Finite Group Databasebeta

← #4387771

Subgroups — #4387771

Order 512.

3426 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic, complemented
2C2×#2598C212yesmaximal, complemented
3C2×#2317C211yesmaximal, characteristic, complemented
4#16684C212yesmaximal, complemented
5#16514C212yesmaximal, complemented
6#15679C214yesmaximal, complemented
7#15439C214yesmaximal, complemented
8#15316C214yesmaximal, complemented
9#15263C214yesmaximal, complemented
10#14886C212yesmaximal, complemented
11#14719C212yesmaximal, complemented
12#14676C211yesmaximal, characteristic, complemented
13#14345C212yesmaximal, complemented
14C22×#121C2214yes—
15C22×#121C2212yes—
16C22×C4×D4C2212yes—
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2598—216no—
21#2537—28no—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#396C2214yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#392C2212yes—
31C2×#392C2212yes—
32#2317—28no—
33C2×#390C2212yes—
34C2×#389C2214yes—
35C2×#388C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#383C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2214yes—
47#1445C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1437C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2314yes—
76C24×C4C2312yes—
77C2×#121—216no—
78C2×#121—48no—
79C2×#121—48no—
80C2×#121—48no—
81C2×#121—44no—
82C2×C4×D4—28no—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—42no—
89C2×#117—42no—
90C2×#117D418yescomplemented
91C22×C4⋊C4—24no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#400—28no—
123#399—28no—
124#396—48no—
125#394—28no—
126#394—28no—
127#394—28no—
128#394—44no—
129#394—48no—
130#392—28no—
131#392—44no—
132#390—44no—
133#389—216no—
134#388—216no—
135#386—216no—
136#386—216no—
137#386—48no—
138#385—216no—
139#385—216no—
140#384—24no—
141#384D418yescomplemented
142#383—28no—
143#382—24no—
144#382—24no—
145#380—216no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C25C2414yes—
165C25C2411yescharacteristic
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—42no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218#121—464no—
219#121—432no—
220C4×D4—432no—
221#117—232no—
222#117—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D4116yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
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—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
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⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C2×D418yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—44no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—24no—
328C24—21no—
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)18yes—
346C24H(5,𝔽2)12yes—
347C242−1+412yes—
348C24C2511yescharacteristic
349C2×D4—216no—
350C2×D4—416no—
351C2×D4—416no—
352C2×D4—48no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—48no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—48no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—84no—
370C2×D4—84no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C4⋊C4—216no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—84no—
467C22⋊C4—216no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C42—28no—
481C42—48no—
482C42#121116yescomplemented
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—28no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—42no—
504C23—42no—
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—44no—
533C23—44no—
534C23#53714yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)14yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23#58514yes—
544D4—432no—
545D4—816no—
546D4—816no—
547D4—832no—
548C2×C4—232no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
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—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—832no—
612C2×C4—816no—
613C2×C4C2×#12114yes—
614C2×C4C2×#12118yes—
615C2×C4#53714yes—
616C22—24no—
617C22—28no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—48no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22C22×#12111yescharacteristic
641C22C2×#53711yescharacteristic
642C22C2×#53711yescharacteristic
643C22#348412yes—
644C22#348411yescharacteristic
645C22#348611yescharacteristic
646C22#348712yes—
647C22#348814yes—
648C22#349211yescharacteristic
649C22#349211yescharacteristic
650C22#349414yes—
651C22#353212yes—
652C22#353514yes—
653C22#353712yes—
654C22#354512yes—
655C22#355112yes—
656C22#355914yes—
657C4—24no—
658C4—24no—
659C4—28no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4—84no—
667C4—84no—
668C2—44no—
669C2—88no—
670C2#3522311yescharacteristic
671C2#3556312yes—
672C2#4198312yes—
673C2#4295212yes—
674C2#4769512yes—
675C2#4770012yes—
676C2#5200411yescharacteristic
677C2#5200612yes—
678C2#5208311yescharacteristic
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×#2537C2semidirect
2. C2×#2598C2semidirect
3. C2×#2317C2semidirect
4. #16684C2semidirect
5. #16514C2semidirect
6. #15679C2semidirect
7. #15439C2semidirect
8. #15316C2semidirect
9. #15263C2semidirect
10. #14886C2semidirect
11. #14719C2semidirect
12. #14676C2semidirect
13. #14345C2semidirect
90. C2×#117D4semidirect
141. #384D4semidirect
482. C42#121semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25982
C2×#23172
#166842
#165142
#156792
#154392
#153162
#152632
#148862
#147192
#146762
#143452