Finite Group Databasebeta

← #6543872

Subgroups — #6543872

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36315C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30043C212yesmaximal, complemented
4#29664C211yesmaximal, characteristic, complemented
5#29395C212yesmaximal, complemented
6#29350C212yesmaximal, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27644C211yesmaximal, characteristic
9#26725C214yesmaximal, complemented
10#26620C214yesmaximal
11#26023C212yesmaximal, complemented
12#24629C212yesmaximal, complemented
13#23679C212yesmaximal, complemented
14#23568C211yesmaximal, characteristic
15#22998C212yesmaximal, complemented
16#22767C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#480C2212yes—
20C2×#462C2212yes—
21C2×C4×Q16C2214yes—
22C2×C4×SD16C2212yes—
23#2861C2212yes—
24#2857C2211yescharacteristic
25#2552C2211yescharacteristic
26#2547C2211yescharacteristic
27#2531—22no—
28#2529—22no—
29#2497—21no—
30#2489C2212yes—
31#2485—21no—
32#2426C2212yes—
33#2365C2211yescharacteristic
34#2360—22no—
35#2360C2214yes—
36#2355—21no—
37#2355C2212yes—
38#2354—21no—
39#2354C2211yescharacteristic
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42#2346—21no—
43#2330—22no—
44#2317—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48#2295C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2083C2212yes—
53#2072C2212yes—
54#2072C2212yes—
55#2071C2212yes—
56#2068C2214yes—
57#2058C2211yescharacteristic
58#2056C2214yes—
59#2043C2211yescharacteristic
60#2035C2212yes—
61#2026C2214yes—
62#2024C2212yes—
63#2023C2211yescharacteristic
64#2016C2212yes—
65#2015C2214yes—
66#2013C2211yescharacteristic
67#2001C2214yes—
68#2000C2212yes—
69#1989C2214yes—
70#1983C2214yes—
71#1956C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1934C2214yes—
75#1933C2214yes—
76#1927C2214yes—
77#1923C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1880C2214yes—
81#1878C2214yes—
82#1872C2214yes—
83#1863C2214yes—
84#1839C2212yes—
85#1834C2212yes—
86#1823C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2212yes—
93C4×#103C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97#533—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—21no—
102C2×#125C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×#121C2311yescharacteristic
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—21no—
117C2×#117D411yescharacteristic
118C2×#117D411yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123#480—44no—
124#462—28no—
125C4×Q16—48no—
126C4×SD16—28no—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#400—22no—
145#398—22no—
146#397—22no—
147#397—22no—
148#396—22no—
149#394—22no—
150#391—22no—
151#391C2314yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#390—22no—
156#390C2311yescharacteristic
157#390C2312yes—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—24no—
166#389—24no—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—21no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#387—22no—
183#386—21no—
184#385—24no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#382—22no—
193#382—22no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—24no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219#376D412yes—
220C43C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×D4—21no—
237C22×D4C2411yescharacteristic
238C23×C4—22no—
239C23×C4—21no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—28no—
244C2×Q16—44no—
245C2×Q16—44no—
246C2×SD16—24no—
247C2×SD16—42no—
248C2×SD16—42no—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255#125—28no—
256#125—44no—
257#124—44no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—44no—
262#121—22no—
263#121—41no—
264C4×Q8—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—42no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×Q8—84no—
271C4×Q8#4118yes—
272C4×D4—24no—
273C4×D4—44no—
274#117—28no—
275#117—24no—
276#117—24no—
277#117—44no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—28no—
345#106—24no—
346C4⋊C8—24no—
347C4⋊C8—44no—
348#103—24no—
349#103—28no—
350#103—24no—
351#103—44no—
352#103—44no—
353#103—42no—
354#103—44no—
355#102—24no—
356#102—44no—
357C22⋊C8—42no—
358C22⋊C8—42no—
359C4×C8—24no—
360C4×C8—44no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—21no—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8C2×#4114yes—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—81no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4#13514yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436Q16—416no—
437SD16—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—28no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4#13714yes—
480C4⋊C4C2×#4114yes—
481C4⋊C4H(5,𝔽2)12yes—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C42—22no—
496C42—22no—
497C42—28no—
498C42—24no—
499C42—24no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—22no—
505C42—42no—
506C42—44no—
507C42—42no—
508C42—44no—
509C42—84no—
510C42—84no—
511C42C2×#4114yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×#13512yes—
526C23C23×D411yescharacteristic
527Q8—24no—
528Q8—24no—
529Q8—48no—
530Q8—42no—
531Q8—42no—
532Q8—44no—
533Q8—44no—
534D4—42no—
535D4—42no—
536D4—84no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—164no—
587C2×C4#52512yes—
588C2×C4#52512yes—
589C2×C4C2×#13512yes—
590C2×C4C2×#13512yes—
591C2×C4C2×#13714yes—
592C2×C4C2×#13712yes—
593C2×C4#57812yes—
594C2×C4C22×#4112yes—
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4#58511yescharacteristic
597C2×C4#58512yes—
598C8—42no—
599C8—42no—
600C8—44no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22C2×#52511yescharacteristic
616C22#348811yescharacteristic
617C22#350011yescharacteristic
618C22C22×#13511yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57811yescharacteristic
621C22#358511yescharacteristic
622C4—24no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4#310814yes—
639C4#349312yes—
640C4#350612yes—
641C2—42no—
642C2—161no—
643C2#3575611yescharacteristic
644C2C2×#310811yescharacteristic
645C2#6058911yescharacteristic
646C2#6194911yescharacteristic
647C2#6198411yescharacteristic
648C2#6252811yescharacteristic
649C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #30043C2semidirect
4. #29664C2semidirect
5. #29395C2semidirect
6. #29350C2semidirect
7. #27972C2semidirect
9. #26725C2semidirect
11. #26023C2semidirect
12. #24629C2semidirect
13. #23679C2semidirect
15. #22998C2semidirect
16. #22767C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#363152
#342142
#300432
#296642
#293952
#293502
#279722
#276442
#267252
#266202
#260232
#246292
#236792
#235682
#229982
#227672
#226392