Finite Group Databasebeta

← #319019

Subgroups — #319019

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic, complemented
2#27972C211yesmaximal, characteristic, complemented
3#27961C211yesmaximal, characteristic, complemented
4#25855C211yesmaximal, characteristic
5#25855C211yesmaximal, characteristic
6C2×#1957C211yesmaximal, characteristic
7#10678C212yesmaximal, complemented
8#10311C212yesmaximal, complemented
9#9822C212yesmaximal, complemented
10#9822C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#487C2211yescharacteristic, complemented
14C2×#486C2212yes—
15C2×#486C2212yes—
16C22×#102—22no—
17C22×C4×C8C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2100—22no—
22#2072—22no—
23#2041—22no—
24#2041—22no—
25#1990C2211yescharacteristic, complemented
26#1990C2212yescomplemented
27#1957—22no—
28#1957C2212yes—
29#1957C2212yes—
30#1957C2212yes—
31#1957C2212yes—
32#1956C2211yescharacteristic, complemented
33#1956C2212yescomplemented
34#1933—22no—
35#1892—22no—
36#1872—22no—
37#1834C2211yescharacteristic, complemented
38#1834C2212yescomplemented
39#1834C2212yescomplemented
40#1824—22no—
41#1786—22no—
42#1786—22no—
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×C82C2212yescomplemented
46#1273C2212yescomplemented
47C23×D4—21no—
48C23×D4—21no—
49C23×D4—21no—
50C23×C8D412yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#127—24no—
54C2×#127—21no—
55C2×#127—21no—
56C2×#127—21no—
57C2×#127—42no—
58C2×#127C2312yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#487—42no—
66#486—24no—
67#486—24no—
68#486—42no—
69#486—42no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—48no—
80C2×C22⋊C8D412yes—
81C2×C22⋊C8D412yes—
82C2×C4×C8—42no—
83C2×C4×C8D412yes—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2312yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2311yescharacteristic
88#391—21no—
89#391—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99#384C2312yes—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—22no—
108#340—22no—
109#336D412yes—
110#336D412yes—
111C82—42no—
112C25—41no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—44no—
120C22×D4—44no—
121C22×D4—44no—
122C22×D4—42no—
123C22×D4—41no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—44no—
127C22×D4—44no—
128C22×D4—44no—
129C22×D4—82no—
130C22×D4—82no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C22×C8—24no—
136C22×C8—24no—
137C22×C8—42no—
138C22×C8C2×D412yes—
139C22×C8C2×D414yes—
140C22×C8C2×D414yes—
141C22×C8C2×D412yes—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146#127—24no—
147#127—22no—
148#127—22no—
149#127—22no—
150#127—44no—
151#127—44no—
152#127—42no—
153#127#4114yes—
154C22⋊Q8—44no—
155C22⋊Q8—44no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—82no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2411yescharacteristic
185#103—44no—
186#103—44no—
187#103—84no—
188#102—48no—
189#102—48no—
190#102—48no—
191#102—48no—
192#102—84no—
193#102—84no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—42no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24—44no—
208C24—42no—
209C24—82no—
210C24—82no—
211C24—82no—
212C24—82no—
213C24—82no—
214C24—82no—
215C24#12711yescharacteristic
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—416no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—816no—
240C2×D4—816no—
241C2×D4—82no—
242C2×D4—88no—
243C2×D4—82no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4#12711yescharacteristic
264C22×C4#12711yescharacteristic
265C22×C4#12711yescharacteristic
266C22×C4#13514yes—
267C22×C4#13514yes—
268C22×C4#13514yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—48no—
284C2×C8AGL(1,ℤ/8)14yes—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—21no—
307C42—21no—
308C42—22no—
309C42—22no—
310C42C2×#4112yes—
311C42H(5,𝔽2)12yes—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—44no—
317C23—88no—
318C23—88no—
319C23—88no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—164no—
327C23—164no—
328C23—164no—
329C23—164no—
330C23#49512yes—
331C23#49512yes—
332C23#49512yes—
333C23C2×#12711yescharacteristic
334Q8—82no—
335Q8—82no—
336Q8—84no—
337D4—48no—
338D4—48no—
339D4—48no—
340D4—88no—
341D4—88no—
342D4—88no—
343D4—84no—
344D4—84no—
345D4—84no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—164no—
372C2×C4#49512yes—
373C2×C4#49512yes—
374C2×C4#49512yes—
375C2×C4#52511yescharacteristic
376C2×C4#53411yescharacteristic
377C2×C4#54612yes—
378C2×C4#54612yes—
379C2×C4C2×#13512yes—
380C2×C4C2×#13512yes—
381C2×C4C2×#13512yes—
382C2×C4C2×AGL(1,ℤ/8)12yes—
383C2×C4#57612yes—
384C2×C4#57712yes—
385C8—24no—
386C8—42no—
387C8—44no—
388C8—44no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—84no—
396C22—84no—
397C22—84no—
398C22—1616no—
399C22—162no—
400C22—162no—
401C22—162no—
402C22—162no—
403C22#171114yes—
404C22#253911yescharacteristic
405C22C2×#49511yescharacteristic
406C22C2×#49511yescharacteristic
407C22C2×#49511yescharacteristic
408C22#315111yescharacteristic
409C22#315411yescharacteristic
410C22#315511yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4—162no—
418C4—161no—
419C4#310512yes—
420C4#318812yes—
421C4#329614yes—
422C4#329714yes—
423C2—24no—
424C2—164no—
425C2C2×#171111yescharacteristic
426C2#1962511yescharacteristic
427C2#1963511yescharacteristic
428C2#1965611yescharacteristic
429C2#2830311yescharacteristic
430C2#2833011yescharacteristic
431C2#2833111yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
1. #33043C2semidirect
2. #27972C2semidirect
3. #27961C2semidirect
7. #10678C2semidirect
8. #10311C2semidirect
9. #9822C2semidirect
10. #9822C2semidirect
11. C8×C22⋊C8C2semidirect
13. C2×#487C22semidirect
25. #1990C22semidirect
26. #1990C22semidirect
32. #1956C22semidirect
33. #1956C22semidirect
37. #1834C22semidirect
38. #1834C22semidirect
39. #1834C22semidirect
45. C2×C82C22semidirect
46. #1273C22semidirect

Maximal subgroups

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

subgroupindex
#330432
#279722
#279612
#258552
C2×#19572
#106782
#103112
#98222
C8×C22⋊C82