Finite Group Databasebeta

← #545504

Subgroups — #545504

Order 512.

2848 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3C2×#2352C211yesmaximal, characteristic
4#14630C212yesmaximal, complemented
5#14430C214yesmaximal, complemented
6#14421C214yesmaximal
7#14313C212yesmaximal, complemented
8#14306C212yesmaximal, complemented
9#14282C211yesmaximal, characteristic, complemented
10#14225C212yesmaximal, complemented
11#14130C212yesmaximal, complemented
12#14129C212yesmaximal, complemented
13#13974C212yesmaximal, complemented
14#13973C212yesmaximal, complemented
15#13954C211yesmaximal, characteristic, complemented
16#13847C211yesmaximal, characteristic
17#13797C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2547C4116yescomplemented
24C2×#397C2212yes—
25C2×#397C2212yes—
26#2369—28no—
27#2352—28no—
28C2×#391C2214yes—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#388C2214yes—
36C2×#387C2212yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×#382C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1446C2214yes—
54#1445C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1444C2212yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1440C2214yes—
64#1439C2211yescharacteristic
65#1439C2211yescharacteristic
66#1439C2211yescharacteristic
67#1439C2211yescharacteristic
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2214yes—
83#1434C2214yes—
84#1434C2212yes—
85C22⋊#95C2214yes—
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8C2×C418yescomplemented
94C2×#124—44no—
95C2×#124—44no—
96C2×#124C2×C4116yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4C2×C418yes—
100C2×#117—24no—
101C2×#117—44no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#397C2×C4116yes—
137#397C2×C4116yescomplemented
138#391—216no—
139#391C2×C4116yescomplemented
140#389—28no—
141#389—28no—
142#389—44no—
143#389—44no—
144#388—28no—
145#388—48no—
146#387—28no—
147#386—28no—
148#385—48no—
149#382—24no—
150#380—24no—
151#380—24no—
152#380C2×C4116yes—
153#380C2×C4116yescomplemented
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4C2×C418yescomplemented
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—44no—
161C4×C22⋊C4—44no—
162#376—24no—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C25C2411yescharacteristic
192C25C2411yescharacteristic
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8C22×C418yes—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4C22×C414yes—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—232no—
238#124—432no—
239C4×D4—232no—
240#117—416no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C22×C414yes—
272C2×C4⋊C4C22×C418yes—
273C2×C4⋊C4C22×C418yes—
274C2×C4⋊C4C22×C418yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C22×C414yes—
305C2×C22⋊C4C22×C414yes—
306C2×C22⋊C4C22×C4116yes—
307C2×C22⋊C4C22×C4116yes—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—44no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C22×C418yes—
323C2×C42C22×C414yes—
324C2×C42C2×D418yes—
325C2×C42C2×D414yes—
326C2×C42#4114yes—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C22×C418yes—
352#95C22×C4116yes—
353#95C22×C4116yes—
354#95C2×D418yes—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—21no—
373C24C23×C412yes—
374C24C23×C412yes—
375C24H(5,𝔽2)12yes—
376C24H(5,𝔽2)12yes—
377C24C2511yescharacteristic
378C2×Q8—216no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—84no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
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—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4C23×C412yes—
479C22×C4C23×C412yes—
480C22×C4C23×C418yes—
481C22×C4C23×C418yes—
482C22×C4C23×C412yes—
483C22×C4C23×C412yes—
484C22×C4C23×C412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C42−1+414yes—
495C22×C42−1+414yes—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—88no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C22⋊C4—232no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—84no—
521C42—48no—
522C42—48no—
523C42—48no—
524C42—48no—
525C42—48no—
526C42—88no—
527C42C4×D4116yes—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23#51814yes—
574C23C24×C411yescharacteristic
575C23C23×D411yescharacteristic
576C23C22×#4111yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)12yes—
580C23C2×2−1+412yes—
581C23C2×2−1+412yes—
582C23#58511yescharacteristic
583C23#58511yescharacteristic
584C23#58512yes—
585Q8—816no—
586D4—416no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—416no—
639C2×C4—416no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—816no—
672C2×C4C2×C4×D414yes—
673C2×C4C2×C4×D414yes—
674C2×C4C2×C4×D414yes—
675C2×C4#52018yes—
676C2×C4#53514yes—
677C2×C4#53614yes—
678C2×C4#54218yes—
679C22—24no—
680C22—28no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—42no—
695C22—81no—
696C22—81no—
697C22—82no—
698C22—82no—
699C22—81no—
700C22—81no—
701C22C22×C4×D411yescharacteristic
702C22C2×#51811yescharacteristic
703C22C2×#52012yes—
704C22#342811yescharacteristic
705C22C4×2−1+412yes—
706C22C2×#53511yescharacteristic
707C22C2×#53611yescharacteristic
708C22C2×#54212yes—
709C22#348511yescharacteristic
710C22#348611yescharacteristic
711C22#348811yescharacteristic
712C22#348811yescharacteristic
713C22#349411yescharacteristic
714C22#349411yescharacteristic
715C22#350712yes—
716C22#353612yes—
717C22#354012yes—
718C22#354112yes—
719C22#354212yes—
720C22#354412yes—
721C22#355512yes—
722C22#355912yes—
723C22#356512yes—
724C4—24no—
725C4—24no—
726C4—42no—
727C4—42no—
728C4—44no—
729C4—48no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4#232618yes—
739C2—44no—
740C2—82no—
741C2C2×#232611yescharacteristic
742C2#3420411yescharacteristic
743C2#3426611yescharacteristic
744C2#3446012yes—
745C2#3446312yes—
746C2#4266612yes—
747C2#4267112yes—
748C2#4267711yescharacteristic
749C2#4268211yescharacteristic
750C2#4281011yescharacteristic
751C2#4281311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
4. #14630C2semidirect
5. #14430C2semidirect
7. #14313C2semidirect
8. #14306C2semidirect
9. #14282C2semidirect
10. #14225C2semidirect
11. #14130C2semidirect
12. #14129C2semidirect
13. #13974C2semidirect
14. #13973C2semidirect
15. #13954C2semidirect
17. #13797C2semidirect
23. #2547C4semidirect
93. C2×C4⋊Q8C2×C4semidirect
137. #397C2×C4semidirect
139. #391C2×C4semidirect
153. #380C2×C4semidirect
157. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#23692
C2×#23522
#146302
#144302
#144212
#143132
#143062
#142822
#142252
#141302
#141292
#139742
#139732
#139542
#138472
#137972