Finite Group Databasebeta

← #719842

Subgroups — #719842

Order 512.

2970 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2421C212yesmaximal, complemented
3C2×#2423C212yesmaximal, complemented
4C2×#2351C211yesmaximal, characteristic, complemented
5#16964C212yesmaximal, complemented
6#16692C212yesmaximal
7#15808C214yesmaximal, complemented
8#15765C214yesmaximal, complemented
9#15223C212yesmaximal, complemented
10#15135C212yesmaximal, complemented
11#15084C212yesmaximal
12#15082C212yesmaximal, complemented
13#14354C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13905C211yesmaximal, characteristic
16#13847C211yesmaximal, characteristic
17#13786C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2538—28no—
24#2423—216no—
25#2421—216no—
26C2×#400C2214yes—
27C2×#399C2214yes—
28C2×#399C2214yes—
29C2×#398C2214yes—
30C2×#397C2214yes—
31C2×#396C2214yes—
32#2351—28no—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#387C2214yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#382C2211yescharacteristic
50C2×#382C2212yes—
51C2×#382C2212yes—
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2212yes—
59C2×C4×C22⋊C4C2212yes—
60C2×#376C2211yescharacteristic
61#1445C2211yescharacteristic
62#1445C2211yescharacteristic
63#1445C2214yes—
64#1445C2212yes—
65#1445C2214yes—
66#1445C2214yes—
67#1444C2212yes—
68#1444C2212yes—
69#1444C2214yes—
70#1444C2214yes—
71#1444C2212yes—
72#1444C2212yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1438C2211yescharacteristic
79#1438C2211yescharacteristic
80#1438C2212yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#400—216no—
141#399—216no—
142#399—48no—
143#398—216no—
144#397—216no—
145#396—216no—
146#391D4116yescomplemented
147#390—28no—
148#389—24no—
149#389—24no—
150#389—28no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—28no—
155#388—216no—
156#388—216no—
157#387—216no—
158#387—216no—
159#384—28no—
160#384—24no—
161#384—44no—
162#382—28no—
163#382—24no—
164#382—28no—
165#380—28no—
166#380—28no—
167#380—28no—
168C4×C4⋊C4—28no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—24no—
171C4×C22⋊C4—28no—
172C4×C22⋊C4—28no—
173#376—24no—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C25C2412yes—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8C2×D414yes—
204C22×Q8C2×D418yes—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—42no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C22⋊Q8—432no—
234C4×Q8—232no—
235C4×Q8—416no—
236C4×Q8—416no—
237C4×D4—416no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D418yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—28no—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42C2×D418yes—
336C2×C42C2×D418yes—
337C2×C42C2×D414yes—
338C2×C42#4114yes—
339C2×C42#4114yes—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—44no—
362#95—44no—
363#95C2×D4116yes—
364#95C2×D4116yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C242−1+414yes—
380C24C2511yescharacteristic
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—216no—
384C2×Q8—216no—
385C2×Q8#127116yes—
386C2×D4—28no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—44no—
390C2×D4—44no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D418yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D418yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C42−1+414yes—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—416no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C22⋊C4—216no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—216no—
522C42—216no—
523C42—48no—
524C42—48no—
525C42—48no—
526C42—44no—
527C42—44no—
528C42#123116yes—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23C23×D411yescharacteristic
565C23C23×D412yes—
566C23C22×#4111yescharacteristic
567C23C22×#4111yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×2−1+412yes—
570C23C2×2−1+411yescharacteristic
571C23C2×2−1+412yes—
572C23#58511yescharacteristic
573C23#58511yescharacteristic
574C23#58512yes—
575Q8—232no—
576D4—416no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—816no—
664C2×C4C2×#12314yes—
665C2×C4C2×#12314yes—
666C2×C4C2×#12314yes—
667C2×C4C2×#12718yes—
668C2×C4C2×#12714yes—
669C2×C4#53614yes—
670C2×C4#54718yes—
671C2×C4#54718yes—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22C22×#12311yescharacteristic
688C22C22×#12711yescharacteristic
689C22C2×#53611yescharacteristic
690C22#346011yescharacteristic
691C22#346111yescharacteristic
692C22C2×#54712yes—
693C22C2×#54712yes—
694C22#348612yes—
695C22#348611yescharacteristic
696C22#348711yescharacteristic
697C22#348712yes—
698C22#348812yes—
699C22#348811yescharacteristic
700C22#348912yes—
701C22#348911yescharacteristic
702C22#349512yes—
703C22#349512yes—
704C22#351111yescharacteristic
705C22#351211yescharacteristic
706C22#355212yes—
707C22#355512yes—
708C22#356312yes—
709C22#356512yes—
710C4—24no—
711C4—24no—
712C4—28no—
713C4—44no—
714C4—42no—
715C4—48no—
716C4—44no—
717C4—44no—
718C4—42no—
719C4—82no—
720C4—81no—
721C4—82no—
722C4—81no—
723C4—82no—
724C4—84no—
725C4#250118yes—
726C2—84no—
727C2C2×#250111yescharacteristic
728C2#3525511yescharacteristic
729C2#3567211yescharacteristic
730C2#3592711yescharacteristic
731C2#3606611yescharacteristic
732C2#3776512yes—
733C2#3776712yes—
734C2#4281411yescharacteristic
735C2#4289111yescharacteristic
736C2#4526412yes—
737C2#4533312yes—
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
2. C2×#2421C2semidirect
3. C2×#2423C2semidirect
4. C2×#2351C2semidirect
5. #16964C2semidirect
7. #15808C2semidirect
8. #15765C2semidirect
9. #15223C2semidirect
10. #15135C2semidirect
12. #15082C2semidirect
13. #14354C2semidirect
14. #13910C2semidirect
17. #13786C2semidirect
146. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24212
C2×#24232
C2×#23512
#169642
#166922
#158082
#157652
#152232
#151352
#150842
#150822
#143542
#139102
#139052
#138472
#137862