Finite Group Databasebeta

← #626299

Subgroups — #626299

Order 512.

3628 conjugacy classes of subgroups.

The subgroup lattice has 775 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic, complemented
3C2×#2436C211yesmaximal, characteristic
4C2×#2434C211yesmaximal, characteristic, complemented
5C2×#2620C211yesmaximal, characteristic
6C2×#2310C211yesmaximal, characteristic, complemented
7#15825C218yesmaximal, complemented
8#15549C218yesmaximal, complemented
9#15243C211yesmaximal, characteristic, complemented
10#15243C211yesmaximal, characteristic, complemented
11#15240C211yesmaximal, characteristic, complemented
12#15199C211yesmaximal, characteristic, complemented
13#14714C211yesmaximal, characteristic, complemented
14#14708C211yesmaximal, characteristic, complemented
15#14274C211yesmaximal, characteristic, complemented
16#14273C211yesmaximal, characteristic, complemented
17#13761C211yesmaximal, characteristic, complemented
18C24×Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#124C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C22×#117C2211yescharacteristic
28#2620—28no—
29#2500—28no—
30#2499—28no—
31#2436—28no—
32#2434—28no—
33C2×#397C2214yes—
34C2×#396C2214yes—
35C2×#395C2214yes—
36C2×#394C2214yes—
37C2×#394C2214yes—
38C2×#393C2214yes—
39#2310—28no—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#391C2214yes—
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#388C2214yes—
56C2×#387C2214yes—
57C2×#386C2214yes—
58C2×#385C2214yes—
59C2×#384C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C2×#384C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#382C2211yescharacteristic
64C2×#381C2214yes—
65C2×C4×C22⋊C4C2211yescharacteristic
66C2×C4×C22⋊C4C2211yescharacteristic
67C2×#376C2211yescharacteristic
68C2×C43C2211yescharacteristic
69#1445C2218yes—
70#1445C2218yes—
71#1444C2218yes—
72#1439C2218yes—
73#1439C2218yes—
74#1439C2218yes—
75#1437C2218yes—
76#1437C2214yes—
77#1436C2214yes—
78C4⋊#95C2218yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8—24no—
82C23×Q8—24no—
83C23×Q8—24no—
84C23×Q8—24no—
85C23×Q8—24no—
86C23×Q8—42no—
87C23×Q8—42no—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×Q8C2312yes—
92C23×D4C2311yescharacteristic
93C23×D4C2312yes—
94C24×C4C2312yes—
95C24×C4C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8—42no—
102C2×#124—24no—
103C2×#124—24no—
104C2×#124—24no—
105C2×#124—24no—
106C2×#124—24no—
107C2×#124—42no—
108C2×#124—42no—
109C2×#124—42no—
110C2×#124—42no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×#121—24no—
115C2×#121—24no—
116C2×#121—24no—
117C2×#117—24no—
118C2×#117—24no—
119C2×#117—24no—
120C2×#117—42no—
121C2×#117—42no—
122C2×#117D418yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148C22×C42C2311yescharacteristic
149C22×C42C2311yescharacteristic
150C22×C42C2311yescharacteristic
151C22×C42C2311yescharacteristic
152C22×C42C2311yescharacteristic
153C22×C42C2314yes—
154#397—216no—
155#396—216no—
156#395—216no—
157#394—216no—
158#394—216no—
159#393—216no—
160#391—216no—
161#391—24no—
162#391—24no—
163#391—24no—
164#391—24no—
165#391—24no—
166#391—24no—
167#391—24no—
168#391—24no—
169#391—24no—
170#391—24no—
171#391—24no—
172#390—24no—
173#390—24no—
174#390—24no—
175#388—216no—
176#387—216no—
177#386—216no—
178#385—216no—
179#384—24no—
180#384—24no—
181#384—24no—
182#382—24no—
183#382—24no—
184#381—216no—
185C4×C22⋊C4D418yes—
186C4×C22⋊C4D418yes—
187#376—42no—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C43—42no—
207C25C2412yes—
208C25C2411yescharacteristic
209C22×Q8—232no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—24no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8—416no—
237C22×Q8—416no—
238C22×Q8—416no—
239C22×Q8—416no—
240C22×Q8—416no—
241C22×Q8—416no—
242C22×Q8#4114yes—
243C22×Q8#4114yes—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—24no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—48no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—24no—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—42no—
265C23×C4—42no—
266C23×C4C2×D414yes—
267C23×C4C2×D414yes—
268C23×C4C2×D414yes—
269C23×C4C2412yes—
270C23×C4C2412yes—
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2412yes—
274C23×C4C2412yes—
275C23×C4C2412yes—
276C23×C4C2412yes—
277C23×C4C2412yes—
278C23×C4C2412yes—
279C23×C4C2412yes—
280C23×C4C2412yes—
281C23×C4C2411yescharacteristic
282C23×C4C2412yes—
283C23×C4C2411yescharacteristic
284C4⋊Q8—416no—
285C4⋊Q8—416no—
286#124—416no—
287#124—416no—
288#124—416no—
289C22⋊Q8—416no—
290#121—416no—
291#117—416no—
292#117—416no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4C2×D418yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—216no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4C2×D418yes—
352C2×C22⋊C4C2×D418yes—
353C2×C22⋊C4C2×D418yes—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—28no—
370C2×C42—44no—
371C2×C42—44no—
372C2×C42C2×D414yes—
373C2×C42C2×D414yes—
374C2×C42C2×D414yes—
375C2×C42C2×D414yes—
376C2×C42C2×D414yes—
377C2×C42C2×D414yes—
378#95—28no—
379#95—28no—
380#95—28no—
381#95—28no—
382#95—28no—
383#95—28no—
384#95—28no—
385#95—28no—
386#95—28no—
387#95—28no—
388#95—28no—
389#95—28no—
390#95—28no—
391#95—28no—
392#95—44no—
393#95—44no—
394#95C2×D418yes—
395#95C2×D418yes—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—24no—
401C24—24no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—28no—
410C24—28no—
411C24C22×D412yes—
412C24C22×D412yes—
413C24C22×D412yes—
414C24C2511yescharacteristic
415C2×Q8—216no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—28no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—4128no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×D4—416no—
445C2×D4—48no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—88no—
454C2×D4—88no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—216no—
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—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—28no—
521C22×C4—28no—
522C22×C4—28no—
523C22×C4—28no—
524C22×C4—48no—
525C22×C4—48no—
526C22×C4—48no—
527C22×C4—48no—
528C22×C4—48no—
529C22×C4—48no—
530C22×C4#127116yes—
531C22×C4C22×D414yes—
532C22×C4C22×D414yes—
533C22×C4C22×D414yes—
534C22×C4C22×D414yes—
535C22×C4C22×D414yes—
536C22×C4C22×D412yes—
537C22×C4C22×D412yes—
538C22×C4C22×D412yes—
539C22×C4C22×D414yes—
540C22×C4C22×D412yes—
541C22×C4C22×D412yes—
542C22×C4C22×D412yes—
543C22×C4C2×#4114yes—
544C22×C4C2×#4114yes—
545C22×C4C2×#4112yes—
546C22×C4C2×#4112yes—
547C22×C4H(5,𝔽2)12yes—
548C22×C4H(5,𝔽2)12yes—
549C22×C42−1+418yes—
550C4⋊C4—48no—
551C4⋊C4—416no—
552C4⋊C4—416no—
553C4⋊C4—416no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—48no—
559C4⋊C4—88no—
560C4⋊C4—88no—
561C22⋊C4—216no—
562C22⋊C4—48no—
563C22⋊C4—48no—
564C22⋊C4—416no—
565C22⋊C4—416no—
566C22⋊C4—416no—
567C22⋊C4—48no—
568C22⋊C4—48no—
569C22⋊C4—48no—
570C22⋊C4—416no—
571C22⋊C4—88no—
572C22⋊C4—88no—
573C42—28no—
574C42—28no—
575C42—28no—
576C42—28no—
577C42—28no—
578C42—28no—
579C42—416no—
580C42—48no—
581C23—22no—
582C23—22no—
583C23—24no—
584C23—24no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—24no—
590C23—24no—
591C23—24no—
592C23—24no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—44no—
601C23—44no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—48no—
611C23—48no—
612C23—48no—
613C23—48no—
614C23C2×#12714yes—
615C23C23×D411yescharacteristic
616C23C23×D411yescharacteristic
617C23C23×D411yescharacteristic
618C23C22×#4111yescharacteristic
619C23C22×#4111yescharacteristic
620C23C2×H(5,𝔽2)11yescharacteristic
621C23C2×H(5,𝔽2)11yescharacteristic
622C23C2×2−1+414yes—
623C23#58514yes—
624Q8—432no—
625Q8—416no—
626Q8—416no—
627Q8—88no—
628D4—816no—
629D4—88no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—232no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
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—432no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
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—42no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4#52514yes—
705C2×C4C2×#12714yes—
706C2×C4C2×#12714yes—
707C2×C4#53414yes—
708C2×C4#53618yes—
709C2×C4#54614yes—
710C2×C4#54614yes—
711C2×C4#54614yes—
712C2×C4#54614yes—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—82no—
723C22—82no—
724C22—84no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22—84no—
730C22—82no—
731C22C2×#52511yescharacteristic
732C22C22×#12711yescharacteristic
733C22C2×#53411yescharacteristic
734C22C2×#53612yes—
735C22#344912yes—
736C22C2×#54611yescharacteristic
737C22C2×#54611yescharacteristic
738C22C2×#54611yescharacteristic
739C22C2×#54611yescharacteristic
740C22#349912yes—
741C22#349912yes—
742C22#350012yes—
743C22#350012yes—
744C22#351614yes—
745C22#351814yes—
746C22#354114yes—
747C22#354714yes—
748C4—24no—
749C4—28no—
750C4—44no—
751C4—44no—
752C4—44no—
753C4—44no—
754C4—44no—
755C4—44no—
756C4—44no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4#253918yes—
763C2—44no—
764C2—162no—
765C2C2×#253911yescharacteristic
766C2#3531711yescharacteristic
767C2#3532911yescharacteristic
768C2#3566912yes—
769C2#3738812yes—
770C2#3808512yes—
771C2#3809112yes—
772C2#3894612yes—
773C2#3895812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2499C2semidirect
4. C2×#2434C2semidirect
6. C2×#2310C2semidirect
7. #15825C2semidirect
8. #15549C2semidirect
9. #15243C2semidirect
10. #15243C2semidirect
11. #15240C2semidirect
12. #15199C2semidirect
13. #14714C2semidirect
14. #14708C2semidirect
15. #14274C2semidirect
16. #14273C2semidirect
17. #13761C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#24992
C2×#24362
C2×#24342
C2×#26202
C2×#23102
#158252
#155492
#152432
#152402
#151992
#147142
#147082
#142742
#142732
#137612