Finite Group Databasebeta

← #6990320

Subgroups — #6990320

Order 512.

1731 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36287C211yesmaximal, characteristic, complemented
2#35285C211yesmaximal, characteristic
3#32818C212yesmaximal, complemented
4#31302C214yesmaximal, complemented
5#31012C214yesmaximal, complemented
6#30227C212yesmaximal, complemented
7#29801C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic, complemented
9#29561C212yesmaximal, complemented
10#28418C212yesmaximal, complemented
11#27758C212yesmaximal, complemented
12#25855C211yesmaximal, characteristic
13#25277C212yesmaximal, complemented
14#24733C212yesmaximal, complemented
15#24211C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic, complemented
17#22636C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C2×#486C2212yes—
21C2×#477C2212yescomplemented
22C2×#464C2214yescomplemented
23C2×#460C2212yes—
24C2×#450C2212yes—
25C2×C4×Q16C2212yescomplemented
26#2859C2212yescomplemented
27#2858C2212yes—
28#2858C2212yes—
29#2857C2211yescharacteristic
30#2641—22no—
31#2612—22no—
32#2567C2211yescharacteristic, complemented
33#2556—22no—
34#2552C2211yescharacteristic, complemented
35#2548—21no—
36#2548—21no—
37#2548C2212yes—
38#2548C2212yes—
39#2506—21no—
40#2497C2212yes—
41#2441—21no—
42#2435—22no—
43#2435C2214yes—
44#2423C2212yes—
45#2370—21no—
46#2369C2211yescharacteristic, complemented
47Q8×C4⋊C4C2211yescharacteristic, complemented
48#2346—21no—
49#2309—21no—
50#2309—21no—
51C4×C22⋊Q8C2211yescharacteristic
52#2295C2212yescomplemented
53C4×#117C2211yescharacteristic
54#2093C2214yescomplemented
55#2071C2214yescomplemented
56#2067C2214yescomplemented
57#2067C2214yescomplemented
58#2057C2211yescharacteristic, complemented
59#2057C2212yescomplemented
60#2043C2211yescharacteristic, complemented
61#2043C2212yescomplemented
62#2042C2214yescomplemented
63#2042C2214yescomplemented
64#2042C2214yescomplemented
65#2041C2212yes—
66#2041C2212yes—
67#2041C2212yes—
68#2036C2212yes—
69#2036C2214yes—
70#2024C2211yescharacteristic, complemented
71#2016C2211yescharacteristic, complemented
72#2015C2214yescomplemented
73#2014C2212yes—
74#2001C2212yes—
75#1983C2214yescomplemented
76#1982C2212yes—
77#1973C2214yescomplemented
78#1959C2214yescomplemented
79#1959C2214yescomplemented
80#1957C2212yes—
81#1939C2212yescomplemented
82#1936C2214yescomplemented
83#1925C2214yescomplemented
84#1924C2214yes—
85#1880C2214yescomplemented
86#1831C2212yescomplemented
87#1803C2212yescomplemented
88#1797C2211yescharacteristic
89#1786C2214yes—
90#1786C2212yes—
91#1770C2212yescomplemented
92C4×#103C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×Q16C2312yes—
99C22×SD16C2314yes—
100#533—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—21no—
105C2×#127—21no—
106C2×#127—44no—
107C2×#127C2312yes—
108C2×#127C2312yes—
109C2×#125—21no—
110C2×#125—21no—
111C2×#125—22no—
112C2×#124—22no—
113C2×#124—22no—
114C2×#124C2314yes—
115C2×#124C2314yes—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2311yescharacteristic
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×#117—21no—
125C2×#117D411yescharacteristic
126C2×#117D411yescharacteristic
127C22×C42C2311yescharacteristic
128#486—44no—
129#477—44no—
130#464—48no—
131#460—44no—
132#450—44no—
133C4×Q16—44no—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#399—22no—
157#398—22no—
158#398—22no—
159#396—22no—
160#395—22no—
161#394—24no—
162#394—24no—
163#394—24no—
164#394—24no—
165#391—21no—
166#391—22no—
167#391C2312yes—
168#391C2311yescharacteristic
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390C2314yes—
180#390C2314yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2311yescharacteristic
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#387—21no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—21no—
202#384—21no—
203#384C2312yes—
204#384C2311yescharacteristic
205#384C2312yes—
206#382—22no—
207#380—21no—
208#380C2312yes—
209#380C2311yescharacteristic
210#379—24no—
211#379—24no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376—22no—
226#376D412yes—
227C43C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235#336C2311yescharacteristic
236C25—22no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—82no—
255C22×D4—82no—
256C22×D4C2414yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×Q16—24no—
261C2×Q16—42no—
262C2×Q16—42no—
263C2×SD16—28no—
264C2×SD16—44no—
265C2×SD16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272#127—24no—
273#127—24no—
274#127—44no—
275#127—44no—
276#127—44no—
277#125—44no—
278#125—44no—
279#125—84no—
280#124—44no—
281#124—44no—
282#124—44no—
283#124—44no—
284#124—84no—
285#124—84no—
286C22⋊Q8—22no—
287C22⋊Q8—41no—
288C22⋊Q8—42no—
289C22⋊Q8—42no—
290C22⋊Q8—44no—
291C22⋊Q8—82no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—44no—
298#117—24no—
299#117—42no—
300#117—44no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
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—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—42no—
362#107—44no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—42no—
366#103—42no—
367#103—42no—
368#103—42no—
369#103—44no—
370#103—44no—
371#102—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378C22⋊C8—42no—
379C22⋊C8—42no—
380#97—44no—
381C4×C8—24no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—24no—
400C24—44no—
401C24—44no—
402C24—44no—
403C24—82no—
404C24—82no—
405C24—82no—
406C24—82no—
407C24C22×D411yescharacteristic
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—81no—
422C2×Q8—81no—
423C2×Q8H(5,𝔽2)12yes—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4—816no—
446C2×D4—88no—
447C2×D4—88no—
448C2×D4—816no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4#13714yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2511yescharacteristic
477Q16—48no—
478SD16—88no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—82no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—81no—
534C22⋊C4—82no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—42no—
548C42—42no—
549C42—82no—
550C42H(5,𝔽2)14yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—88no—
565C23—88no—
566C23—88no—
567C23—164no—
568C23—164no—
569C23—164no—
570C23—164no—
571C23C2×#13712yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—84no—
580Q8—162no—
581D4—44no—
582D4—44no—
583D4—44no—
584D4—44no—
585D4—88no—
586D4—88no—
587D4—88no—
588D4—88no—
589D4—88no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4#53512yes—
640C2×C4#53512yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4#57512yes—
644C2×C4#57612yes—
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)12yes—
647C2×C4#58512yes—
648C2×C4#58511yescharacteristic
649C8—44no—
650C8—42no—
651C8—42no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—1616no—
667C22C2×#53511yescharacteristic
668C22#348811yescharacteristic
669C22#350011yescharacteristic
670C22C22×#13711yescharacteristic
671C22C2×#57511yescharacteristic
672C22C2×#57611yescharacteristic
673C22#358511yescharacteristic
674C4—22no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—41no—
682C4—42no—
683C4—82no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4#321514yes—
691C4#354212yes—
692C4#355112yes—
693C2—42no—
694C2—164no—
695C2#4296211yescharacteristic
696C2C2×#321511yescharacteristic
697C2#6081111yescharacteristic
698C2#6196511yescharacteristic
699C2#6198811yescharacteristic
700C2#6253711yescharacteristic
701C2#6259011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
1. #36287C2semidirect
3. #32818C2semidirect
4. #31302C2semidirect
5. #31012C2semidirect
6. #30227C2semidirect
7. #29801C2semidirect
8. #29691C2semidirect
9. #29561C2semidirect
10. #28418C2semidirect
11. #27758C2semidirect
13. #25277C2semidirect
14. #24733C2semidirect
16. #23571C2semidirect
17. #22636C2semidirect
21. C2×#477C22semidirect
22. C2×#464C22semidirect
25. C2×C4×Q16C22semidirect
26. #2859C22semidirect
32. #2567C22semidirect
34. #2552C22semidirect
46. #2369C22semidirect
47. Q8×C4⋊C4C22semidirect
52. #2295C22semidirect
54. #2093C22semidirect
55. #2071C22semidirect
56. #2067C22semidirect
57. #2067C22semidirect
58. #2057C22semidirect
59. #2057C22semidirect
60. #2043C22semidirect
61. #2043C22semidirect
62. #2042C22semidirect
63. #2042C22semidirect
64. #2042C22semidirect
70. #2024C22semidirect
71. #2016C22semidirect
72. #2015C22semidirect
75. #1983C22semidirect
77. #1973C22semidirect
78. #1959C22semidirect
79. #1959C22semidirect
81. #1939C22semidirect
82. #1936C22semidirect
83. #1925C22semidirect
85. #1880C22semidirect
86. #1831C22semidirect
87. #1803C22semidirect
91. #1770C22semidirect
92. C4×#103C22semidirect
94. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#362872
#352852
#328182
#313022
#310122
#302272
#298012
#296912
#295612
#284182
#277582
#258552
#252772
#247332
#242112
#235712
#226362