Finite Group Databasebeta

← #6992708

Subgroups — #6992708

Order 512.

1825 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36445C211yesmaximal, characteristic, complemented
2#35273C211yesmaximal, characteristic
3#32760C212yesmaximal, complemented
4#31386C214yesmaximal, complemented
5#30705C214yesmaximal, complemented
6#30187C212yesmaximal, complemented
7#30168C212yesmaximal, complemented
8#29663C211yesmaximal, characteristic, complemented
9#28574C212yesmaximal, complemented
10#28141C212yesmaximal, complemented
11#27972C211yesmaximal, characteristic, complemented
12#27647C211yesmaximal, characteristic, complemented
13#25385C212yesmaximal, complemented
14#24992C212yesmaximal, complemented
15#24187C212yesmaximal, complemented
16#23570C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#557C2211yescharacteristic
19C2×#537—21no—
20C2×#532C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#483C2212yes—
23C2×#465C2214yes—
24C2×#461C2212yes—
25C2×C4×SD16C2212yes—
26C2×C8×D4C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2857C2212yes—
30#2856C2212yes—
31#2620—22no—
32#2542C2211yescharacteristic
33#2542C2212yes—
34#2538—21no—
35#2538C2212yes—
36#2538C2212yes—
37#2538C2211yescharacteristic
38#2537—21no—
39#2537C2212yes—
40#2454—22no—
41#2454—21no—
42#2441—21no—
43#2441—22no—
44#2416—22no—
45#2416C2214yes—
46Q8×C4⋊C4—21no—
47Q8×C4⋊C4C2211yescharacteristic
48#2341—21no—
49#2311—21no—
50C4×C22⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C2×#384C2211yescharacteristic
53#2100C2214yes—
54#2084C2212yes—
55#2077C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#2072C2212yes—
59#2071C2214yes—
60#2058C2211yescharacteristic
61#2043C2214yes—
62#2041C2211yescharacteristic
63#2035C2214yes—
64#2034C2212yes—
65#2024C2211yescharacteristic
66#2024C2212yes—
67#2014C2211yescharacteristic
68#2014C2212yes—
69#2013C2214yes—
70#2000C2212yes—
71#2000C2214yes—
72#2000C2214yes—
73#2000C2214yes—
74#1984C2212yes—
75#1956C2212yes—
76#1948C2212yes—
77#1926C2214yes—
78#1919C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1892C2214yes—
82#1880C2214yes—
83#1872C2214yes—
84#1869C2214yes—
85#1865C2214yes—
86#1834C2212yes—
87#1823C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2211yescharacteristic
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41—21no—
96C22×#41D411yescharacteristic
97C22×SD16C2314yes—
98C22×SD16C2312yes—
99C23×C8C2314yes—
100#557—44no—
101#537—48no—
102#532—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#125—22no—
109C2×#124—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×#121—24no—
118C2×#121—22no—
119C2×#121—22no—
120C2×#121—21no—
121C2×#121—22no—
122C2×#121C2314yes—
123C2×#121C2312yes—
124C4×#41—44no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×D4—21no—
132C2×C4×D4D412yes—
133C2×C4×D4D411yescharacteristic
134C2×C4×D4C2312yes—
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2312yes—
137C2×#117D411yescharacteristic
138C2×#117D412yes—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2312yes—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144#483—44no—
145#465—48no—
146#461—44no—
147C4×SD16—44no—
148C8×D4—44no—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2312yes—
164C2×C4×C8C2312yes—
165#399—22no—
166#399—22no—
167#399—22no—
168#396—24no—
169#396—24no—
170#396—22no—
171#396—22no—
172#396—22no—
173#394—24no—
174#394—24no—
175#391—22no—
176#391C2312yes—
177#391C2311yescharacteristic
178#390—21no—
179#390—22no—
180#390—21no—
181#390—22no—
182#390—22no—
183#390—22no—
184#390C2314yes—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—22no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—22no—
194#389C2311yescharacteristic
195#389C2314yes—
196#387—21no—
197#387—22no—
198#386—24no—
199#385—21no—
200#385—22no—
201#384—21no—
202#384—22no—
203#384—21no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#384C2312yes—
209#380—21no—
210#380—22no—
211#380—22no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380C2312yes—
217#380C2314yes—
218#380C2311yescharacteristic
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2311yescharacteristic
236#336C2314yes—
237C2×#41—42no—
238C2×#41—42no—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—44no—
242C2×#41SD1618yescomplemented
243C22×Q8—21no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2411yescharacteristic
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4—24no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D412yes—
253C22×D4C2412yes—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2412yes—
262C2×SD16—42no—
263C2×SD16—42no—
264C2×SD16—42no—
265C2×SD16—44no—
266C2×SD16—44no—
267C2×SD16—44no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8—44no—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276C4⋊Q8—44no—
277#127—42no—
278#125—48no—
279#125—44no—
280#124—44no—
281C22⋊Q8—24no—
282C22⋊Q8—22no—
283C22⋊Q8—41no—
284C22⋊Q8—42no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290#121—42no—
291#121—42no—
292#121—44no—
293#121—48no—
294#121—416no—
295#121—48no—
296#121—44no—
297#121—44no—
298#121—84no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—42no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—42no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—44no—
312#117—44no—
313#117—42no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—44no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#106—28no—
381#106—42no—
382C4⋊C8—24no—
383#103—44no—
384#103—42no—
385#103—44no—
386#103—44no—
387#103—44no—
388#102—24no—
389#102—42no—
390#102—44no—
391#102—42no—
392C22⋊C8—42no—
393C22⋊C8—44no—
394C22⋊C8—42no—
395C4×C8—24no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—22no—
404#95—21no—
405#95—24no—
406#95—24no—
407C24—22no—
408C24C22×D412yes—
409C24C22×D411yescharacteristic
410#41—416no—
411#41—816no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—21no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8C2×SD1614yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—48no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—84no—
447C2×D4—84no—
448C2×D4C2×SD1614yes—
449C2×D4C2×SD1618yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4C2×SD1614yes—
483C22×C4C2×SD1618yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2511yescharacteristic
493SD16—48no—
494SD16—88no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—88no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4H(5,𝔽2)12yes—
541C22⋊C4—24no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—81no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—24no—
561C42—42no—
562C42—42no—
563C42—44no—
564C42—44no—
565C42—42no—
566C42—42no—
567C42H(5,𝔽2)14yes—
568C23—21no—
569C23—22no—
570C23—22no—
571C23—21no—
572C23—21no—
573C23—22no—
574C23—21no—
575C23—22no—
576C23—21no—
577C23—22no—
578C23—21no—
579C23—22no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23C22×SD1614yes—
588C23C22×SD1612yes—
589C23C23×D411yescharacteristic
590Q8—28no—
591Q8—42no—
592Q8—44no—
593Q8—44no—
594Q8—42no—
595Q8—84no—
596D4—28no—
597D4—48no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—84no—
603D4—84no—
604D4—88no—
605D4—168no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—28no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—84no—
655C2×C4—88no—
656C2×C4—164no—
657C2×C4#53412yes—
658C2×C4#53412yes—
659C2×C4C22×SD1614yes—
660C2×C4C22×SD1612yes—
661C2×C4C22×SD1612yes—
662C2×C4#57512yes—
663C2×C4#57812yes—
664C2×C4C2×H(5,𝔽2)12yes—
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4#58511yescharacteristic
667C2×C4#58512yes—
668C8—42no—
669C8—42no—
670C8—44no—
671C22—22no—
672C22—24no—
673C22—24no—
674C22—22no—
675C22—22no—
676C22—24no—
677C22—41no—
678C22—42no—
679C22—41no—
680C22—42no—
681C22—41no—
682C22—42no—
683C22—42no—
684C22—41no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22C2×#53411yescharacteristic
693C22#348811yescharacteristic
694C22#350011yescharacteristic
695C22C23×SD1611yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57811yescharacteristic
698C22#358511yescharacteristic
699C4—22no—
700C4—24no—
701C4—22no—
702C4—22no—
703C4—42no—
704C4—42no—
705C4—41no—
706C4—42no—
707C4—82no—
708C4—81no—
709C4—82no—
710C4—81no—
711C4—81no—
712C4—82no—
713C4—84no—
714C4#318714yes—
715C4#353912yes—
716C4#354612yes—
717C2—42no—
718C2—44no—
719C2—162no—
720C2#4298511yescharacteristic
721C2C2×#318711yescharacteristic
722C2#6076411yescharacteristic
723C2#6195111yescharacteristic
724C2#6198511yescharacteristic
725C2#6252611yescharacteristic
726C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36445C2semidirect
3. #32760C2semidirect
4. #31386C2semidirect
5. #30705C2semidirect
6. #30187C2semidirect
7. #30168C2semidirect
8. #29663C2semidirect
9. #28574C2semidirect
10. #28141C2semidirect
11. #27972C2semidirect
12. #27647C2semidirect
13. #25385C2semidirect
14. #24992C2semidirect
15. #24187C2semidirect
16. #23570C2semidirect
17. #22469C2semidirect
242. C2×#41SD16semidirect

Maximal subgroups

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

subgroupindex
#364452
#352732
#327602
#313862
#307052
#301872
#301682
#296632
#285742
#281412
#279722
#276472
#253852
#249922
#241872
#235702
#224692