Finite Group Databasebeta

← #722919

Subgroups — #722919

Order 512.

2898 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2580C211yesmaximal, characteristic
3C2×#2399C211yesmaximal, characteristic, complemented
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#16550C214yesmaximal, complemented
6#15614C214yesmaximal, complemented
7#15436C214yesmaximal, complemented
8#15134C212yesmaximal
9#15133C212yesmaximal, complemented
10#14877C214yesmaximal, complemented
11#14276C211yesmaximal, characteristic, complemented
12#14045C212yesmaximal, complemented
13#14022C212yesmaximal
14#13925C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2580—28no—
22#2538—28no—
23#2399—28no—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#396C2214yes—
29C2×#396C2214yes—
30C2×#396C2214yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#393C2212yes—
34Q8×C4⋊C4—28no—
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2211yescharacteristic
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#389C2212yes—
44C2×#388C2214yes—
45C2×#385C2214yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×#376C2211yescharacteristic
55#1446C2212yes—
56#1445C2212yes—
57#1444C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—48no—
85C2×C4×Q8—28no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—48no—
92C2×C4×Q8D418yes—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#399—28no—
131#398—28no—
132#398—28no—
133#398—48no—
134#396—216no—
135#396—48no—
136#396—48no—
137#395—28no—
138#395—28no—
139#393—28no—
140#390—28no—
141#390—28no—
142#390—24no—
143#390Q818yescomplemented
144#389—216no—
145#389—28no—
146#389—28no—
147#389—28no—
148#389—28no—
149#388—48no—
150#385—216no—
151#384—28no—
152#384D418yes—
153#382—24no—
154#380—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159#376—24no—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C25C2412yes—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—42no—
191C22×Q8C2×D414yes—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4C2×Q814yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C22⋊Q8—432no—
219C4×Q8—232no—
220C4×Q8—432no—
221C4×Q8—416no—
222C4×D4—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
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⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×Q814yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
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⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×Q8116yes—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C2×D418yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×Q814yes—
320C2×C42#4114yes—
321C2×C42#4114yes—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24C22×Q814yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—216no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—216no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C22×D418yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q812yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C42−1+414yes—
463C22×C42−1+412yes—
464C22×C42−1+412yes—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—216no—
493C42—28no—
494C42—28no—
495C42—416no—
496C42—48no—
497C42—48no—
498C42#121116yes—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C23×Q811yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×2−1+412yes—
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23#58512yes—
533Q8—232no—
534Q8—416no—
535D4—416no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—816no—
615C2×C4C2×#12114yes—
616C2×C4C2×#12114yes—
617C2×C4C2×#12114yes—
618C2×C4#53314yes—
619C2×C4#53718yes—
620C2×C4#54114yes—
621C2×C4#54714yes—
622C2×C4#54814yes—
623C2×C4D4×Q814yes—
624C2×C4D4×Q814yes—
625C22—44no—
626C22—48no—
627C22—44no—
628C22—44no—
629C22—48no—
630C22—84no—
631C22—84no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C22×#12111yescharacteristic
637C22C2×#53311yescharacteristic
638C22C2×#53712yes—
639C22C2×#54111yescharacteristic
640C22C2×#54711yescharacteristic
641C22C2×#54811yescharacteristic
642C22C2×D4×Q811yescharacteristic
643C22C2×D4×Q811yescharacteristic
644C22#348411yescharacteristic
645C22#348411yescharacteristic
646C22#348611yescharacteristic
647C22#348711yescharacteristic
648C22#348812yes—
649C22#349212yes—
650C22#349412yes—
651C22#350112yes—
652C22#350312yes—
653C22#350712yes—
654C22#350812yes—
655C22#354812yes—
656C22#355012yes—
657C22#355412yes—
658C22#355512yes—
659C4—24no—
660C4—28no—
661C4—24no—
662C4—44no—
663C4—44no—
664C4—48no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#242918yes—
673C2—84no—
674C2C2×#242911yescharacteristic
675C2#3522311yescharacteristic
676C2#3556311yescharacteristic
677C2#3593412yes—
678C2#3959812yes—
679C2#3972612yes—
680C2#4250912yes—
681C2#4493111yescharacteristic
682C2#4493411yescharacteristic
683C2#4535711yescharacteristic
684C2#4539411yescharacteristic
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. C2×#2399C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
5. #16550C2semidirect
6. #15614C2semidirect
7. #15436C2semidirect
9. #15133C2semidirect
10. #14877C2semidirect
11. #14276C2semidirect
12. #14045C2semidirect
14. #13925C2semidirect
143. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25802
C2×#23992
C2×Q8×C4⋊C42
#165502
#156142
#154362
#151342
#151332
#148772
#142762
#140452
#140222
#139252
#137852