Finite Group Databasebeta

← #641140

Subgroups — #641140

Order 512.

3670 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, complemented
2C2×#2537C211yesmaximal, characteristic
3C2×#2620C212yesmaximal, complemented
4C2×C4×#124C211yesmaximal, characteristic, complemented
5C2×#2293C211yesmaximal, characteristic, complemented
6#16514C218yesmaximal, complemented
7#15182C212yesmaximal, complemented
8#15131C214yesmaximal, complemented
9#15130C214yesmaximal, complemented
10#15130C212yesmaximal, complemented
11#14692C211yesmaximal, characteristic, complemented
12#14340C212yesmaximal, complemented
13#13926C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic, complemented
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23#2620—216no—
24#2538—28no—
25#2537—28no—
26C2×#396C2218yes—
27C2×#396C2214yes—
28C2×#394C2218yes—
29C2×#394C2214yes—
30C2×#394C2218yes—
31C4×#124—28no—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34#2293—28no—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2214yes—
47C2×#384C2212yes—
48C2×#384C2214yes—
49C2×#382C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2214yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2214yes—
56C2×C4×C22⋊C4C2212yes—
57C2×C43C2211yescharacteristic
58#1441C2218yes—
59#1441C2218yes—
60#1441C2214yes—
61#1439C2218yes—
62#1439C2218yes—
63#1439C2218yes—
64#1439C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C23×Q8C2311yescharacteristic
69C23×D4C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×#124—24no—
75C2×#124—24no—
76C2×#124D418yescomplemented
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—44no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—44no—
84C2×#121—48no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4D418yes—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—28no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2318yes—
110C22×C4⋊C4C2318yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#396—232no—
131#396—216no—
132#394—232no—
133#394—216no—
134#394—416no—
135#391—28no—
136#391—24no—
137#390—28no—
138#390—28no—
139#390—28no—
140#390—24no—
141#389—28no—
142#389—28no—
143#388—216no—
144#388—216no—
145#384—28no—
146#384—24no—
147#384—24no—
148#384—44no—
149#384—48no—
150#384—48no—
151#382—24no—
152#380—28no—
153#380—28no—
154#380—28no—
155#380—48no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—216no—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C43D418yescomplemented
169C25C2412yes—
170C25C2412yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8C2×D414yes—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—44no—
191C22×D4—48no—
192C22×D4—48no—
193C22×D4C2×D414yes—
194C22×D4C2×D414yes—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4C2×D418yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#124—232no—
220C22⋊Q8—432no—
221#121—432no—
222#121—432no—
223C4×Q8—416no—
224C4×D4—232no—
225C4×D4—416no—
226#117—432no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—216no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4C2×D418yes—
298C2×C22⋊C4C2×D4116yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42C2×D418yes—
312C2×C42C2×D414yes—
313C2×C42C2×D418yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42#4114yes—
319C2×C42#4114yes—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—216no—
327#95—48no—
328#95—48no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24C22×D414yes—
350C24C22×D414yes—
351C24C2511yescharacteristic
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—28no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—44no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—88no—
374C2×D4—816no—
375C2×D4—816no—
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—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+414yes—
458C22×C42−1+414yes—
459C22×C42−1+412yes—
460C4⋊C4—28no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C4⋊C4—816no—
473C4⋊C4—84no—
474C4⋊C4—816no—
475C22⋊C4—216no—
476C22⋊C4—232no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—88no—
485C22⋊C4—816no—
486C42—216no—
487C42—216no—
488C42—28no—
489C42—48no—
490C42—48no—
491C42#121116yes—
492C42#121116yescomplemented
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—28no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23C23×D411yescharacteristic
540C23C23×D411yescharacteristic
541C23C23×D411yescharacteristic
542C23C22×#4111yescharacteristic
543C23C22×#4111yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)12yes—
547C23C2×2−1+412yes—
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+412yes—
550Q8—416no—
551D4—416no—
552D4—416no—
553D4—832no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—432no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—816no—
613C2×C4C2×#12114yes—
614C2×C4C2×#12118yes—
615C2×C4C2×#12114yes—
616C2×C4C2×#12114yes—
617C2×C4C2×#12114yes—
618C2×C4#53514yes—
619C2×C4#53614yes—
620C2×C4#53718yes—
621C2×C4#53718yes—
622C2×C4#53714yes—
623C2×C4#54218yes—
624C2×C4D4214yes—
625C2×C4#54714yes—
626C2×C4#54714yes—
627C2×C4#55514yes—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—48no—
638C22—44no—
639C22—48no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22C22×#12111yescharacteristic
651C22C22×#12111yescharacteristic
652C22C2×#53511yescharacteristic
653C22C2×#53611yescharacteristic
654C22C2×#53712yes—
655C22C2×#53711yescharacteristic
656C22C2×#53712yes—
657C22C2×#54212yes—
658C22C2×D4211yescharacteristic
659C22C2×#54711yescharacteristic
660C22C2×#54711yescharacteristic
661C22C2×#55511yescharacteristic
662C22#349811yescharacteristic
663C22#349812yes—
664C22#349911yescharacteristic
665C22#349912yes—
666C22#351112yes—
667C22#351112yes—
668C22#351112yes—
669C22#354312yes—
670C22#354412yes—
671C22#355212yes—
672C22#355212yes—
673C4—24no—
674C4—28no—
675C4—28no—
676C4—44no—
677C4—44no—
678C4—82no—
679C4—84no—
680C4—84no—
681C4—84no—
682C4—82no—
683C4#253718yes—
684C4#262018yescomplemented
685C2—84no—
686C2—84no—
687C2C2×#262011yescharacteristic
688C2C2×#253711yescharacteristic
689C2#3527311yescharacteristic
690C2#3527312yes—
691C2#3762012yes—
692C2#3936011yescharacteristic
693C2#3936111yescharacteristic
694C2#3937712yes—
695C2#3948811yescharacteristic
696C2#3948911yescharacteristic
697C2#3950512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2538C2semidirect
3. C2×#2620C2semidirect
4. C2×C4×#124C2semidirect
5. C2×#2293C2semidirect
6. #16514C2semidirect
7. #15182C2semidirect
8. #15131C2semidirect
9. #15130C2semidirect
10. #15130C2semidirect
11. #14692C2semidirect
12. #14340C2semidirect
13. #13926C2semidirect
14. #13906C2semidirect
76. C2×#124D4semidirect
168. C43D4semidirect
492. C42#121semidirect
684. C4#2620semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25372
C2×#26202
C2×C4×#1242
C2×#22932
#165142
#151822
#151312
#151302
#146922
#143402
#139262
#139062