Finite Group Databasebeta

← #6811853

Subgroups — #6811853

Order 512.

1751 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3#34929C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic, complemented
5#30790C214yesmaximal, complemented
6#30650C214yesmaximal, complemented
7#29691C212yesmaximal, complemented
8#28816C212yesmaximal, complemented
9#28511C211yesmaximal, characteristic, complemented
10#28283C211yesmaximal, characteristic, complemented
11#27777C212yesmaximal, complemented
12#26233C212yesmaximal
13#25017C212yesmaximal
14#24711C212yesmaximal, complemented
15#24303C212yesmaximal, complemented
16#22799C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#533C2212yes—
20#3237—216no—
21C2×#501C2214yes—
22C2×#491C2214yes—
23C2×#485C2214yes—
24C2×#485C2214yes—
25C2×#477C2212yes—
26C2×#477C2212yes—
27C2×#477C2212yes—
28C2×#475C2212yes—
29C2×#475C2211yescharacteristic
30C2×#432C2212yes—
31#2868C2211yescharacteristic
32#2857C2212yes—
33#2556—22no—
34#2552C2212yes—
35#2551C2211yescharacteristic
36#2546C2211yescharacteristic
37#2542—21no—
38#2542C2211yescharacteristic
39#2538—21no—
40#2538C2211yescharacteristic
41#2506—21no—
42#2492C2212yes—
43#2452—21no—
44#2441C2214yes—
45#2429—22no—
46#2429C2212yes—
47#2423—22no—
48#2396—22no—
49#2365—21no—
50#2354C2212yes—
51Q8×C4⋊C4—21no—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C22⋊Q8C2212yes—
56#2293—21no—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2095C2214yes—
60#2075C2214yes—
61#2066C2214yes—
62#2059C2214yes—
63#2057C2214yes—
64#2043C2214yes—
65#2036C2214yes—
66#2036C2214yes—
67#2035C2214yes—
68#2026C2214yes—
69#1988C2211yescharacteristic
70#1987C2212yes—
71#1981C2212yes—
72#1956C2212yes—
73#1954C2212yes—
74#1948C2211yescharacteristic
75#1948C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1944C2214yes—
79#1935C2214yes—
80#1924C2214yes—
81#1919C2212yes—
82#1873C2214yes—
83#1865C2212yes—
84#1854C2214yes—
85#1845C2212yes—
86#1842C2211yescharacteristic
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93#1770C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96#533—28no—
97#533—28no—
98#533—44no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—22no—
109C2×#125C2314yes—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117C2314yes—
130C22×C42C2312yes—
131#501—216no—
132#491—216no—
133#485—216no—
134#485—216no—
135#477—28no—
136#477—28no—
137#477—44no—
138#475—28no—
139#475D418yescomplemented
140#432—28no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97C2314yes—
162C2×C4×C8C2312yes—
163#400—22no—
164#399—22no—
165#399—24no—
166#398—22no—
167#398—22no—
168#398—24no—
169#396—22no—
170#396—22no—
171#396—24no—
172#395—22no—
173#393—22no—
174#391—21no—
175#391—22no—
176#391—22no—
177#391C2312yes—
178#391C2312yes—
179#390—22no—
180#390—21no—
181#390—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389—24no—
193#389C2312yes—
194#389C2314yes—
195#389C2312yes—
196#388—24no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—24no—
201#385—22no—
202#384—21no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2312yes—
209#384C2312yes—
210#380—22no—
211#380C2312yes—
212#380C2314yes—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
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#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2312yes—
240#340C2312yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267C4⋊Q8C2×D414yes—
268#125—28no—
269#125—44no—
270#125—48no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—42no—
275C22⋊Q8—42no—
276C22⋊Q8—42no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—84no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—48no—
288C4×Q8—41no—
289C4×Q8—48no—
290C4×Q8—48no—
291C4×Q8—44no—
292C4×Q8C2×D414yes—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—42no—
297#117—28no—
298#117—48no—
299#117—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—28no—
368#107—28no—
369#107—44no—
370#106—28no—
371#106—24no—
372#106C2×D418yes—
373C4⋊C8—24no—
374C4⋊C8—22no—
375C4⋊C8—24no—
376C4⋊C8—42no—
377C4⋊C8C2×D414yes—
378#103—28no—
379#103—28no—
380#103—28no—
381#103—42no—
382#103—44no—
383#103—44no—
384#103C2×D418yes—
385C22⋊C8—28no—
386C22⋊C8—44no—
387#97—28no—
388C4×C8—24no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24C22×D412yes—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—82no—
426C2×Q8C22×D412yes—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
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—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8C22×D414yes—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—84no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4C22×D414yes—
512C4⋊C4C22×D412yes—
513C4⋊C42−1+412yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—84no—
523C42—24no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—41no—
533C42—48no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—44no—
538C42—42no—
539C42—42no—
540C42—81no—
541C42#13714yes—
542C42C22×D412yes—
543C42H(5,𝔽2)12yes—
544C422−1+412yes—
545C422−1+412yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23C23×D411yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—48no—
557Q8—41no—
558Q8—41no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—88no—
564Q8—88no—
565D4—44no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4#53612yes—
618C2×C4#53612yes—
619C2×C4D4212yes—
620C2×C4D4212yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57512yes—
625C2×C4#57812yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4#58511yescharacteristic
632C8—24no—
633C8—44no—
634C8—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22C2×#53611yescharacteristic
643C22C2×D4211yescharacteristic
644C22#348911yescharacteristic
645C22C22×#13711yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57811yescharacteristic
648C22#358511yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—84no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4#323714yescomplemented
666C4#327614yes—
667C4#349711yescharacteristic
668C4#349911yescharacteristic
669C4#354711yescharacteristic
670C4#355011yescharacteristic
671C2—82no—
672C2#3934311yescharacteristic
673C2C2×#323711yescharacteristic
674C2#6089311yescharacteristic
675C2C2×#327611yescharacteristic
676C2#6107011yescharacteristic
677C2#6211411yescharacteristic
678C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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×#3237C2semidirect
4. #33198C2semidirect
5. #30790C2semidirect
6. #30650C2semidirect
7. #29691C2semidirect
8. #28816C2semidirect
9. #28511C2semidirect
10. #28283C2semidirect
11. #27777C2semidirect
14. #24711C2semidirect
15. #24303C2semidirect
17. C4×#475C2semidirect
139. #475D4semidirect
665. C4#3237semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
#361342
#349292
#331982
#307902
#306502
#296912
#288162
#285112
#282832
#277772
#262332
#250172
#247112
#243032
#227992
C4×#4752