Finite Group Databasebeta

← #737901

Subgroups — #737901

Order 512.

2628 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2592C211yesmaximal, characteristic
3C2×#2640C211yesmaximal, characteristic
4C2×#2369C211yesmaximal, characteristic, complemented
5#17040C212yesmaximal, complemented
6#16821C212yesmaximal, complemented
7#15504C214yesmaximal, complemented
8#15322C214yesmaximal, complemented
9#15189C212yesmaximal, complemented
10#15174C212yesmaximal, complemented
11#15141C212yesmaximal
12#15081C212yesmaximal, complemented
13#14153C211yesmaximal, characteristic
14#14142C212yesmaximal, complemented
15#14141C212yesmaximal
16#13850C211yesmaximal, characteristic, complemented
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#2640—28no—
23#2592—28no—
24#2546—28no—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#397C2212yes—
30C2×#396C2214yes—
31C2×#393C2214yes—
32C2×#393C2212yes—
33#2369—28no—
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#391C2214yes—
39C2×#391C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2212yes—
42C2×#390C2212yes—
43C2×#390C2211yescharacteristic
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#389C2212yes—
49C2×#388C2212yes—
50C2×#388C2212yes—
51C2×#388C2214yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×#376C2211yescharacteristic
58#1446C2212yes—
59#1445C2212yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2214yes—
63#1441C2211yescharacteristic
64#1441C2211yescharacteristic
65#1441C2211yescharacteristic
66#1441C2211yescharacteristic
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2212yes—
76#1439C2212yes—
77#1437C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2214yes—
84#1434C2212yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2312yes—
90C23×D4C2311yescharacteristic
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8D418yes—
97C2×#124—24no—
98C2×#124—24no—
99C2×#124—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#398—28no—
136#398—28no—
137#397—28no—
138#397—28no—
139#397—28no—
140#396—48no—
141#393—216no—
142#393—28no—
143#391—28no—
144#391—24no—
145#391—24no—
146#391—28no—
147#391—28no—
148#391—216no—
149#390—24no—
150#390—24no—
151#390—28no—
152#390—28no—
153#389—28no—
154#389—28no—
155#389—28no—
156#389—28no—
157#389—28no—
158#388—28no—
159#388—28no—
160#388—48no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4D418yes—
166#376—24no—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C25C2412yes—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8—41no—
210C22×Q8—41no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8C2×D418yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—24no—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—232no—
238C4⋊Q8—432no—
239#124—416no—
240C4×Q8—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4#4118yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4114yes—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—42no—
353#95—42no—
354#95—44no—
355#95—42no—
356#95—42no—
357#95—44no—
358#95C2×D418yes—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24C2511yescharacteristic
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×Q8—84no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—48no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D418yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C42−1+414yes—
492C22×C42−1+414yes—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—88no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—48no—
518C42—48no—
519C42—44no—
520C42—44no—
521C42—48no—
522C42—48no—
523C42—88no—
524C42#121116yes—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C22×#4111yescharacteristic
551C23C22×#4112yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×2−1+412yes—
555C23C2×2−1+412yes—
556C23#58512yes—
557C23#58511yescharacteristic
558C23#58511yescharacteristic
559Q8—416no—
560Q8—816no—
561Q8—88no—
562D4—88no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4C2×#12114yes—
655C2×C4C2×#12114yes—
656C2×C4C2×#12114yes—
657C2×C4#53214yes—
658C2×C4#53614yes—
659C2×C4#54818yes—
660C2×C4#55618yes—
661C22—84no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—84no—
668C22—84no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22C22×#12111yescharacteristic
673C22C2×#53211yescharacteristic
674C22C2×#53611yescharacteristic
675C22#346112yes—
676C22C2×#54812yes—
677C22C2×#55612yes—
678C22#348411yescharacteristic
679C22#348511yescharacteristic
680C22#348511yescharacteristic
681C22#348611yescharacteristic
682C22#348811yescharacteristic
683C22#348811yescharacteristic
684C22#349411yescharacteristic
685C22#349411yescharacteristic
686C22#349712yes—
687C22#351412yes—
688C22#351612yes—
689C22#352012yes—
690C22#352412yes—
691C22#355012yes—
692C22#355412yes—
693C22#355512yes—
694C22#355712yes—
695C4—24no—
696C4—24no—
697C4—44no—
698C4—48no—
699C4—44no—
700C4—42no—
701C4—42no—
702C4—48no—
703C4—44no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4#254518yes—
711C2—162no—
712C2C2×#254511yescharacteristic
713C2#3515311yescharacteristic
714C2#3551411yescharacteristic
715C2#3590911yescharacteristic
716C2#3605111yescharacteristic
717C2#3823612yes—
718C2#3824112yes—
719C2#4267811yescharacteristic
720C2#4280911yescharacteristic
721C2#4488312yes—
722C2#4510212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2546C2semidirect
4. C2×#2369C2semidirect
5. #17040C2semidirect
6. #16821C2semidirect
7. #15504C2semidirect
8. #15322C2semidirect
9. #15189C2semidirect
10. #15174C2semidirect
12. #15081C2semidirect
14. #14142C2semidirect
16. #13850C2semidirect
17. #13783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25922
C2×#26402
C2×#23692
#170402
#168212
#155042
#153222
#151892
#151742
#151412
#150812
#141532
#141422
#141412
#138502
#137832