Finite Group Databasebeta

← #4386580

Subgroups — #4386580

Order 512.

3206 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2819C211yesmaximal, characteristic, complemented
2C2×#2610C212yesmaximal, complemented
3C2×#2317C211yesmaximal, characteristic, complemented
4#16684C212yesmaximal, complemented
5#16514C212yesmaximal, complemented
6#15633C214yesmaximal, complemented
7#15617C214yesmaximal, complemented
8#15331C214yesmaximal, complemented
9#15315C214yesmaximal, complemented
10#14886C212yesmaximal, complemented
11#14719C212yesmaximal, complemented
12#14273C212yesmaximal, complemented
13#14168C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×#121C2212yes—
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2819—28no—
20#2610—216no—
21C2×#396C2214yes—
22C2×#395C2212yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#393C2214yes—
29C2×#393C2212yes—
30C2×#392C2214yes—
31C2×#392C2212yes—
32C2×#392C2212yes—
33#2317—28no—
34C2×#391C2214yes—
35C2×#390C2214yes—
36C2×#390C2212yes—
37C2×#390C2214yes—
38C2×#389C2214yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#376C2212yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1434C2212yes—
69C22⋊#95C2212yes—
70C23×Q8C2312yes—
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×#124—28no—
78C2×#124—28no—
79C2×#124—28no—
80C2×#121—48no—
81C2×#121—44no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—42no—
86C2×#117—42no—
87C2×#117D418yescomplemented
88C22×C4⋊C4—24no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#396—48no—
121#395—28no—
122#394—28no—
123#394—28no—
124#394—28no—
125#394—48no—
126#394—44no—
127#393—28no—
128#393—216no—
129#392—28no—
130#392—216no—
131#392—44no—
132#391—48no—
133#390—216no—
134#390—44no—
135#390—48no—
136#389—216no—
137#388—216no—
138#388—216no—
139#386—216no—
140#386—48no—
141#385—216no—
142#384—24no—
143#384—24no—
144#382—24no—
145#382D418yescomplemented
146#376—28no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C25C2414yes—
165C25C2411yescharacteristic
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—28no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—42no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—42no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D414yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216#124—432no—
217#121—432no—
218#117—232no—
219#117—416no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×D418yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95C2×D4116yes—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—24no—
327C24—21no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24C22×D412yes—
345C24H(5,𝔽2)12yes—
346C242−1+412yes—
347C24C2511yescharacteristic
348C2×Q8—416no—
349C2×Q8—48no—
350C2×Q8—88no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—88no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—88no—
374C2×D4—88no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—88no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)18yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C4⋊C4—216no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C22⋊C4—216no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C42—28no—
478C42—416no—
479C42—48no—
480C42#121116yescomplemented
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
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—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23#53714yes—
533C23C23×D411yescharacteristic
534C23C23×D411yescharacteristic
535C23C22×#4111yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)14yes—
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23#58514yes—
542Q8—816no—
543D4—816no—
544D4—832no—
545D4—816no—
546D4—816no—
547C2×C4—232no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
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—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—816no—
611C2×C4C2×#12114yes—
612C2×C4C2×#12118yes—
613C2×C4#53714yes—
614C22—24no—
615C22—28no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22C22×#12111yescharacteristic
639C22C2×#53711yescharacteristic
640C22C2×#53711yescharacteristic
641C22#348411yescharacteristic
642C22#348611yescharacteristic
643C22#348814yes—
644C22#349211yescharacteristic
645C22#349211yescharacteristic
646C22#349812yes—
647C22#350012yes—
648C22#350614yes—
649C22#353212yes—
650C22#353514yes—
651C22#354012yes—
652C22#354512yes—
653C22#354714yes—
654C22#354912yes—
655C4—24no—
656C4—24no—
657C4—28no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—48no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—84no—
666C2—44no—
667C2—164no—
668C2#3522311yescharacteristic
669C2#3553312yes—
670C2#4198312yes—
671C2#4297112yes—
672C2#4780712yes—
673C2#4780912yes—
674C2#5174911yescharacteristic
675C2#5175112yes—
676C2#5190211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2819C2semidirect
2. C2×#2610C2semidirect
3. C2×#2317C2semidirect
4. #16684C2semidirect
5. #16514C2semidirect
6. #15633C2semidirect
7. #15617C2semidirect
8. #15331C2semidirect
9. #15315C2semidirect
10. #14886C2semidirect
11. #14719C2semidirect
12. #14273C2semidirect
13. #14168C2semidirect
87. C2×#117D4semidirect
145. #382D4semidirect
480. C42#121semidirect

Maximal subgroups

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

subgroupindex
C2×#28192
C2×#26102
C2×#23172
#166842
#165142
#156332
#156172
#153312
#153152
#148862
#147192
#142732
#141682