Finite Group Databasebeta

← #316267

Subgroups — #316267

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27680C211yesmaximal, characteristic, complemented
2#24211C211yesmaximal, characteristic
3C8×C22⋊Q8C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic
5#10595C212yesmaximal, complemented
6#10475C212yesmaximal, complemented
7#9564C212yesmaximal, complemented
8#9222C212yesmaximal, complemented
9#8724C211yesmaximal, characteristic, complemented
10#8595C211yesmaximal, characteristic, complemented
11#8339C211yesmaximal, characteristic
12C22×#127C2211yescharacteristic
13C2×#460C2212yes—
14C2×#450—21no—
15C2×C8×Q8C2211yescharacteristic, complemented
16C22×#102—21no—
17C22×#102—21no—
18C22×C4×C8C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2099—21no—
22#2096—21no—
23#2041—21no—
24#2014—21no—
25#1984C2211yescharacteristic, complemented
26#1982C2212yes—
27#1957—21no—
28#1957—21no—
29#1957C2212yes—
30#1957C2212yes—
31#1948C2212yescomplemented
32#1939C2211yescharacteristic, complemented
33#1916—21no—
34#1872—21no—
35#1853—21no—
36#1850—21no—
37#1803C2212yescomplemented
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1786—21no—
42C8×C4⋊C4—21no—
43C8×C4⋊C4C2211yescharacteristic, complemented
44C8×C22⋊C4—21no—
45C2×#334C2212yescomplemented
46C2×#331C2212yes—
47C2×#331C2212yes—
48#1296C2212yescomplemented
49#1296C2212yescomplemented
50#1277C2212yescomplemented
51C23×D4—21no—
52C23×D4—21no—
53C23×D4—21no—
54C22×SD16—22no—
55C23×C8D411yescharacteristic
56C23×C8D411yescharacteristic
57C2×#127—24no—
58C2×#127—21no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—42no—
62C2×#127C2312yes—
63C2×C22⋊Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#460—24no—
70#460—42no—
71#450—44no—
72C8×Q8—42no—
73C2×#106—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×#103—22no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—44no—
89C2×#102—44no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—42no—
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391D411yescharacteristic
101#389D411yescharacteristic
102#387—21no—
103#385—21no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384C2312yes—
108#380D411yescharacteristic
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2311yescharacteristic
111C4×C22⋊C4—21no—
112#340—22no—
113#340—22no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#336—21no—
121#334—42no—
122#331—24no—
123#331—42no—
124C25—41no—
125C22×Q8C2×D411yescharacteristic
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4—41no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—82no—
143C22×D4—82no—
144C22×D4—82no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4#4111yescharacteristic
149C2×SD16—44no—
150C2×SD16—44no—
151C2×SD16—82no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—41no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8—41no—
161C22×C8SD1614yescomplemented
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166#127—24no—
167#127—22no—
168#127—22no—
169#127—22no—
170#127—44no—
171#127—44no—
172#127—42no—
173#127#4114yes—
174C22⋊Q8—22no—
175C22⋊Q8SD1614yes—
176C4×Q8—22no—
177C4×Q8—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42C2411yescharacteristic
204#106—82no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213#103—82no—
214#102—44no—
215#102—44no—
216#102—44no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—82no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—42no—
229C4×C8—42no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235C24—44no—
236C24—42no—
237C24—82no—
238C24—82no—
239C24—82no—
240C24—82no—
241C24—82no—
242C24—82no—
243C24#12111yescharacteristic
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8C2×SD1612yes—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—48no—
252C2×D4—48no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—88no—
268C2×D4—88no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—84no—
273C2×D4—84no—
274C2×D4—816no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C2×SD1612yes—
301C22×C4C2×SD1612yes—
302C22×C4#13512yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307SD16—88no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8C2×SD1612yes—
333C2×C8C2×SD1612yes—
334C4⋊C4—21no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4C2×SD1612yes—
349C22⋊C4—24no—
350C22⋊C4—42no—
351C42—21no—
352C42—21no—
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42C2×#4112yes—
360C42H(5,𝔽2)12yes—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—44no—
366C23—88no—
367C23—88no—
368C23—88no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—164no—
376C23—164no—
377C23—164no—
378C23—164no—
379C23#46012yes—
380C23#46512yes—
381C23#47112yes—
382C23C2×#12111yescharacteristic
383Q8—24no—
384Q8—42no—
385D4—44no—
386D4—44no—
387D4—44no—
388D4—44no—
389D4—84no—
390D4—88no—
391D4—88no—
392D4—84no—
393D4—88no—
394D4—82no—
395D4—82no—
396D4—82no—
397D4—82no—
398C2×C4—28no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—164no—
429C2×C4#46012yes—
430C2×C4#46512yes—
431C2×C4#47212yes—
432C2×C4#52511yescharacteristic
433C2×C4#53411yescharacteristic
434C2×C4#54612yes—
435C2×C4#55112yes—
436C2×C4C22×SD1611yescharacteristic
437C2×C4C22×SD1611yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57511yescharacteristic
442C2×C4#57612yes—
443C8—22no—
444C8—22no—
445C8—41no—
446C8—41no—
447C8—81no—
448C8—81no—
449C8—82no—
450C8—82no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—84no—
458C22—84no—
459C22—84no—
460C22—1616no—
461C22—162no—
462C22—162no—
463C22—162no—
464C22—162no—
465C22#166614yes—
466C22#253411yescharacteristic
467C22C2×#46011yescharacteristic
468C22#304011yescharacteristic
469C22C2×#46511yescharacteristic
470C22#305811yescharacteristic
471C22C2×#47111yescharacteristic
472C22C2×#47211yescharacteristic
473C4—22no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4—162no—
482C4#310111yescharacteristic
483C4#311411yescharacteristic
484C4#318711yescharacteristic
485C4#319411yescharacteristic
486C4#329512yes—
487C4#329712yes—
488C2—24no—
489C2—164no—
490C2C2×#166611yescharacteristic
491C2#1920311yescharacteristic
492C2#1927411yescharacteristic
493C2#1936411yescharacteristic
494C2#2809711yescharacteristic
495C2#2812011yescharacteristic
496C2#2813411yescharacteristic
subgroupG/H
center, socleC23C2×#121
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. #27680C2semidirect
3. C8×C22⋊Q8C2semidirect
5. #10595C2semidirect
6. #10475C2semidirect
7. #9564C2semidirect
8. #9222C2semidirect
9. #8724C2semidirect
10. #8595C2semidirect
15. C2×C8×Q8C22semidirect
25. #1984C22semidirect
31. #1948C22semidirect
32. #1939C22semidirect
37. #1803C22semidirect
43. C8×C4⋊C4C22semidirect
45. C2×#334C22semidirect
48. #1296C22semidirect
49. #1296C22semidirect
50. #1277C22semidirect
161. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#276802
#242112
C8×C22⋊Q82
C2×#19572
#105952
#104752
#95642
#92222
#87242
#85952
#83392