Finite Group Databasebeta

← #816136

Subgroups — #816136

Order 512.

2424 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#16674C212yesmaximal, complemented
4#15700C214yesmaximal, complemented
5#15633C214yesmaximal, complemented
6#15148C211yesmaximal, characteristic
7#15144C211yesmaximal, characteristic
8#15128C211yesmaximal, characteristic, complemented
9#15078C212yesmaximal, complemented
10#14910C212yesmaximal, complemented
11#14166C212yesmaximal, complemented
12#14141C212yesmaximal
13#14141C212yesmaximal
14#14132C212yesmaximal, complemented
15#14005C212yesmaximal, complemented
16C4×#400C211yesmaximal, characteristic, complemented
17#13810C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2212yes—
20#2512—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2212yes—
23C2×#397C2212yes—
24C2×#396C2212yes—
25C2×#395C2212yes—
26C2×#393C2214yes—
27#2369—28no—
28C2×#391C2214yes—
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#388C2212yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×#384C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×#376C2212yes—
51#1446C2212yes—
52#1445C2211yescharacteristic
53#1445C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1443C2212yes—
57#1442C2212yes—
58#1442C2211yescharacteristic
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2211yescharacteristic
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2212yes—
81#1434C2214yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C2×#126—28no—
90C2×#126—28no—
91C2×#126—44no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126#400—24no—
127#399—44no—
128#397—28no—
129#396—28no—
130#395—44no—
131#393—48no—
132#391—216no—
133#390—24no—
134#390—24no—
135#390—28no—
136#390—28no—
137#390—24no—
138#389—28no—
139#389—24no—
140#389—28no—
141#389—28no—
142#389—28no—
143#389—48no—
144#388—28no—
145#388—48no—
146#388—48no—
147#384—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4D418yes—
154#376—44no—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×Q8—24no—
186C22×Q8—28no—
187C22×Q8—42no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C4⋊Q8—232no—
212#126—432no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4#4114yes—
254C2×C4⋊C4#4118yes—
255C2×C4⋊C4#4118yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C2×D418yes—
305C2×C42C2×D414yes—
306#95—22no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95C2×D418yes—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24C2511yescharacteristic
347C2×Q8—216no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—48no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4118yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4118yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+414yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—216no—
460C4⋊C4—44no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—28no—
480C42—216no—
481C42—416no—
482C42—44no—
483C42—48no—
484C42—44no—
485C42—48no—
486C42—88no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23C23×D411yescharacteristic
511C23C22×#4112yes—
512C23C22×#4112yes—
513C23C22×#4111yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+412yes—
520C23#58511yescharacteristic
521Q8—816no—
522D4—88no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4#52514yes—
614C2×C4#53214yes—
615C2×C4#53614yes—
616C2×C4#54614yes—
617C2×C4#55218yes—
618C2×C4#55618yes—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—84no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22C2×#52511yescharacteristic
631C22C2×#53211yescharacteristic
632C22C2×#53611yescharacteristic
633C22C2×#54611yescharacteristic
634C22C2×#55212yes—
635C22C2×#55612yes—
636C22#348212yes—
637C22#348312yes—
638C22#348611yescharacteristic
639C22#348811yescharacteristic
640C22#349811yescharacteristic
641C22#350711yescharacteristic
642C22#351111yescharacteristic
643C22#351312yes—
644C22#351612yes—
645C22#351912yes—
646C22#351912yes—
647C22#352012yes—
648C22#352212yes—
649C22#354412yes—
650C22#354711yescharacteristic
651C22#354811yescharacteristic
652C22#355012yes—
653C4—24no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—48no—
661C4—42no—
662C4—48no—
663C4—44no—
664C4—81no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#260418yes—
669C2—162no—
670C2C2×#260411yescharacteristic
671C2#3590911yescharacteristic
672C2#3599511yescharacteristic
673C2#3641211yescharacteristic
674C2#3642211yescharacteristic
675C2#3947011yescharacteristic
676C2#4278911yescharacteristic
677C2#4499112yes—
678C2#4503212yes—
679C2#5971312yes—
680C2#5996112yes—
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
2. C2×#2369C2semidirect
3. #16674C2semidirect
4. #15700C2semidirect
5. #15633C2semidirect
8. #15128C2semidirect
9. #15078C2semidirect
10. #14910C2semidirect
11. #14166C2semidirect
14. #14132C2semidirect
15. #14005C2semidirect
16. C4×#400C2semidirect
17. #13810C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#23692
#166742
#157002
#156332
#151482
#151442
#151282
#150782
#149102
#141662
#141412
#141322
#140052
C4×#4002
#138102