Finite Group Databasebeta

← #6560056

Subgroups — #6560056

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#30203C211yesmaximal, characteristic, complemented
4#30200C212yesmaximal, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29148C211yesmaximal, characteristic, complemented
7#29103C212yesmaximal, complemented
8#27814C212yesmaximal, complemented
9#27770C212yesmaximal, complemented
10#27725C214yesmaximal, complemented
11#27271C214yesmaximal, complemented
12#25906C212yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23454C212yesmaximal, complemented
15#22502C212yesmaximal, complemented
16#22428C212yesmaximal
17#20567C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×D42C2211yescharacteristic
20C2×#533C2212yescomplemented
21C2×#488C2214yescomplemented
22C2×#436C2212yes—
23C2×#418C2214yes—
24#2868C2211yescharacteristic, complemented
25#2858C2211yescharacteristic
26#2857C2212yescomplemented
27#2850C2212yes—
28#2563C2211yescharacteristic
29#2552C2212yescomplemented
30#2548C2211yescharacteristic
31#2542—21no—
32#2538—21no—
33#2504C2212yes—
34#2490—21no—
35#2490—21no—
36#2472—22no—
37#2472—22no—
38#2441C2214yescomplemented
39#2435C2212yes—
40#2429—22no—
41#2423—22no—
42#2365—21no—
43#2354C2212yes—
44Q8×C4⋊C4—21no—
45D4×C22⋊C4—21no—
46D4×C22⋊C4—21no—
47C4×C4⋊Q8C2211yescharacteristic, complemented
48C4×#124C2211yescharacteristic
49C2×#390C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yescomplemented
52#2101C2214yescomplemented
53#2072C2211yescharacteristic, complemented
54#2071C2211yescharacteristic
55#2071C2214yescomplemented
56#2071C2212yescomplemented
57#2042C2212yes—
58#2042C2214yescomplemented
59#2025C2214yes—
60#2025C2214yescomplemented
61#2024C2211yescharacteristic, complemented
62#2024C2212yescomplemented
63#2023C2211yescharacteristic
64#2023C2212yes—
65#2023C2212yescomplemented
66#2023C2212yescomplemented
67#2023C2212yescomplemented
68#1991C2214yescomplemented
69#1990C2212yescomplemented
70#1987C2212yescomplemented
71#1982C2212yes—
72#1959C2212yes—
73#1957C2212yes—
74#1936C2214yescomplemented
75#1934C2214yes—
76#1934C2214yescomplemented
77#1895C2214yes—
78#1894C2214yescomplemented
79#1875C2214yes—
80#1873C2214yescomplemented
81#1836C2212yescomplemented
82#1823C2214yescomplemented
83#1803C2212yescomplemented
84#1802C2212yes—
85#1786C2214yes—
86C8×C4⋊C4C2212yescomplemented
87#1767C2211yescharacteristic, complemented
88C4×#103C2212yescomplemented
89C4×#102C2212yes—
90#1753C2214yescomplemented
91C4×C22⋊C8C2212yescomplemented
92#1744C2211yescharacteristic, complemented
93#1736C2214yescomplemented
94#1728C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×M16C2314yes—
98#554—28no—
99D42—28no—
100#533—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#125C2314yes—
109C2×#124—21no—
110C2×#124—21no—
111C2×#124C2311yescharacteristic
112C2×#124C2312yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×#120—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2312yes—
135#488—48no—
136#436—28no—
137C2×C4⋊C8C2312yes—
138#418—48no—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2314yes—
150C2×#97C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#399—24no—
154#398—24no—
155#397—22no—
156#397—22no—
157#396—24no—
158#394—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#391—21no—
163#391—22no—
164#391C2311yescharacteristic
165#390—22no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390C2312yes—
173#390C2311yescharacteristic
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—24no—
181#389C2312yes—
182#389C2314yes—
183#388—24no—
184#388—22no—
185#388—22no—
186#388—21no—
187#388—21no—
188#387—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382C2312yes—
200#382C2311yescharacteristic
201#380—22no—
202#380—21no—
203#380—21no—
204#380C2314yes—
205#379—22no—
206#379—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C25—22no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—28no—
245C22×D4—44no—
246C22×D4—42no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C22×D4C2411yescharacteristic
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×M16—28no—
259C2×M16—44no—
260C2×M16—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269#127—24no—
270#125—28no—
271#125—44no—
272#125—44no—
273#124—22no—
274#124—41no—
275#124—44no—
276#124—82no—
277#124—82no—
278C22⋊Q8—28no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—84no—
283#121—28no—
284#121—44no—
285#120—48no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—48no—
289C4×Q8—48no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4#4118yes—
295#117—28no—
296#117—44no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—84no—
343C2×C22⋊C4—84no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C4⋊C8—24no—
362#103—24no—
363#103—42no—
364#103—44no—
365#102—24no—
366#102—24no—
367#102—28no—
368#102—44no—
369#102—44no—
370#102—44no—
371#102—44no—
372C22⋊C8—28no—
373#97—42no—
374C4×C8—24no—
375C4×C8—44no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24—22no—
391C24—44no—
392C24—44no—
393C24—44no—
394C24—44no—
395C24—82no—
396C24—82no—
397C24—82no—
398C24C22×D412yes—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—416no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—88no—
432C2×D4—82no—
433C2×D4—81no—
434C2×D4—81no—
435C2×D4—816no—
436C2×D4#12414yes—
437C2×D42−1+412yes—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4118yes—
470C22×C4C2511yescharacteristic
471M16—88no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4#12414yes—
508C4⋊C4H(5,𝔽2)14yes—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—48no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—84no—
530C22⋊C4#12418yes—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—42no—
541C42—48no—
542C42—42no—
543C42—44no—
544C42C2×#4114yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—88no—
566C23—88no—
567C23—88no—
568C23—88no—
569C23—164no—
570C23—164no—
571C23—164no—
572C23C2×#12414yes—
573C23C23×D411yescharacteristic
574Q8—48no—
575Q8—42no—
576Q8—42no—
577Q8—88no—
578Q8—162no—
579D4—28no—
580D4—24no—
581D4—42no—
582D4—44no—
583D4—44no—
584D4—88no—
585D4—88no—
586D4—84no—
587D4—162no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—164no—
635C2×C4C2×#12412yes—
636C2×C4C2×#12414yes—
637C2×C4C2×#12412yes—
638C2×C4#57512yes—
639C2×C4#57512yes—
640C2×C4#57612yes—
641C2×C4#57712yes—
642C2×C4C22×#4112yes—
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4C2×2−1+411yescharacteristic
645C2×C4#58511yescharacteristic
646C8—44no—
647C8—42no—
648C8—42no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—1616no—
671C22C22×#12411yescharacteristic
672C22#348411yescharacteristic
673C22#348911yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57511yescharacteristic
676C22C2×#57611yescharacteristic
677C22C2×#57711yescharacteristic
678C4—24no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—41no—
686C4—41no—
687C4—81no—
688C4—84no—
689C4—82no—
690C4—81no—
691C4—81no—
692C4—82no—
693C4#314014yes—
694C4#351412yes—
695C4#352012yes—
696C2—44no—
697C2—82no—
698C2—162no—
699C2#3593911yescharacteristic
700C2C2×#314011yescharacteristic
701C2#6062211yescharacteristic
702C2#6160411yescharacteristic
703C2#6160511yescharacteristic
704C2#6214311yescharacteristic
705C2#6214411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36134C2semidirect
3. #30203C2semidirect
4. #30200C2semidirect
5. #29354C2semidirect
6. #29148C2semidirect
7. #29103C2semidirect
8. #27814C2semidirect
9. #27770C2semidirect
10. #27725C2semidirect
11. #27271C2semidirect
12. #25906C2semidirect
13. #23817C2semidirect
14. #23454C2semidirect
15. #22502C2semidirect
17. #20567C2semidirect
20. C2×#533C22semidirect
21. C2×#488C22semidirect
24. #2868C22semidirect
26. #2857C22semidirect
29. #2552C22semidirect
38. #2441C22semidirect
47. C4×C4⋊Q8C22semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2072C22semidirect
55. #2071C22semidirect
56. #2071C22semidirect
58. #2042C22semidirect
60. #2025C22semidirect
61. #2024C22semidirect
62. #2024C22semidirect
65. #2023C22semidirect
66. #2023C22semidirect
67. #2023C22semidirect
68. #1991C22semidirect
69. #1990C22semidirect
70. #1987C22semidirect
74. #1936C22semidirect
76. #1934C22semidirect
78. #1894C22semidirect
80. #1873C22semidirect
81. #1836C22semidirect
82. #1823C22semidirect
83. #1803C22semidirect
86. C8×C4⋊C4C22semidirect
87. #1767C22semidirect
88. C4×#103C22semidirect
90. #1753C22semidirect
91. C4×C22⋊C8C22semidirect
92. #1744C22semidirect
93. #1736C22semidirect
94. #1728C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#361342
#302032
#302002
#293542
#291482
#291032
#278142
#277702
#277252
#272712
#259062
#238172
#234542
#225022
#224282
#205672