Finite Group Databasebeta

← #626424

Subgroups — #626424

Order 512.

3264 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×#2545C211yesmaximal, characteristic, complemented
4C2×#2544C211yesmaximal, characteristic
5C2×#2640C211yesmaximal, characteristic, complemented
6C2×#2326C211yesmaximal, characteristic, complemented
7#15833C218yesmaximal, complemented
8#15526C218yesmaximal, complemented
9#15244C211yesmaximal, characteristic, complemented
10#15243C211yesmaximal, characteristic, complemented
11#15240C211yesmaximal, characteristic, complemented
12#15227C211yesmaximal, characteristic
13#14708C211yesmaximal, characteristic, complemented
14#14708C211yesmaximal, characteristic, complemented
15#14274C211yesmaximal, characteristic, complemented
16#14272C211yesmaximal, characteristic
17#13759C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#127C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×#124C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27#2640—28no—
28#2568—28no—
29#2563—28no—
30#2545—28no—
31#2544—28no—
32C2×#400C2214yes—
33C2×#397C2214yes—
34C2×#396C2214yes—
35C2×#394C2214yes—
36C2×#393C2214yes—
37C2×#393C2214yes—
38#2326—28no—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2214yes—
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#388C2214yes—
55C2×#388C2214yes—
56C2×#386C2214yes—
57C2×#385C2214yes—
58C2×#384C2211yescharacteristic
59C2×#382C2211yescharacteristic
60C2×#382C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#380C2214yes—
64C2×C4×C4⋊C4C2211yescharacteristic
65C2×C4×C4⋊C4C2211yescharacteristic
66C2×#376C2211yescharacteristic
67C2×#376C2211yescharacteristic
68C2×C43C2211yescharacteristic
69#1445C2218yes—
70#1445C2218yes—
71#1439C2218yes—
72#1439C2218yes—
73#1438C2218yes—
74#1438C2218yes—
75#1437C2218yes—
76#1437C2218yes—
77#1436C2214yes—
78#1436C2214yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2314yes—
87C24×C4C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—42no—
94C2×#127—24no—
95C2×#127—24no—
96C2×#127—42no—
97C2×#124—24no—
98C2×#124—24no—
99C2×#124—24no—
100C2×#124—24no—
101C2×#124—24no—
102C2×#124—24no—
103C2×#124—24no—
104C2×#124—24no—
105C2×#124—24no—
106C2×#124—24no—
107C2×#124—42no—
108C2×#124—42no—
109C2×C4×Q8—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2314yes—
146C22×C42C2311yescharacteristic
147#400—216no—
148#397—216no—
149#396—216no—
150#394—216no—
151#393—216no—
152#393—216no—
153#391—216no—
154#391—24no—
155#391—24no—
156#391—24no—
157#391—24no—
158#391—24no—
159#391—24no—
160#391—24no—
161#390—24no—
162#390—24no—
163#390—24no—
164#390—24no—
165#390—24no—
166#390—24no—
167#390—24no—
168#388—216no—
169#388—216no—
170#386—216no—
171#385—216no—
172#384—24no—
173#382—24no—
174#382—24no—
175#382—24no—
176#382—24no—
177#380—216no—
178C4×C4⋊C4D418yes—
179C4×C4⋊C4D418yescomplemented
180#376—42no—
181#376D418yescomplemented
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C43—42no—
201C25C2412yes—
202C25C2412yes—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—48no—
220C22×Q8#4114yes—
221C22×D4—28no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—28no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—44no—
237C22×D4—48no—
238C22×D4—48no—
239C22×D4#4114yes—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—48no—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C4⋊Q8—416no—
260C4⋊Q8—416no—
261#127—416no—
262#124—232no—
263#124—416no—
264#124—416no—
265#124—416no—
266C4×Q8—416no—
267C4×D4—416no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D418yes—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
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—48no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348C2×C42C2×D414yes—
349C2×C42C2×D414yes—
350C2×C42C2×D414yes—
351C2×C42C2×D414yes—
352C2×C42C2×D414yes—
353C2×C42C2×D414yes—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95C2×D418yes—
369#95C2×D418yes—
370#95C2×D418yes—
371#95C2×D418yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—28no—
379C24—28no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—28no—
388C24—28no—
389C24C2×#4114yes—
390C24C2511yescharacteristic
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—416no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—88no—
408C2×Q8—88no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—88no—
426C2×D4—816no—
427C2×D4—816no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
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—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—416no—
484C22×C4—48no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C2×#4114yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—416no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—88no—
517C22⋊C4—232no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C42—28no—
531C42—28no—
532C42—28no—
533C42—28no—
534C42—28no—
535C42—28no—
536C42—416no—
537C42—48no—
538C42#127116yes—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—216no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23C23×D411yescharacteristic
574C23C23×D411yescharacteristic
575C23C23×D411yescharacteristic
576C23C22×#4111yescharacteristic
577C23C22×#4111yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×2−1+411yescharacteristic
580C23#58514yes—
581C23#58514yes—
582Q8—416no—
583Q8—816no—
584Q8—88no—
585D4—416no—
586D4—832no—
587D4—88no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
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—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—816no—
655C2×C4#52514yes—
656C2×C4C2×#12714yes—
657C2×C4C2×#12714yes—
658C2×C4C2×#12714yes—
659C2×C4#53714yes—
660C2×C4#54614yes—
661C2×C4#54614yes—
662C2×C4#54814yes—
663C2×C4#54814yes—
664C22—48no—
665C22—44no—
666C22—48no—
667C22—48no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22C2×#52511yescharacteristic
683C22C22×#12711yescharacteristic
684C22C2×#53711yescharacteristic
685C22#344912yes—
686C22#344912yes—
687C22C2×#54611yescharacteristic
688C22C2×#54611yescharacteristic
689C22C2×#54811yescharacteristic
690C22C2×#54811yescharacteristic
691C22#350012yes—
692C22#350012yes—
693C22#350012yes—
694C22#350012yes—
695C22#352114yes—
696C22#352414yes—
697C22#354014yes—
698C22#355714yes—
699C4—24no—
700C4—24no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#254018yes—
714C2—84no—
715C2—162no—
716C2C2×#254011yescharacteristic
717C2#3532911yescharacteristic
718C2#3532911yescharacteristic
719C2#3562312yes—
720C2#3742112yes—
721C2#3809412yes—
722C2#3809512yes—
723C2#3916312yes—
724C2#3916412yes—
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×#2568C2semidirect
2. C2×#2563C2semidirect
3. C2×#2545C2semidirect
5. C2×#2640C2semidirect
6. C2×#2326C2semidirect
7. #15833C2semidirect
8. #15526C2semidirect
9. #15244C2semidirect
10. #15243C2semidirect
11. #15240C2semidirect
13. #14708C2semidirect
14. #14708C2semidirect
15. #14274C2semidirect
17. #13759C2semidirect
179. C4×C4⋊C4D4semidirect
181. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#25632
C2×#25452
C2×#25442
C2×#26402
C2×#23262
#158332
#155262
#152442
#152432
#152402
#152272
#147082
#142742
#142722
#137592