Finite Group Databasebeta

← #6556245

Subgroups — #6556245

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30199C212yesmaximal, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29107C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#27972C212yesmaximal, complemented
9#27749C212yesmaximal, complemented
10#27727C212yesmaximal, complemented
11#27706C214yesmaximal, complemented
12#27278C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23630C212yesmaximal, complemented
15#22504C212yesmaximal, complemented
16#22431C212yesmaximal
17#20567C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#554—21no—
20C2×#547C2211yescharacteristic
21C2×#489C2212yes—
22C2×#436C2212yes—
23C2×#417C2214yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2857C2212yes—
28#2567—21no—
29#2553—22no—
30#2553—22no—
31#2552—21no—
32#2552C2212yes—
33#2546C2211yescharacteristic
34#2542C2211yescharacteristic
35#2502—22no—
36#2499C2212yes—
37#2440—22no—
38#2440C2214yes—
39#2432C2212yes—
40#2427—21no—
41#2427—21no—
42#2360—21no—
43#2360—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48#2297C2212yes—
49C2×#384C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yes—
52#2099C2214yes—
53#2072C2211yescharacteristic
54#2072C2212yes—
55#2072C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#2072C2212yes—
59#2071C2211yescharacteristic
60#2071C2212yes—
61#2071C2214yes—
62#2043C2214yes—
63#2042C2212yes—
64#2024C2211yescharacteristic
65#2024C2214yes—
66#2024C2212yes—
67#2023C2211yescharacteristic
68#1993C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1981C2212yes—
72#1956C2214yes—
73#1956C2212yes—
74#1935C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1834C2214yes—
82#1825C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1753C2214yes—
91C4×C22⋊C8C2212yes—
92#1744C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C22×M16C2314yes—
97#556—28no—
98#554—48no—
99#547—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#126—22no—
108C2×#125—22no—
109C2×#124—21no—
110C2×#124C2311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117C2312yes—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2314yes—
132C22×C42C2312yes—
133#489—44no—
134#436—28no—
135C2×C4⋊C8C2312yes—
136#417—48no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#399—22no—
152#398—24no—
153#398—24no—
154#398—22no—
155#397—22no—
156#397—22no—
157#395—24no—
158#394—22no—
159#394—22no—
160#391—21no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391C2311yescharacteristic
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#390C2311yescharacteristic
169#390C2312yes—
170#389—22no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2314yes—
180#388—21no—
181#388—21no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#385—24no—
187#385—24no—
188#385—22no—
189#385—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384—22no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#384C2314yes—
198#384C2312yes—
199#382—21no—
200#382—21no—
201#382C2311yescharacteristic
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2314yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234C2×#41—24no—
235C2×#41—28no—
236C2×#41—42no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×D4—22no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2411yescharacteristic
246C23×C4—24no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C2×M16—28no—
253C2×M16—44no—
254C2×M16—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—82no—
266#126—48no—
267#125—48no—
268#124—22no—
269#124—41no—
270#124—44no—
271#123—48no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—416no—
275C22⋊Q8—48no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278#121—44no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—48no—
286C4×D4#4118yes—
287#117—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C4⋊C8—24no—
354#103—42no—
355#103—44no—
356#103—42no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#102—24no—
362#102—28no—
363#102—42no—
364C22⋊C8—28no—
365#97—42no—
366C4×C8—24no—
367C4×C8—44no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D412yes—
387#41—416no—
388#41—88no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4#12414yes—
415C2×D4H(5,𝔽2)12yes—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4118yes—
452C22×C4C2511yescharacteristic
453M16—88no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4#12414yes—
496C4⋊C4AGL(1,ℤ/8)14yes—
497C4⋊C4#13714yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4#12418yes—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—24no—
518C42—22no—
519C42—21no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42—44no—
525C42C2×#4114yes—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C2×#12414yes—
540C23C23×D411yescharacteristic
541Q8—42no—
542Q8—44no—
543Q8—42no—
544Q8—44no—
545Q8—84no—
546Q8—84no—
547Q8—164no—
548D4—28no—
549D4—24no—
550D4—42no—
551D4—88no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4—168no—
601C2×C4C2×#12414yes—
602C2×C4C2×#12412yes—
603C2×C4C2×#12412yes—
604C2×C4C2×AGL(1,ℤ/8)14yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×#13714yes—
607C2×C4C2×#13712yes—
608C2×C4#57712yes—
609C2×C4#57812yes—
610C2×C4C22×#4112yes—
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4#58512yes—
613C2×C4#58511yescharacteristic
614C8—44no—
615C8—42no—
616C8—42no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C22×#12411yescharacteristic
632C22#348511yescharacteristic
633C22#348811yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C22×#13711yescharacteristic
636C22C2×#57711yescharacteristic
637C22C2×#57811yescharacteristic
638C4—24no—
639C4—22no—
640C4—24no—
641C4—22no—
642C4—24no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—82no—
647C4—81no—
648C4—82no—
649C4—82no—
650C4—84no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4#314114yes—
655C4#351312yes—
656C4#352112yes—
657C2—44no—
658C2—82no—
659C2#3584711yescharacteristic
660C2C2×#314111yescharacteristic
661C2#6062311yescharacteristic
662C2#6173511yescharacteristic
663C2#6173711yescharacteristic
664C2#6199911yescharacteristic
665C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
2. #36180C2semidirect
3. #30205C2semidirect
4. #30199C2semidirect
5. #29355C2semidirect
6. #29107C2semidirect
7. #28726C2semidirect
8. #27972C2semidirect
9. #27749C2semidirect
10. #27727C2semidirect
11. #27706C2semidirect
12. #27278C2semidirect
13. #23817C2semidirect
14. #23630C2semidirect
15. #22504C2semidirect
17. #20567C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#361802
#302052
#301992
#293552
#291072
#287262
#279722
#277492
#277272
#277062
#272782
#238172
#236302
#225042
#224312
#205672