Finite Group Databasebeta

← #6811814

Subgroups — #6811814

Order 512.

1835 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3233C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic, complemented
3#34934C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic, complemented
5#30786C214yesmaximal, complemented
6#30642C214yesmaximal, complemented
7#29689C212yesmaximal, complemented
8#28814C212yesmaximal, complemented
9#28508C211yesmaximal, characteristic, complemented
10#28280C211yesmaximal, characteristic, complemented
11#27770C212yesmaximal, complemented
12#26237C212yesmaximal
13#25017C212yesmaximal, complemented
14#24700C212yesmaximal, complemented
15#24300C212yesmaximal, complemented
16#22795C211yesmaximal, characteristic, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#529C2212yes—
20#3233—216no—
21C2×#499C2214yes—
22C2×#488C2214yes—
23C2×#485C2214yes—
24C2×#482C2214yes—
25C2×#477C2211yescharacteristic
26C2×#475C2212yes—
27C2×#476C2212yes—
28C2×#474C2212yes—
29C2×#474C2212yes—
30C2×#431C2212yes—
31#2868C2211yescharacteristic
32#2857C2212yes—
33#2554—22no—
34#2552C2212yes—
35#2552C2211yescharacteristic
36#2548C2211yescharacteristic
37#2542—21no—
38#2541C2211yescharacteristic
39#2538—21no—
40#2538C2211yescharacteristic
41#2512—21no—
42#2490C2212yes—
43#2445—21no—
44#2441C2214yes—
45#2429—22no—
46#2426C2212yes—
47#2423—22no—
48#2392—22no—
49#2365—21no—
50#2354C2212yes—
51Q8×C4⋊C4—21no—
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127—21no—
54C4×#127C2211yescharacteristic
55C4×#121C2212yes—
56#2293—21no—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2093C2214yes—
60#2075C2214yes—
61#2065C2214yes—
62#2059C2214yes—
63#2059C2214yes—
64#2042C2214yes—
65#2036C2214yes—
66#2034C2214yes—
67#2033C2214yes—
68#2025C2214yes—
69#1988C2212yes—
70#1987C2211yescharacteristic
71#1982C2212yes—
72#1957C2212yes—
73#1954C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1947C2211yescharacteristic
78#1942C2214yes—
79#1934C2214yes—
80#1923C2214yes—
81#1921C2212yes—
82#1875C2214yes—
83#1863C2212yes—
84#1854C2214yes—
85#1845C2212yes—
86#1842C2211yescharacteristic
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C8×C4⋊C4C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96#533—28no—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127—21no—
103C2×#127C2311yescharacteristic
104#529—28no—
105#529—44no—
106C2×#126—22no—
107C2×#125C2312yes—
108C2×#125C2314yes—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2312yes—
115C2×#121C2312yes—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×#117—22no—
132C2×#117C2314yes—
133C22×C42C2312yes—
134#499—216no—
135#488—216no—
136#485—216no—
137#482—216no—
138#477D418yescomplemented
139#476—28no—
140#475—28no—
141#474—28no—
142#474—44no—
143#431—28no—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×#107C2314yes—
147C2×#107C2312yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2314yes—
166#399—24no—
167#398—22no—
168#398—24no—
169#397—22no—
170#396—22no—
171#396—24no—
172#396—22no—
173#394—22no—
174#394—22no—
175#394—22no—
176#392—22no—
177#391—21no—
178#391—22no—
179#390—21no—
180#390—22no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—22no—
185#390—22no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#389—22no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—22no—
195#389—24no—
196#389C2312yes—
197#389C2312yes—
198#389C2314yes—
199#388—24no—
200#388—22no—
201#388—22no—
202#388—22no—
203#387—24no—
204#386—22no—
205#384—21no—
206#384—22no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2312yes—
213#380—22no—
214#380C2312yes—
215#380C2314yes—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—22no—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4C2312yes—
232C43—21no—
233C43C2311yescharacteristic
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2312yes—
240#340C2312yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8C2411yescharacteristic
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C4⋊Q8C2×D414yes—
265#127—22no—
266#127—22no—
267#127—44no—
268#126—48no—
269#125—24no—
270#125—28no—
271#125—42no—
272#125—44no—
273#124—24no—
274#124—42no—
275#123—48no—
276C22⋊Q8—24no—
277C22⋊Q8—42no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—84no—
281#121—24no—
282#121—24no—
283#121—42no—
284#121—42no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—41no—
290C4×Q8—48no—
291C4×Q8—48no—
292C4×Q8—44no—
293C4×Q8C2×D414yes—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—42no—
300C4×D4—48no—
301C4×D4—44no—
302C4×D4—82no—
303C4×D4—84no—
304#117—28no—
305#117—48no—
306#117—44no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—28no—
375#107—24no—
376#107—44no—
377#107C2×D418yes—
378#106—28no—
379#106—28no—
380C4⋊C8—24no—
381C4⋊C8—22no—
382C4⋊C8—24no—
383C4⋊C8—42no—
384C4⋊C8C2×D414yes—
385#103—28no—
386#103—42no—
387#103C2×D418yes—
388#102—28no—
389#102—28no—
390#102—44no—
391#102—44no—
392C22⋊C8—28no—
393C22⋊C8—44no—
394C4×C8—28no—
395C4×C8—24no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—22no—
409C24C22×D412yes—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8C22×D412yes—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4—82no—
442C2×D4—82no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8C22×D414yes—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—82no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4C22×D414yes—
521C4⋊C4C22×D412yes—
522C4⋊C42−1+412yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—84no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—21no—
547C42—21no—
548C42—41no—
549C42—48no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—42no—
554C42—42no—
555C42—81no—
556C42#13514yes—
557C42C22×D412yes—
558C42H(5,𝔽2)12yes—
559C422−1+412yes—
560C422−1+412yes—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C23×D411yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—48no—
579Q8—41no—
580Q8—41no—
581Q8—88no—
582D4—44no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—84no—
588D4—88no—
589D4—88no—
590D4—84no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—84no—
639C2×C4—88no—
640C2×C4#53612yes—
641C2×C4#53612yes—
642C2×C4D4212yes—
643C2×C4D4212yes—
644C2×C4C2×#13512yes—
645C2×C4C2×#13512yes—
646C2×C4C2×#13512yes—
647C2×C4#57512yes—
648C2×C4#57612yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C2×C4#58511yescharacteristic
655C8—24no—
656C8—44no—
657C8—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22C2×#53611yescharacteristic
673C22C2×D4211yescharacteristic
674C22#348911yescharacteristic
675C22C22×#13511yescharacteristic
676C22C2×#57511yescharacteristic
677C22C2×#57611yescharacteristic
678C22#358611yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—41no—
689C4—42no—
690C4—82no—
691C4—84no—
692C4—82no—
693C4—81no—
694C4#323314yescomplemented
695C4#328714yes—
696C4#349711yescharacteristic
697C4#349911yescharacteristic
698C4#354711yescharacteristic
699C4#355011yescharacteristic
700C2—82no—
701C2—162no—
702C2#3934311yescharacteristic
703C2C2×#323311yescharacteristic
704C2#6088811yescharacteristic
705C2C2×#328711yescharacteristic
706C2#6106711yescharacteristic
707C2#6210111yescharacteristic
708C2#6214311yescharacteristic
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. C2×#3233C2semidirect
2. #36134C2semidirect
4. #33196C2semidirect
5. #30786C2semidirect
6. #30642C2semidirect
7. #29689C2semidirect
8. #28814C2semidirect
9. #28508C2semidirect
10. #28280C2semidirect
11. #27770C2semidirect
13. #25017C2semidirect
14. #24700C2semidirect
15. #24300C2semidirect
16. #22795C2semidirect
17. C4×#477C2semidirect
138. #477D4semidirect
694. C4#3233semidirect

Maximal subgroups

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

subgroupindex
C2×#32332
#361342
#349342
#331962
#307862
#306422
#296892
#288142
#285082
#282802
#277702
#262372
#250172
#247002
#243002
#227952
C4×#4772