Finite Group Databasebeta

← #6502462

Subgroups — #6502462

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#28123C212yesmaximal
4#27611C211yesmaximal, characteristic, complemented
5#27582C211yesmaximal, characteristic
6#27423C214yesmaximal, complemented
7#27003C214yesmaximal, complemented
8#26903C214yesmaximal, complemented
9#26379C214yesmaximal, complemented
10#24307C212yesmaximal, complemented
11#24064C211yesmaximal, characteristic, complemented
12#23933C211yesmaximal, characteristic
13#22079C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#450C212yesmaximal
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#479C2214yescomplemented
21C2×#450C2212yes—
22C2×C4×SD16C2214yes—
23#2454—22no—
24#2454—22no—
25#2454C2214yes—
26#2454C2214yes—
27C2×#399—21no—
28C2×#396—21no—
29#2370—22no—
30#2369C2212yescomplemented
31#2365C2212yescomplemented
32#2355—22no—
33#2355C2212yescomplemented
34Q8×C4⋊C4C2212yescomplemented
35#2329—21no—
36#2329—21no—
37#2317—21no—
38C4×C22⋊Q8C2212yes—
39C4×C22⋊Q8C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45#2285C2211yescharacteristic
46C2×#380—21no—
47#2274—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2099C2212yescomplemented
50#2093C2212yes—
51#2084C2212yescomplemented
52#2071C2212yes—
53#2067C2214yes—
54#2059C2214yescomplemented
55#2057C2212yescomplemented
56#2055C2214yes—
57#2043C2212yescomplemented
58#2026C2214yescomplemented
59#2023C2212yes—
60#2013C2212yes—
61#1989C2214yescomplemented
62#1942C2214yescomplemented
63#1933C2214yescomplemented
64#1925C2214yes—
65#1925C2214yes—
66#1921C2214yescomplemented
67#1920C2214yescomplemented
68#1916C2214yescomplemented
69#1916C2214yescomplemented
70#1880C2214yes—
71#1880C2214yes—
72#1877C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1864C2214yes—
76#1863C2214yescomplemented
77#1863C2214yescomplemented
78#1853C2212yes—
79#1852C2212yes—
80#1824C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1786C2212yes—
85C4×#102C2214yes—
86C4×C22⋊C8C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C22×SD16C2314yes—
90#533—28no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#125—21no—
94C2×#125—22no—
95C2×#125—44no—
96C2×#125C2314yes—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—21no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—44no—
105C2×C4×D4C2314yes—
106C2×#117—21no—
107C2×#117—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#484—28no—
126#479—48no—
127#450—28no—
128C4×SD16—48no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2314yes—
142#399—24no—
143#399—24no—
144#396—24no—
145#396—24no—
146#391—22no—
147#391—22no—
148#391C2312yes—
149#391C2312yes—
150#390—21no—
151#390—21no—
152#390—22no—
153#390C2314yes—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#385—24no—
177#385—24no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#382—22no—
182#382—21no—
183#380—21no—
184#380—24no—
185#380—24no—
186#380C2312yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C25—21no—
220C22×Q8—21no—
221C22×Q8C2412yes—
222C22×D4—24no—
223C22×D4—21no—
224C22×D4—24no—
225C22×D4—42no—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—82no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×SD16—28no—
242C2×SD16—28no—
243C2×SD16—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248#125—28no—
249#125—28no—
250#125—44no—
251#125—48no—
252C22⋊Q8—28no—
253C22⋊Q8—24no—
254C22⋊Q8#4118yes—
255C4×Q8—28no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×D4—28no—
261C4×D4—44no—
262C4×D4—48no—
263#117—28no—
264#117—44no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336#107—28no—
337#107—44no—
338#106—28no—
339C4⋊C8—44no—
340#103—28no—
341#103—28no—
342#103—44no—
343#102—28no—
344#102—28no—
345#102—44no—
346C22⋊C8—24no—
347C22⋊C8—24no—
348C4×C8—44no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—24no—
367C24—22no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24C22×D411yescharacteristic
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—81no—
380C2×Q8—81no—
381C2×Q8C2×#4114yes—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—416no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D42−1+414yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—84no—
427C22×C4—816no—
428C22×C4#13514yes—
429C22×C4#13714yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2511yescharacteristic
441SD16—416no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C2×#4114yes—
480C4⋊C4H(5,𝔽2)14yes—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—84no—
487C22⋊C4—82no—
488C22⋊C4—84no—
489C22⋊C4—82no—
490C42—28no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—82no—
501C42—82no—
502C42—82no—
503C42—82no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—24no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23#52512yes—
523C23C2×#13512yes—
524C23C2×#13712yes—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—44no—
528Q8—84no—
529D4—28no—
530D4—48no—
531D4—44no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—816no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4#52512yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4C2×#13714yes—
591C2×C4C22×#4112yes—
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4C2×2−1+412yes—
594C8—44no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—816no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22#310814yes—
611C22C2×#52511yescharacteristic
612C22#348411yescharacteristic
613C22#348611yescharacteristic
614C22C22×#13511yescharacteristic
615C22C22×#13711yescharacteristic
616C22#358511yescharacteristic
617C22#358511yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—41no—
628C4—41no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4#352014yes—
635C2—24no—
636C2—84no—
637C2#3535711yescharacteristic
638C2C2×#310811yescharacteristic
639C2#6056811yescharacteristic
640C2#6156611yescharacteristic
641C2#6157011yescharacteristic
642C2#6181911yescharacteristic
643C2#6182611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34213C2semidirect
4. #27611C2semidirect
6. #27423C2semidirect
7. #27003C2semidirect
8. #26903C2semidirect
9. #26379C2semidirect
10. #24307C2semidirect
11. #24064C2semidirect
13. #22079C2semidirect
14. #21912C2semidirect
20. C2×#479C22semidirect
30. #2369C22semidirect
31. #2365C22semidirect
33. #2355C22semidirect
34. Q8×C4⋊C4C22semidirect
49. #2099C22semidirect
51. #2084C22semidirect
54. #2059C22semidirect
55. #2057C22semidirect
57. #2043C22semidirect
58. #2026C22semidirect
61. #1989C22semidirect
62. #1942C22semidirect
63. #1933C22semidirect
66. #1921C22semidirect
67. #1920C22semidirect
68. #1916C22semidirect
69. #1916C22semidirect
76. #1863C22semidirect
77. #1863C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
#342132
#281232
#276112
#275822
#274232
#270032
#269032
#263792
#243072
#240642
#239332
#220792
#219122
C4×#4502