Finite Group Databasebeta

← #6499772

Subgroups — #6499772

Order 512.

4126 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic, complemented
2C2×#3007C212yesmaximal, complemented
3C2×#2407C211yesmaximal, characteristic
4#23981C218yesmaximal, complemented
5#23934C212yesmaximal, complemented
6#23933C212yesmaximal, complemented
7#23921C212yesmaximal, complemented
8#23919C212yesmaximal, complemented
9#23868C218yesmaximal, complemented
10#21765C212yesmaximal, complemented
11#21742C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×#137C2214yescomplemented
14#3501—216no—
15C2×#554—24no—
16C2×#554C2214yescomplemented
17C2×D4×Q8C2214yescomplemented
18C2×#544—21no—
19C2×#543—21no—
20C2×#543—21no—
21C2×#533C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C2×#518—21no—
27C22×C4×D4C2211yescharacteristic
28#3007—216no—
29C2×#452C2218yescomplemented
30C2×#451C2214yes—
31C2×#451C2214yescomplemented
32C2×#448C2214yescomplemented
33C2×#450C2214yes—
34C2×#417C2214yescomplemented
35C2×#407C2212yes—
36#2407—24no—
37#2407C2218yes—
38C2×#394—21no—
39C2×#392—21no—
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#386—21no—
43C2×#385—21no—
44C2×#380C2211yescharacteristic
45#2024C2214yescomplemented
46#2023C2214yescomplemented
47#2016C2214yescomplemented
48#2013C2214yescomplemented
49#2001C2218yescomplemented
50#2000C2218yescomplemented
51#1894C2218yescomplemented
52#1880C2218yescomplemented
53#1878C2218yescomplemented
54#1874C2214yescomplemented
55#1872C2214yes—
56#1872C2214yes—
57#1872C2214yescomplemented
58#1870C2218yescomplemented
59#1823C2214yescomplemented
60#1817C2212yes—
61#1787C2214yescomplemented
62#1779C2212yes—
63C22×#41—22no—
64C22×#41—28no—
65C22×#41—44no—
66C22×#41—44no—
67C22×#41C2314yes—
68C23×Q8C2311yescharacteristic
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×D4—21no—
72C23×D4—21no—
73C23×D4D411yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4—21no—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×#137—216no—
79C2×#137—48no—
80C2×#137—48no—
81C22×Q16C2318yes—
82C22×SD16C2318yes—
83C22×M16C2314yes—
84#554—264no—
85#554—216no—
86#554—216no—
87D4×Q8—216no—
88D4×Q8—216no—
89#544—216no—
90#543—216no—
91#543—216no—
92#533—216no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×#125—24no—
97C2×#125—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—216no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—44no—
111C2×C22⋊Q8D4116yescomplemented
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2314yes—
115C2×C22⋊Q8C2314yes—
116C2×#121—22no—
117C2×#121—44no—
118C2×#121C2314yes—
119#518—216no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2314yes—
122C2×C4×D4—24no—
123C2×C4×D4—24no—
124C2×C4×D4—28no—
125C2×C4×D4D414yes—
126C2×C4×D4C2314yes—
127C2×#117—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4—21no—
143C22×C22⋊C4D411yescharacteristic
144C22×C22⋊C4D411yescharacteristic
145C22×C42C2311yescharacteristic
146#452—232no—
147#451—216no—
148#451—48no—
149#450—216no—
150#448—48no—
151#417—216no—
152C2×#103C2318yes—
153C2×#103C2318yes—
154C2×#103C2318yes—
155C2×#102C2318yes—
156#407—28no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161#394—28no—
162#392—28no—
163#391—22no—
164#391C2314yes—
165#390—22no—
166#390C2314yes—
167#386—28no—
168#385—28no—
169#380—22no—
170#380C2314yes—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174#340C2318yes—
175#340C2318yes—
176#336C2314yes—
177#336C2314yes—
178C25—21no—
179C25C2×D411yescharacteristic
180C25C2×D411yescharacteristic
181C2×#41—28no—
182C2×#41—28no—
183C2×#41—28no—
184C2×#41—28no—
185C2×#41—44no—
186C2×#41—44no—
187C2×#41—48no—
188C2×#41—48no—
189C2×#41—464no—
190C2×#41—48no—
191C22×Q8—24no—
192C22×Q8—28no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—42no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8C2×D418yescomplemented
205C22×Q8C2414yes—
206C22×Q8C2412yes—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—24no—
215C22×D4—42no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—82no—
222C22×D4C2×D414yes—
223C22×D4C2412yes—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2×D418yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250#137—464no—
251C2×Q16—216no—
252C2×Q16—216no—
253C2×Q16—48no—
254C2×SD16—216no—
255C2×SD16—216no—
256C2×SD16—48no—
257C2×M16—28no—
258C2×M16—44no—
259C2×M16—44no—
260C22×C8C2418yes—
261C4⋊Q8—232no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264#125—216no—
265#125—232no—
266C22⋊Q8—216no—
267C22⋊Q8—216no—
268C22⋊Q8—216no—
269C22⋊Q8—232no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—232no—
273C22⋊Q8—28no—
274C22⋊Q8—216no—
275C22⋊Q8—216no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—44no—
281C22⋊Q8—48no—
282C22⋊Q8C2×D4116yes—
283C22⋊Q8C2×D4116yescomplemented
284#121—28no—
285#121—44no—
286#121—48no—
287C4×Q8—216no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×D4—232no—
291C4×D4—232no—
292C4×D4—216no—
293C4×D4—28no—
294C4×D4—28no—
295#117—216no—
296C2×C4⋊C4—216no—
297C2×C4⋊C4—216no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D418yescomplemented
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C22⋊C4—216no—
322C2×C22⋊C4—216no—
323C2×C22⋊C4—24no—
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—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344#103—216no—
345#103—48no—
346#103—48no—
347#102—216no—
348C22⋊C8—28no—
349C22⋊C8—28no—
350C22⋊C8—44no—
351C22⋊C8—44no—
352#95—24no—
353#95—24no—
354#95—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—41no—
362C24—41no—
363C24—41no—
364C24—41no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24#12014yes—
376C24#13714yescomplemented
377C24C22×D412yes—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380#41—232no—
381#41—416no—
382#41—416no—
383#41—416no—
384#41—464no—
385C2×Q8—24no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—24no—
395C2×Q8—416no—
396C2×Q8—42no—
397C2×Q8—416no—
398C2×Q8—42no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—88no—
406C2×Q8—816no—
407C2×Q8C22×D418yes—
408C2×Q8H(5,𝔽2)14yes—
409C2×D4—232no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—416no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—48no—
421C2×D4—432no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—42no—
426C2×D4—84no—
427C2×D4—816no—
428C2×D4H(5,𝔽2)14yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—416no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—48no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—432no—
466C22×C4—42no—
467C22×C4—48no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—88no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4#12014yes—
484C22×C4#12018yes—
485C22×C4#13714yescomplemented
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C2511yescharacteristic
495Q16—432no—
496SD16—432no—
497M16—416no—
498C2×C8—28no—
499C2×C8—28no—
500C2×C8—28no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4C22×D418yes—
518C22⋊C4—216no—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4—28no—
522C22⋊C4—28no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—216no—
529C42—24no—
530C42—24no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—28no—
549C23—24no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23C2×#12014yes—
576C23C2×#12012yes—
577C23C2×#13714yes—
578C23C2×#13712yes—
579C23C2×#13712yes—
580C23C23×D411yescharacteristic
581Q8—216no—
582Q8—28no—
583Q8—48no—
584Q8—44no—
585Q8—48no—
586Q8—88no—
587Q8—88no—
588Q8—88no—
589D4—232no—
590D4—28no—
591D4—48no—
592D4—432no—
593D4—44no—
594D4—816no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—432no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—88no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4C2×#12012yes—
641C2×C4C2×#12014yes—
642C2×C4C2×#13714yes—
643C2×C4C23×D412yes—
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4C2×H(5,𝔽2)12yes—
646C8—48no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—88no—
671C22—88no—
672C22—816no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22#300414yes—
678C22C22×#12011yescharacteristic
679C22#348411yescharacteristic
680C22#348411yescharacteristic
681C22C22×#13711yescharacteristic
682C22C22×#13711yescharacteristic
683C22#358511yescharacteristic
684C22#358611yescharacteristic
685C4—24no—
686C4—22no—
687C4—24no—
688C4—42no—
689C4—44no—
690C4—48no—
691C4—42no—
692C4—48no—
693C4—82no—
694C4—82no—
695C4#348414yes—
696C2—24no—
697C2—44no—
698C2—44no—
699C2—84no—
700C2#3534611yescharacteristic
701C2C2×#300411yescharacteristic
702C2#6037111yescharacteristic
703C2#6157011yescharacteristic
704C2#6157311yescharacteristic
705C2#6158611yescharacteristic
706C2#6158811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2407C2
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×#3501C2semidirect
2. C2×#3007C2semidirect
4. #23981C2semidirect
5. #23934C2semidirect
6. #23933C2semidirect
7. #23921C2semidirect
8. #23919C2semidirect
9. #23868C2semidirect
10. #21765C2semidirect
11. #21742C2semidirect
13. C22×#137C22semidirect
16. C2×#554C22semidirect
17. C2×D4×Q8C22semidirect
29. C2×#452C22semidirect
31. C2×#451C22semidirect
32. C2×#448C22semidirect
34. C2×#417C22semidirect
45. #2024C22semidirect
46. #2023C22semidirect
47. #2016C22semidirect
48. #2013C22semidirect
49. #2001C22semidirect
50. #2000C22semidirect
51. #1894C22semidirect
52. #1880C22semidirect
53. #1878C22semidirect
54. #1874C22semidirect
57. #1872C22semidirect
58. #1870C22semidirect
59. #1823C22semidirect
61. #1787C22semidirect
111. C2×C22⋊Q8D4semidirect
204. C22×Q8C2×D4semidirect
283. C22⋊Q8C2×D4semidirect
317. C2×C4⋊C4C2×D4semidirect
376. C24#137semidirect
485. C22×C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#30072
C2×#24072
#239812
#239342
#239332
#239212
#239192
#238682
#217652
#217422