Finite Group Databasebeta

← #6472777

Subgroups — #6472777

Order 512.

3098 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3#32985C214yesmaximal, complemented
4#29894C214yesmaximal, complemented
5#27604C214yesmaximal, complemented
6#24554C218yesmaximal, complemented
7#24010C214yesmaximal, complemented
8#21940C211yesmaximal, characteristic, complemented
9#21938C211yesmaximal, characteristic, complemented
10#21759C211yesmaximal, characteristic, complemented
11#21759C211yesmaximal, characteristic, complemented
12C2×#1779C211yesmaximal, characteristic
13C25×C4C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×#117C2212yes—
17C2×#485C2218yescomplemented
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2214yes—
21#2848—21no—
22#2848—21no—
23#2848—22no—
24#2848C2212yes—
25#2848C2212yes—
26#2848C2214yes—
27C2×#385—21no—
28C2×#385—22no—
29#2280—22no—
30#2280—22no—
31#2280C2214yes—
32#2280C2214yes—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#379—21no—
40C2×#379—21no—
41C2×C4×C22⋊C4—21no—
42C2×C4×C22⋊C4—21no—
43#2077C2218yescomplemented
44#2075C2218yescomplemented
45#2036C2218yescomplemented
46#2035C2218yescomplemented
47#1892C2214yescomplemented
48#1892C2214yescomplemented
49#1873C2218yescomplemented
50#1872C2214yescomplemented
51#1872C2214yescomplemented
52#1854C2218yescomplemented
53#1853C2214yescomplemented
54#1853C2214yescomplemented
55#1852C2214yescomplemented
56#1852C2214yescomplemented
57#1825C2218yescomplemented
58#1824C2212yescomplemented
59#1824C2212yescomplemented
60#1808C2214yescomplemented
61#1807C2214yescomplemented
62#1787C2212yescomplemented
63#1787C2212yescomplemented
64#1779C2212yes—
65#1779C2214yes—
66#1779C2212yes—
67C26D411yescharacteristic
68C23×Q8C2311yescharacteristic
69C24×C4—24no—
70C24×C4—24no—
71C24×C4—48no—
72C24×C4D411yescharacteristic
73C24×C4D411yescharacteristic
74C24×C4D412yes—
75C24×C4D411yescharacteristic
76C24×C4D412yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C23×C8C2312yes—
83C23×C8C2312yes—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8C2314yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C22⋊Q8C2314yes—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117—48no—
98C2×#117C2314yes—
99C2×#117C2314yes—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4—21no—
114C22×C42—22no—
115#485—416no—
116C2×#107C2318yescomplemented
117C2×#106C2318yescomplemented
118C2×#103C2318yescomplemented
119C2×#103C2318yescomplemented
120C2×C22⋊C8—24no—
121C2×C22⋊C8—24no—
122C2×C22⋊C8—48no—
123C2×C22⋊C8C2318yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2318yes—
129#385—24no—
130#385—22no—
131#385—24no—
132#385—24no—
133#385—24no—
134#385—22no—
135#385—24no—
136#381—24no—
137#381—24no—
138#380—22no—
139#380—22no—
140#380—21no—
141#380—21no—
142#380—22no—
143#380—22no—
144#380—21no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380C2312yes—
149#380C2312yes—
150#380C2312yes—
151#380C2314yes—
152#380C2314yes—
153#380C2314yes—
154#380C2312yes—
155#380C2312yes—
156#380C2314yes—
157#380C2312yes—
158#379—22no—
159#379—22no—
160#379—24no—
161#379—24no—
162#379—24no—
163C4×C22⋊C4—24no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C2×#95—22no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—22no—
171C2×#95—21no—
172#340C2318yescomplemented
173#340C2318yescomplemented
174#340C2318yescomplemented
175#340C2318yescomplemented
176C25—24no—
177C25—24no—
178C25—24no—
179C25—44no—
180C25—44no—
181C25C2×D411yescharacteristic
182C25C2×D411yescharacteristic
183C25C2×D412yes—
184C25C2×D412yes—
185C25C2×D411yescharacteristic
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—44no—
190C22×Q8C2412yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4#4114yes—
239C23×C4#4114yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C22×C8—24no—
246C22×C8—24no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—216no—
256C22⋊Q8—48no—
257C22⋊Q8—48no—
258C22⋊Q8—48no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261#117—216no—
262#117—48no—
263#117—48no—
264#117—48no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—48no—
311#107—48no—
312#106—48no—
313#103—48no—
314#103—48no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C22⋊C8—216no—
320C22⋊C8—48no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—416no—
358C24—416no—
359C24—432no—
360C24—432no—
361C24—416no—
362C24#12314yes—
363C24#13514yes—
364C24#13714yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C22×D411yescharacteristic
368C24C22×D411yescharacteristic
369C24C22×D411yescharacteristic
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—28no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—84no—
376C2×Q8—84no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—432no—
433C22×C4—432no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4#12314yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)18yes—
466C22×C4C2511yescharacteristic
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—28no—
470C2×C8—28no—
471C2×C8—84no—
472C2×C8—84no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C42—48no—
503C42—48no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—28no—
511C23—216no—
512C23—216no—
513C23—28no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—44no—
527C23—44no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—44no—
543C23—48no—
544C23—48no—
545C23—44no—
546C23—4128no—
547C23—44no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23#485116yescomplemented
552C23C2×#12312yes—
553C23C2×#12312yes—
554C23C2×#12312yes—
555C23#53414yes—
556C23C2×#13512yes—
557C23C2×#13512yes—
558C23C2×#13512yes—
559C23C2×#13712yes—
560C23C2×#13712yes—
561C23C2×#13712yes—
562C23#57514yes—
563C23C23×D411yescharacteristic
564Q8—44no—
565Q8—44no—
566Q8—84no—
567C2×C4—216no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
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—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4C2×#12312yes—
617C2×C4C2×#12312yes—
618C2×C4C2×#12312yes—
619C2×C4#53414yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)14yes—
623C8—44no—
624C8—44no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—416no—
644C22—416no—
645C22—432no—
646C22—432no—
647C22—416no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22C2×#48514yes—
657C22C2×#48514yes—
658C22C2×#48514yes—
659C22C22×#12311yescharacteristic
660C22C2×#53412yes—
661C22C22×#13511yescharacteristic
662C22C22×#13711yescharacteristic
663C22C2×#57512yes—
664C4—22no—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—42no—
669C4—84no—
670C4—84no—
671C4—84no—
672C4—82no—
673C4—82no—
674C4#349012yes—
675C4#349012yes—
676C2—24no—
677C2—24no—
678C2—24no—
679C2—44no—
680C2—44no—
681C2#3499211yescharacteristic
682C2C22×#48511yescharacteristic
683C2#6051111yescharacteristic
684C2#6073412yes—
685C2#6075112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32985C2semidirect
4. #29894C2semidirect
5. #27604C2semidirect
6. #24554C2semidirect
7. #24010C2semidirect
8. #21940C2semidirect
9. #21938C2semidirect
10. #21759C2semidirect
11. #21759C2semidirect
17. C2×#485C22semidirect
43. #2077C22semidirect
44. #2075C22semidirect
45. #2036C22semidirect
46. #2035C22semidirect
47. #1892C22semidirect
48. #1892C22semidirect
49. #1873C22semidirect
50. #1872C22semidirect
51. #1872C22semidirect
52. #1854C22semidirect
53. #1853C22semidirect
54. #1853C22semidirect
55. #1852C22semidirect
56. #1852C22semidirect
57. #1825C22semidirect
58. #1824C22semidirect
59. #1824C22semidirect
60. #1808C22semidirect
61. #1807C22semidirect
62. #1787C22semidirect
63. #1787C22semidirect
116. C2×#107C23semidirect
117. C2×#106C23semidirect
118. C2×#103C23semidirect
119. C2×#103C23semidirect
172. #340C23semidirect
173. #340C23semidirect
174. #340C23semidirect
175. #340C23semidirect
551. C23#485semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#22802
#329852
#298942
#276042
#245542
#240102
#219402
#219382
#217592
C2×#17792