Finite Group Databasebeta

← #1395882

Subgroups — #1395882

Order 512.

3570 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#2453C212yesmaximal, complemented
3C2×#2454C211yesmaximal, characteristic
4#16674C214yesmaximal, complemented
5#16659C214yesmaximal, complemented
6#16366C212yesmaximal, complemented
7#16264C212yesmaximal, complemented
8#15226C212yesmaximal
9#15217C212yesmaximal
10#15202C211yesmaximal, characteristic, complemented
11#15188C211yesmaximal, characteristic, complemented
12#15180C211yesmaximal, characteristic, complemented
13#15114C212yesmaximal, complemented
14#14264C212yesmaximal, complemented
15#13935C212yesmaximal, complemented
16#13920C211yesmaximal, characteristic
17C4×#395C211yesmaximal, characteristic, complemented
18C2×#556—216no—
19C2×#556—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2214yes—
22C22×#125C2211yescharacteristic
23C22×#124C2212yes—
24C22×C22⋊Q8C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2212yes—
27C22×C4×D4C2211yescharacteristic
28C22×#117C2212yes—
29#2454—28no—
30#2453—216no—
31C2×#400C2214yes—
32C2×#399C2212yes—
33C2×#398C2214yes—
34C2×#396C2212yes—
35C2×#395C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2212yes—
43C2×#389C2214yes—
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#389C2211yescharacteristic
48C2×#387C2214yes—
49C2×#385C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54#1446C2211yescharacteristic
55#1446C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1444C2211yescharacteristic
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1440C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83#1434C2214yes—
84C22⋊#95C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2312yes—
91#556—2128no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×#126—216no—
95C2×#126—216no—
96C2×#126—216no—
97C2×#125—28no—
98C2×#125—24no—
99C2×#124—28no—
100C2×#124—28no—
101C2×#124—28no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—216no—
104C2×C4×Q8—28no—
105C2×C4×Q8—28no—
106C2×C4×Q8—28no—
107C2×C4×Q8—24no—
108C2×C4×Q8D4116yes—
109C2×C4×D4—28no—
110C2×C4×D4D418yes—
111C2×#117—28no—
112C2×#117—216no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2314yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144C22×C42C2314yes—
145C22×C42C2311yescharacteristic
146C22×C42C2312yes—
147#400—216no—
148#399—28no—
149#398—216no—
150#396—28no—
151#395—24no—
152#391—28no—
153#391—28no—
154#391—44no—
155#391—44no—
156#390—24no—
157#390—24no—
158#390—44no—
159#389—28no—
160#389—24no—
161#389—216no—
162#389—28no—
163#389—24no—
164#387—216no—
165#385—28no—
166#380—28no—
167#380—28no—
168C4×C4⋊C4—28no—
169C4×C4⋊C4—24no—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C25C2412yes—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—42no—
209C22×Q8C2×D418yes—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4C2×D414yes—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4C2×D418yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—232no—
237#126—2128no—
238#125—416no—
239#124—264no—
240C22⋊Q8—264no—
241C4×Q8—232no—
242C4×Q8—264no—
243C4×D4—232no—
244#117—264no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2×D4116yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D414yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D418yes—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42C2×D4116yes—
330C2×C42C2×D414yes—
331C2×C42C2×D418yes—
332#95—22no—
333#95—24no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—22no—
348#95—42no—
349#95—44no—
350#95—42no—
351#95—42no—
352#95—44no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95#4114yes—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24C22×D414yes—
370C24C2511yescharacteristic
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—216no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—48no—
387C2×D4—48no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4C22×D418yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D418yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C22×C42−1+414yes—
486C22×C42−1+412yes—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—232no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4#120116yes—
506C22⋊C4—232no—
507C22⋊C4—232no—
508C22⋊C4—216no—
509C22⋊C4—48no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C42—232no—
516C42—28no—
517C42—216no—
518C42—44no—
519C42—44no—
520C42—48no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C23×D411yescharacteristic
545C23C23×D412yes—
546C23C22×#4111yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+412yes—
549C23C2×2−1+411yescharacteristic
550C23C2×2−1+411yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23#58512yes—
553C23#58511yescharacteristic
554C23#58512yes—
555Q8—416no—
556Q8—432no—
557D4—416no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—432no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4C2×#12018yes—
647C2×C4C2×#12014yes—
648C2×C4#53614yes—
649C2×C4#53618yes—
650C2×C4#53714yes—
651C2×C4#53718yes—
652C2×C4#54714yes—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22C22×#12011yescharacteristic
665C22C2×#53612yes—
666C22C2×#53611yescharacteristic
667C22C2×#53711yescharacteristic
668C22C2×#53712yes—
669C22C2×#54711yescharacteristic
670C22D4×#4112yes—
671C22#348511yescharacteristic
672C22#348712yes—
673C22#348911yescharacteristic
674C22#348912yes—
675C22#349712yes—
676C22#350011yescharacteristic
677C22#350712yes—
678C22#351411yescharacteristic
679C22#352311yescharacteristic
680C22#355211yescharacteristic
681C22#355511yescharacteristic
682C22#355512yes—
683C22#355512yes—
684C22#355712yes—
685C22#355712yes—
686C22#356212yes—
687C4—24no—
688C4—28no—
689C4—42no—
690C4—48no—
691C4—44no—
692C4—44no—
693C4—42no—
694C4—44no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4—82no—
699C4—81no—
700C4—82no—
701C4—84no—
702C4#261618yes—
703C2—84no—
704C2C2×#261611yescharacteristic
705C2#3575711yescharacteristic
706C2#3621311yescharacteristic
707C2#3892311yescharacteristic
708C2#3949011yescharacteristic
709C2#3988212yes—
710C2#3995112yes—
711C2#4264511yescharacteristic
712C2#4267611yescharacteristic
713C2#4280212yes—
714C2#4288412yes—
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×#2453C2semidirect
4. #16674C2semidirect
5. #16659C2semidirect
6. #16366C2semidirect
7. #16264C2semidirect
10. #15202C2semidirect
11. #15188C2semidirect
12. #15180C2semidirect
13. #15114C2semidirect
14. #14264C2semidirect
15. #13935C2semidirect
17. C4×#395C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#24532
C2×#24542
#166742
#166592
#163662
#162642
#152262
#152172
#152022
#151882
#151802
#151142
#142642
#139352
#139202
C4×#3952