Finite Group Databasebeta

← #5956128

Subgroups — #5956128

Order 512.

4662 conjugacy classes of subgroups.

The subgroup lattice has 807 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic, complemented
2C2×#2324C212yesmaximal, complemented
3C2×#2293C211yesmaximal, characteristic, complemented
4#15218C211yesmaximal, characteristic, complemented
5#15216C211yesmaximal, characteristic, complemented
6#15127C211yesmaximal, characteristic, complemented
7#15126C212yesmaximal, complemented
8#15124C211yesmaximal, characteristic, complemented
9#15123C212yesmaximal, complemented
10#14764C214yesmaximal, complemented
11#14762C214yesmaximal, complemented
12#14761C214yesmaximal, complemented
13#14760C214yesmaximal, complemented
14#13777C211yesmaximal, characteristic, complemented
15C4×#390C212yesmaximal, complemented
16C2×#540—216no—
17C2×#540—28no—
18C22×#127C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#123C2214yes—
21C22×#121C2212yes—
22C22×#121C2212yes—
23C22×#121C2214yes—
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2212yes—
26C22×#117C2211yescharacteristic
27C22×#117C2211yescharacteristic
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#392C2214yes—
31C2×#392C2214yes—
32#2324—216no—
33#2293—28no—
34C2×#390C2212yes—
35C2×#390C2214yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#388C2214yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#386C2214yes—
45C2×#385C2214yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2214yes—
53C2×C4×C22⋊C4C2214yes—
54C2×C43C2211yescharacteristic
55#1444C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2214yes—
66#1438C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2314yes—
78C23×D4C2314yes—
79C23×D4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2314yes—
83#540—2128no—
84C2×#127—28no—
85C2×#127—24no—
86C2×#125—28no—
87C2×#125—24no—
88C2×#123—216no—
89C2×#123—216no—
90C2×#123—216no—
91C2×#121—28no—
92C2×#121—28no—
93C2×#121—216no—
94C2×#121—28no—
95C2×#121—28no—
96C2×#121—28no—
97C2×#121—28no—
98C2×#121—216no—
99C2×#121—216no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—24no—
105C2×C4×D4D4116yescomplemented
106C2×#117—28no—
107C2×#117—28no—
108C2×#117—24no—
109C2×#117D418yescomplemented
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2314yes—
143C22×C42C2311yescharacteristic
144#394—48no—
145#394—48no—
146#392—48no—
147#392—48no—
148#390—28no—
149#390—28no—
150#390—216no—
151#390—28no—
152#390—28no—
153#389—28no—
154#389—28no—
155#388—28no—
156#388—216no—
157#388—28no—
158#386—216no—
159#385—216no—
160#380—28no—
161#380—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—48no—
167C4×C22⋊C4—48no—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C43—24no—
182C25C2411yescharacteristic
183C25C2414yes—
184C25C2411yescharacteristic
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—28no—
203C22×D4—24no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4C2×D418yes—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—28no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D414yes—
233C23×C4C2×D4116yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247#127—232no—
248#125—232no—
249#123—2128no—
250#121—264no—
251#121—264no—
252#121—2128no—
253C4×D4—232no—
254C4×D4—264no—
255#117—232no—
256#117—416no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4#4118yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4C2×D418yes—
334C2×C22⋊C4C2×D4116yes—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—28no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D414yes—
352C2×C42C2×D418yes—
353C2×C42C2×D414yes—
354C2×C42#4114yes—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—24no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—24no—
380C24—24no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—22no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24C22×D418yes—
399C24C22×D412yes—
400C24C2×#4112yes—
401C24C2511yescharacteristic
402C2×D4—28no—
403C2×D4—232no—
404C2×D4—232no—
405C2×D4—232no—
406C2×D4—216no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—44no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—88no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—88no—
431C2×D4—88no—
432C2×D4—88no—
433C2×D4—88no—
434C2×D4—88no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—216no—
482C22×C4—28no—
483C22×C4—232no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—416no—
499C22×C4—416no—
500C22×C4—44no—
501C22×C4—48no—
502C22×C4—44no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—88no—
506C22×C4—88no—
507C22×C4—88no—
508C22×C4—88no—
509C22×C4C22×D414yes—
510C22×C4C22×D412yes—
511C22×C4C22×D418yes—
512C22×C4C22×D414yes—
513C22×C4C22×D414yes—
514C22×C4C22×D414yes—
515C22×C4C22×D412yes—
516C22×C4C22×D414yes—
517C22×C4C2×#4118yes—
518C22×C4C2×#4112yes—
519C22×C4C2×#4112yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4C2×#4114yes—
523C22×C4C2×#4112yes—
524C22×C4H(5,𝔽2)14yes—
525C22×C4H(5,𝔽2)14yes—
526C4⋊C4—28no—
527C4⋊C4—232no—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—216no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—88no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—232no—
538C22⋊C4—232no—
539C22⋊C4—216no—
540C22⋊C4—232no—
541C22⋊C4—48no—
542C22⋊C4—416no—
543C22⋊C4—416no—
544C22⋊C4—48no—
545C22⋊C4—416no—
546C22⋊C4—48no—
547C22⋊C4—416no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C42—28no—
553C42—28no—
554C42—48no—
555C42—48no—
556C42—44no—
557C42—44no—
558C42—88no—
559C42#120116yes—
560C23—28no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—24no—
566C23—22no—
567C23—28no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—28no—
575C23—22no—
576C23—22no—
577C23—41no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—44no—
601C23—44no—
602C23—42no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—44no—
608C23—42no—
609C23—44no—
610C23—42no—
611C23—44no—
612C23—42no—
613C23—42no—
614C23—44no—
615C23—42no—
616C23—42no—
617C23—44no—
618C23—41no—
619C23—44no—
620C23—41no—
621C23—44no—
622C23—42no—
623C23—44no—
624C23—42no—
625C23—44no—
626C23—42no—
627C23—44no—
628C23—42no—
629C23—44no—
630C23—42no—
631C23—42no—
632C23—44no—
633C23—44no—
634C23—42no—
635C23—44no—
636C23—42no—
637C23C23×D411yescharacteristic
638C23C23×D412yes—
639C23C22×#4111yescharacteristic
640C23C22×#4111yescharacteristic
641C23C22×#4112yes—
642C23C2×H(5,𝔽2)12yes—
643C23C2×H(5,𝔽2)12yes—
644C23#58512yes—
645C23#58512yes—
646D4—416no—
647D4—432no—
648D4—832no—
649D4—832no—
650D4—832no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—22no—
663C2×C4—28no—
664C2×C4—22no—
665C2×C4—24no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—28no—
669C2×C4—22no—
670C2×C4—24no—
671C2×C4—24no—
672C2×C4—24no—
673C2×C4—24no—
674C2×C4—28no—
675C2×C4—28no—
676C2×C4—24no—
677C2×C4—28no—
678C2×C4—24no—
679C2×C4—28no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—42no—
699C2×C4—41no—
700C2×C4—44no—
701C2×C4—41no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—416no—
705C2×C4—416no—
706C2×C4—42no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—42no—
711C2×C4—42no—
712C2×C4—44no—
713C2×C4—42no—
714C2×C4—42no—
715C2×C4—44no—
716C2×C4—42no—
717C2×C4—42no—
718C2×C4—44no—
719C2×C4—44no—
720C2×C4—832no—
721C2×C4C2×#12018yes—
722C2×C4C2×#12014yes—
723C2×C4#52514yes—
724C2×C4#52518yes—
725C2×C4#53214yes—
726C2×C4#53218yes—
727C2×C4#54014yescomplemented
728C22—48no—
729C22—44no—
730C22—48no—
731C22—48no—
732C22—48no—
733C22—44no—
734C22—48no—
735C22—42no—
736C22—42no—
737C22—42no—
738C22—42no—
739C22—44no—
740C22—48no—
741C22—42no—
742C22—42no—
743C22—48no—
744C22—44no—
745C22—82no—
746C22—84no—
747C22—82no—
748C22—82no—
749C22—84no—
750C22—82no—
751C22—82no—
752C22—84no—
753C22—82no—
754C22—84no—
755C22—84no—
756C22—82no—
757C22—84no—
758C22—82no—
759C22—84no—
760C22—84no—
761C22C22×#12011yescharacteristic
762C22C2×#52512yes—
763C22C2×#52511yescharacteristic
764C22C2×#53212yes—
765C22C2×#53211yescharacteristic
766C22C2×#54011yescharacteristic
767C22#345511yescharacteristic
768C22#346012yes—
769C22#348812yes—
770C22#348812yes—
771C22#349412yes—
772C22#349612yes—
773C22#349612yes—
774C22#349812yes—
775C22#350012yes—
776C22#350811yescharacteristic
777C22#351211yescharacteristic
778C22#351312yes—
779C22#351712yes—
780C22#351912yes—
781C22#352412yes—
782C4—28no—
783C4—24no—
784C4—48no—
785C4—44no—
786C4—42no—
787C4—48no—
788C4—42no—
789C4—42no—
790C4—44no—
791C4—84no—
792C4—82no—
793C4#238818yes—
794C2—82no—
795C2—88no—
796C2—82no—
797C2C2×#238811yescharacteristic
798C2#3516912yes—
799C2#3517112yes—
800C2#3525911yescharacteristic
801C2#3526111yescharacteristic
802C2#3551312yes—
803C2#3553812yes—
804C2#3556712yes—
805C2#3560112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#540C2semidirect
2. C2×#2324C2semidirect
3. C2×#2293C2semidirect
4. #15218C2semidirect
5. #15216C2semidirect
6. #15127C2semidirect
7. #15126C2semidirect
8. #15124C2semidirect
9. #15123C2semidirect
10. #14764C2semidirect
11. #14762C2semidirect
12. #14761C2semidirect
13. #14760C2semidirect
14. #13777C2semidirect
15. C4×#390C2semidirect
105. C2×C4×D4D4semidirect
109. C2×#117D4semidirect
727. C2×C4#540semidirect

Maximal subgroups

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

subgroupindex
C22×#5402
C2×#23242
C2×#22932
#152182
#152162
#151272
#151262
#151242
#151232
#147642
#147622
#147612
#147602
#137772
C4×#3902