Finite Group Databasebeta

← #4456949

Subgroups — #4456949

Order 512.

2804 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic, complemented
2C2×#2486C211yesmaximal, characteristic
3C2×#2415C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#17070C212yesmaximal, complemented
6#17040C212yesmaximal, complemented
7#16992C212yesmaximal, complemented
8#16821C212yesmaximal, complemented
9#16396C214yesmaximal, complemented
10#16257C214yesmaximal
11#15189C212yesmaximal, complemented
12#15179C212yesmaximal, complemented
13#15137C212yesmaximal, complemented
14#15131C212yesmaximal, complemented
15#14258C211yesmaximal, characteristic, complemented
16#13951C211yesmaximal, characteristic, complemented
17#13839C211yesmaximal, characteristic
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2502—28no—
26#2486—28no—
27#2415—28no—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#398C2214yes—
32C2×#397C2212yes—
33C2×#397C2212yes—
34C2×#396C2214yes—
35C2×#394C2212yes—
36#2316—28no—
37C2×#391C2212yes—
38C2×#391C2211yescharacteristic
39C2×#391C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#388C2212yes—
49C2×#385C2214yes—
50C2×#384C2212yes—
51C2×#381C2212yes—
52C2×#380C2214yes—
53C2×#380C2211yescharacteristic
54C2×#380C2212yes—
55C2×#380C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58#1444C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2211yescharacteristic
68#1441C2211yescharacteristic
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2212yes—
73#1440C2212yes—
74#1440C2214yes—
75#1440C2214yes—
76#1439C2214yes—
77#1439C2211yescharacteristic
78#1439C2211yescharacteristic
79#1439C2214yes—
80#1439C2214yes—
81#1438C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#123—28no—
93C2×#123—44no—
94C2×#123—44no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—44no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×#117—24no—
104C2×#117—44no—
105C2×#117—44no—
106C2×#117D418yes—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4D418yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143C22×C42C2312yes—
144#398—28no—
145#398—28no—
146#398—28no—
147#398—48no—
148#397—28no—
149#397—28no—
150#396—48no—
151#394—28no—
152#391—24no—
153#391—28no—
154#391—44no—
155#389—28no—
156#389—28no—
157#389—24no—
158#389—28no—
159#389—28no—
160#389—28no—
161#389—28no—
162#389—44no—
163#388—28no—
164#385—216no—
165#384—28no—
166#381—28no—
167#380—24no—
168#380—24no—
169#380—28no—
170#380—216no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—44no—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C25C2411yescharacteristic
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—41no—
212C22×D4—41no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D418yes—
229C23×C4#4114yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#123—432no—
246C22⋊Q8—416no—
247C22⋊Q8—416no—
248C4×D4—416no—
249#117—416no—
250#117—416no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
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⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4C2×D418yes—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42C2×D418yes—
346#95—22no—
347#95—22no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—28no—
361#95—24no—
362#95—24no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24C22×D412yes—
391C24C22×D412yes—
392C24C2×#4112yes—
393C24H(5,𝔽2)12yes—
394C24C2511yescharacteristic
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×D4—28no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—216no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—84no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4—84no—
500C22×C4#121116yes—
501C22×C4C22×D414yes—
502C22×C4C22×D418yes—
503C22×C4C22×D414yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4H(5,𝔽2)12yes—
511C22×C4H(5,𝔽2)14yes—
512C22×C42−1+414yes—
513C22×C42−1+414yes—
514C22×C42−1+414yes—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—48no—
520C4⋊C4—416no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—88no—
529C4⋊C4—88no—
530C4⋊C4—88no—
531C4⋊C4—88no—
532C4⋊C4—88no—
533C22⋊C4—216no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—416no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C42—48no—
546C42—44no—
547C42—48no—
548C42—48no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—24no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23C2×#12114yes—
595C23C23×D411yescharacteristic
596C23C23×D411yescharacteristic
597C23C22×#4111yescharacteristic
598C23C2×H(5,𝔽2)12yes—
599C23C2×H(5,𝔽2)11yescharacteristic
600C23C2×H(5,𝔽2)11yescharacteristic
601C23C2×2−1+412yes—
602C23C2×2−1+412yes—
603C23C2×2−1+412yes—
604C23#58511yescharacteristic
605C23#58511yescharacteristic
606Q8—88no—
607D4—416no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—232no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—816no—
695C2×C4—88no—
696C2×C4C2×#12114yes—
697C2×C4C2×#12114yes—
698C2×C4#53414yes—
699C2×C4#53614yes—
700C2×C4#53718yes—
701C2×C4#53918yes—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—42no—
709C22—42no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—44no—
717C22—44no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22—82no—
722C22—82no—
723C22—81no—
724C22C22×#12111yescharacteristic
725C22C2×#53411yescharacteristic
726C22C2×#53611yescharacteristic
727C22C2×#53712yes—
728C22C2×#53912yes—
729C22#348411yescharacteristic
730C22#348511yescharacteristic
731C22#348612yes—
732C22#348811yescharacteristic
733C22#348811yescharacteristic
734C22#349411yescharacteristic
735C22#349411yescharacteristic
736C22#349911yescharacteristic
737C22#349911yescharacteristic
738C22#350912yes—
739C22#353612yes—
740C22#354112yes—
741C22#354412yes—
742C22#354712yes—
743C22#354812yes—
744C22#355212yes—
745C22#355512yes—
746C22#356312yes—
747C4—24no—
748C4—44no—
749C4—44no—
750C4—48no—
751C4—42no—
752C4—44no—
753C4—44no—
754C4—81no—
755C4—82no—
756C4—82no—
757C4—82no—
758C4—84no—
759C4—81no—
760C4—82no—
761C4#261818yes—
762C2—44no—
763C2—82no—
764C2C2×#261811yescharacteristic
765C2#3551411yescharacteristic
766C2#3566711yescharacteristic
767C2#3736412yes—
768C2#3736912yes—
769C2#4247712yes—
770C2#4267512yes—
771C2#4291611yescharacteristic
772C2#4291811yescharacteristic
773C2#4296411yescharacteristic
774C2#4298411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2502C2semidirect
4. C2×#2316C2semidirect
5. #17070C2semidirect
6. #17040C2semidirect
7. #16992C2semidirect
8. #16821C2semidirect
9. #16396C2semidirect
11. #15189C2semidirect
12. #15179C2semidirect
13. #15137C2semidirect
14. #15131C2semidirect
15. #14258C2semidirect
16. #13951C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#24862
C2×#24152
C2×#23162
#170702
#170402
#169922
#168212
#163962
#162572
#151892
#151792
#151372
#151312
#142582
#139512
#138392