Finite Group Databasebeta

← #367355

Subgroups — #367355

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1959C211yesmaximal, characteristic
2C2×#1959C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic, complemented
4#14708C211yesmaximal, characteristic
5C4×#384C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10843C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C43C2211yescharacteristic
30#1959—24no—
31#1959C418yescomplemented
32#1836—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1437—21no—
40#1437—21no—
41#1437—21no—
42#1437—21no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124C2×C414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—22no—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103C2×C418yes—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102C2×C418yes—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97C2×C414yes—
96#391—24no—
97#391—42no—
98#390—24no—
99#390—42no—
100#384—22no—
101#384—22no—
102#384C2×C414yes—
103#384D414yes—
104#382—22no—
105#382—22no—
106C4×C4⋊C4—42no—
107C4×C4⋊C4—42no—
108C4×C4⋊C4—44no—
109C4×C4⋊C4—44no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C43—22no—
120C43—22no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—24no—
126#336—24no—
127C25—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C22×C412yes—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C22×C412yes—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C22×C414yes—
173C22×C8#4114yes—
174#124—28no—
175#124—28no—
176#124—44no—
177#124—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C22×C412yes—
251C2×C42C2×D412yes—
252C2×C42#4114yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257#103—48no—
258#103—48no—
259#102—48no—
260#102—48no—
261#97—28no—
262#97—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—44no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24C22×D411yescharacteristic
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4#12112yes—
388C22×C4#12112yes—
389C22×C4#12312yes—
390C22×C4#12312yes—
391C22×C4#12412yes—
392C22×C4#12412yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4#13712yes—
396C22×C4#13712yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C4H(5,𝔽2)12yes—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C42—28no—
444C42—28no—
445C42—24no—
446C42—24no—
447C42—44no—
448C42—44no—
449C42—42no—
450C42—42no—
451C42—88no—
452C42#11714yes—
453C42C4×D414yes—
454C42#12414yes—
455C42#12614yes—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23C2×C4×D411yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12311yescharacteristic
494C23C2×#12411yescharacteristic
495C23#53211yescharacteristic
496C23#53511yescharacteristic
497C23#53511yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×#13711yescharacteristic
501C23C2×#13711yescharacteristic
502C23#57511yescharacteristic
503C23#57511yescharacteristic
504C23#57511yescharacteristic
505C23#57511yescharacteristic
506Q8—44no—
507Q8—44no—
508Q8—82no—
509Q8—82no—
510D4—44no—
511D4—44no—
512D4—82no—
513D4—82no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4#38814yes—
589C2×C4#38814yes—
590C2×C4#48914yes—
591C2×C4C2×#11711yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C4×#4112yes—
594C2×C4#51812yes—
595C2×C4C2×#12411yescharacteristic
596C2×C4C2×#12611yescharacteristic
597C2×C4#52912yes—
598C2×C4#54012yes—
599C2×C4#54612yes—
600C2×C4#55012yes—
601C2×C4#55312yes—
602C2×C4#55312yes—
603C2×C4#55312yes—
604C2×C4#55512yes—
605C2×C4#55912yes—
606C2×C4#55912yes—
607C8—44no—
608C8—44no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—161no—
617C22—161no—
618C22—161no—
619C22—161no—
620C22—161no—
621C22—161no—
622C22—161no—
623C22—161no—
624C22C2×#38811yescharacteristic
625C22C4×#12411yescharacteristic
626C22#232811yescharacteristic
627C22#250011yescharacteristic
628C22#250411yescharacteristic
629C22#254411yescharacteristic
630C22#254811yescharacteristic
631C22C4×AGL(1,ℤ/8)11yescharacteristic
632C22C4×AGL(1,ℤ/8)11yescharacteristic
633C22C4×#13711yescharacteristic
634C22C4×#13711yescharacteristic
635C22#303611yescharacteristic
636C22#303611yescharacteristic
637C22#303711yescharacteristic
638C22#303711yescharacteristic
639C22#309211yescharacteristic
640C22#309211yescharacteristic
641C22#309311yescharacteristic
642C22#309311yescharacteristic
643C22C2×#48911yescharacteristic
644C22#313511yescharacteristic
645C22#314011yescharacteristic
646C22#314111yescharacteristic
647C22#317111yescharacteristic
648C22#317211yescharacteristic
649C22#318011yescharacteristic
650C22#318111yescharacteristic
651C22#321311yescharacteristic
652C22#321411yescharacteristic
653C22#321511yescharacteristic
654C22#321611yescharacteristic
655C22#321711yescharacteristic
656C22#321811yescharacteristic
657C22#321911yescharacteristic
658C22#321911yescharacteristic
659C4—24no—
660C4—24no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—84no—
666C4—82no—
667C4—84no—
668C4—161no—
669C4—161no—
670C4#235712yes—
671C4#236312yes—
672C4#249012yes—
673C4#251312yes—
674C2—162no—
675C2#1400211yescharacteristic
676C2#2105911yescharacteristic
677C2#2105911yescharacteristic
678C2C4×#48911yescharacteristic
679C2#2165511yescharacteristic
680C2#2227211yescharacteristic
681C2#2227311yescharacteristic
682C2#2789411yescharacteristic
683C2#2789511yescharacteristic
684C2#2797811yescharacteristic
685C2#2797911yescharacteristic
686C2#2861811yescharacteristic
687C2#2861911yescharacteristic
688C2#2875311yescharacteristic
689C2#2875411yescharacteristic
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
3. C2×#1836C2semidirect
5. C4×#384C2semidirect
6. #11190C2semidirect
8. #10932C2semidirect
10. #10843C2semidirect
11. #10693C2semidirect
31. #1959C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19592
C2×#18362
#147082
C4×#3842
#111902
#111892
#109322
#109312
#108432
#106932