Finite Group Databasebeta

← #381432

Subgroups — #381432

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2013C211yesmaximal, characteristic
3C2×#1880C212yesmaximal
4#14700C211yesmaximal, characteristic
5#11267C211yesmaximal, characteristic, complemented
6#11067C212yesmaximal, complemented
7#11049C211yesmaximal, characteristic
8#11023C212yesmaximal
9#11010C211yesmaximal, characteristic, complemented
10#10795C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C4×C4⋊Q8—28no—
20C2×#391C2212yes—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#388—21no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C43C2211yescharacteristic
33#2013C418yescomplemented
34#1880—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1437—21no—
43#1437—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C23×Q8C2312yes—
47C23×D4C2311yescharacteristic
48C22×SD16—24no—
49C22×SD16—24no—
50C22×SD16—42no—
51C22×SD16—42no—
52C22×SD16—42no—
53C22×SD16—42no—
54C22×SD16C2×C418yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C2×C4×Q8—42no—
66C2×C4×Q8D418yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42—22no—
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103C2×C418yes—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102—42no—
103#391—24no—
104#391—24no—
105#390—24no—
106#390—22no—
107#390—42no—
108#390C2×C414yes—
109#389—28no—
110#389—28no—
111#389—42no—
112#388—44no—
113#388—44no—
114#384—24no—
115C4×C4⋊C4—28no—
116C4×C4⋊C4—28no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C43—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—22no—
130#336—42no—
131#336C2×C414yes—
132C25—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8C22×C414yes—
145C22×Q8C2×D414yes—
146C22×Q8#4114yes—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22×C412yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C2×SD16—216no—
170C2×SD16—48no—
171C2×SD16—48no—
172C2×SD16—48no—
173C2×SD16—48no—
174C2×SD16—48no—
175C2×SD16—48no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C22×C414yes—
190C4⋊Q8—216no—
191C4⋊Q8—216no—
192C4×Q8—48no—
193C4×Q8—48no—
194C4×Q8—48no—
195C4×Q8—48no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—28no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42SD1614yescomplemented
274C2×C42C22×C412yes—
275C2×C42C2×D414yes—
276C2×C42#4112yes—
277#103—48no—
278#103—48no—
279#103—48no—
280#102—48no—
281#95—28no—
282#95—28no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—42no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24C22×D411yescharacteristic
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8C4×D418yes—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—41no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
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—22no—
382C22×C4—24no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4C4×D412yes—
414C22×C4C4×D412yes—
415C22×C4#12012yes—
416C22×C4#12012yes—
417C22×C4C2×SD1612yes—
418C22×C4C2×SD1612yes—
419C22×C4C2×SD1612yes—
420C22×C4#13512yes—
421C22×C4AGL(1,ℤ/8)12yes—
422C22×C4#13712yes—
423C22×C4#13712yes—
424C22×C4#13712yes—
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C42−1+411yescharacteristic
435SD16—432no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—84no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C42—28no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—24no—
477C42—28no—
478C42—44no—
479C42—44no—
480C42—416no—
481C42—42no—
482C42—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23C2×C4×D411yescharacteristic
519C23C2×#12011yescharacteristic
520C23#52511yescharacteristic
521C23#53511yescharacteristic
522C23#53511yescharacteristic
523C23#53511yescharacteristic
524C23#53711yescharacteristic
525C23C22×SD1611yescharacteristic
526C23C2×#13511yescharacteristic
527C23C2×AGL(1,ℤ/8)11yescharacteristic
528C23C2×#13711yescharacteristic
529C23C2×#13711yescharacteristic
530C23C2×#13711yescharacteristic
531C23#57511yescharacteristic
532C23#57511yescharacteristic
533Q8—28no—
534Q8—28no—
535Q8—44no—
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—44no—
541D4—44no—
542D4—44no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4C4×SD1614yes—
620C2×C4#44014yes—
621C2×C4#44814yes—
622C2×C4C2×C4×D412yes—
623C2×C4#51811yescharacteristic
624C2×C4#51812yes—
625C2×C4#52011yescharacteristic
626C2×C4#52512yes—
627C2×C4#53412yes—
628C2×C4#53412yes—
629C2×C4#53512yes—
630C2×C4#53712yes—
631C2×C4#53812yes—
632C2×C4#53812yes—
633C2×C4#53812yes—
634C2×C4#53912yes—
635C2×C4#54012yes—
636C2×C4#54011yescharacteristic
637C2×C4#54211yescharacteristic
638C8—44no—
639C8—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—161no—
648C22—161no—
649C22—161no—
650C22—161no—
651C22—161no—
652C22—161no—
653C22—161no—
654C22—161no—
655C22C4×#12011yescharacteristic
656C22#232411yescharacteristic
657C22#232811yescharacteristic
658C22#261311yescharacteristic
659C22#261511yescharacteristic
660C22#263111yescharacteristic
661C22#263611yescharacteristic
662C22C2×C4×SD1611yescharacteristic
663C22C2×#44011yescharacteristic
664C22C4×#13711yescharacteristic
665C22C4×#13711yescharacteristic
666C22C2×#44811yescharacteristic
667C22#300211yescharacteristic
668C22#300411yescharacteristic
669C22#300611yescharacteristic
670C22#310311yescharacteristic
671C22#310811yescharacteristic
672C22#311211yescharacteristic
673C22#311711yescharacteristic
674C22#320611yescharacteristic
675C22#320611yescharacteristic
676C22#320711yescharacteristic
677C22#320811yescharacteristic
678C22#321011yescharacteristic
679C22#321211yescharacteristic
680C22#321411yescharacteristic
681C22#321511yescharacteristic
682C22#321611yescharacteristic
683C22#321811yescharacteristic
684C22#321811yescharacteristic
685C22#321911yescharacteristic
686C22#324611yescharacteristic
687C22#324811yescharacteristic
688C22#325211yescharacteristic
689C22#325411yescharacteristic
690C4—24no—
691C4—24no—
692C4—24no—
693C4—42no—
694C4—48no—
695C4—42no—
696C4—42no—
697C4—84no—
698C4—82no—
699C4—161no—
700C4—161no—
701C4#232112yes—
702C4#232512yes—
703C4#263012yes—
704C4#263512yes—
705C2—162no—
706C2#1429911yescharacteristic
707C2C4×#44811yescharacteristic
708C2#2145411yescharacteristic
709C2#2214611yescharacteristic
710C2#2215211yescharacteristic
711C2#2226811yescharacteristic
712C2#2227311yescharacteristic
713C2#2962411yescharacteristic
714C2#2962811yescharacteristic
715C2#2966111yescharacteristic
716C2#2966811yescharacteristic
717C2#3004111yescharacteristic
718C2#3004311yescharacteristic
719C2#3012911yescharacteristic
720C2#3013011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊Q8C2semidirect
5. #11267C2semidirect
6. #11067C2semidirect
9. #11010C2semidirect
10. #10795C2semidirect
11. #10789C2semidirect
33. #2013C4semidirect
273. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20132
C2×#18802
#147002
#112672
#110672
#110492
#110232
#110102
#107952
#107892