Finite Group Databasebeta

← #6616263

Subgroups — #6616263

Order 512.

1737 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33107C211yesmaximal, characteristic
4#33094C212yesmaximal, complemented
5#27018C214yesmaximal, complemented
6#26710C214yesmaximal, complemented
7#26441C212yesmaximal, complemented
8#25903C211yesmaximal, characteristic
9#25277C212yesmaximal, complemented
10#24607C212yesmaximal, complemented
11#24302C211yesmaximal, characteristic, complemented
12#24067C212yesmaximal, complemented
13#23223C212yesmaximal, complemented
14#22971C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#484C211yesmaximal, characteristic, complemented
17#21391C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C2×#486C2212yes—
21C2×#484C2211yescharacteristic
22C2×#481C2212yes—
23C2×#477C2212yes—
24C2×#432C2212yes—
25#2856C2211yescharacteristic
26#2854C2212yes—
27#2552C2211yescharacteristic
28#2548C2211yescharacteristic
29#2512—21no—
30#2501—21no—
31#2479—22no—
32#2454C2212yes—
33#2448C2212yes—
34#2425—22no—
35#2369—21no—
36#2369—22no—
37#2369—21no—
38#2369C2212yes—
39#2369C2214yes—
40#2369C2212yes—
41#2366C2211yescharacteristic
42#2361—21no—
43#2354—21no—
44Q8×C4⋊C4C2211yescharacteristic
45C4×C22⋊Q8C2211yescharacteristic
46C2×#391—22no—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2088C2214yes—
54#2067C2214yes—
55#2059C2214yes—
56#2057C2212yes—
57#2042C2214yes—
58#2024C2212yes—
59#2023C2214yes—
60#1990C2212yes—
61#1989C2214yes—
62#1987C2212yes—
63#1983C2214yes—
64#1982C2212yes—
65#1948C2212yes—
66#1943C2214yes—
67#1939C2212yes—
68#1936C2214yes—
69#1928C2212yes—
70#1926C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2212yes—
74#1918C2214yes—
75#1918C2214yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1865C2214yes—
83#1862C2214yes—
84#1852C2212yes—
85#1852C2211yescharacteristic
86#1824C2211yescharacteristic
87#1817C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1762C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102#529—28no—
103#529—44no—
104C2×#126—22no—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×#123—22no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×#121—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42—22no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#486—28no—
131#484—24no—
132#481—44no—
133#477—28no—
134#432—44no—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2311yescharacteristic
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#399—22no—
154#399—22no—
155#398—22no—
156#397—22no—
157#396—22no—
158#394—22no—
159#391—24no—
160#391—24no—
161#391—24no—
162#391—24no—
163#391C2311yescharacteristic
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2311yescharacteristic
180#389C2314yes—
181#389C2312yes—
182#389C2312yes—
183#389C2314yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—21no—
189#386—22no—
190#386—22no—
191#385—21no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#380—22no—
197#380C2311yescharacteristic
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—24no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—21no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—82no—
240C22×Q8—82no—
241C22×Q8C2411yescharacteristic
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8#4118yes—
263#127—24no—
264#126—48no—
265#125—24no—
266#125—42no—
267#124—24no—
268#124—42no—
269#123—48no—
270#123—84no—
271C22⋊Q8—22no—
272C22⋊Q8#4114yes—
273#121—24no—
274#121—24no—
275#121—44no—
276#121—82no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—42no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—48no—
283C4×D4—44no—
284#117—48no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
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—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—44no—
346C2×C42—84no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—28no—
356#107—24no—
357#107—24no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#103—24no—
362#103—28no—
363#103—24no—
364#102—28no—
365#102—28no—
366#102—28no—
367#102—44no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—24no—
371#97—44no—
372C4×C8—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—22no—
391C24C22×D411yescharacteristic
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—816no—
402C2×Q8C2×#4112yes—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—82no—
449C22×C4#13714yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4C2511yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C2×#4118yes—
504C4⋊C4C2×#4112yes—
505C4⋊C42−1+414yes—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—44no—
528C42—44no—
529C42—42no—
530C42—44no—
531C42—42no—
532C42—84no—
533C42—84no—
534C42C2×#4114yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23#54612yes—
547C23C2×#13712yes—
548C23#57612yes—
549C23C23×D411yescharacteristic
550Q8—24no—
551Q8—42no—
552Q8—88no—
553Q8—88no—
554D4—44no—
555D4—44no—
556D4—88no—
557D4—84no—
558D4—162no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—416no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—84no—
614C2×C4#52512yes—
615C2×C4#52512yes—
616C2×C4#54612yes—
617C2×C4C2×#13712yes—
618C2×C4C2×#13712yes—
619C2×C4#57512yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4112yes—
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+412yes—
624C8—42no—
625C8—42no—
626C8—44no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22#333314yes—
641C22C2×#52511yescharacteristic
642C22C2×#54611yescharacteristic
643C22#348611yescharacteristic
644C22C22×#13711yescharacteristic
645C22C2×#57511yescharacteristic
646C22C2×#57611yescharacteristic
647C22#358511yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—42no—
652C4—44no—
653C4—41no—
654C4—48no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—81no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4#311714yes—
665C4#348212yes—
666C4#352212yes—
667C2—24no—
668C2—162no—
669C2#3642211yescharacteristic
670C2C2×#311711yescharacteristic
671C2#6057811yescharacteristic
672C2C2×#333311yescharacteristic
673C2#6116611yescharacteristic
674C2#6183011yescharacteristic
675C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
4. #33094C2semidirect
5. #27018C2semidirect
6. #26710C2semidirect
7. #26441C2semidirect
9. #25277C2semidirect
10. #24607C2semidirect
11. #24302C2semidirect
12. #24067C2semidirect
13. #23223C2semidirect
14. #22971C2semidirect
15. #22603C2semidirect
16. C4×#484C2semidirect
17. #21391C2semidirect

Maximal subgroups

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

subgroupindex
#361172
C2×#23692
#331072
#330942
#270182
#267102
#264412
#259032
#252772
#246072
#243022
#240672
#232232
#229712
#226032
C4×#4842
#213912