Finite Group Databasebeta

← #315868

Subgroups — #315868

Order 512.

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

normal subgroups only

This group has 732 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#33174C211yesmaximal, characteristic
2#27949C212yesmaximal, complemented
3#27891C212yesmaximal
4#20824C211yesmaximal, characteristic, complemented
5#10686C211yesmaximal, characteristic, complemented
6#10683C211yesmaximal, characteristic, complemented
7#10674C211yesmaximal, characteristic, complemented
8#10673C211yesmaximal, characteristic, complemented
9#9829C212yesmaximal, complemented
10#8820C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C2×#492C2211yescharacteristic
13C2×#468—22no—
14C2×C4×M16C2211yescharacteristic
15#2857C2211yescharacteristic
16#2856C2211yescharacteristic
17#2285C2212yes—
18#2085—24no—
19#2033—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1973—22no—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2211yescharacteristic
26#1956C2211yescharacteristic
27#1925—24no—
28#1854—24no—
29#1845C2212yes—
30#1845C2211yescharacteristic
31#1845C2211yescharacteristic
32#1845C2212yes—
33#1844C2212yes—
34#1844C2211yescharacteristic
35#1834C2212yes—
36#1834C2212yes—
37#1809—22no—
38#1776—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—22no—
44C22×M16D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×#121—21no—
49C2×#121—22no—
50C2×#117—22no—
51C22×C42C2311yescharacteristic
52#492—24no—
53#468—48no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#107—24no—
57C2×#107—24no—
58C2×#106—22no—
59C2×#106—24no—
60C2×#103—22no—
61C2×#102—24no—
62C2×#102—22no—
63C2×#102—24no—
64C2×C22⋊C8D414yes—
65C4×M16—24no—
66C2×#97D412yes—
67C2×C4×C8C2312yes—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2312yes—
70C2×C4×C8C2311yescharacteristic
71#391—21no—
72#390—22no—
73#390—21no—
74#389—22no—
75#384C2311yescharacteristic
76#384C2311yescharacteristic
77#384C2312yes—
78#384C2312yes—
79#382—22no—
80#380—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—24no—
91C22×Q8—21no—
92C22×D4—21no—
93C22×D4—22no—
94C22×D4—41no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C2×SD16—44no—
98C2×SD16—44no—
99C2×SD16—44no—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—42no—
103C22×C8C2×D414yes—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106C22×C8C2×D414yes—
107C4⋊Q8—22no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110#127—22no—
111#127—21no—
112#127—21no—
113C22⋊Q8—44no—
114#121—44no—
115#121—84no—
116#117—84no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—41no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—42no—
133C2×C42—22no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—42no—
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2411yescharacteristic
140#107—44no—
141#107—44no—
142#107—84no—
143#107—84no—
144#106—44no—
145#106—84no—
146#103—44no—
147#102—44no—
148#102—48no—
149#102—84no—
150C22⋊C8—28no—
151#97—24no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8#4114yes—
158#95—42no—
159#95—42no—
160#95—42no—
161C24—41no—
162C24#12711yescharacteristic
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—44no—
173C2×D4—42no—
174C2×D4—44no—
175C2×D4—82no—
176C2×D4—84no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4#12712yes—
191C22×C4#12711yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D412yes—
194SD16—88no—
195M16—48no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8#12314yes—
206C2×C8AGL(1,ℤ/8)14yes—
207C2×C8#13718yes—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—82no—
221C22⋊C4—82no—
222C22⋊C4—84no—
223C22⋊C4—84no—
224C22⋊C4—82no—
225C42—21no—
226C42—21no—
227C42—22no—
228C42—22no—
229C42—84no—
230C42C2×#4112yes—
231C422−1+411yescharacteristic
232C422−1+411yescharacteristic
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—81no—
239C23—81no—
240C23—82no—
241C23—82no—
242C23—81no—
243C23C2×#12711yescharacteristic
244Q8—82no—
245Q8—82no—
246D4—84no—
247D4—82no—
248D4—84no—
249D4—82no—
250D4—164no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4C2×#12311yescharacteristic
275C2×C4#53611yescharacteristic
276C2×C4#54712yes—
277C2×C4#54712yes—
278C2×C4C2×AGL(1,ℤ/8)12yes—
279C2×C4C2×#13714yes—
280C2×C4#57512yes—
281C2×C4#57614yes—
282C8—28no—
283C8—24no—
284C8—24no—
285C8—44no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—161no—
292C22—161no—
293C22—161no—
294C22—162no—
295C22—162no—
296C22#250111yescharacteristic
297C22#315011yescharacteristic
298C22#315111yescharacteristic
299C22#315212yes—
300C22#315512yes—
301C4—41no—
302C4—42no—
303C4—161no—
304C4—161no—
305C4—162no—
306C4—161no—
307C4—162no—
308C4#309412yes—
309C4#323512yes—
310C4#334614yes—
311C4#335714yes—
312C2—81no—
313C2—321no—
314C2#1968812yes—
315C2#1972711yescharacteristic
316C2#1974611yescharacteristic
317C2#2791711yescharacteristic
318C2#2794512yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
2. #27949C2semidirect
4. #20824C2semidirect
5. #10686C2semidirect
6. #10683C2semidirect
7. #10674C2semidirect
8. #10673C2semidirect
9. #9829C2semidirect
10. #8820C2semidirect
11. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#331742
#279492
#278912
#208242
#106862
#106832
#106742
#106732
#98292
#88202
#81472