Finite Group Databasebeta

← #6499896

Subgroups — #6499896

Order 512.

4812 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3496C211yesmaximal, characteristic, complemented
2C2×#3186C212yesmaximal, complemented
3C2×#2416C211yesmaximal, characteristic, complemented
4#25533C218yesmaximal, complemented
5#24424C218yesmaximal, complemented
6#24060C212yesmaximal, complemented
7#24002C212yesmaximal, complemented
8#23930C212yesmaximal, complemented
9#23918C212yesmaximal, complemented
10#21738C211yesmaximal, characteristic, complemented
11C2×#1874C212yesmaximal, complemented
12C22×H(5,𝔽2)—21no—
13#3496—216no—
14C2×#547C2214yescomplemented
15C2×#546—24no—
16C2×D42C2214yescomplemented
17C2×#537—21no—
18C2×#534—21no—
19C2×#534—21no—
20C2×#532C2212yes—
21C22×#121—21no—
22C22×#121C2212yes—
23C22×#121C2212yes—
24C22×#121C2211yescharacteristic, complemented
25C22×#121C2211yescharacteristic, complemented
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic, complemented
28#3186—216no—
29C2×#480C2214yescomplemented
30C2×#480C2214yescomplemented
31C2×#466C2218yescomplemented
32C2×#447C2214yescomplemented
33C2×#447C2214yescomplemented
34C2×#447C2214yescomplemented
35C22×#102C2214yescomplemented
36C22×C22⋊C8C2212yescomplemented
37#2416—24no—
38#2416C2218yescomplemented
39C2×#396—21no—
40C2×#394—21no—
41C2×#390C2211yescharacteristic, complemented
42C2×#389C2211yescharacteristic, complemented
43C2×#386—21no—
44C2×#380C2211yescharacteristic, complemented
45#2057C2214yescomplemented
46#2041C2214yescomplemented
47#2033C2218yescomplemented
48#2023C2214yescomplemented
49#2011C2214yescomplemented
50#1999C2218yescomplemented
51#1879C2218yescomplemented
52#1874C2218yescomplemented
53#1874C2214yescomplemented
54#1874C2214yescomplemented
55#1874C2214yescomplemented
56#1869C2218yescomplemented
57#1862C2218yescomplemented
58#1852C2214yescomplemented
59#1817C2212yescomplemented
60#1786C2214yescomplemented
61#1779C2212yescomplemented
62C2×#340C2214yescomplemented
63C2×H(5,𝔽2)—216no—
64C2×H(5,𝔽2)—48no—
65C2×H(5,𝔽2)—48no—
66C22×#41—24no—
67C22×#41—22no—
68C23×D4—22no—
69C23×D4—22no—
70C23×D4—24no—
71C23×D4—21no—
72C23×D4—21no—
73C23×D4D411yescharacteristic
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4—21no—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C22×D8C2318yes—
81C23×C8C2314yes—
82#547—216no—
83#547—216no—
84#546—264no—
85D42—216no—
86D42—216no—
87#537—216no—
88#534—216no—
89#534—216no—
90#532—216no—
91C2×C4⋊Q8C2312yes—
92C2×#127C2312yes—
93C2×#124—22no—
94C2×#123—24no—
95C2×#123—24no—
96C2×C22⋊Q8—24no—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121—22no—
100C2×#121—22no—
101C2×#121—28no—
102C2×#121—22no—
103C2×#121—216no—
104C2×#121—28no—
105C2×#121—22no—
106C2×#121—22no—
107C2×#121—24no—
108C2×#121—28no—
109C2×#121—44no—
110C2×#121—48no—
111C2×#121D4116yescomplemented
112C2×#121C2314yes—
113C2×#121C2314yes—
114C2×#121C2314yes—
115C2×#121C2314yes—
116C2×#121C2314yes—
117C2×#120—24no—
118C2×#120—24no—
119C2×C4×D4—28no—
120C2×C4×D4—22no—
121C2×C4×D4D414yes—
122C2×C4×D4C2314yes—
123C2×C4×D4C2314yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—44no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4D411yescharacteristic
143C22×C22⋊C4D411yescharacteristic
144C22×C42C2311yescharacteristic
145#480—216no—
146#480—216no—
147#466—232no—
148#447—216no—
149#447—216no—
150#447—48no—
151C2×#107C2318yes—
152C2×#102—48no—
153C2×#102C2318yes—
154C2×#102C2318yes—
155C2×#102C2318yes—
156C2×#102C2318yes—
157C2×C22⋊C8—28no—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162#396—28no—
163#394—28no—
164#390—22no—
165#390C2314yes—
166#389—22no—
167#389C2314yes—
168#386—28no—
169#380—22no—
170#380C2314yes—
171C2×#95—21no—
172C2×#95—21no—
173#340C2318yes—
174#340C2318yes—
175#336C2314yes—
176#336C2314yes—
177C25—22no—
178C25—22no—
179C25—21no—
180C25C2×D411yescharacteristic
181C25C2×D411yescharacteristic
182H(5,𝔽2)—4128no—
183C2×#41—216no—
184C2×#41—232no—
185C2×#41—48no—
186C2×#41—416no—
187C2×#41—416no—
188C2×#41—48no—
189C22×Q8—22no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—216no—
193C22×D4—232no—
194C22×D4—216no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—416no—
216C22×D4—48no—
217C22×D4—48no—
218C22×D4—48no—
219C22×D4—48no—
220C22×D4—416no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—44no—
225C22×D4—82no—
226C22×D4D818yescomplemented
227C22×D4C2×D414yes—
228C22×D4C2×D418yescomplemented
229C22×D4C2412yes—
230C22×D4C2414yes—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—24no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—82no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D418yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C2×D8—216no—
257C2×D8—48no—
258C2×D8—48no—
259C22×C8—28no—
260C22×C8—44no—
261C22×C8—44no—
262C22×C8C2418yes—
263C4⋊Q8—28no—
264#127—28no—
265#124—216no—
266#123—232no—
267#123—232no—
268C22⋊Q8—232no—
269#121—216no—
270#121—216no—
271#121—216no—
272#121—232no—
273#121—28no—
274#121—216no—
275#121—216no—
276#121—28no—
277#121—28no—
278#121—28no—
279#121—28no—
280#121—28no—
281#121—28no—
282#121—44no—
283#121—48no—
284#121—432no—
285#121—48no—
286#121C2×D4116yes—
287#121C2×D4116yescomplemented
288#120—232no—
289#120—232no—
290C4×D4—216no—
291C4×D4—28no—
292C4×D4—216no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D418yescomplemented
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—216no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—216no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C42—24no—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344#107—216no—
345#102—216no—
346#102—216no—
347#102—216no—
348#102—48no—
349C22⋊C8—28no—
350C22⋊C8—28no—
351C22⋊C8—44no—
352C22⋊C8—44no—
353#95—24no—
354#95—24no—
355C24—22no—
356C24—22no—
357C24—28no—
358C24—24no—
359C24—28no—
360C24—22no—
361C24—41no—
362C24—41no—
363C24—41no—
364C24—41no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—44no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—44no—
375C24—44no—
376C24—44no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—82no—
381C24—82no—
382C24—82no—
383C24—82no—
384C24—82no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24#12014yes—
389C24C2×D814yes—
390C24C2×D814yes—
391C24AGL(1,ℤ/8)14yes—
392C24C22×D412yes—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—464no—
396#41—4128no—
397C2×Q8—28no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—232no—
403C2×D4—216no—
404C2×D4—28no—
405C2×D4—216no—
406C2×D4—216no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—416no—
417C2×D4—44no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—4128no—
422C2×D4—48no—
423C2×D4—464no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—464no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—88no—
437C2×D4—88no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—816no—
445C2×D4—84no—
446C2×D4—816no—
447C2×D4—816no—
448C2×D4C2×D8116yes—
449C2×D4C22×D418yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—216no—
459C22×C4—28no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—416no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—48no—
496C22×C4—42no—
497C22×C4—88no—
498C22×C4—84no—
499C22×C4—816no—
500C22×C4#12018yes—
501C22×C4#12014yes—
502C22×C4C2×D814yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D414yes—
510C22×C4C22×D412yes—
511C22×C4C2511yescharacteristic
512D8—816no—
513C2×C8—28no—
514C2×C8—28no—
515C2×C8—28no—
516C2×C8—88no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4C22×D418yes—
531C4⋊C4H(5,𝔽2)14yescomplemented
532C22⋊C4—216no—
533C22⋊C4—216no—
534C22⋊C4—28no—
535C22⋊C4—216no—
536C22⋊C4—216no—
537C22⋊C4—28no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C42—24no—
544C42—24no—
545C23—21no—
546C23—22no—
547C23—21no—
548C23—22no—
549C23—21no—
550C23—22no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—28no—
560C23—28no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—416no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—416no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—82no—
585C23—82no—
586C23—82no—
587C23—82no—
588C23—88no—
589C23—88no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—88no—
597C23—88no—
598C23—84no—
599C23—164no—
600C23—164no—
601C23—164no—
602C23—164no—
603C23—164no—
604C23—164no—
605C23—164no—
606C23—164no—
607C23#447116yescomplemented
608C23C2×#12014yes—
609C23C2×#12012yes—
610C23C22×D814yes—
611C23C22×D812yes—
612C23C22×D814yes—
613C23C2×AGL(1,ℤ/8)12yes—
614C23C2×AGL(1,ℤ/8)14yes—
615C23C23×D411yescharacteristic
616Q8—416no—
617D4—232no—
618D4—216no—
619D4—416no—
620D4—432no—
621D4—432no—
622D4—432no—
623D4—48no—
624D4—48no—
625D4—88no—
626D4—84no—
627D4—816no—
628D4—88no—
629D4—88no—
630D4—816no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—28no—
637C2×C4—22no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—28no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—48no—
650C2×C4—44no—
651C2×C4—432no—
652C2×C4—48no—
653C2×C4—416no—
654C2×C4—42no—
655C2×C4—416no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—48no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—84no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—816no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4C2×#12012yes—
673C2×C4C2×#12014yes—
674C2×C4C22×D814yes—
675C2×C4C23×D412yes—
676C2×C4C2×H(5,𝔽2)12yes—
677C2×C4#58512yes—
678C8—48no—
679C22—22no—
680C22—24no—
681C22—22no—
682C22—24no—
683C22—22no—
684C22—24no—
685C22—22no—
686C22—24no—
687C22—22no—
688C22—24no—
689C22—22no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—416no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—48no—
700C22—48no—
701C22—48no—
702C22—88no—
703C22—84no—
704C22—84no—
705C22—84no—
706C22—84no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—1616no—
715C22—168no—
716C22—1616no—
717C22C2×#44714yes—
718C22C2×#44718yes—
719C22C22×#12011yescharacteristic
720C22#348411yescharacteristic
721C22#348511yescharacteristic
722C22C23×D811yescharacteristic
723C22C22×AGL(1,ℤ/8)11yescharacteristic
724C22#358511yescharacteristic
725C22#358611yescharacteristic
726C4—22no—
727C4—24no—
728C4—24no—
729C4—42no—
730C4—44no—
731C4—48no—
732C4—42no—
733C4—82no—
734C4—82no—
735C4#349614yes—
736C2—24no—
737C2—28no—
738C2—44no—
739C2—88no—
740C2—162no—
741C2#3534711yescharacteristic
742C2C22×#44711yescharacteristic
743C2#6037111yescharacteristic
744C2#6155911yescharacteristic
745C2#6157211yescharacteristic
746C2#6169411yescharacteristic
747C2#6170711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3496C2semidirect
2. C2×#3186C2semidirect
3. C2×#2416C2semidirect
4. #25533C2semidirect
5. #24424C2semidirect
6. #24060C2semidirect
7. #24002C2semidirect
8. #23930C2semidirect
9. #23918C2semidirect
10. #21738C2semidirect
11. C2×#1874C2semidirect
14. C2×#547C22semidirect
16. C2×D42C22semidirect
24. C22×#121C22semidirect
25. C22×#121C22semidirect
27. C23×C4⋊C4C22semidirect
29. C2×#480C22semidirect
30. C2×#480C22semidirect
31. C2×#466C22semidirect
32. C2×#447C22semidirect
33. C2×#447C22semidirect
34. C2×#447C22semidirect
35. C22×#102C22semidirect
36. C22×C22⋊C8C22semidirect
38. #2416C22semidirect
41. C2×#390C22semidirect
42. C2×#389C22semidirect
44. C2×#380C22semidirect
45. #2057C22semidirect
46. #2041C22semidirect
47. #2033C22semidirect
48. #2023C22semidirect
49. #2011C22semidirect
50. #1999C22semidirect
51. #1879C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
54. #1874C22semidirect
55. #1874C22semidirect
56. #1869C22semidirect
57. #1862C22semidirect
58. #1852C22semidirect
59. #1817C22semidirect
60. #1786C22semidirect
61. #1779C22semidirect
62. C2×#340C22semidirect
111. C2×#121D4semidirect
226. C22×D4D8semidirect
228. C22×D4C2×D4semidirect
287. #121C2×D4semidirect
316. C2×C4⋊C4C2×D4semidirect
531. C4⋊C4H(5,𝔽2)semidirect
607. C23#447semidirect

Maximal subgroups

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

subgroupindex
C2×#34962
C2×#31862
C2×#24162
#255332
#244242
#240602
#240022
#239302
#239182
#217382
C2×#18742