Finite Group Databasebeta

← #6376068

Subgroups — #6376068

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#34222C211yesmaximal, characteristic, complemented
3C2×#2319C211yesmaximal, characteristic, complemented
4#22158C214yesmaximal, complemented
5#22127C214yesmaximal, complemented
6#22081C212yesmaximal, complemented
7#22075C212yesmaximal, complemented
8#22046C212yesmaximal, complemented
9#22040C212yesmaximal, complemented
10#21395C211yesmaximal, characteristic, complemented
11#21361C211yesmaximal, characteristic, complemented
12#21284C212yesmaximal, complemented
13#21282C212yesmaximal, complemented
14#21134C212yesmaximal, complemented
15#21129C212yesmaximal, complemented
16#20739C211yesmaximal, characteristic, complemented
17#20469C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3100C4116yescomplemented
21C2×#480C2211yescharacteristic
22C2×#480C2211yescharacteristic
23C2×#474C2212yes—
24C2×#459C2212yes—
25C2×#447C2211yescharacteristic
26C2×#447C2211yescharacteristic
27C2×C4×D8C2212yes—
28C2×C4×D8C2212yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2372—22no—
32#2364—22no—
33#2353—21no—
34#2353—21no—
35#2339—21no—
36#2339—21no—
37#2331C2212yescomplemented
38#2328C2212yescomplemented
39#2319—21no—
40#2319—21no—
41#2319—22no—
42#2319C2212yes—
43#2319C2214yes—
44#2319C2212yes—
45#2319C2211yescharacteristic, complemented
46#2319C2211yescharacteristic, complemented
47C4×#121C2212yes—
48C4×#121C2212yes—
49C2×#390C2212yes—
50C2×#389C2212yes—
51C2×#388—22no—
52#2283C2211yescharacteristic
53C2×#381—21no—
54C2×#380—21no—
55#2275C2211yescharacteristic, complemented
56#2275C2211yescharacteristic, complemented
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59#1987C2212yescomplemented
60#1982C2212yescomplemented
61#1972C2214yes—
62#1970C2214yes—
63#1949C2214yescomplemented
64#1943C2214yescomplemented
65#1942C2214yescomplemented
66#1936C2214yes—
67#1936C2214yescomplemented
68#1934C2214yes—
69#1934C2214yescomplemented
70#1926C2214yes—
71#1925C2214yes—
72#1921C2214yescomplemented
73#1918C2214yescomplemented
74#1879C2214yes—
75#1879C2214yes—
76#1874C2212yes—
77#1874C2212yes—
78#1864C2214yescomplemented
79#1862C2214yescomplemented
80#1853C2212yes—
81#1852C2212yes—
82#1823C2212yes—
83#1804C2214yes—
84#1786C2212yes—
85C4×#107C2212yescomplemented
86C4×#106C2212yescomplemented
87#1768C2214yescomplemented
88#1762C2212yes—
89#1762C2212yes—
90#1762C2212yescomplemented
91#1762C2212yescomplemented
92#1749C2212yes—
93C4×C22⋊C8C2212yes—
94#1746C2212yescomplemented
95#1726C2212yescomplemented
96C22×#41—21no—
97C23×D4C2311yescharacteristic
98C24×C4—21no—
99C22×D8C2312yes—
100C22×D8C2312yes—
101#532—24no—
102#532C2×C418yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×#121—21no—
107C2×#121C2311yescharacteristic
108C2×#121C2311yescharacteristic
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4C2×C418yescomplemented
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42—21no—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#480C2×C418yescomplemented
132#480C2×C418yescomplemented
133#474C2×C4116yescomplemented
134#459C2×C4116yescomplemented
135#447C2×C418yescomplemented
136#447C2×C418yescomplemented
137C4×D8C2×C4116yescomplemented
138C4×D8C2×C4116yescomplemented
139#432—24no—
140#431C2×C418yescomplemented
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×#97C2312yes—
159C2×#97C2314yes—
160C2×C4×C8C2314yes—
161C2×C4×C8C2312yes—
162#391—22no—
163#391—22no—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#390C2314yes—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—24no—
179#388—24no—
180#388—24no—
181#388—24no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#383—21no—
193#382C2312yes—
194#381—21no—
195#381—22no—
196#381—22no—
197#381—24no—
198#381—21no—
199#381—21no—
200#380—22no—
201#380—22no—
202#380—24no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221#376—21no—
222#376D411yescharacteristic
223#376D411yescharacteristic
224#376D412yes—
225#376C2311yescharacteristic
226#376C2312yes—
227#376C2311yescharacteristic
228C2×#95—21no—
229C2×#95—22no—
230C2×#95—21no—
231C2×#95—21no—
232C2×#95D411yescharacteristic
233C2×#95D411yescharacteristic
234C2×#95D412yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C25—21no—
244C2×#41—28no—
245C2×#41—84no—
246C22×Q8—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—44no—
254C22×D4C22×C414yes—
255C22×D4C2411yescharacteristic
256C22×D4C2411yescharacteristic
257C22×D4C2412yes—
258C23×C4—21no—
259C23×C4—28no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—84no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D412yes—
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×D8—44no—
274C2×D8—44no—
275C2×D8C22×C418yes—
276C2×D8C22×C418yes—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—22no—
281C4⋊Q8C22×C414yes—
282#127—22no—
283#127C22×C414yes—
284#124—28no—
285#121—28no—
286#121—22no—
287#121—22no—
288#121C22×C414yes—
289#121C22×C414yes—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—84no—
300C4×D4—84no—
301C4×D4C22×C418yes—
302C4×D4C22×C418yes—
303C4×D4C22×C414yes—
304C4×D4C22×C414yes—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—84no—
315C2×C4⋊C4C22×C414yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—28no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—84no—
341C2×C22⋊C4—84no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42—82no—
350C2×C42C22×C414yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42#4114yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#107C22×C418yes—
362#107C22×C418yes—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8C22×C418yes—
366#102—44no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102C22×C418yes—
371#102C22×C418yes—
372#102C22×C418yes—
373#102C22×C418yes—
374C22⋊C8—24no—
375C22⋊C8C22×C414yes—
376C22⋊C8C22×C414yes—
377#97—24no—
378#97—44no—
379C4×C8—44no—
380C4×C8C22×C418yes—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D411yescharacteristic
395#95C2×D411yescharacteristic
396#95C2×D412yes—
397#95C2×D411yescharacteristic
398#95C2×D412yes—
399#95C2×D411yescharacteristic
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—24no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24—42no—
409C24—44no—
410C24—42no—
411C24—44no—
412C24C22×D411yescharacteristic
413#41—816no—
414C2×Q8—24no—
415C2×Q8—82no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—81no—
434C2×D4—81no—
435C2×D4—82no—
436C2×D4C23×C414yes—
437C2×D4C23×C412yes—
438C2×D4C23×C412yes—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—816no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4C4×D414yes—
476C22×C4C4×D414yes—
477C22×C4C23×C412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2511yescharacteristic
489D8—48no—
490D8—48no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8C23×C414yes—
497C2×C8C23×C414yes—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—44no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4C23×C414yes—
513C4⋊C4C23×C412yes—
514C4⋊C4C23×C412yes—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C42—24no—
533C42—24no—
534C42—42no—
535C42—42no—
536C42—81no—
537C42—81no—
538C42—82no—
539C42—82no—
540C42—82no—
541C42—82no—
542C42C4×D414yes—
543C42C4×D414yes—
544C42C23×C412yes—
545C42C23×C412yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—48no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—88no—
570C23—88no—
571C23C2×C4×D412yes—
572C23#57614yes—
573C23C23×D411yescharacteristic
574Q8—84no—
575D4—24no—
576D4—24no—
577D4—24no—
578D4—24no—
579D4—48no—
580D4—48no—
581D4—48no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4—84no—
586D4—84no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—88no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4C2×C4×D412yes—
636C2×C4C2×C4×D412yes—
637C2×C4C2×C4×D412yes—
638C2×C4C2×C4×D412yes—
639C2×C4C2×C4×D412yes—
640C2×C4C24×C411yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4#58514yes—
643C8—24no—
644C8—24no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—816no—
660C22—82no—
661C22—82no—
662C22—84no—
663C22—82no—
664C22—82no—
665C22—84no—
666C22#294814yes—
667C22C22×C4×D411yescharacteristic
668C22#344912yes—
669C22C2×#57612yes—
670C22#358511yescharacteristic
671C22#358611yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4#342812yes—
688C4#345512yes—
689C2—24no—
690C2—82no—
691C2—82no—
692C2—84no—
693C2#3406511yescharacteristic
694C2C2×#294811yescharacteristic
695C2#6017011yescharacteristic
696C2#6097612yes—
697C2#6097812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34222C2semidirect
3. C2×#2319C2semidirect
4. #22158C2semidirect
5. #22127C2semidirect
6. #22081C2semidirect
7. #22075C2semidirect
8. #22046C2semidirect
9. #22040C2semidirect
10. #21395C2semidirect
11. #21361C2semidirect
12. #21284C2semidirect
13. #21282C2semidirect
14. #21134C2semidirect
15. #21129C2semidirect
16. #20739C2semidirect
17. #20469C2semidirect
20. #3100C4semidirect
37. #2331C22semidirect
38. #2328C22semidirect
45. #2319C22semidirect
46. #2319C22semidirect
55. #2275C22semidirect
56. #2275C22semidirect
59. #1987C22semidirect
60. #1982C22semidirect
63. #1949C22semidirect
64. #1943C22semidirect
65. #1942C22semidirect
67. #1936C22semidirect
69. #1934C22semidirect
72. #1921C22semidirect
73. #1918C22semidirect
78. #1864C22semidirect
79. #1862C22semidirect
85. C4×#107C22semidirect
86. C4×#106C22semidirect
87. #1768C22semidirect
90. #1762C22semidirect
91. #1762C22semidirect
94. #1746C22semidirect
95. #1726C22semidirect
115. C2×C4×D4C2×C4semidirect
131. #480C2×C4semidirect
132. #480C2×C4semidirect
133. #474C2×C4semidirect
134. #459C2×C4semidirect
135. #447C2×C4semidirect
136. #447C2×C4semidirect
137. C4×D8C2×C4semidirect
138. C4×D8C2×C4semidirect
140. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#342222
C2×#23192
#221582
#221272
#220812
#220752
#220462
#220402
#213952
#213612
#212842
#212822
#211342
#211292
#207392
#204692