Finite Group Databasebeta

← #6488718

Subgroups — #6488718

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#33016C214yesmaximal, complemented
4#30144C214yesmaximal, complemented
5#29383C214yesmaximal, complemented
6#29191C211yesmaximal, characteristic, complemented
7#29190C211yesmaximal, characteristic, complemented
8#28616C211yesmaximal, characteristic, complemented
9#28613C211yesmaximal, characteristic, complemented
10#28133C214yesmaximal, complemented
11#27961C211yesmaximal, characteristic, complemented
12#27905C211yesmaximal, characteristic, complemented
13#25857C211yesmaximal, characteristic
14#25719C211yesmaximal, characteristic
15#22270C211yesmaximal, characteristic, complemented
16#22268C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic, complemented
18#22218C211yesmaximal, characteristic, complemented
19#20782C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#520C2212yes—
22C2×#494C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#483C2214yes—
27C2×#452C2214yes—
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2645C2212yes—
33#2625C2212yes—
34#2514—21no—
35#2509—21no—
36#2508—21no—
37#2495—21no—
38#2494—21no—
39#2489—21no—
40#2453—21no—
41#2421—21no—
42#2361—21no—
43#2361—21no—
44#2361—21no—
45#2361—21no—
46#2354—21no—
47#2354—21no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50#2326C2212yes—
51#2326C2212yes—
52#2323C2212yes—
53#2320C2212yes—
54#2311C2212yes—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C2×#384C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2100C2214yes—
64#2096C2214yes—
65#2066C2214yes—
66#2065C2214yes—
67#2056C2214yes—
68#2055C2214yes—
69#2042C2214yes—
70#2015C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1927C2214yes—
80#1926C2214yes—
81#1880C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1877C2214yes—
86#1877C2214yes—
87#1865C2214yes—
88#1865C2214yes—
89#1864C2214yes—
90#1864C2214yes—
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1840C2212yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1832C2212yes—
97#1824C2214yes—
98#1818C2212yes—
99#1798C2212yes—
100#1788C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×#41C2312yes—
108C22×Q16C2314yes—
109C22×SD16C2314yes—
110#532—24no—
111#532—24no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×#127C2311yescharacteristic
116C2×#125—21no—
117C2×#124C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—21no—
122#520—48no—
123C2×C4×Q8—21no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×#117—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—22no—
143C22×C4⋊C4C2312yes—
144C22×C42C2311yescharacteristic
145#494—24no—
146#492—24no—
147#487—24no—
148#486—24no—
149#483—48no—
150#452—48no—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#401—22no—
170#400—22no—
171#400—22no—
172#400—22no—
173#400—22no—
174#398—22no—
175#397—22no—
176#397—22no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#386—22no—
220#386—22no—
221#385—22no—
222#385—22no—
223#385—22no—
224#385—22no—
225#384—21no—
226#384—21no—
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2312yes—
232#383—22no—
233#382—21no—
234#382—21no—
235#382—21no—
236#382—22no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—22no—
241#380—22no—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259#376C2311yescharacteristic
260#376C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C2×#41—24no—
276C2×#41—24no—
277C2×#41—28no—
278C2×#41—44no—
279C22×Q8—21no—
280C22×Q8C2412yes—
281C22×Q8C2412yes—
282C22×D4—22no—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4C2412yes—
286C23×C4—22no—
287C23×C4—22no—
288C23×C4—22no—
289C23×C4—22no—
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D411yescharacteristic
292C23×C4C2411yescharacteristic
293C2×Q16—28no—
294C2×Q16—44no—
295C2×Q16—44no—
296C2×SD16—28no—
297C2×SD16—44no—
298C2×SD16—44no—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8#4114yes—
309#127—22no—
310#127#4114yes—
311#125—82no—
312#124—24no—
313#124—24no—
314C22⋊Q8—28no—
315C22⋊Q8—82no—
316#121—28no—
317#121—28no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—82no—
322C4×D4—28no—
323C4×D4—28no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—48no—
327C4×D4—48no—
328C4×D4—82no—
329C4×D4—82no—
330C4×D4—82no—
331C4×D4—82no—
332#117—24no—
333#117—42no—
334#117—48no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—48no—
361C2×C4⋊C4—82no—
362C2×C4⋊C4—82no—
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408#106—44no—
409#106—44no—
410C4⋊C8—42no—
411C4⋊C8—42no—
412#103—28no—
413#103—44no—
414#103—44no—
415#103—44no—
416#103—44no—
417#102—28no—
418#102—44no—
419#102—44no—
420C22⋊C8—24no—
421C22⋊C8—24no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452C24—22no—
453C24C22×D411yescharacteristic
454#41—416no—
455#41—416no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8C2×#4114yes—
467C2×D4—28no—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4C2×#4114yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—28no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4H(5,𝔽2)14yes—
522C22×C4C2511yescharacteristic
523Q16—88no—
524SD16—88no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C22⋊C4—28no—
572C22⋊C4—28no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—41no—
576C22⋊C4—41no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—21no—
596C42—21no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—44no—
602C42—44no—
603C42—44no—
604C42—82no—
605C42C2×#4112yes—
606C42C2×#4112yes—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—24no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23#53512yes—
621C23C23×D411yescharacteristic
622Q8—28no—
623Q8—44no—
624Q8—44no—
625Q8—44no—
626Q8—84no—
627D4—28no—
628D4—416no—
629D4—48no—
630D4—48no—
631D4—44no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—416no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—164no—
691C2×C4—164no—
692C2×C4—164no—
693C2×C4#52512yes—
694C2×C4#52512yes—
695C2×C4#53512yes—
696C2×C4#57712yes—
697C2×C4#57712yes—
698C2×C4#57712yes—
699C2×C4#57812yes—
700C2×C4C22×#4111yescharacteristic
701C2×C4C22×#4111yescharacteristic
702C2×C4C2×H(5,𝔽2)12yes—
703C2×C4#58512yes—
704C8—42no—
705C8—42no—
706C8—42no—
707C8—42no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22C2×#52511yescharacteristic
723C22C2×#53511yescharacteristic
724C22#344911yescharacteristic
725C22C2×#57711yescharacteristic
726C22C2×#57711yescharacteristic
727C22C2×#57711yescharacteristic
728C22C2×#57811yescharacteristic
729C4—22no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—44no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4#312014yes—
748C4#312514yes—
749C4#351312yes—
750C4#351312yes—
751C2—42no—
752C2—84no—
753C2#3523311yescharacteristic
754C2C2×#312011yescharacteristic
755C2C2×#312511yescharacteristic
756C2#6081811yescharacteristic
757C2#6082411yescharacteristic
758C2#6100811yescharacteristic
759C2#6101511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #35291C2semidirect
3. #33016C2semidirect
4. #30144C2semidirect
5. #29383C2semidirect
6. #29191C2semidirect
7. #29190C2semidirect
8. #28616C2semidirect
9. #28613C2semidirect
10. #28133C2semidirect
11. #27961C2semidirect
12. #27905C2semidirect
15. #22270C2semidirect
16. #22268C2semidirect
17. #22219C2semidirect
18. #22218C2semidirect
19. #20782C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#342072
#330162
#301442
#293832
#291912
#291902
#286162
#286132
#281332
#279612
#279052
#258572
#257192
#222702
#222682
#222192
#222182
#207822