Finite Group Databasebeta

← #4468245

Subgroups — #4468245

Order 512.

3362 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic
2C2×#2640C211yesmaximal, characteristic, complemented
3#17107C214yesmaximal, complemented
4#16887C212yesmaximal, complemented
5#16839C214yesmaximal, complemented
6#16255C214yesmaximal, complemented
7#15833C212yesmaximal, complemented
8#15761C212yesmaximal, complemented
9#15281C212yesmaximal, complemented
10#15259C214yesmaximal, complemented
11#15259C212yesmaximal, complemented
12#15120C212yesmaximal, complemented
13#14503C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic, complemented
15C22×#124C2211yescharacteristic
16C22×#124C2212yes—
17C22×#120C2214yes—
18C22×#120C2212yes—
19#2679—28no—
20#2640—28no—
21C2×#397C2214yescomplemented
22C2×#394C2212yes—
23C2×#394C2214yes—
24C2×#394C2212yes—
25C2×#394C2214yes—
26C2×#393C2214yescomplemented
27C2×#393C2214yescomplemented
28C2×#393C2214yescomplemented
29C2×#391C2211yescharacteristic, complemented
30C2×#391C2211yescharacteristic, complemented
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#388C2214yescomplemented
35C2×#379C2212yescomplemented
36C2×#379C2214yescomplemented
37C2×#379C2212yescomplemented
38C2×#376C2211yescharacteristic, complemented
39#1446C2212yescomplemented
40#1445C2214yescomplemented
41#1445C2214yescomplemented
42#1444C2212yescomplemented
43#1442C2212yescomplemented
44#1440C2214yescomplemented
45#1440C2214yescomplemented
46#1440C2214yescomplemented
47#1440C2212yescomplemented
48#1440C2214yescomplemented
49#1440C2212yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1438C2212yes—
56#1438C2212yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2214yes—
63#1437C2214yescomplemented
64#1437C2214yescomplemented
65#1437C2214yescomplemented
66#1436C2212yescomplemented
67#1436C2212yescomplemented
68#1434C2212yescomplemented
69C26C2312yes—
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2314yes—
75C23×D4C2311yescharacteristic
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—42no—
79C2×#124—24no—
80C2×#124—42no—
81C2×#124—42no—
82C2×#124—48no—
83C2×#124—44no—
84C2×#120—28no—
85C2×#120—216no—
86C2×#120—48no—
87C2×#120—48no—
88C2×#120—48no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#397—216no—
118#394—28no—
119#394—216no—
120#394—44no—
121#394—48no—
122#393—216no—
123#393—48no—
124#393—48no—
125#391—24no—
126#391—24no—
127#390—24no—
128#390—28no—
129#390D418yes—
130#388—48no—
131#379—44no—
132#379—44no—
133#379—48no—
134#376—24no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25—24no—
154C25—28no—
155C25—44no—
156C25—44no—
157C25—44no—
158C25—44no—
159C25—44no—
160C25—42no—
161C25—42no—
162C25C2412yes—
163C25C2414yes—
164C22×Q8—22no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—44no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—42no—
195C22×D4C2×D414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C4⋊Q8—416no—
208#124—416no—
209#124—432no—
210#120—464no—
211#120—432no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C2×D414yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D4116yes—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—44no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C2×D414yes—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—28no—
295#95—42no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—44no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—416no—
330C24—48no—
331C24—48no—
332C24—416no—
333C24—48no—
334C24—84no—
335C24—84no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24—88no—
340C24—88no—
341C24—88no—
342C24—84no—
343C24—84no—
344C24—88no—
345C24—88no—
346C24—88no—
347C24—88no—
348C24—88no—
349C24—88no—
350C24C22×D414yes—
351C24H(5,𝔽2)18yes—
352C24C2511yescharacteristic
353C2×Q8—48no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8—88no—
361C2×Q8—84no—
362C2×Q8—88no—
363C2×D4—216no—
364C2×D4—28no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—84no—
383C2×D4—88no—
384C2×D4—88no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)14yes—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C42—28no—
469C42—44no—
470C42—48no—
471C42—44no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—816no—
529C23—832no—
530C23—832no—
531C23—832no—
532C23—816no—
533C23—816no—
534C23—832no—
535C23—832no—
536C23—832no—
537C23C23×D411yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)14yes—
541C23C2×H(5,𝔽2)11yescharacteristic
542C23#58512yes—
543C23#58514yes—
544Q8—88no—
545Q8—816no—
546D4—88no—
547D4—816no—
548D4—816no—
549D4—832no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4#53514yes—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—48no—
613C22—48no—
614C22—44no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—864no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22C2×#53511yescharacteristic
635C22#348411yescharacteristic
636C22#348412yes—
637C22#348511yescharacteristic
638C22#350012yes—
639C22#353212yes—
640C22#353314yes—
641C22#353712yes—
642C22#354012yes—
643C22#354214yes—
644C22#354512yes—
645C22#354514yes—
646C22#355812yes—
647C22#355814yes—
648C22#355812yes—
649C4—24no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—84no—
660C2—88no—
661C2—84no—
662C2#3696711yescharacteristic
663C2#4242612yes—
664C2#4681412yes—
665C2#5174911yescharacteristic
666C2#5200411yescharacteristic
667C2#5629514yes—
668C2#5764314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2679C2
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×#2640C2semidirect
3. #17107C2semidirect
4. #16887C2semidirect
5. #16839C2semidirect
6. #16255C2semidirect
7. #15833C2semidirect
8. #15761C2semidirect
9. #15281C2semidirect
10. #15259C2semidirect
11. #15259C2semidirect
12. #15120C2semidirect
13. #14503C2semidirect
14. C22×C4⋊Q8C22semidirect
21. C2×#397C22semidirect
26. C2×#393C22semidirect
27. C2×#393C22semidirect
28. C2×#393C22semidirect
29. C2×#391C22semidirect
30. C2×#391C22semidirect
34. C2×#388C22semidirect
35. C2×#379C22semidirect
36. C2×#379C22semidirect
37. C2×#379C22semidirect
38. C2×#376C22semidirect
39. #1446C22semidirect
40. #1445C22semidirect
41. #1445C22semidirect
42. #1444C22semidirect
43. #1442C22semidirect
44. #1440C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1437C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#26402
#171072
#168872
#168392
#162552
#158332
#157612
#152812
#152592
#151202
#145032