Finite Group Databasebeta

← #4389782

Subgroups — #4389782

Order 512.

3188 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic, complemented
2C2×#2536C211yesmaximal, characteristic, complemented
3C2×#2584C211yesmaximal, characteristic, complemented
4C2×#2544C211yesmaximal, characteristic
5C2×#2632C211yesmaximal, characteristic, complemented
6C2×#2633C211yesmaximal, characteristic, complemented
7#16811C212yesmaximal, complemented
8#16510C212yesmaximal, complemented
9#15696C214yesmaximal, complemented
10#15696C214yesmaximal, complemented
11#15505C214yesmaximal, complemented
12#15325C214yesmaximal, complemented
13#15173C212yesmaximal, complemented
14#15140C212yesmaximal
15#13781C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2820—28no—
24#2633—28no—
25#2632—28no—
26#2584—28no—
27#2544—28no—
28#2536—28no—
29C2×#399C2212yes—
30C2×#399C2212yes—
31C2×#399C2212yes—
32C2×#399C2214yes—
33C2×#399C2214yes—
34C2×#398C2212yes—
35C2×#398C2212yes—
36C2×#398C2212yes—
37C2×#397C2212yes—
38C2×#397C2212yes—
39C2×#396C2214yes—
40C2×#395C2212yes—
41C2×#395C2212yes—
42C2×#395C2212yes—
43C2×#394C2212yes—
44C2×#394C2212yes—
45C2×#394C2212yes—
46C2×#394C2212yes—
47C2×#394C2214yes—
48C2×#393C2212yes—
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2214yes—
52C2×#390C2214yes—
53C2×#389C2212yes—
54C2×#389C2212yes—
55C2×#389C2212yes—
56C2×#389C2212yes—
57C2×#388C2214yes—
58C2×#388C2214yes—
59C2×#386C2214yes—
60C2×#385C2214yes—
61C2×#383C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#380C2212yes—
64C2×C4×C4⋊C4C2211yescharacteristic
65#1444C2214yes—
66#1444C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2214yes—
74#1438C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80C4×#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2314yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#127—24no—
88C2×#127—44no—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—24no—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124—24no—
95C2×#124—24no—
96C2×#124—44no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—48no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#399—28no—
141#399—28no—
142#399—28no—
143#399—48no—
144#399—48no—
145#398—28no—
146#398—28no—
147#398—28no—
148#397—28no—
149#397—28no—
150#396—216no—
151#395—28no—
152#395—28no—
153#395—28no—
154#394—216no—
155#394—28no—
156#394—28no—
157#394—28no—
158#394—28no—
159#393—28no—
160#391—24no—
161#390—24no—
162#390—48no—
163#390—48no—
164#389—28no—
165#389—28no—
166#389—28no—
167#389—28no—
168#388—216no—
169#388—216no—
170#386—216no—
171#385—216no—
172#383D418yescomplemented
173#382D418yescomplemented
174#380—28no—
175C4×C4⋊C4D418yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C25C2412yes—
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×D4—24no—
207C22×D4—28no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—28no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—28no—
216C22×D4—28no—
217C22×D4—42no—
218C22×D4—44no—
219C22×D4—42no—
220C22×D4—44no—
221C22×D4—42no—
222C22×D4—44no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241#127—416no—
242#125—416no—
243#124—416no—
244#124—416no—
245C22⋊Q8—432no—
246C4×Q8—416no—
247C4×D4—416no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D418yes—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
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⋊C4—44no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—28no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—44no—
358#95—44no—
359#95C2×D418yes—
360#95C2×D418yes—
361#95C2×D418yes—
362#95C2×D418yes—
363#95C2×D418yes—
364#95C2×D418yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24H(5,𝔽2)14yes—
385C24C2511yescharacteristic
386C2×Q8—28no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×D4—28no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—416no—
408C2×D4—88no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—88no—
412C2×D4—88no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4C22×D418yes—
476C22×C4C22×D418yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+418yes—
490C4⋊C4—216no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—88no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C42—44no—
521C42—44no—
522C42—48no—
523C42—88no—
524C42#127116yes—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23C23×D411yescharacteristic
570C23C23×D411yescharacteristic
571C23C23×D411yescharacteristic
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)12yes—
574C23C2×2−1+414yes—
575C23#58512yes—
576C23#58512yes—
577Q8—416no—
578Q8—88no—
579D4—416no—
580D4—832no—
581D4—88no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—816no—
651C2×C4C2×#12714yes—
652C2×C4C2×#12714yes—
653C2×C4C2×#12714yes—
654C2×C4#53618yes—
655C22—44no—
656C22—48no—
657C22—44no—
658C22—48no—
659C22—44no—
660C22—82no—
661C22—84no—
662C22—82no—
663C22—84no—
664C22—82no—
665C22—82no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22C22×#12711yescharacteristic
676C22C2×#53612yes—
677C22#348411yescharacteristic
678C22#348411yescharacteristic
679C22#348711yescharacteristic
680C22#348711yescharacteristic
681C22#348812yes—
682C22#348812yes—
683C22#348812yes—
684C22#348812yes—
685C22#349912yes—
686C22#349912yes—
687C22#353514yes—
688C22#353614yes—
689C22#355414yes—
690C22#355714yes—
691C4—28no—
692C4—24no—
693C4—24no—
694C4—24no—
695C4—48no—
696C4—48no—
697C4—44no—
698C4—44no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C2—84no—
705C2—162no—
706C2#3531711yescharacteristic
707C2#3556511yescharacteristic
708C2#3556511yescharacteristic
709C2#4059212yes—
710C2#4288612yes—
711C2#4768214yes—
712C2#5068912yes—
713C2#5106612yes—
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×#2820C2semidirect
2. C2×#2536C2semidirect
3. C2×#2584C2semidirect
5. C2×#2632C2semidirect
6. C2×#2633C2semidirect
7. #16811C2semidirect
8. #16510C2semidirect
9. #15696C2semidirect
10. #15696C2semidirect
11. #15505C2semidirect
12. #15325C2semidirect
13. #15173C2semidirect
15. #13781C2semidirect
172. #383D4semidirect
173. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#25362
C2×#25842
C2×#25442
C2×#26322
C2×#26332
#168112
#165102
#156962
#155052
#153252
#151732
#151402
#137812