Finite Group Databasebeta

← #674406

Subgroups — #674406

Order 512.

4154 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2415C214yesmaximal, complemented
4C2×#2354C211yesmaximal, characteristic, complemented
5C2×C4×#127C211yesmaximal, characteristic
6#15696C218yesmaximal, complemented
7#15177C214yesmaximal, complemented
8#15136C212yesmaximal, complemented
9#15127C212yesmaximal
10#14714C211yesmaximal, characteristic, complemented
11#14001C212yesmaximal, complemented
12#13910C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic, complemented
14C4×#389C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2214yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2214yes—
22C22×C4×D4C2211yescharacteristic
23C22×#117C2212yes—
24#2542—28no—
25#2542—28no—
26#2415—232no—
27C2×#399C2214yes—
28C2×#399C2214yes—
29C2×#397C2218yes—
30C2×#394C2218yes—
31#2354—28no—
32C4×#127—28no—
33C2×#391C2214yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2214yes—
40C2×#390C2211yescharacteristic
41C2×#389C2218yes—
42C2×#389C2212yes—
43C2×#388C2214yes—
44C2×#388C2218yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#380C2212yes—
50C2×#380C2214yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C43C2211yescharacteristic
56#1444C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1439C2214yes—
60#1438C2218yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4×#95C2212yes—
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×#127—24no—
78C2×#127—24no—
79C2×#127—24no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—48no—
87C2×C4×D4—216no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—216no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—48no—
95C2×C4×D4D418yes—
96C2×#117—28no—
97C2×#117—216no—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#399—216no—
132#399—216no—
133#397—232no—
134#394—232no—
135#391—216no—
136#390—24no—
137#390—28no—
138#390—28no—
139#390—24no—
140#390—216no—
141#390—28no—
142#390—28no—
143#389—232no—
144#389D4116yescomplemented
145#388—216no—
146#388—232no—
147#384—24no—
148#384—28no—
149#384—24no—
150#384—28no—
151#380—28no—
152#380—216no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4D418yescomplemented
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C43—24no—
167C25C2412yes—
168C25C2412yes—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—28no—
172C22×Q8—24no—
173C22×Q8—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—28no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—28no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4C2×D414yes—
194C22×D4#4114yes—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4C2×D418yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C4⋊Q8—416no—
218#127—232no—
219C22⋊Q8—464no—
220C22⋊Q8—432no—
221C22⋊Q8—432no—
222C4×D4—232no—
223C4×D4—464no—
224C4×D4—416no—
225#117—432no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—216no—
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—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D4116yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4C2×D418yes—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—216no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—42no—
308C2×C42C2×D418yes—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42#4114yes—
313C2×C42#4114yes—
314#95—28no—
315#95—24no—
316#95—216no—
317#95—216no—
318#95—216no—
319#95—28no—
320#95—44no—
321#95C2×D418yes—
322#95C2×D4116yes—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24C22×D414yes—
344C24C2×#4114yes—
345C24C2511yescharacteristic
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×D4—232no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—416no—
366C2×D4—44no—
367C2×D4—44no—
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—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—216no—
402C22×C4—232no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D418yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D418yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+414yes—
448C22×C42−1+412yes—
449C22×C42−1+412yes—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—232no—
454C4⋊C4—416no—
455C4⋊C4—432no—
456C4⋊C4—416no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—84no—
464C4⋊C4#127116yescomplemented
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—432no—
468C22⋊C4—416no—
469C22⋊C4—432no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—88no—
475C42—28no—
476C42—216no—
477C42—216no—
478C42—48no—
479C42—48no—
480C42—416no—
481C42—44no—
482C42#121116yes—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C23×D411yescharacteristic
530C23C23×D411yescharacteristic
531C23C23×D412yes—
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23C2×2−1+412yes—
539C23#58512yes—
540Q8—816no—
541D4—464no—
542D4—416no—
543D4—416no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—432no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—816no—
599C2×C4C2×#12114yes—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C2×C4#52514yes—
603C2×C4C2×#12718yes—
604C2×C4C2×#12714yes—
605C2×C4#53514yes—
606C2×C4#53614yes—
607C2×C4D4218yes—
608C2×C4#54718yes—
609C2×C4#54714yes—
610C2×C4#54814yes—
611C2×C4#55018yes—
612C2×C4#55214yes—
613C2×C4#55614yes—
614C22—48no—
615C22—48no—
616C22—44no—
617C22—48no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—48no—
623C22—44no—
624C22—82no—
625C22—84no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—82no—
636C22C22×#12111yescharacteristic
637C22C2×#52511yescharacteristic
638C22C22×#12711yescharacteristic
639C22C2×#53511yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×D4212yes—
642C22C2×#54712yes—
643C22C2×#54711yescharacteristic
644C22C2×#54811yescharacteristic
645C22C2×#55012yes—
646C22D4×#4112yes—
647C22C2×#55211yescharacteristic
648C22C2×#55611yescharacteristic
649C22#348912yes—
650C22#349712yes—
651C22#349911yescharacteristic
652C22#349912yes—
653C22#350011yescharacteristic
654C22#350612yes—
655C22#350712yes—
656C22#352012yes—
657C22#354712yes—
658C22#355012yes—
659C4—28no—
660C4—24no—
661C4—28no—
662C4—48no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—82no—
667C4—84no—
668C4—84no—
669C4#241518yescomplemented
670C4#254118yes—
671C2—84no—
672C2—84no—
673C2C2×#241511yescharacteristic
674C2C2×#254111yescharacteristic
675C2#3562111yescharacteristic
676C2#3610012yes—
677C2#3617911yescharacteristic
678C2#3646511yescharacteristic
679C2#3916711yescharacteristic
680C2#3934312yes—
681C2#3949111yescharacteristic
682C2#5944412yes—
683C2#5982112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2542C2semidirect
2. C2×#2542C2semidirect
3. C2×#2415C2semidirect
4. C2×#2354C2semidirect
6. #15696C2semidirect
7. #15177C2semidirect
8. #15136C2semidirect
10. #14714C2semidirect
11. #14001C2semidirect
12. #13910C2semidirect
13. #13906C2semidirect
14. C4×#389C2semidirect
144. #389D4semidirect
156. C4×C4⋊C4D4semidirect
464. C4⋊C4#127semidirect
669. C4#2415semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24152
C2×#23542
C2×C4×#1272
#156962
#151772
#151362
#151272
#147142
#140012
#139102
#139062
C4×#3892