Finite Group Databasebeta

← #415466

Subgroups — #415466

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1847C211yesmaximal, characteristic, complemented
2C2×#1749C211yesmaximal, characteristic
3#14686C211yesmaximal, characteristic
4#14682C211yesmaximal, characteristic
5#11257C212yesmaximal, complemented
6#11256C212yesmaximal, complemented
7#11096C212yesmaximal
8#11095C212yesmaximal
9#10843C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10843C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27#1847—24no—
28#1749—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41#1434—21no—
42#1434—21no—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—42no—
75C2×#107—42no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#106—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8Q818yes—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—22no—
87C2×#97D414yes—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92#382—22no—
93#382—22no—
94#382—22no—
95#382—22no—
96#380—42no—
97#380—42no—
98#380—42no—
99#380—42no—
100#380—44no—
101#380—44no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336—24no—
122C25C2×D411yescharacteristic
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×Q814yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
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×C8C2×D414yes—
170C22×C8C2×Q814yes—
171C22×C8C2×Q814yes—
172C22×C8#4114yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
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⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42C2×D412yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244#107—48no—
245#106—48no—
246C22⋊C8—216no—
247#97—28no—
248#97—44no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—44no—
261#95—42no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24C4⋊Q812yes—
275C24AGL(1,ℤ/8)12yes—
276C24#13712yes—
277C24C22×D411yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12312yes—
379C22×C4#12312yes—
380C22×C4#12312yes—
381C22×C4#12312yes—
382C22×C4C4⋊Q812yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q812yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4H(5,𝔽2)12yes—
394C22×C4H(5,𝔽2)12yes—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8C22⋊Q818yes—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—88no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—44no—
437C42—44no—
438C42#12314yes—
439C42#12414yes—
440C42#12614yes—
441C42#12614yes—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23#50114yes—
477C23C2×#12111yescharacteristic
478C23C2×#12311yescharacteristic
479C23C2×#12311yescharacteristic
480C23C2×C4⋊Q811yescharacteristic
481C23#53211yescharacteristic
482C23#53511yescharacteristic
483C23#53511yescharacteristic
484C23C2×AGL(1,ℤ/8)11yescharacteristic
485C23C2×AGL(1,ℤ/8)11yescharacteristic
486C23C2×#13711yescharacteristic
487C23C2×#13711yescharacteristic
488C23#57511yescharacteristic
489C23#57511yescharacteristic
490C23#57511yescharacteristic
491C23#57511yescharacteristic
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—416no—
514C2×C4—416no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4#39714yes—
574C2×C4#39714yes—
575C2×C4C2×C22⋊Q812yes—
576C2×C4C2×#12311yescharacteristic
577C2×C4C2×#12411yescharacteristic
578C2×C4C2×#12611yescharacteristic
579C2×C4C2×#12611yescharacteristic
580C2×C4#53312yes—
581C2×C4#53512yes—
582C2×C4#54012yes—
583C2×C4#55312yes—
584C2×C4#55312yes—
585C2×C4#55412yes—
586C2×C4#55412yes—
587C2×C4#55412yes—
588C2×C4#55412yes—
589C2×C4#55512yes—
590C2×C4#55512yes—
591C8—28no—
592C8—44no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22C2×#39711yescharacteristic
609C22#250011yescharacteristic
610C22#250011yescharacteristic
611C22#250211yescharacteristic
612C22#250211yescharacteristic
613C22#261911yescharacteristic
614C22#285711yescharacteristic
615C22#305511yescharacteristic
616C22#305511yescharacteristic
617C22#305611yescharacteristic
618C22#305611yescharacteristic
619C22#309211yescharacteristic
620C22#309211yescharacteristic
621C22#309311yescharacteristic
622C22#309311yescharacteristic
623C22#309411yescharacteristic
624C22#309411yescharacteristic
625C22#309511yescharacteristic
626C22#309511yescharacteristic
627C22C2×#50111yescharacteristic
628C22#316211yescharacteristic
629C22#316511yescharacteristic
630C22#316611yescharacteristic
631C22#317111yescharacteristic
632C22#317211yescharacteristic
633C22#318011yescharacteristic
634C22#318111yescharacteristic
635C22#321311yescharacteristic
636C22#321411yescharacteristic
637C22#321511yescharacteristic
638C22#321611yescharacteristic
639C22#321711yescharacteristic
640C22#321811yescharacteristic
641C22#321911yescharacteristic
642C22#321911yescharacteristic
643C4—24no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—161no—
649C4—162no—
650C4—161no—
651C4—161no—
652C4—161no—
653C4—162no—
654C4#248712yes—
655C4#249712yes—
656C4#249712yes—
657C4#256312yes—
658C2—44no—
659C2#1519511yescharacteristic
660C2#2364211yescharacteristic
661C2#2364211yescharacteristic
662C2#2789411yescharacteristic
663C2#2789511yescharacteristic
664C2#2789611yescharacteristic
665C2#2789711yescharacteristic
666C2#2795311yescharacteristic
667C2#2795411yescharacteristic
668C2#2795511yescharacteristic
669C2#2795611yescharacteristic
670C2#2981411yescharacteristic
671C2#2981511yescharacteristic
672C2#3321711yescharacteristic
673C2#3321811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
1. C2×#1847C2semidirect
5. #11257C2semidirect
6. #11256C2semidirect
9. #10843C2semidirect
10. #10843C2semidirect
11. #10843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18472
C2×#17492
#146862
#146822
#112572
#112562
#110962
#110952
#108432