Finite Group Databasebeta

← #367182

Subgroups — #367182

Order 512.

1507 conjugacy classes of subgroups.

The subgroup lattice has 787 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1929C211yesmaximal, characteristic, complemented
2C2×#1896C211yesmaximal, characteristic
3#20967C211yesmaximal, characteristic
4#20895C211yesmaximal, characteristic
5C2×#1810C211yesmaximal, characteristic, complemented
6#20385C211yesmaximal, characteristic, complemented
7C22×#336C211yesmaximal, characteristic, complemented
8#11317C211yesmaximal, characteristic, complemented
9#11258C211yesmaximal, characteristic, complemented
10#11191C211yesmaximal, characteristic, complemented
11#11125C211yesmaximal, characteristic, complemented
12#10934C211yesmaximal, characteristic, complemented
13#10910C211yesmaximal, characteristic, complemented
14#10809C211yesmaximal, characteristic, complemented
15#10719C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C2×#525C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#407C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×#405C2211yescharacteristic
32#2280C2211yescharacteristic
33#1929—24no—
34#1896—24no—
35#1893C412yescomplemented
36#1893C412yescomplemented
37#1892C412yescomplemented
38#1892C412yescomplemented
39#1869—22no—
40#1869—22no—
41#1810—24no—
42#1752—22no—
43C8×C22⋊C4—22no—
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46#1735C2211yescharacteristic
47#1735C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58C2×#336—24no—
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C23×C8—22no—
71C23×C8—22no—
72C23×C8—22no—
73C23×C8—22no—
74C23×C8—22no—
75C23×C8—22no—
76C23×C8—22no—
77C23×C8—22no—
78C23×C8—41no—
79C23×C8—41no—
80C23×C8—41no—
81C23×C8—41no—
82C23×C8C2×C412yes—
83C23×C8C2×C412yes—
84C23×C8D412yes—
85C23×C8D412yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90#525—22no—
91#525—22no—
92#525—22no—
93#525—22no—
94C2×#123—21no—
95C2×C22⋊Q8C2×C412yes—
96C2×#121C2×C412yes—
97C2×#120—21no—
98C2×C4×D4—22no—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C42—21no—
103C22×C42—22no—
104C22×C42—22no—
105C22×C42—22no—
106C22×C42—22no—
107C22×C42—41no—
108C22×C42—41no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C2×#108—22no—
112C2×#108—22no—
113C2×#108—22no—
114C2×#108—22no—
115C2×#108—41no—
116C2×#108—41no—
117C2×#104—22no—
118C2×#104—22no—
119C2×#104—22no—
120C2×#104—22no—
121C2×#104—41no—
122C2×#104—41no—
123#416—22no—
124#416—22no—
125#416—41no—
126#416—41no—
127C2×#103C2×C411yescharacteristic, complemented
128C2×#103C2×C411yescharacteristic, complemented
129C2×#103C2×C412yescomplemented
130C2×#102C2×C411yescharacteristic, complemented
131C2×#102C2×C411yescharacteristic, complemented
132C2×#102C2×C412yescomplemented
133#408—22no—
134#408—22no—
135#408—41no—
136#408—41no—
137#407—22no—
138#407—22no—
139#407—22no—
140#407D414yes—
141C2×C22⋊C8—22no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8—22no—
144C2×C22⋊C8—21no—
145C2×C22⋊C8—21no—
146#405—22no—
147#405—22no—
148#405—41no—
149#405—41no—
150C2×#97—21no—
151C2×#97—21no—
152C2×C4×C8—21no—
153C2×C4×C8—21no—
154#381—21no—
155#380C2×C412yes—
156#380C2×C412yes—
157#379—21no—
158C4×C22⋊C4—22no—
159#341—22no—
160#341—22no—
161#340C2×C412yescomplemented
162#340C2×C412yescomplemented
163#340C2×C412yescomplemented
164#340C2×C412yescomplemented
165#339—22no—
166#339—22no—
167#339—22no—
168#339—22no—
169#339—41no—
170#339—41no—
171#336—22no—
172#336—21no—
173#336—21no—
174#336—22no—
175#336—22no—
176#336—22no—
177#336—22no—
178#336—22no—
179#336D414yescomplemented
180C25#4111yescharacteristic
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C22×Q8C22×C411yescharacteristic
194C22×D4—22no—
195C22×D4—21no—
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4C22×C412yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4#4112yes—
217C23×C4#4112yes—
218C23×C4#4112yes—
219C23×C4#4111yescharacteristic
220C23×C4C2411yescharacteristic
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16C2×D412yes—
244C2×M16C2×D412yes—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8—21no—
276C22×C8—21no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8—42no—
280C22×C8—42no—
281C22×C8—44no—
282C22×C8—44no—
283C22×C8—44no—
284C22×C8—44no—
285C22×C8—44no—
286C22×C8—44no—
287C22×C8—44no—
288C22×C8—44no—
289C22×C8C22×C412yes—
290C22×C8C22×C411yescharacteristic
291C22×C8C22×C411yescharacteristic
292C22×C8C2×D412yes—
293C22×C8C2×D412yes—
294C22×C8C2×D412yes—
295C22×C8C2×D411yescharacteristic
296C22×C8C2×D411yescharacteristic
297C22×C8#4112yes—
298C22×C8#4112yes—
299#123—42no—
300#123—42no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303#121—24no—
304#121—24no—
305#120—42no—
306#120—42no—
307C4×D4—44no—
308C4×D4—44no—
309#117—21no—
310#117—21no—
311#117—21no—
312#117—21no—
313#117—21no—
314#117—21no—
315#117—21no—
316#117—21no—
317#117—21no—
318#117—21no—
319#117—21no—
320#117—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4C22×C412yes—
323C2×C4⋊C4C22×C411yescharacteristic
324C2×C4⋊C4C22×C411yescharacteristic
325C2×C4⋊C4C22×C412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42C2×D412yes—
354C2×C42#4112yes—
355#108—44no—
356#108—44no—
357#104—44no—
358#104—44no—
359#103—22no—
360#103—22no—
361#103—42no—
362#102—22no—
363#102—22no—
364#102—42no—
365C22⋊C8—28no—
366C22⋊C8—44no—
367C22⋊C8—44no—
368C22⋊C8—42no—
369C22⋊C8—42no—
370#97—42no—
371#97—42no—
372C4×C8—42no—
373C4×C8—42no—
374#95—22no—
375#95—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—24no—
381C24—24no—
382C24—21no—
383C24C4×D412yes—
384C24#12412yes—
385C24#12612yes—
386C24C2×#4111yescharacteristic
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—216no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—416no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4#11714yes—
479C22×C4C4×D414yes—
480C22×C4C4×D412yes—
481C22×C4#12114yes—
482C22×C4#12314yes—
483C22×C4#12412yes—
484C22×C4#12612yes—
485C22×C4C23×C411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4111yescharacteristic
490C22×C4C2×#4111yescharacteristic
491C22×C4C2×#4111yescharacteristic
492C22×C4C2×#4111yescharacteristic
493C22×C4C2×#4111yescharacteristic
494C22×C4C2×#4111yescharacteristic
495C22×C4H(5,𝔽2)11yescharacteristic
496C22×C4H(5,𝔽2)11yescharacteristic
497C22×C4H(5,𝔽2)11yescharacteristic
498C22×C42−1+411yescharacteristic
499M16—22no—
500M16—22no—
501M16—22no—
502M16—22no—
503M16—41no—
504M16—41no—
505M16—41no—
506M16—41no—
507M16—41no—
508M16—41no—
509M16—41no—
510M16—41no—
511M16—41no—
512M16—41no—
513M16—41no—
514M16—41no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—48no—
540C2×C8—48no—
541C2×C8—48no—
542C2×C8—48no—
543C2×C8—41no—
544C2×C8—41no—
545C2×C8—41no—
546C2×C8—41no—
547C2×C8—41no—
548C2×C8—41no—
549C2×C8—41no—
550C2×C8—41no—
551C2×C8—41no—
552C2×C8—41no—
553C2×C8—41no—
554C2×C8—41no—
555C2×C8—41no—
556C2×C8—41no—
557C2×C8—41no—
558C2×C8—41no—
559C2×C8—41no—
560C2×C8—41no—
561C2×C8C4×D412yes—
562C2×C8C4×D412yes—
563C2×C8#12112yes—
564C2×C8#12112yes—
565C4⋊C4—22no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—44no—
574C22⋊C4—41no—
575C22⋊C4—41no—
576C22⋊C4—41no—
577C22⋊C4—41no—
578C22⋊C4—41no—
579C22⋊C4—41no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—48no—
589C42—41no—
590C42—41no—
591C42—41no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—41no—
599C23—24no—
600C23—24no—
601C23—28no—
602C23—24no—
603C23—24no—
604C23—28no—
605C23—22no—
606C23—28no—
607C23—28no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—22no—
612C23—22no—
613C23—22no—
614C23—22no—
615C23—22no—
616C23—22no—
617C23—41no—
618C23—42no—
619C23—42no—
620C23—41no—
621C23—41no—
622C23#38814yes—
623C23#44314yes—
624C23#44414yes—
625C23C2×C4×D411yescharacteristic
626C23#52511yescharacteristic
627C23C2×#12411yescharacteristic
628C23C2×#12611yescharacteristic
629C23#52911yescharacteristic
630C23#53811yescharacteristic
631C23#53911yescharacteristic
632Q8—41no—
633Q8—41no—
634Q8—41no—
635Q8—41no—
636D4—41no—
637D4—41no—
638D4—41no—
639D4—41no—
640D4—88no—
641D4—81no—
642D4—81no—
643D4—81no—
644D4—81no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—432no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—81no—
684C2×C4—81no—
685C2×C4—81no—
686C2×C4—81no—
687C2×C4#38814yes—
688C2×C4C2×#11711yescharacteristic
689C2×C4C2×C4×D411yescharacteristic
690C2×C4C2×C4×D411yescharacteristic
691C2×C4C4×#4111yescharacteristic
692C2×C4#51811yescharacteristic
693C2×C4#52011yescharacteristic
694C2×C4C2×#12111yescharacteristic
695C2×C4C2×#12111yescharacteristic
696C2×C4C2×#12311yescharacteristic
697C2×C4#53211yescharacteristic
698C2×C4#53411yescharacteristic
699C2×C4#53711yescharacteristic
700C2×C4D4211yescharacteristic
701C2×C4#54611yescharacteristic
702C2×C4#54611yescharacteristic
703C2×C4#54611yescharacteristic
704C2×C4#54611yescharacteristic
705C2×C4#54711yescharacteristic
706C2×C4#54811yescharacteristic
707C2×C4#55111yescharacteristic
708C2×C4#55111yescharacteristic
709C2×C4#55111yescharacteristic
710C2×C4#55111yescharacteristic
711C2×C4#55211yescharacteristic
712C2×C4#55211yescharacteristic
713C2×C4#55311yescharacteristic
714C2×C4#55311yescharacteristic
715C2×C4#55311yescharacteristic
716C8—22no—
717C8—22no—
718C8—22no—
719C8—22no—
720C8—42no—
721C8—42no—
722C8—42no—
723C8—42no—
724C8—81no—
725C8—81no—
726C22—28no—
727C22—28no—
728C22—28no—
729C22—28no—
730C22—21no—
731C22—21no—
732C22—24no—
733C22—21no—
734C22—21no—
735C22—24no—
736C22—22no—
737C22—22no—
738C22—24no—
739C22—24no—
740C22—22no—
741C22—22no—
742C22—24no—
743C22—24no—
744C22—81no—
745C22—81no—
746C22—82no—
747C22—81no—
748C22—82no—
749C22#192918yescomplemented
750C22C2×#38811yescharacteristic
751C22C4×#12111yescharacteristic
752C22#232511yescharacteristic
753C22#247211yescharacteristic
754C22#248111yescharacteristic
755C22#248811yescharacteristic
756C22#251011yescharacteristic
757C22C2×#44312yes—
758C22C2×#44412yes—
759C4—24no—
760C4—44no—
761C4—44no—
762C4—81no—
763C4—81no—
764C4—82no—
765C4—81no—
766C4—82no—
767C4#235211yescharacteristic
768C4#235311yescharacteristic
769C4#235811yescharacteristic
770C4#235911yescharacteristic
771C4#241211yescharacteristic
772C4#241611yescharacteristic
773C4#249811yescharacteristic
774C4#254311yescharacteristic
775C2—21no—
776C2—28no—
777C2—21no—
778C2—21no—
779C2—21no—
780C2—161no—
781C2#1399911yescharacteristic
782C2C2×#192912yes—
783C2#2154411yescharacteristic
784C2#2154611yescharacteristic
785C2#2220412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1929C2semidirect
5. C2×#1810C2semidirect
6. #20385C2semidirect
7. C22×#336C2semidirect
8. #11317C2semidirect
9. #11258C2semidirect
10. #11191C2semidirect
11. #11125C2semidirect
12. #10934C2semidirect
13. #10910C2semidirect
14. #10809C2semidirect
15. #10719C2semidirect
35. #1893C4semidirect
36. #1893C4semidirect
37. #1892C4semidirect
38. #1892C4semidirect
127. C2×#103C2×C4semidirect
128. C2×#103C2×C4semidirect
129. C2×#103C2×C4semidirect
130. C2×#102C2×C4semidirect
131. C2×#102C2×C4semidirect
132. C2×#102C2×C4semidirect
161. #340C2×C4semidirect
162. #340C2×C4semidirect
163. #340C2×C4semidirect
164. #340C2×C4semidirect
179. #336D4semidirect
749. C22#1929semidirect

Maximal subgroups

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

subgroupindex
C2×#19292
C2×#18962
#209672
#208952
C2×#18102
#203852
C22×#3362
#113172
#112582
#111912
#111252
#109342
#109102
#108092
#107192