Finite Group Databasebeta

← #6556185

Subgroups — #6556185

Order 512.

1633 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36415C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30188C212yesmaximal, complemented
5#29347C211yesmaximal, characteristic, complemented
6#29107C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#27976C212yesmaximal, complemented
9#27748C212yesmaximal, complemented
10#27728C212yesmaximal, complemented
11#27698C214yesmaximal, complemented
12#27257C214yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23627C212yesmaximal, complemented
15#22489C212yesmaximal, complemented
16#22417C212yesmaximal, complemented
17#20555C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#486C2212yes—
21C2×C8×D4C2212yes—
22C2×#416C2214yes—
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2857C2212yes—
26#2850C2212yes—
27#2571C2212yes—
28#2563—21no—
29#2563C2211yescharacteristic
30#2557—22no—
31#2557—22no—
32#2547C2211yescharacteristic
33#2538—21no—
34#2535—21no—
35#2535—21no—
36#2499C2212yes—
37#2497—22no—
38#2492—22no—
39#2439C2214yes—
40#2431C2212yes—
41#2369—21no—
42#2355—21no—
43#2354C2212yes—
44#2352—21no—
45#2352—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2297C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2099C2212yes—
53#2074C2214yes—
54#2073C2214yes—
55#2072C2212yes—
56#2072C2212yes—
57#2072C2212yes—
58#2071C2211yescharacteristic
59#2071C2211yescharacteristic
60#2071C2212yes—
61#2071C2212yes—
62#2042C2214yes—
63#2041C2212yes—
64#2024C2214yes—
65#2024C2212yes—
66#2023C2211yescharacteristic
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1987C2212yes—
71#1981C2212yes—
72#1958C2214yes—
73#1956C2212yes—
74#1934C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1875C2214yes—
80#1873C2214yes—
81#1835C2214yes—
82#1825C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86C8×C4⋊C4C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#102C2212yes—
89C4×#102C2212yes—
90C8×C22⋊C4C2214yes—
91C4×C22⋊C8C2212yes—
92#1744C2211yescharacteristic
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#555—28no—
98#547—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#124—22no—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117C2312yes—
124C2×#117C2314yes—
125C22×C4⋊C4—21no—
126C22×C42C2312yes—
127#486—44no—
128C8×D4—28no—
129C2×C4⋊C8C2312yes—
130#416—48no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#397—22no—
147#397—22no—
148#397—22no—
149#396—24no—
150#393—24no—
151#393—22no—
152#392—22no—
153#392—22no—
154#391—21no—
155#391—22no—
156#391—21no—
157#391—21no—
158#391—24no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#390—22no—
163#390—21no—
164#390—21no—
165#390C2311yescharacteristic
166#390C2311yescharacteristic
167#390C2312yes—
168#389—22no—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2314yes—
173#388—24no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—24no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—21no—
193#382—21no—
194#382C2312yes—
195#382C2314yes—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2314yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2312yes—
233C2×#41—24no—
234C2×#41—28no—
235C2×#41—42no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C22×C8—28no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267#127—22no—
268#127—41no—
269#124—44no—
270#124—48no—
271#123—48no—
272C22⋊Q8—28no—
273C22⋊Q8—84no—
274#121—28no—
275#121—44no—
276#121—44no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4#4118yes—
285#117—28no—
286#117—44no—
287#117—44no—
288#117—44no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C4⋊C8—24no—
348#103—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352#102—24no—
353#102—28no—
354#102—42no—
355#102—42no—
356#102—42no—
357#102—44no—
358C22⋊C8—28no—
359C4×C8—24no—
360C4×C8—44no—
361C4×C8—42no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24—21no—
381C24C22×D412yes—
382#41—416no—
383#41—88no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4#12414yes—
419C2×D4H(5,𝔽2)12yes—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
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—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—88no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4#12414yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4#12418yes—
512C42—21no—
513C42—24no—
514C42—22no—
515C42—21no—
516C42—42no—
517C42—42no—
518C42—41no—
519C42—41no—
520C42—44no—
521C42—44no—
522C42—84no—
523C42#13514yes—
524C42#13514yes—
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—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C2×#12414yes—
547C23C23×D411yescharacteristic
548Q8—44no—
549Q8—44no—
550Q8—84no—
551Q8—162no—
552Q8—164no—
553D4—28no—
554D4—24no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—84no—
559D4—84no—
560D4—88no—
561D4—164no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—88no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4C2×#12414yes—
609C2×C4C2×#12412yes—
610C2×C4C2×#12412yes—
611C2×C4C2×#13512yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C22×#4112yes—
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4#58512yes—
620C2×C4#58511yescharacteristic
621C8—44no—
622C8—42no—
623C8—42no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C22×#12411yescharacteristic
646C22#348511yescharacteristic
647C22#348811yescharacteristic
648C22C22×#13511yescharacteristic
649C22C22×#13511yescharacteristic
650C22#358511yescharacteristic
651C22#358611yescharacteristic
652C4—22no—
653C4—22no—
654C4—24no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4#313114yes—
668C4#351312yes—
669C4#352112yes—
670C2—44no—
671C2—82no—
672C2—161no—
673C2#3584711yescharacteristic
674C2C2×#313111yescharacteristic
675C2#6062511yescharacteristic
676C2#6170011yescharacteristic
677C2#6170111yescharacteristic
678C2#6194911yescharacteristic
679C2#6195011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36415C2
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. #36194C2semidirect
3. #30205C2semidirect
4. #30188C2semidirect
5. #29347C2semidirect
6. #29107C2semidirect
7. #28716C2semidirect
8. #27976C2semidirect
9. #27748C2semidirect
10. #27728C2semidirect
11. #27698C2semidirect
12. #27257C2semidirect
13. #23815C2semidirect
14. #23627C2semidirect
15. #22489C2semidirect
16. #22417C2semidirect
17. #20555C2semidirect

Maximal subgroups

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

subgroupindex
#364152
#361942
#302052
#301882
#293472
#291072
#287162
#279762
#277482
#277282
#276982
#272572
#238152
#236272
#224892
#224172
#205552