Finite Group Databasebeta

← #729140

Subgroups — #729140

Order 512.

3778 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic, complemented
2C2×#2617C211yesmaximal, characteristic
3C22×#399C211yesmaximal, characteristic, complemented
4C2×#2364C211yesmaximal, characteristic, complemented
5#16468C214yesmaximal, complemented
6#15739C214yesmaximal, complemented
7#15493C214yesmaximal, complemented
8#14839C214yesmaximal, complemented
9#14776C212yesmaximal, complemented
10#14725C212yesmaximal, complemented
11#13967C212yesmaximal, complemented
12#13924C211yesmaximal, characteristic, complemented
13#13908C211yesmaximal, characteristic, complemented
14#13872C212yesmaximal, complemented
15#13775C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#120C2212yes—
18C22×C4×D4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23#2617—28no—
24#2504—28no—
25C2×#399—28no—
26C2×#399C2212yes—
27C2×#399C2214yes—
28C2×#399C2212yes—
29C2×#397C2212yes—
30C2×#397C2214yescomplemented
31C2×#397C2214yescomplemented
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#392C2212yes—
35#2364—28no—
36C2×#390C2212yescomplemented
37C2×#388C2212yescomplemented
38C2×#387C2212yescomplemented
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2214yes—
44C2×#386C2212yescomplemented
45C2×#382C2212yescomplemented
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2214yescomplemented
49C2×#379C2211yescharacteristic
50C2×#379C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yescomplemented
54C22×#95C2211yescharacteristic
55#1445C2214yescomplemented
56#1445C2214yescomplemented
57#1444C2212yescomplemented
58#1442C2212yescomplemented
59#1442C2214yescomplemented
60#1441C2214yescomplemented
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1438C2214yes—
67#1438C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72C4⋊#95C2214yescomplemented
73C4⋊#95C2214yescomplemented
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C26C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2314yes—
82C23×D4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×#121—24no—
89C2×#121—24no—
90C2×#121D418yes—
91C2×#120—28no—
92C2×#120—28no—
93C2×#120—44no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—44no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—48no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—48no—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#399—28no—
138#399—216no—
139#399—28no—
140#397—28no—
141#397—216no—
142#397—48no—
143#394—216no—
144#394—48no—
145#392—28no—
146#390—28no—
147#388—28no—
148#387—28no—
149#386—28no—
150#386—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—48no—
155#382—28no—
156#380—24no—
157#380—44no—
158#380—48no—
159#379—24no—
160#379—24no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163C4×C22⋊C4—28no—
164C2×#95D418yescomplemented
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C25—22no—
186C25—22no—
187C25—22no—
188C25—22no—
189C25—22no—
190C25—22no—
191C25—44no—
192C25—44no—
193C25#4114yes—
194C25C2412yes—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—28no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4C2×D418yes—
214C22×D4C2×D414yes—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—42no—
233C23×C4—44no—
234C23×C4C2×D418yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250#121—232no—
251#120—432no—
252C4×D4—432no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—432no—
321C2×C22⋊C4—416no—
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⋊C4#4114yes—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—42no—
340C2×C42—44no—
341C2×C42—44no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95C2×D414yes—
360#95C2×D418yes—
361#95C2×D414yes—
362C24—24no—
363C24—24no—
364C24—28no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—28no—
386C24—28no—
387C24—416no—
388C24—48no—
389C24—48no—
390C24—416no—
391C24—44no—
392C24—44no—
393C24—44no—
394C24—44no—
395C24—84no—
396C24—84no—
397C24—84no—
398C24—84no—
399C24—88no—
400C24—88no—
401C24C22×D414yes—
402C24C22×D412yes—
403C24C22×D412yes—
404C24C2×#4114yes—
405C24C2×#4112yes—
406C24C2×#4112yes—
407C24C2×#4112yes—
408C24H(5,𝔽2)12yes—
409C242−1+412yes—
410C24C2511yescharacteristic
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—216no—
414C2×D4—28no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—216no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—216no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—416no—
490C22×C4—416no—
491C22×C4—416no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—44no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—84no—
499C22×C4—88no—
500C22×C4—84no—
501C22×C4—88no—
502C22×C4—84no—
503C22×C4—88no—
504C22×C4#121116yes—
505C22×C4C22×D414yes—
506C22×C4C22×D414yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D414yes—
512C22×C4C2×#4114yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4H(5,𝔽2)18yes—
516C4⋊C4—28no—
517C4⋊C4—48no—
518C4⋊C4—416no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—416no—
522C4⋊C4—84no—
523C4⋊C4—88no—
524C22⋊C4—28no—
525C22⋊C4—216no—
526C22⋊C4—48no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—416no—
537C22⋊C4—84no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C42—48no—
544C42—48no—
545C42—84no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—28no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—28no—
566C23—432no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—416no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—42no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—816no—
605C23—816no—
606C23—816no—
607C23—88no—
608C23—816no—
609C23—816no—
610C23—88no—
611C23C2×#12114yes—
612C23#52914yes—
613C23#54214yes—
614C23#54614yes—
615C23#54614yes—
616C23#54714yes—
617C23#54814yes—
618C23C23×D411yescharacteristic
619C23C23×D411yescharacteristic
620C23C22×#4111yescharacteristic
621C23C22×#4111yescharacteristic
622C23C22×#4111yescharacteristic
623C23C2×H(5,𝔽2)14yes—
624C23C2×H(5,𝔽2)11yescharacteristic
625C23C2×2−1+411yescharacteristic
626C23#58514yes—
627D4—432no—
628D4—88no—
629D4—832no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—416no—
649C2×C4—432no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—816no—
694C2×C4—816no—
695C2×C4—88no—
696C2×C4C2×#12114yes—
697C2×C4C2×#12114yes—
698C2×C4#53518yes—
699C22—28no—
700C22—28no—
701C22—24no—
702C22—28no—
703C22—28no—
704C22—24no—
705C22—28no—
706C22—28no—
707C22—48no—
708C22—48no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—48no—
714C22—48no—
715C22—44no—
716C22—44no—
717C22—832no—
718C22—82no—
719C22—82no—
720C22—84no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—84no—
725C22—84no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—84no—
730C22#242718yes—
731C22C22×#12111yescharacteristic
732C22C2×#52911yescharacteristic
733C22C2×#53512yes—
734C22C2×#54211yescharacteristic
735C22C2×#54611yescharacteristic
736C22C2×#54611yescharacteristic
737C22C2×#54711yescharacteristic
738C22C2×#54811yescharacteristic
739C22#348512yes—
740C22#348512yes—
741C22#349112yes—
742C22#349812yes—
743C22#350514yes—
744C22#350812yes—
745C22#351314yes—
746C22#353414yes—
747C22#354114yes—
748C4—24no—
749C4—24no—
750C4—28no—
751C4—44no—
752C4—44no—
753C4—44no—
754C4—82no—
755C4—82no—
756C4—82no—
757C4—84no—
758C4—82no—
759C4—84no—
760C2—28no—
761C2—84no—
762C2—84no—
763C2C2×#242711yescharacteristic
764C2#3527211yescharacteristic
765C2#3544711yescharacteristic
766C2#3583712yes—
767C2#3802312yes—
768C2#3972212yes—
769C2#4139012yes—
770C2#4349112yes—
771C2#4408212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
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×#2504C2semidirect
3. C22×#399C2semidirect
4. C2×#2364C2semidirect
5. #16468C2semidirect
6. #15739C2semidirect
7. #15493C2semidirect
8. #14839C2semidirect
9. #14776C2semidirect
10. #14725C2semidirect
11. #13967C2semidirect
12. #13924C2semidirect
13. #13908C2semidirect
14. #13872C2semidirect
15. #13775C2semidirect
30. C2×#397C22semidirect
31. C2×#397C22semidirect
36. C2×#390C22semidirect
37. C2×#388C22semidirect
38. C2×#387C22semidirect
44. C2×#386C22semidirect
45. C2×#382C22semidirect
48. C2×#380C22semidirect
53. C2×C4×C22⋊C4C22semidirect
55. #1445C22semidirect
56. #1445C22semidirect
57. #1444C22semidirect
58. #1442C22semidirect
59. #1442C22semidirect
60. #1441C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
71. #1436C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
164. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#26172
C22×#3992
C2×#23642
#164682
#157392
#154932
#148392
#147762
#147252
#139672
#139242
#139082
#138722
#137752