Finite Group Databasebeta

← #315861

Subgroups — #315861

Order 512.

681 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 174 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#33176C211yesmaximal, characteristic
2#27949C212yesmaximal
3#27875C212yesmaximal
4#20824C211yesmaximal, characteristic
5#10685C211yesmaximal, characteristic, complemented
6#10685C211yesmaximal, characteristic, complemented
7#10676C211yesmaximal, characteristic, complemented
8#10674C211yesmaximal, characteristic, complemented
9#9835C212yesmaximal, complemented
10#8819C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#470—22no—
14C2×C4×M16C2211yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2285C2212yes—
18#2085—24no—
19#2077—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1971—22no—
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1920—24no—
28#1854—24no—
29#1845C2212yes—
30#1845C2211yescharacteristic
31#1844C2212yes—
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1844C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1809—22no—
38#1776—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×Q16—22no—
44C22×M16D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×#117—22no—
50C22×C42C2311yescharacteristic
51#494—24no—
52#470—48no—
53C2×#107—22no—
54C2×#107—24no—
55C2×#106—22no—
56C2×#106—24no—
57C2×#106—22no—
58C2×#106—24no—
59C2×#103—22no—
60C2×#103—22no—
61C2×C22⋊C8D414yes—
62C4×M16—24no—
63C2×#97D412yes—
64C2×C4×C8C2312yes—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2312yes—
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#391—21no—
70#389—22no—
71#389—22no—
72#384C2311yescharacteristic
73#384C2311yescharacteristic
74#384C2312yes—
75#384C2312yes—
76#382—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—24no—
91C22×Q8—21no—
92C22×Q8—21no—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C2×Q16—44no—
96C2×Q16—44no—
97C2×Q16—44no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—42no—
101C22×C8C2×D414yes—
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104C22×C8C2×D414yes—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113#117—84no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C42—22no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—42no—
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2411yescharacteristic
138#107—44no—
139#107—84no—
140#106—44no—
141#106—44no—
142#106—48no—
143#106—84no—
144#103—44no—
145#103—44no—
146C22⋊C8—28no—
147#97—24no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8#4114yes—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158C24#12711yescharacteristic
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4#12712yes—
182C22×C4#12711yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D412yes—
185Q16—88no—
186M16—48no—
187C2×C8—22no—
188C2×C8—24no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8#12314yes—
197C2×C8AGL(1,ℤ/8)14yes—
198C2×C8#13718yes—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C22⋊C4—84no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42—21no—
219C42—21no—
220C42—22no—
221C42—22no—
222C42—84no—
223C42C2×#4112yes—
224C422−1+411yescharacteristic
225C422−1+411yescharacteristic
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23C2×#12711yescharacteristic
232Q8—82no—
233Q8—82no—
234Q8—82no—
235Q8—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4C2×#12311yescharacteristic
263C2×C4#53611yescharacteristic
264C2×C4#54712yes—
265C2×C4#54712yes—
266C2×C4C2×AGL(1,ℤ/8)12yes—
267C2×C4C2×#13714yes—
268C2×C4#57512yes—
269C2×C4#57714yes—
270C8—28no—
271C8—24no—
272C8—24no—
273C8—44no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22#250111yescharacteristic
280C22#315011yescharacteristic
281C22#315111yescharacteristic
282C22#315212yes—
283C22#315412yes—
284C4—41no—
285C4—42no—
286C4—161no—
287C4—162no—
288C4—161no—
289C4—161no—
290C4—161no—
291C4—162no—
292C4#309412yes—
293C4#323512yes—
294C4#334414yes—
295C4#335614yes—
296C2—81no—
297C2#1969312yes—
298C2#1972111yescharacteristic
299C2#1974611yescharacteristic
300C2#2791711yescharacteristic
301C2#2794012yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10685C2semidirect
6. #10685C2semidirect
7. #10676C2semidirect
8. #10674C2semidirect
9. #9835C2semidirect
10. #8819C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#279492
#278752
#208242
#106852
#106762
#106742
#98352
#88192
#81472