Finite Group Databasebeta

← #244043

Subgroups — #244043

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C212yesmaximal
2#21834C212yesmaximal
3#21202C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic, complemented
6#10685C211yesmaximal, characteristic, complemented
7#10682C212yesmaximal, complemented
8#10680C212yesmaximal, complemented
9#10676C211yesmaximal, characteristic, complemented
10#10674C211yesmaximal, characteristic, complemented
11#7850C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#431C2212yes—
14C2×#419—22no—
15C2×C4×M16C2211yescharacteristic
16#2285C2212yes—
17#2285C2211yescharacteristic
18#1990C2212yes—
19#1990C2212yes—
20#1990C2212yes—
21#1990C2212yes—
22#1987—24no—
23#1960—22no—
24#1956C2211yescharacteristic
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1956C2211yescharacteristic
28#1949—24no—
29#1854—24no—
30#1845C2211yescharacteristic
31#1845C2212yes—
32#1844C2211yescharacteristic
33#1844C2212yes—
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1824—24no—
38#1809—22no—
39#1776—22no—
40#1273C2211yescharacteristic
41#1273C2211yescharacteristic
42#1273C2212yes—
43C23×Q8—21no—
44C22×M16D412yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8Q814yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#117—21no—
53C2×#117—22no—
54C22×C4⋊C4—22no—
55C22×C42C2311yescharacteristic
56#431—24no—
57#431—24no—
58C2×#107—22no—
59C2×#107—24no—
60C2×#106—22no—
61C2×#106—24no—
62C2×C4⋊C8D414yes—
63#419—48no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×C22⋊C8D414yes—
69C4×M16—24no—
70C2×#97D412yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2312yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75#384C2311yescharacteristic
76#384C2312yes—
77#384C2312yes—
78#384C2311yescharacteristic
79#382—22no—
80#382—21no—
81#380—22no—
82#380—22no—
83#380—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94C22×Q8—24no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—44no—
98C22×Q8—44no—
99C22×Q8—42no—
100C22×Q8—41no—
101C23×C4—41no—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—42no—
107C22×C8C2×D412yes—
108C22×C8C2×D414yes—
109C22×C8C2×D414yes—
110C22×C8C2×D412yes—
111C4⋊Q8—28no—
112C4⋊Q8—24no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—42no—
118C4⋊Q8—44no—
119C4⋊Q8C2×Q812yes—
120C4⋊Q8C2×Q812yes—
121#117—44no—
122#117—84no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—42no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C42—41no—
141C2×C42—42no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D412yes—
144C2×C42C2×Q812yes—
145C2×C42#4114yes—
146C2×C42#4112yes—
147C2×C42C2411yescharacteristic
148#107—44no—
149#107—84no—
150#106—44no—
151#106—84no—
152C4⋊C8—28no—
153#103—44no—
154#103—44no—
155#103—44no—
156#103—44no—
157C22⋊C8—28no—
158#97—24no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165#95—42no—
166#95—42no—
167#95—42no—
168C24#12711yescharacteristic
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—416no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—44no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8—88no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—44no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4C22⋊Q814yes—
196C22×C4C22⋊Q812yes—
197C22×C4#12314yes—
198C22×C4#12312yes—
199C22×C4#12712yes—
200C22×C4#12711yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D412yes—
203M16—48no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8AGL(1,ℤ/8)14yes—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—48no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C22⋊C4—82no—
232C22⋊C4—84no—
233C42—82no—
234C42—84no—
235C42C22⋊Q812yes—
236C42C22⋊Q814yes—
237C42#12312yes—
238C42#12314yes—
239C42C22×Q811yescharacteristic
240C42C2×#4111yescharacteristic
241C42C2×#4112yes—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23#39912yes—
247C23C2×#12711yescharacteristic
248Q8—44no—
249Q8—44no—
250Q8—88no—
251Q8—82no—
252Q8—82no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—28no—
259C2×C4—24no—
260C2×C4—28no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—88no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#39914yes—
279C2×C4#39912yes—
280C2×C4C2×C22⋊Q811yescharacteristic
281C2×C4C2×C22⋊Q812yes—
282C2×C4C2×#12311yescharacteristic
283C2×C4C2×#12312yes—
284C2×C4C2×AGL(1,ℤ/8)12yes—
285C2×C4#57512yes—
286C2×C4#57714yes—
287C2×C4#57814yes—
288C8—24no—
289C8—42no—
290C8—44no—
291C8—44no—
292C22—22no—
293C22—24no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22C2×#39911yescharacteristic
298C22#315111yescharacteristic
299C22#315311yescharacteristic
300C22#315412yes—
301C22#315612yes—
302C4—22no—
303C4—24no—
304C4—82no—
305C4—82no—
306C4—161no—
307C4—162no—
308C4—162no—
309C4—161no—
310C4#307712yes—
311C4#308214yes—
312C4#309412yes—
313C4#309714yes—
314C2—42no—
315C2#1972111yescharacteristic
316C2#1972312yes—
317C2#1973111yescharacteristic
318C2#2373111yescharacteristic
319C2#2373612yes—
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. #10686C2semidirect
6. #10685C2semidirect
7. #10682C2semidirect
8. #10680C2semidirect
9. #10676C2semidirect
10. #10674C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#218342
#212022
#208242
#106862
#106852
#106822
#106802
#106762
#106742
#78502