Finite Group Databasebeta

← #6568681

Subgroups — #6568681

Order 512.

1644 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36261C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic, complemented
4#30197C212yesmaximal, complemented
5#29334C212yesmaximal, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29121C211yesmaximal, characteristic, complemented
8#27968C212yesmaximal, complemented
9#27821C212yesmaximal, complemented
10#27735C212yesmaximal, complemented
11#27720C214yesmaximal, complemented
12#27704C214yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23747C212yesmaximal, complemented
15#22503C212yesmaximal, complemented
16#22503C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#555C2212yes—
19C2×#554—21no—
20C2×#436C2212yes—
21#2868C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2212yes—
24#2856C2212yes—
25#2584—21no—
26#2583—22no—
27#2563C2212yes—
28#2552C2211yescharacteristic
29#2551C2211yescharacteristic
30#2550—21no—
31#2549—22no—
32#2537—21no—
33#2537—21no—
34#2500C2214yes—
35#2491—22no—
36#2491—22no—
37#2440C2212yes—
38#2440C2212yes—
39#2354—21no—
40#2354—21no—
41#2349—21no—
42D4×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C2×#389C2212yes—
47#2285C2212yes—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2101C2214yes—
51#2100C2214yes—
52#2100C2212yes—
53#2100C2212yes—
54#2100C2212yes—
55#2100C2212yes—
56#2099C2211yescharacteristic
57#2099C2211yescharacteristic
58#2099C2212yes—
59#2071C2212yes—
60#2071C2214yes—
61#2041C2211yescharacteristic
62#2041C2211yescharacteristic
63#2041C2212yes—
64#2023C2212yes—
65#2023C2214yes—
66#1993C2214yes—
67#1990C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1987C2212yes—
71#1987C2212yes—
72#1936C2214yes—
73#1936C2214yes—
74#1934C2214yes—
75#1895C2214yes—
76#1895C2214yes—
77#1874C2214yes—
78#1874C2214yes—
79#1836C2214yes—
80#1834C2212yes—
81#1825C2214yes—
82#1825C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1770C2212yes—
86C4×C4⋊C8C2211yescharacteristic
87C4×#102C2212yes—
88C4×#102C2212yes—
89#1753C2214yes—
90#1749C2212yes—
91#1744C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×M16C2314yes—
96#555—28no—
97#555—28no—
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125—21no—
108C2×#124—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121C2314yes—
116C2×C4×Q8—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117C2314yes—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C42C2312yes—
131#436—28no—
132C2×C4⋊C8C2312yes—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143#398—22no—
144#398—22no—
145#398—22no—
146#395—22no—
147#395—22no—
148#395—22no—
149#394—24no—
150#394—24no—
151#393—24no—
152#391—21no—
153#391—22no—
154#390—21no—
155#390—21no—
156#390—22no—
157#390—22no—
158#390C2312yes—
159#390C2314yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#388—24no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#382—21no—
192#382C2314yes—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211C2×#95—22no—
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×D4—24no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2412yes—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×M16—28no—
247C2×M16—44no—
248C2×M16—44no—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—41no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8—82no—
260#127—42no—
261#125—44no—
262#125—44no—
263#124—416no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267#121—28no—
268#121—28no—
269#121—48no—
270#121—84no—
271#121—84no—
272C4×Q8—44no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4#4118yes—
282#117—28no—
283#117—28no—
284#117—48no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C4⋊C8—24no—
352#102—24no—
353#102—44no—
354#102—42no—
355#102—42no—
356#102—44no—
357#102—42no—
358C22⋊C8—28no—
359#97—44no—
360C4×C8—24no—
361C4×C8—42no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—22no—
378C24C22×D412yes—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—81no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4#12414yes—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4118yes—
442C22×C4C2511yescharacteristic
443M16—88no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4#12414yes—
493C4⋊C42−1+412yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4#12418yes—
506C42—21no—
507C42—21no—
508C42—24no—
509C42—22no—
510C42—42no—
511C42—42no—
512C42—41no—
513C42—41no—
514C42—48no—
515C42—42no—
516C42—42no—
517C42AGL(1,ℤ/8)14yes—
518C42#13714yes—
519C42C2×#4114yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23C2×#12414yes—
534C23C23×D411yescharacteristic
535Q8—84no—
536Q8—164no—
537Q8—164no—
538D4—28no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—84no—
544D4—84no—
545D4—168no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4—164no—
595C2×C4C2×#12414yes—
596C2×C4C2×#12412yes—
597C2×C4C2×#12412yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C22×#4112yes—
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58512yes—
607C2×C4#58511yescharacteristic
608C8—42no—
609C8—44no—
610C8—42no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C22×#12411yescharacteristic
626C22#348511yescharacteristic
627C22#348911yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C22×#13711yescharacteristic
630C22#358511yescharacteristic
631C22#358511yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—24no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4#314114yes—
649C4#351512yes—
650C4#352112yes—
651C2—44no—
652C2—162no—
653C2#3604211yescharacteristic
654C2C2×#314111yescharacteristic
655C2#6062611yescharacteristic
656C2#6172011yescharacteristic
657C2#6172111yescharacteristic
658C2#6211011yescharacteristic
659C2#6211411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36261C2semidirect
3. #30230C2semidirect
4. #30197C2semidirect
5. #29334C2semidirect
6. #29218C2semidirect
7. #29121C2semidirect
8. #27968C2semidirect
9. #27821C2semidirect
10. #27735C2semidirect
11. #27720C2semidirect
12. #27704C2semidirect
13. #23815C2semidirect
14. #23747C2semidirect
15. #22503C2semidirect
16. #22503C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#362612
#361872
#302302
#301972
#293342
#292182
#291212
#279682
#278212
#277352
#277202
#277042
#238152
#237472
#225032
#205682