Finite Group Databasebeta

← #220004

Subgroups — #220004

Order 512.

1093 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36150C211yesmaximal, characteristic
2#22748C211yesmaximal, characteristic, complemented
3#22748C211yesmaximal, characteristic, complemented
4C2×#1542C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic, complemented
6#9699C211yesmaximal, characteristic, complemented
7#9699C211yesmaximal, characteristic, complemented
8#9134C211yesmaximal, characteristic, complemented
9#9133C211yesmaximal, characteristic, complemented
10#9128C211yesmaximal, characteristic, complemented
11#9127C211yesmaximal, characteristic, complemented
12#8593C211yesmaximal, characteristic, complemented
13#8343C211yesmaximal, characteristic, complemented
14#8341C211yesmaximal, characteristic, complemented
15C4×#332C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2858C2211yescharacteristic
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2540C412yes—
26#2490—21no—
27#2454—21no—
28#2448—21no—
29#2427C412yes—
30#2426—21no—
31#2426C412yes—
32#2414—21no—
33#2407—21no—
34#2367C412yes—
35#2363—21no—
36#2341—21no—
37C4×C4⋊Q8C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1923—21no—
40#1923—21no—
41#1916—21no—
42#1916—21no—
43#1869—21no—
44#1869—21no—
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C8×C22⋊C4—21no—
54C8×C22⋊C4—21no—
55#1542—24no—
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60#1540—24no—
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C2×#331C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×#41—21no—
74C23×C8—21no—
75C23×C8—21no—
76#533—22no—
77#533—22no—
78#533—41no—
79#533—41no—
80#532—22no—
81#532—22no—
82#532—41no—
83#532—41no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#125—21no—
88C2×#124—21no—
89C2×#124C2×C412yes—
90C2×#124C2×C412yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8C2×C412yes—
93C2×C22⋊Q8C2×C412yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121C2×C412yes—
98C2×#121C2×C412yes—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8C2×C412yes—
102C2×C4×D4—21no—
103C2×C4×D4C2×C412yes—
104C2×C4×D4C2×C411yescharacteristic
105C2×C4×D4D411yescharacteristic
106C2×C4×D4D411yescharacteristic
107C2×#117—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#431—22no—
111#431—22no—
112#431—41no—
113#431—41no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—41no—
119C2×C4⋊C8—41no—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138#399—22no—
139#397—22no—
140#396—22no—
141#394—22no—
142#394—22no—
143#392—22no—
144#391—21no—
145#391C2×C412yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2×C412yes—
156#389—21no—
157#389—21no—
158#389—21no—
159#389C2×C412yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#386—22no—
164#385—22no—
165#384C2311yescharacteristic
166#380—21no—
167#380—21no—
168#380—22no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C2×C412yes—
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2×C411yescharacteristic
176C4×C22⋊C4C2×C411yescharacteristic
177C4×C22⋊C4D411yescharacteristic
178C4×C22⋊C4D411yescharacteristic
179C4×C22⋊C4D411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181C43C2311yescharacteristic
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#333—22no—
193#333—22no—
194#333—41no—
195#333—41no—
196#332—22no—
197#332—22no—
198#332—22no—
199#332D414yescomplemented
200#331—22no—
201#331—22no—
202#331—41no—
203#331—41no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—42no—
207C2×#41—42no—
208C2×#41—81no—
209C2×#41—81no—
210C22×Q8—21no—
211C22×Q8C22×C412yes—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4C22⋊C411yescharacteristic
217C22×D4C22×C412yes—
218C23×C4—22no—
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8—81no—
238C22×C8—81no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8C2×D412yes—
246C4⋊Q8#4112yes—
247#127—21no—
248#127—21no—
249#127—21no—
250#127—21no—
251#125—42no—
252#125—42no—
253#124—42no—
254#124—42no—
255#124—44no—
256#124—82no—
257C22⋊Q8—24no—
258C22⋊Q8—42no—
259C22⋊Q8—42no—
260C22⋊Q8—42no—
261C22⋊Q8—82no—
262#121—24no—
263#121—42no—
264#121—42no—
265#121—42no—
266#121—42no—
267#121—42no—
268#121—81no—
269#121—81no—
270#121—82no—
271C4×Q8—24no—
272C4×Q8—42no—
273C4×Q8—42no—
274C4×Q8—42no—
275C4×Q8—81no—
276C4×Q8—81no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—42no—
286#117—42no—
287#117—42no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C22⋊C411yescharacteristic
298C2×C4⋊C4C22⋊C411yescharacteristic
299C2×C4⋊C4C22⋊C411yescharacteristic
300C2×C4⋊C4C22×C412yes—
301C2×C4⋊C4C22×C412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4C22⋊C411yescharacteristic
314C2×C22⋊C4C22⋊C411yescharacteristic
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C22×C411yescharacteristic
332C2×C42C22×C411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—42no—
357C4⋊C8—42no—
358C4⋊C8C2×D412yes—
359C4⋊C8C2×D412yes—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C22⋊C8—81no—
365C22⋊C8—81no—
366C4×C8—42no—
367C4×C8—42no—
368C4×C8—81no—
369C4×C8—81no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C22⋊C411yescharacteristic
375#95C22⋊C411yescharacteristic
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380C24—22no—
381C24C2×C22⋊C411yescharacteristic
382C24C2×C22⋊C411yescharacteristic
383#41—84no—
384#41—84no—
385#41—84no—
386#41—84no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—41no—
394C2×Q8—41no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—41no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4#13512yes—
467C22×C4#13512yes—
468C22×C4#13512yes—
469C22×C4#13512yes—
470C22×C4C23×C411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—84no—
498C2×C8—84no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—41no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4C4×D414yes—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—22no—
557C42—41no—
558C42—44no—
559C42—82no—
560C42C22×D411yescharacteristic
561C42C2×#4111yescharacteristic
562C42H(5,𝔽2)11yescharacteristic
563C42H(5,𝔽2)11yescharacteristic
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23#41612yes—
584C23#41612yes—
585C23C22×C22⋊C411yescharacteristic
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—82no—
591Q8—82no—
592D4—44no—
593D4—82no—
594D4—82no—
595D4—88no—
596D4—81no—
597D4—81no—
598D4—81no—
599D4—81no—
600D4—81no—
601D4—81no—
602D4—81no—
603D4—81no—
604D4—164no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—24no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—88no—
664C2×C4#41312yes—
665C2×C4#41612yes—
666C2×C4#41612yes—
667C2×C4C2×C4×D411yescharacteristic
668C2×C4#51811yescharacteristic
669C2×C4D4211yescharacteristic
670C2×C4D4211yescharacteristic
671C2×C4#54611yescharacteristic
672C2×C4#54611yescharacteristic
673C2×C4C2×#13511yescharacteristic
674C2×C4C2×#13511yescharacteristic
675C2×C4C2×#13511yescharacteristic
676C2×C4C2×#13511yescharacteristic
677C2×C4#57611yescharacteristic
678C2×C4#57711yescharacteristic
679C2×C4#57711yescharacteristic
680C2×C4#57811yescharacteristic
681C8—42no—
682C8—42no—
683C8—81no—
684C8—81no—
685C8—81no—
686C8—81no—
687C8—81no—
688C8—81no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22D4×C22⋊C411yescharacteristic
711C22C2×#41311yescharacteristic
712C22#288811yescharacteristic
713C22C2×#41611yescharacteristic
714C22C2×#41611yescharacteristic
715C22#289711yescharacteristic
716C22#289811yescharacteristic
717C4—22no—
718C4—21no—
719C4—21no—
720C4—22no—
721C4—22no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—44no—
729C4—41no—
730C4—82no—
731C4—82no—
732C4#328011yescharacteristic
733C4#328211yescharacteristic
734C4#328211yescharacteristic
735C4#328511yescharacteristic
736C4#329611yescharacteristic
737C4#329711yescharacteristic
738C4#330011yescharacteristic
739C4#330111yescharacteristic
740C2—42no—
741C2—42no—
742C2—162no—
743C2#1770011yescharacteristic
744C2#1772411yescharacteristic
745C2#1772611yescharacteristic
746C2#1777511yescharacteristic
747C2#2241211yescharacteristic
748C2#2241411yescharacteristic
749C2#2241611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36150C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22748C2semidirect
3. #22748C2semidirect
4. C2×#1542C2semidirect
5. C2×#1540C2semidirect
6. #9699C2semidirect
7. #9699C2semidirect
8. #9134C2semidirect
9. #9133C2semidirect
10. #9128C2semidirect
11. #9127C2semidirect
12. #8593C2semidirect
13. #8343C2semidirect
14. #8341C2semidirect
15. C4×#332C2semidirect
199. #332D4semidirect

Maximal subgroups

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

subgroupindex
#361502
#227482
C2×#15422
C2×#15402
#96992
#91342
#91332
#91282
#91272
#85932
#83432
#83412
C4×#3322