Finite Group Databasebeta

← #636438

Subgroups — #636438

Order 512.

3670 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2537C211yesmaximal, characteristic
4C2×#2620C212yesmaximal, complemented
5C2×#2293C211yesmaximal, characteristic, complemented
6#16467C218yesmaximal, complemented
7#15215C212yesmaximal, complemented
8#15182C212yesmaximal, complemented
9#15131C214yesmaximal, complemented
10#15126C214yesmaximal, complemented
11#14714C211yesmaximal, characteristic, complemented
12#14258C212yesmaximal, complemented
13#13910C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2212yes—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#117C2212yes—
24#2620—216no—
25#2547—28no—
26#2542—28no—
27#2537—28no—
28C2×#399C2214yes—
29C2×#397C2214yes—
30C2×#396C2218yes—
31C2×#394C2218yes—
32C2×#394C2214yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36#2293—28no—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#388C2214yes—
44C2×#384C2212yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#380C2214yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2214yes—
54C2×C43C2211yescharacteristic
55#1444C2214yes—
56#1441C2218yes—
57#1441C2218yes—
58#1440C2218yes—
59#1439C2218yes—
60#1439C2218yes—
61#1438C2218yes—
62#1437C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×#124—28no—
78C2×#124—28no—
79C2×#124—44no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8—44no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—28no—
89C2×#121—48no—
90C2×#121—44no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4D418yes—
97C2×#117—28no—
98C2×#117—28no—
99C2×#117—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2318yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#399—216no—
133#397—216no—
134#396—232no—
135#394—232no—
136#394—216no—
137#391—28no—
138#391—28no—
139#391—44no—
140#390—28no—
141#390—28no—
142#390—24no—
143#390D418yescomplemented
144#389—28no—
145#389—48no—
146#388—216no—
147#384—28no—
148#384—28no—
149#384—24no—
150#380—28no—
151#380—28no—
152#380—216no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D418yescomplemented
155C4×C22⋊C4—28no—
156C4×C22⋊C4—48no—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C2×#95C2312yes—
167C43—24no—
168C25C2412yes—
169C25C2412yes—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—44no—
190C22×D4—48no—
191C22×D4—48no—
192C22×D4C2×D414yes—
193C22×D4C2×D414yes—
194C23×C4—24no—
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—28no—
204C23×C4—28no—
205C23×C4C2×D418yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—416no—
219#124—432no—
220C22⋊Q8—432no—
221C22⋊Q8—432no—
222#121—432no—
223#121—432no—
224C4×D4—232no—
225C4×D4—416no—
226#117—432no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4#4114yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D418yes—
298C2×C22⋊C4C2×D4116yes—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42C2×D414yes—
315C2×C42C2×D418yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42#4114yes—
319#95—28no—
320#95—24no—
321#95—28no—
322#95—216no—
323#95—216no—
324#95—44no—
325#95—44no—
326#95—48no—
327#95—48no—
328#95C2×D418yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24C22×D414yes—
350C24C22×D414yes—
351C24C2511yescharacteristic
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
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—816no—
374C2×D4—88no—
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—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
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—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4C22×D414yes—
440C22×C4C22×D418yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+414yes—
459C22×C42−1+414yes—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—816no—
474C22⋊C4—216no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—432no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C42—28no—
486C42—216no—
487C42—48no—
488C42—48no—
489C42—48no—
490C42—44no—
491C42#121116yes—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23C23×D411yescharacteristic
539C23C23×D411yescharacteristic
540C23C23×D411yescharacteristic
541C23C22×#4111yescharacteristic
542C23C22×#4111yescharacteristic
543C23C2×H(5,𝔽2)12yes—
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+412yes—
547C23C2×2−1+412yes—
548C23#58512yes—
549Q8—816no—
550D4—416no—
551D4—416no—
552D4—832no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
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—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—432no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—816no—
614C2×C4C2×#12114yes—
615C2×C4C2×#12114yes—
616C2×C4C2×#12114yes—
617C2×C4#53214yes—
618C2×C4#53514yes—
619C2×C4#53614yes—
620C2×C4#53718yes—
621C2×C4#54014yes—
622C2×C4#54218yes—
623C2×C4D4214yes—
624C2×C4D4214yes—
625C2×C4#54614yes—
626C2×C4#55014yes—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22C22×#12111yescharacteristic
650C22C2×#53211yescharacteristic
651C22C2×#53511yescharacteristic
652C22C2×#53611yescharacteristic
653C22C2×#53712yes—
654C22C2×#54011yescharacteristic
655C22C2×#54212yes—
656C22#346112yes—
657C22C2×D4211yescharacteristic
658C22C2×D4211yescharacteristic
659C22C2×#54611yescharacteristic
660C22C2×#55011yescharacteristic
661C22#348912yes—
662C22#349612yes—
663C22#349911yescharacteristic
664C22#349912yes—
665C22#350011yescharacteristic
666C22#351812yes—
667C22#352012yes—
668C22#354612yes—
669C22#354712yes—
670C22#354812yes—
671C22#355212yes—
672C4—24no—
673C4—44no—
674C4—44no—
675C4—48no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—82no—
680C4—84no—
681C4—84no—
682C4#242818yes—
683C4#262018yescomplemented
684C2—84no—
685C2—84no—
686C2C2×#262011yescharacteristic
687C2C2×#242811yescharacteristic
688C2#3517012yes—
689C2#3530811yescharacteristic
690C2#3762712yes—
691C2#3895111yescharacteristic
692C2#3895811yescharacteristic
693C2#3913612yes—
694C2#3934012yes—
695C2#3936011yescharacteristic
696C2#3936211yescharacteristic
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×#2547C2semidirect
2. C2×#2542C2semidirect
4. C2×#2620C2semidirect
5. C2×#2293C2semidirect
6. #16467C2semidirect
7. #15215C2semidirect
8. #15182C2semidirect
9. #15131C2semidirect
10. #15126C2semidirect
11. #14714C2semidirect
12. #14258C2semidirect
13. #13910C2semidirect
14. C4×#390C2semidirect
143. #390D4semidirect
154. C4×C4⋊C4D4semidirect
683. C4#2620semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25422
C2×#25372
C2×#26202
C2×#22932
#164672
#152152
#151822
#151312
#151262
#147142
#142582
#139102
C4×#3902