Finite Group Databasebeta

← #6559993

Subgroups — #6559993

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36419C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#30203C211yesmaximal, characteristic, complemented
4#30188C212yesmaximal, complemented
5#29346C211yesmaximal, characteristic, complemented
6#29141C211yesmaximal, characteristic, complemented
7#29107C212yesmaximal, complemented
8#27816C212yesmaximal, complemented
9#27770C212yesmaximal, complemented
10#27702C214yesmaximal, complemented
11#27255C214yesmaximal, complemented
12#25906C212yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23451C212yesmaximal, complemented
15#22490C212yesmaximal, complemented
16#22418C212yesmaximal
17#20555C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#550C2211yescharacteristic
20C2×#533C2212yes—
21C2×#488C2214yes—
22C2×C8×D4C2212yes—
23C2×#416C2214yes—
24#2868C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2212yes—
27#2850C2212yes—
28#2568C2211yescharacteristic
29#2559—22no—
30#2559—22no—
31#2552C2212yes—
32#2548C2211yescharacteristic
33#2544—21no—
34#2544—21no—
35#2542—21no—
36#2538—21no—
37#2500C2212yes—
38#2441C2214yes—
39#2432C2212yes—
40#2429—22no—
41#2423—22no—
42#2367—21no—
43#2367—21no—
44#2365—21no—
45#2354C2212yes—
46Q8×C4⋊C4—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49#2297C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2212yes—
53#2072C2211yescharacteristic
54#2072C2211yescharacteristic
55#2071C2214yes—
56#2071C2212yes—
57#2043C2212yes—
58#2042C2214yes—
59#2026C2214yes—
60#2025C2214yes—
61#2024C2211yescharacteristic
62#2024C2211yescharacteristic
63#2024C2212yes—
64#2024C2212yes—
65#2023C2212yes—
66#2023C2212yes—
67#2023C2212yes—
68#1991C2214yes—
69#1990C2212yes—
70#1988C2212yes—
71#1982C2212yes—
72#1957C2212yes—
73#1956C2212yes—
74#1934C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1875C2214yes—
80#1873C2214yes—
81#1834C2212yes—
82#1825C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1788C2214yes—
86C8×C4⋊C4C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#103C2212yes—
89C4×#103C2212yes—
90C8×C22⋊C4C2214yes—
91C4×C22⋊C8C2212yes—
92#1744C2211yescharacteristic
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#556—28no—
98#550—28no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127—21no—
107C2×#127C2311yescharacteristic
108C2×#126—22no—
109C2×#125C2314yes—
110C2×#124—21no—
111C2×#124—21no—
112C2×#124—21no—
113C2×#124C2312yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C2312yes—
118C2×#121—22no—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2314yes—
132C22×C42C2312yes—
133#488—48no—
134C8×D4—28no—
135C2×C4⋊C8C2312yes—
136#416—48no—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#399—24no—
153#398—24no—
154#396—24no—
155#394—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—22no—
164#391C2311yescharacteristic
165#391C2311yescharacteristic
166#391C2312yes—
167#390—22no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—24no—
181#389C2312yes—
182#389C2314yes—
183#388—24no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382C2312yes—
203#380—22no—
204#380C2314yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—28no—
236C2×#41—42no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2411yescharacteristic
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—41no—
264C4⋊Q8—44no—
265C4⋊Q8—82no—
266#127—24no—
267#127—44no—
268#126—48no—
269#125—28no—
270#125—44no—
271#124—44no—
272#124—44no—
273#124—44no—
274#124—44no—
275C22⋊Q8—28no—
276C22⋊Q8—48no—
277C22⋊Q8—48no—
278C22⋊Q8—84no—
279#121—28no—
280#121—48no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—48no—
284C4×Q8—48no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4#4118yes—
291#117—28no—
292#117—28no—
293#117—44no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C4⋊C8—24no—
356#103—24no—
357#103—24no—
358#103—24no—
359#103—42no—
360#103—44no—
361#103—44no—
362#102—28no—
363#102—44no—
364#102—44no—
365#102—44no—
366C22⋊C8—28no—
367C4×C8—24no—
368C4×C8—42no—
369C4×C8—44no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24C22×D412yes—
387#41—416no—
388#41—88no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q82−1+412yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4#12414yes—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4118yes—
455C22×C4C2511yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—88no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4#12414yes—
497C4⋊C4H(5,𝔽2)14yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—84no—
513C22⋊C4#12418yes—
514C42—24no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—42no—
520C42—41no—
521C42—41no—
522C42—48no—
523C42—44no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42#13514yes—
528C42#13514yes—
529C42C2×#4114yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C2×#12414yes—
544C23C23×D411yescharacteristic
545Q8—24no—
546Q8—48no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—88no—
551Q8—84no—
552D4—28no—
553D4—44no—
554D4—44no—
555D4—88no—
556D4—84no—
557D4—162no—
558D4—164no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—88no—
606C2×C4C2×#12414yes—
607C2×C4C2×#12412yes—
608C2×C4C2×#12412yes—
609C2×C4C2×#13512yes—
610C2×C4C2×#13512yes—
611C2×C4C2×#13512yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4C22×#4112yes—
616C2×C4C2×H(5,𝔽2)12yes—
617C2×C4C2×2−1+411yescharacteristic
618C2×C4#58511yescharacteristic
619C8—44no—
620C8—42no—
621C8—42no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C22×#12411yescharacteristic
637C22#348411yescharacteristic
638C22#348911yescharacteristic
639C22C22×#13511yescharacteristic
640C22C22×#13511yescharacteristic
641C22#358511yescharacteristic
642C22#358611yescharacteristic
643C4—24no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—81no—
654C4—84no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4#313114yes—
660C4#351412yes—
661C4#352012yes—
662C2—44no—
663C2—162no—
664C2#3593911yescharacteristic
665C2C2×#313111yescharacteristic
666C2#6062511yescharacteristic
667C2#6156511yescharacteristic
668C2#6156611yescharacteristic
669C2#6210111yescharacteristic
670C2#6210211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #36134C2semidirect
3. #30203C2semidirect
4. #30188C2semidirect
5. #29346C2semidirect
6. #29141C2semidirect
7. #29107C2semidirect
8. #27816C2semidirect
9. #27770C2semidirect
10. #27702C2semidirect
11. #27255C2semidirect
12. #25906C2semidirect
13. #23816C2semidirect
14. #23451C2semidirect
15. #22490C2semidirect
17. #20555C2semidirect

Maximal subgroups

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

subgroupindex
#364192
#361342
#302032
#301882
#293462
#291412
#291072
#278162
#277702
#277022
#272552
#259062
#238162
#234512
#224902
#224182
#205552