Finite Group Databasebeta

← #7230577

Subgroups — #7230577

Order 512.

1647 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46004C211yesmaximal, characteristic, complemented
2#36406C211yesmaximal, characteristic
3#33032C212yesmaximal, complemented
4#32808C212yesmaximal, complemented
5#32644C212yesmaximal, complemented
6#30910C214yesmaximal, complemented
7#30507C212yesmaximal, complemented
8#30469C214yesmaximal, complemented
9#26743C214yesmaximal, complemented
10#25643C212yesmaximal, complemented
11#25082C214yesmaximal, complemented
12#23961C212yesmaximal
13#23352C211yesmaximal, characteristic, complemented
14C2×#556C2211yescharacteristic
15C2×#547C2211yescharacteristic
16C2×#497C2212yes—
17C2×#474C2212yes—
18C2×#467C2214yes—
19C2×#462C2212yes—
20C2×#461C2212yes—
21C2×#451C2214yes—
22C2×#448C2212yes—
23C22×#103C2212yes—
24#2862C2212yes—
25#2862C2211yescharacteristic
26#2857—21no—
27#2844—21no—
28#2811—21no—
29#2804—21no—
30#2763—24no—
31#2568C2212yes—
32#2553—24no—
33#2546C2211yescharacteristic
34#2542C2211yescharacteristic
35#2499—21no—
36#2492C2214yes—
37#2434C2214yes—
38#2432—21no—
39#2427C2214yes—
40#2409C2212yes—
41#2379—21no—
42#2360C2214yes—
43C4×#124C2211yescharacteristic
44#2297—21no—
45C42×D4C2211yescharacteristic
46#2068C2214yes—
47#2056C2214yes—
48#2056C2214yes—
49#2043C2214yes—
50#2036C2214yes—
51#2024C2214yes—
52#2024C2214yes—
53#2016C2214yes—
54#2016C2214yes—
55#2015C2214yes—
56#2001C2214yes—
57#2000C2214yes—
58#1988C2212yes—
59#1985C2212yes—
60#1983C2212yes—
61#1981C2212yes—
62#1935C2214yes—
63#1928C2214yes—
64#1928C2214yes—
65#1924C2214yes—
66#1918C2214yes—
67#1893C2212yes—
68#1878C2214yes—
69#1877C2214yes—
70#1877C2214yes—
71#1872C2214yes—
72#1869C2212yes—
73#1850C2214yes—
74#1831C2212yes—
75#1823C2212yes—
76#1763C2212yes—
77C22×#41—21no—
78C23×Q8C2312yes—
79C22×Q16C2314yes—
80C22×Q16C2314yes—
81C22×SD16C2314yes—
82C23×C8C2312yes—
83#556—42no—
84#556—42no—
85#547—42no—
86#547—42no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2312yes—
93C2×#126—22no—
94C2×#124D411yescharacteristic
95C2×#124C2312yes—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2314yes—
101C2×#121C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×C4×D4C2312yes—
106C2×C4×D4C2311yescharacteristic
107C2×#117—21no—
108C22×C4⋊C4C2312yes—
109C22×C42—21no—
110#497—44no—
111#474—44no—
112#467—48no—
113#462—44no—
114#461—44no—
115#451—48no—
116#448—44no—
117C2×#107C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#103—44no—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2312yes—
133#400—22no—
134#400—22no—
135#400—24no—
136#399—22no—
137#398—22no—
138#397—24no—
139#395—22no—
140#395—24no—
141#395—24no—
142#394—24no—
143#393—22no—
144#393—24no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—22no—
150#391—22no—
151#391—24no—
152#391D411yescharacteristic
153#391C2314yes—
154#391C2314yes—
155#391C2314yes—
156#391C2314yes—
157#390—21no—
158#390D411yescharacteristic
159#389—24no—
160#389C2314yes—
161#388—24no—
162#388D414yes—
163#387—24no—
164#386—22no—
165#386—24no—
166#385—24no—
167#385—24no—
168#385—22no—
169#385—24no—
170#384—21no—
171#382—22no—
172#382—22no—
173#382—22no—
174#382—22no—
175#382D411yescharacteristic
176#380—24no—
177#380C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2311yescharacteristic
181C4×C22⋊C4—24no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4C2314yes—
185C43C2311yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193C2×#41—44no—
194C2×#41—44no—
195C2×#41—82no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—44no—
203C22×Q8C2×D411yescharacteristic
204C22×Q8C2412yes—
205C22×Q8C2414yes—
206C22×D4—22no—
207C22×D4C2×D411yescharacteristic
208C22×D4C2411yescharacteristic
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C2412yes—
214C2×Q16—44no—
215C2×Q16—44no—
216C2×Q16—44no—
217C2×Q16—44no—
218C2×Q16—44no—
219C2×Q16—44no—
220C2×SD16—28no—
221C2×SD16—44no—
222C2×SD16—44no—
223C22×C8—42no—
224C22×C8—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—41no—
228C4⋊Q8—41no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231C4⋊Q8—41no—
232C4⋊Q8—41no—
233C4⋊Q8—42no—
234C4⋊Q8—42no—
235C4⋊Q8—42no—
236C4⋊Q8—42no—
237C4⋊Q8—82no—
238#126—48no—
239#124—24no—
240#124—24no—
241#124—42no—
242#123—48no—
243C22⋊Q8—28no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246C22⋊Q8—44no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249#121—24no—
250#121—42no—
251C4×Q8—24no—
252C4×Q8—41no—
253C4×Q8—41no—
254C4×Q8—42no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—24no—
258C4×D4—42no—
259C4×D4—41no—
260C4×D4—41no—
261#117—44no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D414yes—
294C2×C22⋊C4C2×D412yes—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—82no—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2412yes—
305C2×C42C2412yes—
306#107—28no—
307C4⋊C8—42no—
308C4⋊C8—42no—
309C4⋊C8—42no—
310#103—28no—
311#103—28no—
312#103—28no—
313#103—44no—
314#103—44no—
315#103—44no—
316#103—44no—
317#102—44no—
318C22⋊C8—42no—
319C22⋊C8—44no—
320#97—42no—
321#95—22no—
322#95—21no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—21no—
337C24C22×D412yes—
338#41—816no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—82no—
358C2×Q8—84no—
359C2×Q8—82no—
360C2×Q8—82no—
361C2×Q8—84no—
362C2×Q8—88no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—82no—
371C2×D4#13714yes—
372C2×D4H(5,𝔽2)12yes—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—84no—
393C22×C4—82no—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C2511yescharacteristic
402Q16—88no—
403Q16—88no—
404SD16—88no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—84no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—84no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—84no—
434C4⋊C4H(5,𝔽2)14yes—
435C4⋊C4H(5,𝔽2)12yes—
436C22⋊C4—24no—
437C22⋊C4—28no—
438C22⋊C4—28no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—82no—
446C22⋊C4—84no—
447C42—22no—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—21no—
453C42—21no—
454C42—22no—
455C42—41no—
456C42—44no—
457C42—42no—
458C42—82no—
459C42—82no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—42no—
469C23C2×#13714yes—
470C23C23×D411yescharacteristic
471Q8—42no—
472Q8—44no—
473Q8—42no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—82no—
479Q8—84no—
480Q8—88no—
481Q8—164no—
482D4—24no—
483D4—44no—
484D4—41no—
485D4—41no—
486D4—88no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—88no—
525C2×C4—88no—
526C2×C4—88no—
527C2×C4C2×#13712yes—
528C2×C4#57814yes—
529C2×C4C2×H(5,𝔽2)11yescharacteristic
530C2×C4C2×H(5,𝔽2)12yes—
531C2×C4C2×H(5,𝔽2)11yescharacteristic
532C2×C4#58512yes—
533C8—44no—
534C8—44no—
535C22—28no—
536C22—24no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22#348411yescharacteristic
546C22#348812yes—
547C22C22×#13711yescharacteristic
548C22C2×#57812yes—
549C22#358511yescharacteristic
550C4—24no—
551C4—22no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—84no—
560C4—82no—
561C4—84no—
562C4—82no—
563C4#352911yescharacteristic
564C4#354911yescharacteristic
565C4#355812yes—
566C2—44no—
567C2—82no—
568C2#4622511yescharacteristic
569C2#6157011yescharacteristic
570C2#6166511yescharacteristic
571C2#6198412yes—
572C2#6201712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #46004C2semidirect
3. #33032C2semidirect
4. #32808C2semidirect
5. #32644C2semidirect
6. #30910C2semidirect
7. #30507C2semidirect
8. #30469C2semidirect
9. #26743C2semidirect
10. #25643C2semidirect
11. #25082C2semidirect
13. #23352C2semidirect

Maximal subgroups

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

subgroupindex
#460042
#364062
#330322
#328082
#326442
#309102
#305072
#304692
#267432
#256432
#250822
#239612
#233522