Finite Group Databasebeta

← #6447227

Subgroups — #6447227

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3#26303C212yesmaximal, complemented
4#26302C212yesmaximal, complemented
5#26301C212yesmaximal, complemented
6#26300C212yesmaximal, complemented
7#26299C212yesmaximal, complemented
8#26298C212yesmaximal, complemented
9#26297C212yesmaximal, complemented
10#26296C212yesmaximal, complemented
11#23643C212yesmaximal, complemented
12#23641C212yesmaximal, complemented
13C2×#1990C211yesmaximal, characteristic, complemented
14#21313C211yesmaximal, characteristic, complemented
15#21187C211yesmaximal, characteristic
16#21171C211yesmaximal, characteristic, complemented
17#20849C212yesmaximal, complemented
18#20703C212yesmaximal, complemented
19C4×#427C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#520C2212yes—
23C2×#428C2212yes—
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#408C2212yes—
28C2×#408C2212yes—
29C22×C4×C8C2211yescharacteristic
30#2862—21no—
31#2858—21no—
32#2640—22no—
33#2563C2212yes—
34#2563C2212yes—
35#2547C2212yes—
36#2547C2212yes—
37#2504—22no—
38#2436—22no—
39#2361—22no—
40#2361—22no—
41#2354C2212yes—
42#2354C2212yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45#2326—22no—
46#2311C2212yes—
47#2311C2212yes—
48#2311C2212yes—
49C4×C4⋊Q8—21no—
50C4×#127—21no—
51C2×#384C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2077C2214yes—
55#2077C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2076C2214yes—
59#2076C2214yes—
60#2075C2214yes—
61#2075C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1990C2212yes—
65#1990C2212yes—
66#1988C2212yes—
67#1988C2212yes—
68#1987C2212yes—
69#1987C2212yes—
70#1954C2212yes—
71#1954C2212yes—
72#1954C2212yes—
73#1954C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1854C2214yes—
79#1854C2214yes—
80#1853C2214yes—
81#1852C2214yes—
82#1825C2214yes—
83#1825C2214yes—
84#1824C2214yes—
85#1823C2214yes—
86C4×#107C2211yescharacteristic
87C4×#107C2211yescharacteristic
88C4×#107C2211yescharacteristic
89C4×#107C2211yescharacteristic
90C4×#106C2211yescharacteristic
91C4×#106C2211yescharacteristic
92C4×#106C2211yescharacteristic
93C4×#106C2211yescharacteristic
94#1736C2212yes—
95#1736C2212yes—
96#1736C2212yes—
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99C2×#340C2212yes—
100C2×#340C2212yes—
101C22×#41C2312yes—
102C23×Q8—21no—
103C22×M16C2312yes—
104C23×C8C2311yescharacteristic
105C23×C8C2311yescharacteristic
106#532—24no—
107#532—24no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—24no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8D411yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#127D411yescharacteristic
116C2×#124—22no—
117C2×#124—22no—
118C2×#124D412yes—
119C2×#121D412yes—
120C2×#121D412yes—
121#520—28no—
122#520—44no—
123C2×C4×Q8—22no—
124C2×C4×D4—22no—
125C2×C4×D4—24no—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2314yes—
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2314yes—
141C22×C42C2311yescharacteristic
142#428—44no—
143#427—24no—
144#427—24no—
145C2×#107C2311yescharacteristic
146C2×#107C2311yescharacteristic
147C2×#107C2311yescharacteristic
148C2×#107C2311yescharacteristic
149C2×#107C2314yes—
150C2×#106C2311yescharacteristic
151C2×#106C2311yescharacteristic
152C2×#106C2311yescharacteristic
153C2×#106C2311yescharacteristic
154C2×#106C2314yes—
155#423—24no—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158#408—28no—
159#408—44no—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8—24no—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#397—24no—
170#397—24no—
171#393—24no—
172#393—24no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390—22no—
184#390—22no—
185#389—22no—
186#389—22no—
187#388—24no—
188#388—24no—
189#386—24no—
190#386—24no—
191#386—24no—
192#385—24no—
193#385—24no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#382—22no—
200#380C2314yes—
201#380C2314yes—
202#380C2314yes—
203#380C2314yes—
204#380C2314yes—
205#380C2314yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220#376—21no—
221#376—21no—
222C43—21no—
223C43—21no—
224#340C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2311yescharacteristic
234#340C2311yescharacteristic
235#340C2314yes—
236#340C2311yescharacteristic
237#340C2314yes—
238#340C2311yescharacteristic
239#340C2311yescharacteristic
240#340C2311yescharacteristic
241#340C2314yes—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—24no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—44no—
251C22×Q8—82no—
252C22×Q8—82no—
253C22×Q8C2×D412yes—
254C22×D4C2×D412yes—
255C22×D4C2×D412yes—
256C22×D4C2×D414yes—
257C22×D4C2×D412yes—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—24no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2414yes—
264C23×C4C2411yescharacteristic
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8#4114yes—
274C22×C8C2411yescharacteristic
275C22×C8C2411yescharacteristic
276C22×C8C2414yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286#127—24no—
287#124—28no—
288#124—84no—
289#124—84no—
290#121—28no—
291#121—28no—
292C4×Q8—48no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—48no—
298C4×D4—416no—
299C4×D4—84no—
300C4×D4—84no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—28no—
309#117—44no—
310#117#4114yes—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4C2×D414yes—
353C2×C22⋊C4C2×D414yes—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42#4114yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—22no—
374#107—22no—
375#107—22no—
376#107—22no—
377#107—44no—
378#106—22no—
379#106—22no—
380#106—22no—
381#106—22no—
382#106—44no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C22⋊C8—28no—
386C22⋊C8—28no—
387C22⋊C8—44no—
388C22⋊C8—44no—
389C4×C8—22no—
390C4×C8—22no—
391C4×C8—22no—
392C4×C8—22no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407C24C22×D414yes—
408C24C22×D411yescharacteristic
409#41—48no—
410#41—48no—
411#41—48no—
412#41—48no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8—816no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4#12314yes—
465C22×C4#12314yes—
466C22×C4C22×D414yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+412yes—
478C22×C4C2511yescharacteristic
479M16—48no—
480C2×C8—28no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—24no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—44no—
495C2×C8C2×#4112yes—
496C2×C8C2×#4112yes—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4C2×#4112yes—
527C4⋊C4C2×#4112yes—
528C22⋊C4—24no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—48no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—24no—
542C42—48no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42#12314yes—
555C42#12314yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23C2×#12312yes—
572C23#54612yes—
573C23#54712yes—
574C23C23×D411yescharacteristic
575Q8—48no—
576Q8—88no—
577Q8—88no—
578D4—416no—
579D4—48no—
580D4—48no—
581D4—48no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—88no—
626C2×C4—168no—
627C2×C4—164no—
628C2×C4C2×#12312yes—
629C2×C4C2×#12312yes—
630C2×C4C2×#12312yes—
631C2×C4C2×#12312yes—
632C2×C4C2×#12312yes—
633C2×C4#52914yes—
634C2×C4#54214yes—
635C2×C4#54612yes—
636C2×C4#54712yes—
637C2×C4#55614yes—
638C2×C4#55914yes—
639C2×C4#57612yes—
640C2×C4#57712yes—
641C2×C4#57712yes—
642C2×C4#57812yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C8—22no—
650C8—22no—
651C8—41no—
652C8—41no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22#244714yes—
668C22C22×#12311yescharacteristic
669C22C2×#54611yescharacteristic
670C22C2×#54711yescharacteristic
671C22C2×#57611yescharacteristic
672C22C2×#57711yescharacteristic
673C22C2×#57711yescharacteristic
674C22C2×#57811yescharacteristic
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—81no—
681C4—84no—
682C4—81no—
683C4—84no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4#244714yes—
689C4#309614yes—
690C4#309714yes—
691C4C2×#52911yescharacteristic
692C4C2×#54211yescharacteristic
693C4C2×#55611yescharacteristic
694C4C2×#55911yescharacteristic
695C2—24no—
696C2—84no—
697C2C2×#244711yescharacteristic
698C2C2×#309611yescharacteristic
699C2C2×#309711yescharacteristic
700C2#6121511yescharacteristic
701C2#6121711yescharacteristic
702C2#6128711yescharacteristic
703C2#6128811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26303C2semidirect
4. #26302C2semidirect
5. #26301C2semidirect
6. #26300C2semidirect
7. #26299C2semidirect
8. #26298C2semidirect
9. #26297C2semidirect
10. #26296C2semidirect
11. #23643C2semidirect
12. #23641C2semidirect
13. C2×#1990C2semidirect
14. #21313C2semidirect
16. #21171C2semidirect
17. #20849C2semidirect
18. #20703C2semidirect
19. C4×#427C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#342072
#263032
#263022
#263012
#263002
#262992
#262982
#262972
#262962
#236432
#236412
C2×#19902
#213132
#211872
#211712
#208492
#207032
C4×#4272