Finite Group Databasebeta

← #614419

Subgroups — #614419

Order 512.

3146 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic
3C2×#2283C211yesmaximal, characteristic
4C22×C4×C4⋊C4C211yesmaximal, characteristic
5#15237C212yesmaximal, complemented
6#15221C212yesmaximal, complemented
7#15076C214yesmaximal, complemented
8#15073C212yesmaximal
9#14919C212yesmaximal
10#14702C211yesmaximal, characteristic
11#14697C211yesmaximal, characteristic
12#14685C211yesmaximal, characteristic
13#14081C214yesmaximal, complemented
14#14046C212yesmaximal, complemented
15#14043C212yesmaximal
16#14021C212yesmaximal, complemented
17#13972C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2855—28no—
25#2855—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#389C2214yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#387C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41#2283—28no—
42C2×#383C2212yes—
43C2×#382C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4—28no—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C22⋊C4C2212yes—
58C2×#376C2212yes—
59C22×#95C2211yescharacteristic
60C22×#95C2211yescharacteristic
61#1445C2214yes—
62#1444C2214yes—
63#1443C2214yes—
64#1443C2214yes—
65#1443C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42—24no—
122C22×C42—24no—
123C22×C42—24no—
124C22×C42—28no—
125C22×C42—24no—
126C22×C42—24no—
127C22×C42—24no—
128C22×C42D418yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#391—28no—
139#391—24no—
140#391—24no—
141#391—28no—
142#389—28no—
143#389—216no—
144#389—24no—
145#389—24no—
146#389—28no—
147#387—28no—
148#387—28no—
149#387—216no—
150#385—28no—
151#385—28no—
152#385—28no—
153#383—28no—
154#382—28no—
155#381—28no—
156#381—28no—
157#381—28no—
158#381—28no—
159#380—24no—
160#380—24no—
161#380D418yes—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—28no—
167C4×C22⋊C4—28no—
168#376—28no—
169C2×#95—24no—
170C2×#95—24no—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C25C2411yescharacteristic
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4#4114yes—
244C23×C4#4114yes—
245C23×C4#4114yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2414yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C22⋊Q8—416no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—232no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
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—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4#4118yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4C2×D418yes—
315C2×C42—232no—
316C2×C42—232no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—28no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—28no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42—416no—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42#4118yes—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—24no—
359#95—24no—
360#95—28no—
361#95—24no—
362#95—24no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—28no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95C2×D418yes—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C2×#4112yes—
382C24C2×#4112yes—
383C24C2×#4112yes—
384C242−1+412yes—
385C242−1+412yes—
386C24C2511yescharacteristic
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—48no—
391C2×Q8—48no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—28no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—28no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
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—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4#121116yes—
507C22×C4C22×D418yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C2×#4114yes—
517C22×C4C2×#4114yes—
518C22×C4C2×#4114yes—
519C22×C4C2×#4112yes—
520C22×C4C2×#4114yes—
521C22×C4C2×#4114yes—
522C22×C4C2×#4114yes—
523C22×C4C2×#4114yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4114yes—
526C22×C42−1+414yes—
527C4⋊C4—216no—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—216no—
531C4⋊C4—216no—
532C4⋊C4—416no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—416no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C22⋊C4—216no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C42—216no—
546C42—28no—
547C42—28no—
548C42—28no—
549C42—28no—
550C42—216no—
551C42—416no—
552C42—48no—
553C42—48no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23C2×#12114yes—
571C23#52514yes—
572C23#53614yes—
573C23#54714yes—
574C23#54814yes—
575C23#55214yes—
576C23#55614yes—
577C23C23×D411yescharacteristic
578C23C23×D411yescharacteristic
579C23C22×#4111yescharacteristic
580C23C22×#4111yescharacteristic
581C23C22×#4112yes—
582C23C22×#4112yes—
583C23C22×#4111yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23C2×2−1+412yes—
586C23C2×2−1+411yescharacteristic
587C23#58512yes—
588Q8—88no—
589C2×C4—232no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—232no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—416no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—416no—
632C2×C4—416no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
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—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4C2×#12114yes—
685C2×C4C2×#12114yes—
686C2×C4#52514yes—
687C2×C4#52514yes—
688C2×C4#52918yes—
689C2×C4#53614yes—
690C2×C4#54718yes—
691C2×C4#55218yes—
692C22—24no—
693C22—28no—
694C22—24no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22#241518yes—
703C22C22×#12111yescharacteristic
704C22C2×#52511yescharacteristic
705C22C2×#52511yescharacteristic
706C22C2×#52511yescharacteristic
707C22C2×#52912yes—
708C22#344412yes—
709C22C2×#53611yescharacteristic
710C22C2×#53611yescharacteristic
711C22#344911yescharacteristic
712C22C2×#54711yescharacteristic
713C22C2×#54712yes—
714C22C2×#54811yescharacteristic
715C22D4×#4112yes—
716C22C2×#55212yes—
717C22C2×#55211yescharacteristic
718C22C2×#55611yescharacteristic
719C22#347512yes—
720C22#349512yes—
721C22#349512yes—
722C22#350712yes—
723C22#350712yes—
724C22#352312yes—
725C22#352312yes—
726C4—28no—
727C4—24no—
728C4—24no—
729C4—48no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—44no—
735C4—44no—
736C4—82no—
737C4—82no—
738C4—84no—
739C4—82no—
740C4—82no—
741C4#241318yes—
742C2—28no—
743C2C2×#241511yescharacteristic
744C2C2×#241311yescharacteristic
745C2#3473711yescharacteristic
746C2#3511111yescharacteristic
747C2#3524811yescharacteristic
748C2#3532511yescharacteristic
749C2#3611612yes—
750C2#3614012yes—
751C2#3617712yes—
752C2#3619912yes—
753C2#3662111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #15237C2semidirect
6. #15221C2semidirect
7. #15076C2semidirect
13. #14081C2semidirect
14. #14046C2semidirect
16. #14021C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22832
C22×C4×C4⋊C42
#152372
#152212
#150762
#150732
#149192
#147022
#146972
#146852
#140812
#140462
#140432
#140212
#139722