Finite Group Databasebeta

← #4389097

Subgroups — #4389097

Order 512.

3534 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic, complemented
2C2×#2631C212yesmaximal, complemented
3C2×#2326C211yesmaximal, characteristic, complemented
4#16652C212yesmaximal, complemented
5#16551C212yesmaximal, complemented
6#15687C214yesmaximal, complemented
7#15390C214yesmaximal, complemented
8#15320C214yesmaximal, complemented
9#15263C214yesmaximal, complemented
10#14883C212yesmaximal, complemented
11#14722C212yesmaximal, complemented
12#14676C211yesmaximal, characteristic, complemented
13#13959C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×#121C2214yes—
16C22×#121C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2631—216no—
21#2537—28no—
22C2×#397C2214yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2214yes—
26C2×#394C2212yes—
27C2×#393C2214yes—
28C2×#392C2214yes—
29C2×#392C2214yes—
30C2×#392C2212yes—
31C2×#392C2214yes—
32#2326—28no—
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#388C2214yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#383C2212yes—
43C2×#380C2214yes—
44C2×#376C2211yescharacteristic
45#1445C2212yes—
46#1445C2212yes—
47#1443C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2212yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2314yes—
75C23×D4C2314yes—
76C23×D4C2311yescharacteristic
77C24×C4C2314yes—
78C24×C4C2312yes—
79C2×#124—24no—
80C2×#124—24no—
81C2×#124D418yescomplemented
82C2×#121—216no—
83C2×#121—44no—
84C2×#121—48no—
85C2×#121—48no—
86C2×#121—48no—
87C2×C4×Q8—24no—
88C2×C4×Q8—42no—
89C2×C4×Q8—42no—
90C2×C4×D4—24no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
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×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#397—216no—
122#394—28no—
123#394—28no—
124#394—44no—
125#394—48no—
126#393—216no—
127#392—216no—
128#392—216no—
129#392—44no—
130#392—48no—
131#391—24no—
132#390—24no—
133#390—44no—
134#388—216no—
135#386—216no—
136#386—216no—
137#386—48no—
138#385—216no—
139#384D418yescomplemented
140#383—28no—
141#380—216no—
142#376—24no—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C25C2412yes—
163C25C2414yes—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8C2×D414yes—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4C2×D414yes—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230#124—232no—
231#121—464no—
232#121—432no—
233C4×Q8—416no—
234C4×D4—416no—
235C4×D4—432no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D4116yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
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—42no—
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⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C2×D4116yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—42no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42#4114yes—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95—44no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24C22×D414yes—
353C24H(5,𝔽2)18yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×D4—216no—
363C2×D4—28no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—48no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—88no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
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×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—84no—
480C22⋊C4—232no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—28no—
495C42—48no—
496C42—44no—
497C42#121116yescomplemented
498C23—24no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
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—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C22×#4111yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)14yes—
562C23#58511yescharacteristic
563C23#58511yescharacteristic
564C23#58514yes—
565Q8—416no—
566D4—416no—
567D4—432no—
568D4—816no—
569D4—816no—
570D4—832no—
571D4—832no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
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—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—832no—
634C2×C4—816no—
635C2×C4C2×#12118yes—
636C2×C4C2×#12114yes—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—44no—
641C22—48no—
642C22—44no—
643C22—48no—
644C22—48no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—82no—
650C22—84no—
651C22—84no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—82no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22C22×#12111yescharacteristic
664C22#346011yescharacteristic
665C22#346011yescharacteristic
666C22#348412yes—
667C22#348411yescharacteristic
668C22#348711yescharacteristic
669C22#348712yes—
670C22#348814yes—
671C22#349414yes—
672C22#349611yescharacteristic
673C22#349611yescharacteristic
674C22#353212yes—
675C22#353314yes—
676C22#353712yes—
677C22#354512yes—
678C22#355112yes—
679C22#356014yes—
680C4—24no—
681C4—28no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—82no—
688C4—84no—
689C4—84no—
690C4—84no—
691C2—88no—
692C2—84no—
693C2#3516811yescharacteristic
694C2#3556312yes—
695C2#4203512yes—
696C2#4240112yes—
697C2#4767412yes—
698C2#4767812yes—
699C2#5200411yescharacteristic
700C2#5201012yes—
701C2#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×#2631C2semidirect
3. C2×#2326C2semidirect
4. #16652C2semidirect
5. #16551C2semidirect
6. #15687C2semidirect
7. #15390C2semidirect
8. #15320C2semidirect
9. #15263C2semidirect
10. #14883C2semidirect
11. #14722C2semidirect
12. #14676C2semidirect
13. #13959C2semidirect
81. C2×#124D4semidirect
139. #384D4semidirect
497. 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×#26312
C2×#23262
#166522
#165512
#156872
#153902
#153202
#152632
#148832
#147222
#146762
#139592