Finite Group Databasebeta

← #366298

Subgroups — #366298

Order 512.

2332 conjugacy classes of subgroups.

The subgroup lattice has 785 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#391C211yesmaximal, characteristic
2C2×#1927C212yesmaximal, complemented
3C2×#1853C211yesmaximal, characteristic
4C2×#1787C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6C4×#380C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#10932C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#391—21no—
23C2×#391—21no—
24C2×#391—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32C22×#95—21no—
33C22×#95—21no—
34#1927—28no—
35#1853—24no—
36#1787—24no—
37#1787C418yescomplemented
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434—21no—
46C22⋊#95—21no—
47C4×#95—21no—
48C23×Q8—24no—
49C23×Q8—24no—
50C23×Q8—21no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2×C414yes—
59C23×Q8C2312yes—
60C24×C4—21no—
61C24×C4—21no—
62C24×C4D411yescharacteristic
63C24×C4D411yescharacteristic
64C24×C4C2311yescharacteristic
65C23×C8C2312yes—
66C23×C8C2312yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—24no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—42no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103C2×C418yescomplemented
103C2×#103C2×C418yescomplemented
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8C2×C414yes—
110#391—28no—
111#391—28no—
112#391—24no—
113#391—24no—
114#391—24no—
115#391D418yes—
116#380—22no—
117#380—22no—
118#380C2×C414yes—
119#380D414yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—42no—
122C4×C22⋊C4—44no—
123C2×#95—28no—
124C2×#95—28no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137#340—24no—
138#340—24no—
139#340—42no—
140#336—24no—
141#336—42no—
142C25C2×D411yescharacteristic
143C22×Q8—24no—
144C22×Q8—216no—
145C22×Q8—24no—
146C22×Q8—216no—
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—28no—
152C22×Q8—28no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—48no—
162C22×Q8—48no—
163C22×Q8—48no—
164C22×Q8—48no—
165C22×Q8—48no—
166C22×Q8—48no—
167C22×Q8—48no—
168C22×Q8—48no—
169C22×Q8—48no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—48no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8C22×C414yes—
178C22×Q8C2×D414yes—
179C22×Q8#4114yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—41no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4SD1614yes—
218C23×C4C22×C412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4#4112yes—
226C23×C4#4112yes—
227C23×C4#4112yes—
228C23×C4#4112yes—
229C23×C4C2411yescharacteristic
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8C22×C414yes—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—48no—
289C2×C42—48no—
290C2×C42—41no—
291C2×C42—41no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C2×D414yes—
299#106—48no—
300#103—216no—
301#103—48no—
302#103—48no—
303#103—48no—
304C22⋊C8—28no—
305C22⋊C8—44no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24C4×D412yes—
327C24#12112yes—
328C24#12312yes—
329C24C2×SD1612yes—
330C24#13512yes—
331C24#13712yes—
332C24#13712yes—
333C24C22×D411yescharacteristic
334C2×Q8—216no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—432no—
351C2×Q8—432no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—432no—
358C2×Q8—432no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—48no—
372C2×Q8C4×D418yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4#11714yes—
482C22×C4C4×D414yes—
483C22×C4C4×D412yes—
484C22×C4#12012yes—
485C22×C4#12012yes—
486C22×C4#12112yes—
487C22×C4#12312yes—
488C22×C4#12414yes—
489C22×C4#12614yes—
490C22×C4C2×SD1612yes—
491C22×C4C2×SD1612yes—
492C22×C4#13712yes—
493C22×C4#13712yes—
494C22×C4C23×C411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4111yescharacteristic
498C22×C4C2×#4111yescharacteristic
499C22×C4C2×#4112yes—
500C22×C4C2×#4111yescharacteristic
501C22×C4C2×#4111yescharacteristic
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—42no—
508C2×C8—42no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—42no—
512C2×C8—42no—
513C2×C8—42no—
514C2×C8—42no—
515C2×C8—42no—
516C2×C8—42no—
517C2×C8—42no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C42—24no—
534C42—24no—
535C42—44no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—84no—
540C42—84no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23#38814yes—
570C23C4×SD1614yes—
571C23#44114yes—
572C23#46114yes—
573C23#46414yes—
574C23#48314yes—
575C23#48514yes—
576C23C2×C4×D411yescharacteristic
577C23C2×#12011yescharacteristic
578C23C2×#12111yescharacteristic
579C23C2×#12311yescharacteristic
580C23#52511yescharacteristic
581C23#53411yescharacteristic
582C23#53511yescharacteristic
583C23C22×SD1611yescharacteristic
584C23C2×#13511yescharacteristic
585C23C2×#13711yescharacteristic
586C23C2×#13711yescharacteristic
587C23C2×#13711yescharacteristic
588C23C2×#13711yescharacteristic
589C23#57511yescharacteristic
590C23#57511yescharacteristic
591Q8—28no—
592Q8—28no—
593Q8—432no—
594Q8—44no—
595Q8—44no—
596Q8—44no—
597Q8—44no—
598Q8—44no—
599Q8—44no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—28no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—432no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—816no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—81no—
674C2×C4—82no—
675C2×C4—82no—
676C2×C4—81no—
677C2×C4—81no—
678C2×C4—81no—
679C2×C4—81no—
680C2×C4—81no—
681C2×C4—82no—
682C2×C4—82no—
683C2×C4#38814yes—
684C2×C4#44814yes—
685C2×C4C2×#11711yescharacteristic
686C2×C4C2×C4×D412yes—
687C2×C4C2×C4×D411yescharacteristic
688C2×C4#51812yes—
689C2×C4#52512yes—
690C2×C4C2×#12411yescharacteristic
691C2×C4C2×#12611yescharacteristic
692C2×C4#53812yes—
693C2×C4D4212yes—
694C2×C4#54612yes—
695C2×C4#54612yes—
696C2×C4#54612yes—
697C2×C4#54612yes—
698C2×C4#55112yes—
699C2×C4#55112yes—
700C2×C4#55112yes—
701C8—44no—
702C8—82no—
703C22—24no—
704C22—24no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—24no—
712C22—24no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—24no—
717C22#192718yescomplemented
718C22C2×#38811yescharacteristic
719C22C4×#12011yescharacteristic
720C22#232111yescharacteristic
721C22#240511yescharacteristic
722C22#240811yescharacteristic
723C22#248211yescharacteristic
724C22#253411yescharacteristic
725C22C2×C4×SD1611yescharacteristic
726C22C2×#44111yescharacteristic
727C22C4×#13711yescharacteristic
728C22C4×#13711yescharacteristic
729C22C2×#44811yescharacteristic
730C22#300211yescharacteristic
731C22#300411yescharacteristic
732C22#300711yescharacteristic
733C22C2×#46111yescharacteristic
734C22C2×#46411yescharacteristic
735C22#303911yescharacteristic
736C22#304011yescharacteristic
737C22C2×#48311yescharacteristic
738C22C2×#48511yescharacteristic
739C22#309511yescharacteristic
740C22#309511yescharacteristic
741C22#310311yescharacteristic
742C22#310811yescharacteristic
743C22#311311yescharacteristic
744C22#311711yescharacteristic
745C22#318711yescharacteristic
746C22#319011yescharacteristic
747C22#319511yescharacteristic
748C22#319611yescharacteristic
749C22#320611yescharacteristic
750C22#321111yescharacteristic
751C22#321511yescharacteristic
752C22#321811yescharacteristic
753C4—24no—
754C4—24no—
755C4—48no—
756C4—42no—
757C4—42no—
758C4—42no—
759C4—82no—
760C4—82no—
761C4—84no—
762C4—82no—
763C4—82no—
764C4#234212yes—
765C4#235812yes—
766C4#247212yes—
767C4#248012yes—
768C2—28no—
769C2#1399111yescharacteristic
770C2C2×#192711yescharacteristic
771C2#2106111yescharacteristic
772C2C4×#44811yescharacteristic
773C2#2145511yescharacteristic
774C2#2204111yescharacteristic
775C2#2205011yescharacteristic
776C2#2400311yescharacteristic
777C2#2401011yescharacteristic
778C2#2420911yescharacteristic
779C2#2421711yescharacteristic
780C2#2758311yescharacteristic
781C2#2758911yescharacteristic
782C2#2811511yescharacteristic
783C2#2812411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1927C2semidirect
7. #11190C2semidirect
8. #11122C2semidirect
9. #10932C2semidirect
37. #1787C4semidirect
102. C2×#103C2×C4semidirect
103. C2×#103C2×C4semidirect
717. C22#1927semidirect

Maximal subgroups

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

subgroupindex
C22×#3912
C2×#19272
C2×#18532
C2×#17872
C4×#3802
#111902
#111222
#109322
#107302
#107122