Finite Group Databasebeta

← #2736853

Subgroups — #2736853

Order 512.

2438 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×#2538C211yesmaximal, characteristic
2C2×#2452C211yesmaximal, characteristic
3#17124C214yesmaximal, complemented
4#17122C214yesmaximal, complemented
5#16815C212yesmaximal, complemented
6#16262C212yesmaximal, complemented
7#16254C212yesmaximal, complemented
8#15197C211yesmaximal, characteristic, complemented
9#15187C212yesmaximal, complemented
10#15158C211yesmaximal, characteristic, complemented
11#15152C211yesmaximal, characteristic, complemented
12#15147C212yesmaximal
13#15138C212yesmaximal
14#15128C212yesmaximal, complemented
15#14660C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic
17C4×#395C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2538—28no—
24#2452—28no—
25C2×#400C2212yes—
26C2×#397C2214yes—
27C2×#397C2214yes—
28C2×#396C2214yes—
29C2×#396C2212yes—
30C2×#395C2211yescharacteristic
31C2×#395C2214yes—
32C2×#395C2212yes—
33C2×#393C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#385C2212yes—
43C2×#384C2211yescharacteristic
44C2×#381C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48#1446C2214yes—
49#1446C2211yescharacteristic
50#1445C2212yes—
51#1445C2214yes—
52#1444C2214yes—
53#1444C2211yescharacteristic
54#1444C2212yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2211yescharacteristic
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2212yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—44no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—48no—
95C2×C22⋊Q8—44no—
96C2×C4×Q8—24no—
97C2×C4×Q8—44no—
98C2×C4×D4—44no—
99C2×C4×D4D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#400—44no—
133#397—48no—
134#397—48no—
135#396—28no—
136#396—216no—
137#395—24no—
138#395—44no—
139#395—48no—
140#393—28no—
141#390—24no—
142#390—24no—
143#390—28no—
144#390—44no—
145#389—24no—
146#389—28no—
147#389—24no—
148#389—44no—
149#385—28no—
150#384—24no—
151#381—44no—
152#380—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—44no—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4C2×D414yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D418yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8—416no—
222C22⋊Q8—432no—
223C22⋊Q8—432no—
224C4×Q8—416no—
225C4×D4—416no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4#4114yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×D418yes—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42C2×D414yes—
313#95—22no—
314#95—22no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—42no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—42no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24C22×D414yes—
353C24C2511yescharacteristic
354C2×Q8—28no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—44no—
361C2×Q8—84no—
362C2×Q8—88no—
363C2×Q8—88no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—48no—
368C2×D4—48no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—216no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C42−1+414yes—
467C22×C42−1+412yes—
468C22×C42−1+412yes—
469C22×C42−1+414yes—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—416no—
493C22⋊C4—88no—
494C22⋊C4—84no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C42—28no—
498C42—48no—
499C42—44no—
500C42—44no—
501C42—48no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23C23×D411yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+412yes—
534C23C2×2−1+412yes—
535C23#58511yescharacteristic
536Q8—416no—
537Q8—816no—
538D4—416no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—816no—
630C2×C4#53414yes—
631C2×C4#53614yes—
632C2×C4#53614yes—
633C2×C4#54614yes—
634C2×C4#56318yes—
635C2×C4#56318yes—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22C2×#53411yescharacteristic
648C22C2×#53611yescharacteristic
649C22C2×#53611yescharacteristic
650C22C2×#54611yescharacteristic
651C22#348411yescharacteristic
652C22#348811yescharacteristic
653C22#349911yescharacteristic
654C22#350711yescharacteristic
655C22#351111yescharacteristic
656C22#351912yes—
657C22#352312yes—
658C22C2×#56312yes—
659C22C2×#56312yes—
660C22#353612yes—
661C22#353712yes—
662C22#354712yes—
663C22#354711yescharacteristic
664C22#354811yescharacteristic
665C22#354812yes—
666C22#355012yes—
667C22#355012yes—
668C22#355212yes—
669C22#355412yes—
670C4—24no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—82no—
679C4—84no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4#281718yes—
686C2—84no—
687C2C2×#281711yescharacteristic
688C2#3890411yescharacteristic
689C2#3895411yescharacteristic
690C2#3947011yescharacteristic
691C2#4242711yescharacteristic
692C2#4247911yescharacteristic
693C2#4298411yescharacteristic
694C2#4565412yes—
695C2#4634112yes—
696C2#5297412yes—
697C2#5309912yes—
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
3. #17124C2semidirect
4. #17122C2semidirect
5. #16815C2semidirect
6. #16262C2semidirect
7. #16254C2semidirect
8. #15197C2semidirect
9. #15187C2semidirect
10. #15158C2semidirect
11. #15152C2semidirect
14. #15128C2semidirect
15. #14660C2semidirect
17. C4×#395C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24522
#171242
#171222
#168152
#162622
#162542
#151972
#151872
#151582
#151522
#151472
#151382
#151282
#146602
#139162
C4×#3952