Finite Group Databasebeta

← #242836

Subgroups — #242836

Order 512.

681 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21890C212yesmaximal
2#21375C211yesmaximal, characteristic
3#20760C211yesmaximal, characteristic, complemented
4#10626C212yesmaximal, complemented
5#10625C212yesmaximal, complemented
6#10624C211yesmaximal, characteristic, complemented
7#10623C211yesmaximal, characteristic, complemented
8#10622C211yesmaximal, characteristic, complemented
9#10621C211yesmaximal, characteristic, complemented
10#8042C212yesmaximal, complemented
11#7874C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#423C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2285C2212yes—
17#1989—22no—
18#1984C2211yescharacteristic
19#1984C2211yescharacteristic
20#1983—22no—
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1895—22no—
24#1894—22no—
25#1869—22no—
26#1835—22no—
27#1834C2211yescharacteristic
28#1834C2211yescharacteristic
29#1834C2212yes—
30#1803C2212yes—
31#1803C2212yes—
32#1802C2212yes—
33#1802C2212yes—
34#1750—22no—
35#1735—22no—
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1277C2211yescharacteristic
41#1277C2211yescharacteristic
42#1277C2212yes—
43C22×M16—22no—
44C23×C8D412yes—
45#529—24no—
46#529—24no—
47C2×#126—21no—
48C2×#125D411yescharacteristic
49C2×#124D411yescharacteristic
50C2×#123—21no—
51C2×C22⋊Q8D411yescharacteristic
52C2×#121D411yescharacteristic
53C2×C4×Q8C2311yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4C2311yescharacteristic
56C2×#117—21no—
57C2×#117—22no—
58C22×C42C2311yescharacteristic
59#432—24no—
60#423—24no—
61#423—24no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×#103—22no—
68C2×#103—22no—
69C2×#102—22no—
70C2×#102—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×#97—21no—
75C2×C4×C8—24no—
76C2×C4×C8D412yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#384C2312yes—
80#384C2312yes—
81#382—22no—
82#380—22no—
83#380—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86#340—22no—
87#340—22no—
88#340—24no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95C22×Q8C2×D411yescharacteristic
96C22×D4—21no—
97C22×D4C2×D411yescharacteristic
98C23×C4—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—44no—
103C2×M16—82no—
104C2×M16—82no—
105C22×C8—24no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114#126—44no—
115#125—24no—
116#124—24no—
117#123—44no—
118C22⋊Q8—24no—
119#121—24no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—42no—
125C4×D4—42no—
126#117—42no—
127#117—42no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—42no—
148C2×C42—21no—
149C2×C42—42no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42#4114yes—
153C2×C42#4114yes—
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—44no—
163#103—82no—
164#103—82no—
165#102—82no—
166#102—82no—
167C22⋊C8—44no—
168C22⋊C8—44no—
169C22⋊C8—42no—
170#97—42no—
171C4×C8—24no—
172C4×C8—22no—
173C4×C8—22no—
174#95—42no—
175#95—42no—
176C24—21no—
177C24#12111yescharacteristic
178C2×Q8—22no—
179C2×Q8—21no—
180C2×Q8—22no—
181C2×Q8—21no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—44no—
187C2×D4—42no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12312yes—
207C22×C4#12314yes—
208C22×C4#12412yes—
209C22×C4#12614yes—
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C2×#4111yescharacteristic
213M16—88no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—48no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—84no—
253C42—41no—
254C42—41no—
255C42—84no—
256C42#12312yes—
257C42#12314yes—
258C42#12412yes—
259C42#12614yes—
260C42C2×#4112yes—
261C42C2×#4112yes—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23#39712yes—
272C23C2×#12111yescharacteristic
273Q8—42no—
274Q8—42no—
275D4—42no—
276D4—42no—
277D4—84no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—44no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4#39712yes—
306C2×C4#39712yes—
307C2×C4#39712yes—
308C2×C4C2×#12311yescharacteristic
309C2×C4C2×#12312yes—
310C2×C4C2×#12411yescharacteristic
311C2×C4C2×#12612yes—
312C2×C4#57512yes—
313C2×C4#57512yes—
314C2×C4#57611yescharacteristic
315C2×C4#57712yes—
316C2×C4#57811yescharacteristic
317C8—42no—
318C8—42no—
319C8—82no—
320C8—82no—
321C8—81no—
322C8—81no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—82no—
332C22—82no—
333C22C2×#39711yescharacteristic
334C22#304012yes—
335C22#306011yescharacteristic
336C22#306111yescharacteristic
337C22#306712yes—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—162no—
346C4—162no—
347C4#309312yes—
348C4#309612yes—
349C4#309712yes—
350C4#314211yescharacteristic
351C4#314211yescharacteristic
352C2—42no—
353C2—161no—
354C2#1936612yes—
355C2#1936712yes—
356C2#2363111yescharacteristic
357C2#2364111yescharacteristic
358C2#2364611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
3. #20760C2semidirect
4. #10626C2semidirect
5. #10625C2semidirect
6. #10624C2semidirect
7. #10623C2semidirect
8. #10622C2semidirect
9. #10621C2semidirect
10. #8042C2semidirect
11. #7874C2semidirect

Maximal subgroups

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

subgroupindex
#218902
#213752
#207602
#106262
#106252
#106242
#106232
#106222
#106212
#80422
#78742