Finite Group Databasebeta

← #377033

Subgroups — #377033

Order 512.

2146 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic, complemented
2C2×#1954C212yesmaximal, complemented
3C2×#1923C211yesmaximal, characteristic, complemented
4C2×#1869C212yesmaximal, complemented
5C2×#1850C212yesmaximal, complemented
6C4×#392C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic, complemented
8#11042C211yesmaximal, characteristic, complemented
9#10964C211yesmaximal, characteristic, complemented
10#10921C212yesmaximal, complemented
11C8×#95C211yesmaximal, characteristic, complemented
12C24×C8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2212yes—
15C2×C8×D4—28no—
16C2×C8×D4—24no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2212yes—
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26C22×C4×C8C2212yes—
27C22×C4×C8C2211yescharacteristic
28C2×#392C411yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31#1954—28no—
32#1923—24no—
33#1869—28no—
34#1850—28no—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1444C411yescharacteristic
42#1439C412yes—
43#1439C411yescharacteristic
44#1438C412yes—
45#1438C411yescharacteristic
46C4×#95C2211yescharacteristic
47C23×D4C2×C411yescharacteristic
48C23×D4C2×C412yes—
49C24×C4C2311yescharacteristic
50C24×C4C2312yes—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8—42no—
57C23×C8—42no—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—22no—
69C2×C4×D4—24no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2312yes—
86C8×D4—232no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—42no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—42no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8—42no—
109C2×C22⋊C8D418yescomplemented
110C2×C4×C8—24no—
111C2×C4×C8—24no—
112C2×C4×C8—24no—
113C2×C4×C8—42no—
114C2×C4×C8D414yescomplemented
115#392C818yescomplemented
116C4×C22⋊C4—24no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—24no—
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C412yes—
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C412yes—
124C2×#95C2×C411yescharacteristic
125C2×#95C2×C412yes—
126C2×#95C2×C411yescharacteristic
127#336—24no—
128#336—22no—
129#336—24no—
130#336—22no—
131#336—22no—
132#336D418yescomplemented
133C25C22×C411yescharacteristic
134C25C22×C412yes—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4C2×C814yes—
146C22×D4C2×C818yescomplemented
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C2×D414yes—
178C23×C4#4114yes—
179C23×C4C2411yescharacteristic
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—216no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—48no—
210C22×C8—48no—
211C22×C8—48no—
212C22×C8—48no—
213C22×C8—48no—
214C22×C8—48no—
215C22×C8C2×D414yes—
216C22×C8C2×D414yes—
217C22×C8C2×D412yes—
218C22×C8C2×D414yes—
219C22×C8C2×D414yes—
220C22×C8C2×D412yes—
221C4×D4—28no—
222C4×D4—216no—
223C4×D4—28no—
224C4×D4—48no—
225C4×D4—48no—
226C4×D4—48no—
227C4×D4—48no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—42no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4C2×C814yescomplemented
253C2×C22⋊C4C2×C818yescomplemented
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42C2×D414yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267C2×C42#4114yes—
268C4⋊C8—28no—
269C4⋊C8—48no—
270C4⋊C8—44no—
271C22⋊C8—216no—
272C22⋊C8—48no—
273C22⋊C8—48no—
274C22⋊C8—48no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277C4×C8—28no—
278C4×C8—48no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95C2×C814yescomplemented
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—21no—
308C24C4×D414yes—
309C24C22×C812yes—
310C24C22×C814yes—
311C24#13114yes—
312C24C23×C411yescharacteristic
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—216no—
316C2×D4—216no—
317C2×D4—216no—
318C2×D4—28no—
319C2×D4—216no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4C4×D414yes—
408C22×C4C4×D412yes—
409C22×C4C4×D414yes—
410C22×C4C4×D412yes—
411C22×C4C4×D414yes—
412C22×C4C22×C812yes—
413C22×C4C22×C812yes—
414C22×C4C22×C814yes—
415C22×C4#13112yes—
416C22×C4#13114yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C42−1+411yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—432no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8#12014yes—
461C2×C8#12714yes—
462C4⋊C4—28no—
463C4⋊C4—48no—
464C22⋊C4—216no—
465C22⋊C4—28no—
466C22⋊C4—216no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—24no—
478C42—41no—
479C42—41no—
480C42—42no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23C8×D418yes—
539C23C2×C4×D411yescharacteristic
540C23C2×C4×D412yes—
541C23C2×C4×D412yes—
542C23#51811yescharacteristic
543C23#52011yescharacteristic
544C23C23×C811yescharacteristic
545C23C2×#13111yescharacteristic
546C23C2×#13112yes—
547C23C2×#13112yes—
548C23#56811yescharacteristic
549C23#56811yescharacteristic
550D4—232no—
551D4—432no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4C8×D414yes—
611C2×C4C8×D418yescomplemented
612C2×C4C2×#12011yescharacteristic
613C2×C4#52511yescharacteristic
614C2×C4#52512yes—
615C2×C4C2×#12711yescharacteristic
616C2×C4#53211yescharacteristic
617C2×C4#53212yes—
618C2×C4#53511yescharacteristic
619C2×C4#53711yescharacteristic
620C2×C4#54011yescharacteristic
621C2×C4#54211yescharacteristic
622C2×C4D4212yes—
623C2×C4D4212yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4#54712yes—
627C2×C4#54812yes—
628C2×C4#55112yes—
629C2×C4#55312yes—
630C8—24no—
631C8—24no—
632C8—44no—
633C8—44no—
634C8—81no—
635C8—81no—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—82no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—82no—
662C22C4×#12011yescharacteristic
663C22C4×#12711yescharacteristic
664C22#232411yescharacteristic
665C22#235812yes—
666C22#236112yes—
667C22C2×C8×D412yes—
668C22C2×C8×D412yes—
669C22C2×C8×D411yescharacteristic
670C22#292912yes—
671C22#292911yescharacteristic
672C22#293011yescharacteristic
673C22#293112yes—
674C22#293111yescharacteristic
675C22#293312yes—
676C22#293312yes—
677C22#293312yes—
678C22#293512yes—
679C22#297211yescharacteristic
680C22#297712yes—
681C22#298111yescharacteristic
682C22#298811yescharacteristic
683C22#299311yescharacteristic
684C22#299412yes—
685C4—28no—
686C4—24no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—42no—
691C4—81no—
692C4—81no—
693C4#243011yescharacteristic
694C4#243311yescharacteristic
695C4#246112yes—
696C4#246412yes—
697C4#254011yescharacteristic
698C4#260311yescharacteristic
699C2—48no—
700C2—82no—
701C2#1418911yescharacteristic
702C2C8×#12011yescharacteristic
703C2#2144511yescharacteristic
704C2C8×#12711yescharacteristic
705C2#2168511yescharacteristic
706C2#2213311yescharacteristic
707C2#2213411yescharacteristic
708C2#2284212yes—
709C2#2284712yes—
710C2#2298512yes—
711C2#2299212yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C8×D4C2semidirect
2. C2×#1954C2semidirect
3. C2×#1923C2semidirect
4. C2×#1869C2semidirect
5. C2×#1850C2semidirect
7. #11322C2semidirect
8. #11042C2semidirect
9. #10964C2semidirect
10. #10921C2semidirect
11. C8×#95C2semidirect
109. C2×C22⋊C8D4semidirect
114. C2×C4×C8D4semidirect
115. #392C8semidirect
132. #336D4semidirect
146. C22×D4C2×C8semidirect
252. C2×C22⋊C4C2×C8semidirect
253. C2×C22⋊C4C2×C8semidirect
285. #95C2×C8semidirect
611. C2×C4C8×D4semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#19542
C2×#19232
C2×#18692
C2×#18502
C4×#3922
#113222
#110422
#109642
#109212
C8×#952