Finite Group Databasebeta

← #6848667

Subgroups — #6848667

Order 512.

1609 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32209C214yesmaximal, complemented
5#31500C214yesmaximal, complemented
6#29689C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#28599C212yesmaximal
9#28279C211yesmaximal, characteristic, complemented
10#27626C212yesmaximal, complemented
11#26445C212yesmaximal, complemented
12#25340C212yesmaximal, complemented
13#25277C212yesmaximal, complemented
14#24835C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21887C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#486C2212yes—
20C2×#477C2212yes—
21C2×#475C2211yescharacteristic
22C2×#474C2212yes—
23C2×C8×D4C2212yes—
24C22×C4⋊C8C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2571—21no—
28#2563C2212yes—
29#2554—22no—
30#2552C2211yescharacteristic
31#2548C2211yescharacteristic
32#2541C2211yescharacteristic
33#2538—21no—
34#2538C2211yescharacteristic
35#2512—21no—
36#2497C2214yes—
37#2492—22no—
38#2490C2212yes—
39#2445—21no—
40#2439—22no—
41#2426C2212yes—
42#2392—22no—
43#2369C2212yes—
44#2355—21no—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127—21no—
48C4×#127C2211yescharacteristic
49C4×#121C2212yes—
50#2293—21no—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2086C2214yes—
55#2076C2214yes—
56#2071C2214yes—
57#2067C2214yes—
58#2067C2214yes—
59#2059C2214yes—
60#2042C2214yes—
61#2036C2214yes—
62#2034C2214yes—
63#2034C2214yes—
64#1990C2212yes—
65#1989C2214yes—
66#1988C2211yescharacteristic
67#1987C2212yes—
68#1983C2214yes—
69#1982C2212yes—
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1925C2214yes—
76#1920C2214yes—
77#1919C2212yes—
78#1893C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1865C2212yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1808C2214yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×C8C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127—21no—
101C2×#127C2311yescharacteristic
102#529—28no—
103#529—44no—
104C2×#126—22no—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—22no—
126C22×C42C2312yes—
127#486—44no—
128#477—44no—
129#475—24no—
130#474—44no—
131C8×D4—44no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×C4⋊C8—44no—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#400—24no—
153#398—22no—
154#397—22no—
155#396—22no—
156#396—24no—
157#396—22no—
158#394—22no—
159#394—22no—
160#394—22no—
161#393—24no—
162#392—22no—
163#391—21no—
164#391—22no—
165#391—24no—
166#390—21no—
167#390—22no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389C2312yes—
183#389C2314yes—
184#389C2312yes—
185#388—24no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—24no—
190#386—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#382—22no—
200#380—22no—
201#380—22no—
202#380C2314yes—
203#380C2314yes—
204#380C2312yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43—21no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C22×C8—44no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—82no—
260C4⋊Q8—82no—
261#127—22no—
262#127—22no—
263#127—44no—
264#126—48no—
265#125—24no—
266#125—42no—
267#124—24no—
268#124—42no—
269#123—48no—
270C22⋊Q8—24no—
271C22⋊Q8—42no—
272C22⋊Q8—84no—
273#121—24no—
274#121—24no—
275#121—42no—
276#121—42no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—41no—
282C4×Q8—44no—
283C4×Q8#4114yes—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—82no—
293C4×D4—84no—
294#117—48no—
295#117—84no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
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—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—21no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—44no—
362#107—44no—
363#106—24no—
364#106—24no—
365C4⋊C8—22no—
366C4⋊C8—22no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370C4⋊C8—42no—
371#103—24no—
372#103—42no—
373#103—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378C22⋊C8—44no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—24no—
389#95—22no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24C22×D412yes—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8C2×#4112yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4#13714yes—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2511yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—88no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C2×#4112yes—
507C4⋊C42−1+412yes—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C42—24no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—41no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—81no—
540C42—84no—
541C42#13514yes—
542C42C2×#4112yes—
543C42H(5,𝔽2)12yes—
544C42H(5,𝔽2)12yes—
545C422−1+412yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C2×#13714yes—
560C23C23×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565Q8—84no—
566Q8—164no—
567D4—28no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—84no—
573D4—88no—
574D4—88no—
575D4—84no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—88no—
625C2×C4#53712yes—
626C2×C4#53712yes—
627C2×C4#54612yes—
628C2×C4#54612yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13512yes—
631C2×C4C2×#13512yes—
632C2×C4C2×#13712yes—
633C2×C4#57612yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58511yescharacteristic
640C8—42no—
641C8—44no—
642C8—42no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22C2×#53711yescharacteristic
658C22C2×#54611yescharacteristic
659C22#348911yescharacteristic
660C22C22×#13511yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57611yescharacteristic
663C22#358511yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4—84no—
679C4#324814yes—
680C4#330514yes—
681C4#350711yescharacteristic
682C4#351111yescharacteristic
683C4#354611yescharacteristic
684C4#355011yescharacteristic
685C2—44no—
686C2—162no—
687C2#3946811yescharacteristic
688C2C2×#324811yescharacteristic
689C2#6092211yescharacteristic
690C2C2×#330511yescharacteristic
691C2#6116211yescharacteristic
692C2#6210211yescharacteristic
693C2#6212711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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
1. #36194C2semidirect
3. #33198C2semidirect
4. #32209C2semidirect
5. #31500C2semidirect
6. #29689C2semidirect
7. #28716C2semidirect
9. #28279C2semidirect
10. #27626C2semidirect
11. #26445C2semidirect
12. #25340C2semidirect
13. #25277C2semidirect
14. #24835C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21887C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349342
#331982
#322092
#315002
#296892
#287162
#285992
#282792
#276262
#264452
#253402
#252772
#248352
D4×C4⋊C82
#218872
C4×#4752