Finite Group Databasebeta

← #2737098

Subgroups — #2737098

Order 512.

2264 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×#2567C211yesmaximal, characteristic, complemented
2C2×#2448C211yesmaximal, characteristic
3#17122C214yesmaximal, complemented
4#17104C214yesmaximal, complemented
5#17006C212yesmaximal, complemented
6#16984C212yesmaximal, complemented
7#16821C212yesmaximal, complemented
8#15226C212yesmaximal
9#15192C212yesmaximal, complemented
10#15147C211yesmaximal, characteristic
11#15144C211yesmaximal, characteristic
12#15137C212yesmaximal, complemented
13#15134C212yesmaximal
14#15128C211yesmaximal, characteristic, complemented
15#14630C212yesmaximal, complemented
16#13935C211yesmaximal, characteristic, complemented
17C4×#398C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2567—28no—
22#2448—28no—
23C2×#399C2212yes—
24C2×#398C2211yescharacteristic
25C2×#398C2214yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#396C2214yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#393C2214yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#384C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51#1446C2212yes—
52#1445C2211yescharacteristic
53#1445C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2211yescharacteristic
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—44no—
90C2×#124—24no—
91C2×#124—44no—
92C2×C4×Q8—44no—
93C2×C4×Q8D418yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#399—28no—
128#398—24no—
129#398—48no—
130#397—28no—
131#397—28no—
132#396—48no—
133#396—48no—
134#395—216no—
135#395—44no—
136#393—48no—
137#391—28no—
138#391—24no—
139#391—44no—
140#390—24no—
141#390—24no—
142#390—44no—
143#390—44no—
144#389—24no—
145#389—28no—
146#389—28no—
147#389—44no—
148#389—44no—
149#388—28no—
150#388—28no—
151#384—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—44no—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C25C2412yes—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—44no—
191C22×Q8—42no—
192C22×Q8—44no—
193C22×Q8—42no—
194C22×Q8C2×D414yes—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—22no—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—416no—
217#124—416no—
218C4×Q8—416no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4#4114yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
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×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—44no—
303C2×C42C2×D414yes—
304C2×C42C2×D418yes—
305#95—22no—
306#95—24no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24C2511yescharacteristic
346C2×Q8—28no—
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×Q8—88no—
355C2×Q8—88no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—48no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+414yes—
452C22×C42−1+412yes—
453C22×C42−1+414yes—
454C22×C42−1+412yes—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C4⋊C4—84no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—28no—
483C42—48no—
484C42—44no—
485C42—48no—
486C42—44no—
487C42—48no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23C23×D411yescharacteristic
512C23C22×#4111yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)12yes—
517C23C2×2−1+412yes—
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+412yes—
521C23#58511yescharacteristic
522Q8—416no—
523Q8—816no—
524D4—88no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4#53414yes—
614C2×C4#53614yes—
615C2×C4#53714yes—
616C2×C4#54814yes—
617C2×C4#56218yes—
618C2×C4#56318yes—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—84no—
627C22—84no—
628C22—82no—
629C22—82no—
630C22C2×#53411yescharacteristic
631C22C2×#53611yescharacteristic
632C22C2×#53711yescharacteristic
633C22C2×#54811yescharacteristic
634C22#348611yescharacteristic
635C22#348811yescharacteristic
636C22#349811yescharacteristic
637C22#351112yes—
638C22#351511yescharacteristic
639C22#351911yescharacteristic
640C22#352212yes—
641C22C2×#56212yes—
642C22C2×#56312yes—
643C22#353612yes—
644C22#354312yes—
645C22#354411yescharacteristic
646C22#354612yes—
647C22#354711yescharacteristic
648C22#355012yes—
649C22#355012yes—
650C22#355212yes—
651C22#355212yes—
652C22#355512yes—
653C4—24no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—82no—
662C4—82no—
663C4—81no—
664C4—81no—
665C4—84no—
666C4—81no—
667C4—84no—
668C4#281418yes—
669C2—162no—
670C2C2×#281411yescharacteristic
671C2#3912611yescharacteristic
672C2#3916811yescharacteristic
673C2#3972111yescharacteristic
674C2#4279411yescharacteristic
675C2#4280911yescharacteristic
676C2#4296911yescharacteristic
677C2#4565312yes—
678C2#4611312yes—
679C2#5302112yes—
680C2#5311812yes—
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×#2567C2semidirect
3. #17122C2semidirect
4. #17104C2semidirect
5. #17006C2semidirect
6. #16984C2semidirect
7. #16821C2semidirect
9. #15192C2semidirect
12. #15137C2semidirect
14. #15128C2semidirect
15. #14630C2semidirect
16. #13935C2semidirect
17. C4×#398C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#24482
#171222
#171042
#170062
#169842
#168212
#152262
#151922
#151472
#151442
#151372
#151342
#151282
#146302
#139352
C4×#3982