
107C4×Q8—22no—
108C4×Q8—42no—
109#117—21no—
110#117—21no—
111#117—22no—
112#117—22no—
113#117—21no—
114#117—21no—
115#117—41no—
116#117—41no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4#4112yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4#4112yes—
123C2×C42—22no—
124C2×C42—41no—
125C2×C42C2×D412yes—
126C2×C42C2×D412yes—
127C2×C42C2×D411yescharacteristic
128C2×C42C2411yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—82no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—42no—
141C22⋊C8—82no—
142#97—41no—
143#97—82no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—41no—
147#95—22no—
148#95—22no—
149#95—22no—
150C24#12311yescharacteristic
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—42no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—42no—
161C22×C4#12311yescharacteristic
162C22×C4#12312yes—
163C22×C4C22×D411yescharacteristic
164C22×C4C2×#4112yes—
165M16—42no—
166M16—42no—
167C2×C8—42no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—82no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—82no—
196C42—21no—
197C42—21no—
198C42—22no—
199C42—22no—
200C42—22no—
201C42—22no—
202C42—42no—
203C42—41no—
204C42—41no—
205C42—82no—
206C42C2×#4111yescharacteristic
207C42C2×#4111yescharacteristic
208C42H(5,𝔽2)12yes—
209C23—21no—
210C23—22no—
211C23—42no—
212C23C2×#12311yescharacteristic
213Q8—82no—
214Q8—82no—
215Q8—162no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—162no—
233C2×C4#52511yescharacteristic
234C2×C4#53511yescharacteristic
235C2×C4#55112yes—
236C2×C4#55112yes—
237C2×C4#57712yes—
238C2×C4#57812yes—
239C8—82no—
240C8—82no—
241C8—82no—
242C8—82no—
243C8—81no—
244C8—81no—
245C22—22no—
246C22—41no—
247C22—42no—
248C22—82no—
249C22#248211yescharacteristic
250C22#309712yes—
251C4—22no—
252C4—22no—
253C4—41no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—82no—
258C4—82no—
259C4#312112yes—
260C4#322111yescharacteristic
261C4#322311yescharacteristic
262C2—22no—
263C2—81no—
264C2#2759611yescharacteristic
265C2#2759912yes—
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. #10670C2semidirect
4. #10668C2semidirect
5. #10630C2semidirect
6. #10630C2semidirect
7. #9584C2semidirect
8. #9583C2semidirect

Maximal subgroups

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

subgroupindex
#219272
#218462
#106702
#106682
#106302
#95842
#95832
#88632
#87322
#84102