Finite Group Databasebeta

← #6512619

Subgroups — #6512619

Order 512.

3414 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3506C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3C2×#3103C211yesmaximal, characteristic, complemented
4C2×#2538C211yesmaximal, characteristic, complemented
5#24242C214yesmaximal, complemented
6#24241C214yesmaximal, complemented
7#24123C214yesmaximal, complemented
8#24121C214yesmaximal, complemented
9#24007C214yesmaximal, complemented
10#23922C214yesmaximal, complemented
11#21860C211yesmaximal, characteristic, complemented
12#21859C211yesmaximal, characteristic, complemented
13#21786C211yesmaximal, characteristic, complemented
14#3506—28no—
15#3506—28no—
16C2×#559—22no—
17C2×#556C2212yes—
18C2×#555C2212yes—
19C2×#553—22no—
20C2×#550C2212yes—
21C2×#547C2212yes—
22C2×#538—22no—
23C2×#535—21no—
24C2×#533C2211yescharacteristic
25C2×#532C2211yescharacteristic
26C2×#525C2214yes—
27C22×C22⋊Q8C2212yes—
28C2×#520—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31#3104—28no—
32#3103—28no—
33C2×#484C2212yes—
34C2×#481C2212yes—
35C2×#477C2212yescomplemented
36C2×#475C2212yescomplemented
37C2×#462C2212yescomplemented
38C2×#460C2212yescomplemented
39C2×#452C2218yescomplemented
40C2×#451C2212yes—
41C2×#448C2212yes—
42C2×C4×Q16C2214yescomplemented
43C2×C4×SD16C2214yescomplemented
44C2×#431C2211yescharacteristic
45C2×#431C2211yescharacteristic
46#2538—22no—
47#2538—22no—
48#2538C2214yes—
49#2538C2214yes—
50C2×#396—22no—
51C2×#390—21no—
52C2×#389—21no—
53C2×#384C2211yescharacteristic
54C2×#380C2212yes—
55#2035C2218yescomplemented
56#2033C2218yescomplemented
57#2001C2218yescomplemented
58#2000C2218yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1957C2212yescomplemented
62#1956C2212yescomplemented
63#1954C2214yes—
64#1954C2214yes—
65#1874C2214yes—
66#1873C2218yescomplemented
67#1872C2214yes—
68#1854C2218yescomplemented
69#1803C2214yescomplemented
70#1802C2214yescomplemented
71#1788C2218yescomplemented
72#1779C2214yes—
73C22×#41—21no—
74C22×#41C2314yes—
75C23×Q8C2311yescharacteristic
76C23×D4D411yescharacteristic
77C24×C4C2312yes—
78C22×Q16C2314yes—
79C22×SD16C2314yes—
80#559—232no—
81#556—28no—
82#556—28no—
83#555—28no—
84#555—28no—
85#553—232no—
86#550—28no—
87#550—28no—
88#547—28no—
89#547—28no—
90#538—232no—
91#535—216no—
92#533—24no—
93#533D418yes—
94#532—24no—
95#532D418yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127—22no—
100C2×#127C2311yescharacteristic
101C2×#126—24no—
102C2×#125—21no—
103C2×#124—22no—
104C2×#124—24no—
105C2×#124—21no—
106#525—216no—
107#525—216no—
108C2×#123—24no—
109C2×#123—24no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—48no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2318yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121—21no—
119C2×#121—24no—
120C2×#121C2314yes—
121C2×#121C2312yes—
122C2×#120—24no—
123#520—216no—
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—24no—
129C2×C4×Q8—24no—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—22no—
136C2×C4×D4—24no—
137C2×C4×D4—24no—
138C2×C4×D4—21no—
139C2×C4×D4D412yes—
140C2×C4×D4D412yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×#117—21no—
145C2×#117—22no—
146C2×#117C2314yes—
147C22×C4⋊C4—22no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—22no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4D411yescharacteristic
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2312yes—
155C22×C22⋊C4—22no—
156C22×C22⋊C4—22no—
157C22×C22⋊C4D412yes—
158C22×C42—21no—
159C22×C42C2311yescharacteristic
160#484—28no—
161#481—28no—
162#477—28no—
163#475—28no—
164#462—28no—
165#460—28no—
166#452—416no—
167#451—28no—
168#448—28no—
169C4×Q16—216no—
170C4×SD16—216no—
171#431—24no—
172#431—24no—
173C2×#107C2314yes—
174C2×#106C2314yes—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2312yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#103C2314yes—
180C2×#103C2318yes—
181C2×#102C2314yes—
182C2×#102C2318yes—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2318yes—
186C2×C22⋊C8C2318yes—
187C2×C4×C8C2312yes—
188C2×C4×C8C2312yes—
189#396—28no—
190#396—28no—
191#390—24no—
192#390—24no—
193#389—24no—
194#389—24no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2312yes—
199#380—24no—
200#380C2318yes—
201C2×#95—22no—
202#340C2318yes—
203#340C2318yes—
204C25C2×D412yes—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—28no—
208C2×#41—48no—
209C2×#41—48no—
210C2×#41—44no—
211C2×#41—44no—
212C22×Q8—24no—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—42no—
219C22×Q8C2412yes—
220C22×Q8C2412yes—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2×D412yes—
230C22×D4C2×D412yes—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D412yes—
242C23×C4C2×D414yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C2×Q16—28no—
248C2×Q16—28no—
249C2×Q16—44no—
250C2×SD16—28no—
251C2×SD16—28no—
252C2×SD16—44no—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8C2×D414yes—
259C4⋊Q8C2×D414yes—
260C4⋊Q8C2×D414yes—
261#127—216no—
262#127—22no—
263#127C2×D414yes—
264#126—232no—
265#125—28no—
266#124—216no—
267#124—232no—
268#124—28no—
269#123—232no—
270#123—232no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—28no—
275C22⋊Q8—216no—
276C22⋊Q8—48no—
277C22⋊Q8—416no—
278C22⋊Q8C2×D418yes—
279#121—216no—
280#121—216no—
281#121—232no—
282#121—28no—
283#121—28no—
284#121—24no—
285#121—28no—
286#121C2×D418yes—
287#120—232no—
288C4×Q8—216no—
289C4×Q8—28no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8C2×D418yes—
298C4×D4—232no—
299C4×D4—216no—
300C4×D4—232no—
301C4×D4—28no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4C2×D418yes—
310#117—216no—
311#117—28no—
312#117—28no—
313#117—28no—
314C2×C4⋊C4—216no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—28no—
353C2×C22⋊C4—28no—
354C2×C22⋊C4—48no—
355C2×C22⋊C4—84no—
356C2×C22⋊C4C2×D414yes—
357C2×C22⋊C4C2×D414yes—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D414yes—
366C2×C42C2×D414yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—28no—
370#106—28no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373#103—28no—
374#103—28no—
375#103—28no—
376#103—48no—
377#102—28no—
378#102—48no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—48no—
382C22⋊C8—48no—
383C4×C8—24no—
384C4×C8—24no—
385#95—28no—
386C24—24no—
387C24—22no—
388C24—24no—
389C24—24no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22×D414yes—
395C24C22×D411yescharacteristic
396#41—432no—
397#41—816no—
398#41—816no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—81no—
417C2×Q8—81no—
418C2×Q8C22×D414yes—
419C2×Q8H(5,𝔽2)14yes—
420C2×D4—28no—
421C2×D4—216no—
422C2×D4—216no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—416no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4—84no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4C22×D414yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—216no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4#12014yes—
484C22×C4#12014yes—
485C22×C4#13514yes—
486C22×C4#13714yes—
487C22×C4C22×D414yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2511yescharacteristic
496Q16—416no—
497SD16—416no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C4⋊C4—216no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—28no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—28no—
518C4⋊C4—416no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4C22×D414yes—
528C4⋊C4C22×D414yes—
529C22⋊C4—216no—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—416no—
538C22⋊C4—88no—
539C42—216no—
540C42—28no—
541C42—44no—
542C42—44no—
543C42#12014yes—
544C42#12014yes—
545C42C22×D412yes—
546C42C22×D412yes—
547C42C22×D412yes—
548C42C22×D412yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—48no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—48no—
568C23—42no—
569C23—42no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23C2×#12012yes—
575C23C2×#13512yes—
576C23C2×#13712yes—
577C23C23×D411yescharacteristic
578Q8—24no—
579Q8—24no—
580Q8—28no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—44no—
585Q8—48no—
586Q8—84no—
587D4—28no—
588D4—416no—
589D4—44no—
590D4—48no—
591D4—48no—
592D4—48no—
593D4—84no—
594D4—816no—
595D4—816no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—416no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4—816no—
633C2×C4—88no—
634C2×C4—84no—
635C2×C4#452116yescomplemented
636C2×C4C2×#12012yes—
637C2×C4C2×#12012yes—
638C2×C4C2×#12012yes—
639C2×C4C2×#12012yes—
640C2×C4C2×#12012yes—
641C2×C4C2×#13512yes—
642C2×C4C2×#13512yes—
643C2×C4C2×#13712yes—
644C2×C4C2×#13712yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C23×D411yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4#58512yes—
649C8—44no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22C2×#45214yes—
673C22C22×#12011yescharacteristic
674C22#348411yescharacteristic
675C22#348511yescharacteristic
676C22C22×#13511yescharacteristic
677C22C22×#13711yescharacteristic
678C22#358511yescharacteristic
679C22#358611yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—84no—
689C4—82no—
690C4C2×#45214yes—
691C4C2×#45214yes—
692C4#348812yes—
693C4#348812yes—
694C2—24no—
695C2—84no—
696C2—84no—
697C2#3551211yescharacteristic
698C2C22×#45211yescharacteristic
699C2#6037111yescharacteristic
700C2#6156611yescharacteristic
701C2#6158411yescharacteristic
702C2#6170111yescharacteristic
703C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3506C2
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
2. C2×#3104C2semidirect
3. C2×#3103C2semidirect
4. C2×#2538C2semidirect
5. #24242C2semidirect
6. #24241C2semidirect
7. #24123C2semidirect
8. #24121C2semidirect
9. #24007C2semidirect
10. #23922C2semidirect
11. #21860C2semidirect
12. #21859C2semidirect
13. #21786C2semidirect
35. C2×#477C22semidirect
36. C2×#475C22semidirect
37. C2×#462C22semidirect
38. C2×#460C22semidirect
39. C2×#452C22semidirect
42. C2×C4×Q16C22semidirect
43. C2×C4×SD16C22semidirect
55. #2035C22semidirect
56. #2033C22semidirect
57. #2001C22semidirect
58. #2000C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1957C22semidirect
62. #1956C22semidirect
66. #1873C22semidirect
68. #1854C22semidirect
69. #1803C22semidirect
70. #1802C22semidirect
71. #1788C22semidirect
635. C2×C4#452semidirect

Maximal subgroups

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

subgroupindex
C2×#35062
C2×#31042
C2×#31032
C2×#25382
#242422
#242412
#241232
#241212
#240072
#239222
#218602
#218592
#217862