Finite Group Databasebeta

← #6650286

Subgroups — #6650286

Order 512.

2101 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35031C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic, complemented
3#33098C211yesmaximal, characteristic, complemented
4#33094C211yesmaximal, characteristic, complemented
5#32034C214yesmaximal, complemented
6#30712C214yesmaximal, complemented
7#30710C214yesmaximal, complemented
8#30504C214yesmaximal, complemented
9#29627C211yesmaximal, characteristic, complemented
10#29624C211yesmaximal, characteristic, complemented
11#24381C212yesmaximal, complemented
12#24380C212yesmaximal, complemented
13#24010C212yesmaximal, complemented
14#23922C212yesmaximal, complemented
15#21738C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#485C2214yescomplemented
22C2×#467C2214yescomplemented
23C2×#465C2214yescomplemented
24C2×#452C2214yescomplemented
25C2×#451C2212yes—
26C2×#448C2212yes—
27C22×C22⋊C8C2212yes—
28#2855C2211yescharacteristic
29#2854C2211yescharacteristic
30#2848C2211yescharacteristic
31#2847C2211yescharacteristic
32#2622—21no—
33#2613—21no—
34#2602—22no—
35#2600—22no—
36#2542—22no—
37#2542—22no—
38#2542C2214yes—
39#2542C2214yes—
40#2485C2212yes—
41#2415C2212yes—
42#2414C2212yes—
43#2407C2212yes—
44C2×#399—21no—
45C2×#399—21no—
46#2322—21no—
47#2321—21no—
48C2×#390—21no—
49C2×#384—21no—
50#2280C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2043C2214yescomplemented
53#2043C2214yescomplemented
54#2036C2214yescomplemented
55#2036C2214yescomplemented
56#2035C2214yescomplemented
57#2035C2214yescomplemented
58#2016C2214yescomplemented
59#2013C2214yescomplemented
60#2001C2214yescomplemented
61#2001C2214yescomplemented
62#2000C2214yescomplemented
63#2000C2214yescomplemented
64#1927C2214yescomplemented
65#1925C2214yescomplemented
66#1920C2214yescomplemented
67#1918C2214yescomplemented
68#1892C2214yescomplemented
69#1892C2214yescomplemented
70#1874C2212yes—
71#1873C2214yescomplemented
72#1873C2214yescomplemented
73#1872C2212yes—
74#1869C2214yes—
75#1869C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1817C2211yescharacteristic
79#1817C2211yescharacteristic
80#1787C2212yes—
81#1787C2212yes—
82#1779C2212yes—
83C22×#41C2311yescharacteristic
84C23×Q8C2312yes—
85C23×D4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C22×Q16C2314yes—
89C22×SD16C2314yes—
90C23×C8C2314yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2314yes—
94#525—44no—
95C2×#123—21no—
96C2×#123—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—48no—
102C2×C22⋊Q8—48no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2311yescharacteristic
110C2×#121C2311yescharacteristic
111C2×#121C2312yes—
112C2×#121C2311yescharacteristic
113C2×#120—21no—
114C2×#120—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—44no—
118C2×C4×D4D412yes—
119C2×C4×D4D412yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#485—48no—
137#467—48no—
138#465—48no—
139#452—48no—
140#451—44no—
141#448—44no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8—44no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#399—24no—
159#399—24no—
160#399—24no—
161#399—24no—
162#397—22no—
163#397—22no—
164#394—22no—
165#394—22no—
166#394—22no—
167#394—22no—
168#392—22no—
169#392—22no—
170#391C2312yes—
171#390—24no—
172#390—24no—
173#390—22no—
174#390C2312yes—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#386—21no—
185#386—21no—
186#386—22no—
187#386—22no—
188#385—21no—
189#385—21no—
190#385—22no—
191#385—22no—
192#384—24no—
193#384—24no—
194#381—21no—
195#381—21no—
196#380C2311yescharacteristic
197#380C2312yes—
198#380C2311yescharacteristic
199#380C2312yes—
200#380C2312yes—
201#379—21no—
202#379—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2314yes—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C2×#95—21no—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
215C25C2×D411yescharacteristic
216C2×#41—42no—
217C2×#41—44no—
218C22×Q8—22no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8C2414yes—
224C22×Q8C2411yescharacteristic
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4C2×D412yes—
233C22×D4C2×D412yes—
234C22×D4C2411yescharacteristic
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×Q16—44no—
254C2×Q16—44no—
255C2×Q16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×SD16—44no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8—44no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267#123—44no—
268#123—44no—
269C22⋊Q8—24no—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—42no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—44no—
284#121—24no—
285#121—42no—
286#121—44no—
287#120—44no—
288#120—44no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—44no—
296C4×D4—44no—
297#117—42no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
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—22no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358#107—28no—
359#106—28no—
360#103—28no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367#102—28no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—44no—
371C22⋊C8—44no—
372C22⋊C8—44no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24#13714yes—
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398#41—88no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—88no—
411C2×Q8—88no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—416no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—82no—
472C22×C4#13514yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4C2511yescharacteristic
481Q16—88no—
482SD16—88no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—88no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4H(5,𝔽2)18yes—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—42no—
537C42—48no—
538C42—82no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—48no—
559C23—48no—
560C23—44no—
561C23—42no—
562C23—41no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23—82no—
570C23—84no—
571C23—84no—
572C23#53412yes—
573C23C2×#13512yes—
574C23C2×#13714yes—
575C23C2×#13712yes—
576C23C23×D411yescharacteristic
577Q8—44no—
578Q8—44no—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582Q8—88no—
583D4—48no—
584D4—48no—
585D4—44no—
586D4—84no—
587D4—84no—
588D4—84no—
589D4—164no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—44no—
607C2×C4—416no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4#53412yes—
638C2×C4C2×#13514yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×H(5,𝔽2)14yes—
641C8—44no—
642C8—44no—
643C22—22no—
644C22—22no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—88no—
665C22—88no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—82no—
670C22—82no—
671C22#319014yes—
672C22C2×#53411yescharacteristic
673C22#348412yes—
674C22C22×#13511yescharacteristic
675C22C22×#13711yescharacteristic
676C22#358612yes—
677C4—24no—
678C4—22no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—41no—
683C4—48no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—84no—
688C4—82no—
689C4—82no—
690C4#354314yes—
691C2—24no—
692C2—44no—
693C2—81no—
694C2—81no—
695C2—161no—
696C2#3723211yescharacteristic
697C2C2×#319011yescharacteristic
698C2#6075411yescharacteristic
699C2#6156512yes—
700C2#6157412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35031C2
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×#2542C2semidirect
3. #33098C2semidirect
4. #33094C2semidirect
5. #32034C2semidirect
6. #30712C2semidirect
7. #30710C2semidirect
8. #30504C2semidirect
9. #29627C2semidirect
10. #29624C2semidirect
11. #24381C2semidirect
12. #24380C2semidirect
13. #24010C2semidirect
14. #23922C2semidirect
15. #21738C2semidirect
21. C2×#485C22semidirect
22. C2×#467C22semidirect
23. C2×#465C22semidirect
24. C2×#452C22semidirect
52. #2043C22semidirect
53. #2043C22semidirect
54. #2036C22semidirect
55. #2036C22semidirect
56. #2035C22semidirect
57. #2035C22semidirect
58. #2016C22semidirect
59. #2013C22semidirect
60. #2001C22semidirect
61. #2001C22semidirect
62. #2000C22semidirect
63. #2000C22semidirect
64. #1927C22semidirect
65. #1925C22semidirect
66. #1920C22semidirect
67. #1918C22semidirect
68. #1892C22semidirect
69. #1892C22semidirect
71. #1873C22semidirect
72. #1873C22semidirect

Maximal subgroups

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

subgroupindex
#350312
C2×#25422
#330982
#330942
#320342
#307122
#307102
#305042
#296272
#296242
#243812
#243802
#240102
#239222
#217382