Finite Group Databasebeta

← #412102

Subgroups — #412102

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#2056C212yesmaximal
3C2×#1956C211yesmaximal, characteristic
4C2×#1939C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic
6#14155C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic
8#11190C212yesmaximal
9#11077C212yesmaximal
10#11024C212yesmaximal
11#10849C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#462—24no—
16C2×#462—24no—
17C2×#462—24no—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2056—28no—
33#1956—24no—
34#1939—24no—
35C4×#103—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1445—22no—
42#1441—22no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8C2312yes—
49C23×Q8C2311yescharacteristic
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—24no—
55C22×Q16—42no—
56C22×Q16—42no—
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8D414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#462—232no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—42no—
93C2×C4⋊C8D414yes—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103—22no—
100C2×#103—24no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—42no—
107C2×#103—42no—
108C2×#103D414yes—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112#391—24no—
113#391—24no—
114#391—24no—
115#391—42no—
116#391—42no—
117#391—42no—
118#384—22no—
119#384—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D412yes—
134#340—24no—
135#340—22no—
136#340—22no—
137#340—42no—
138#336—24no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8C2×D414yes—
164C22×Q8C2×D412yes—
165C22×Q8C2×D412yes—
166C22×Q8#4112yes—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×Q16—216no—
179C2×Q16—48no—
180C2×Q16—48no—
181C2×Q16—48no—
182C2×Q16—48no—
183C2×Q16—48no—
184C2×Q16—48no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8C2×D414yes—
204C22×C8C2×D412yes—
205C22×C8#4112yes—
206C4⋊Q8—28no—
207C4⋊Q8—28no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210C4×Q8—28no—
211C4×Q8—28no—
212C4×Q8—44no—
213C4×Q8—44no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—81no—
254C2×C4⋊C4—81no—
255C2×C4⋊C4Q1614yescomplemented
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4#4112yes—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C2×C42#4112yes—
277C4⋊C8—28no—
278C4⋊C8—48no—
279#103—28no—
280#103—216no—
281#103—28no—
282#103—48no—
283#103—48no—
284C4×C8—44no—
285#95—24no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—82no—
294#95—82no—
295C24C22×D411yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8#12114yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
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—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4#12112yes—
412C22×C4#12112yes—
413C22×C4#12112yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4C2×Q1614yes—
418C22×C4C2×Q1612yes—
419C22×C4#13512yes—
420C22×C4#13712yes—
421C22×C4#13712yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436Q16—432no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—42no—
449C2×C8—41no—
450C2×C8—42no—
451C2×C8—41no—
452C2×C8—42no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—42no—
456C2×C8—41no—
457C2×C8—41no—
458C4⋊C4—28no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—48no—
486C42—84no—
487C42#12014yes—
488C42#12314yes—
489C42#12314yes—
490C42#12414yes—
491C23C2×#12111yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12111yescharacteristic
494C23D4211yescharacteristic
495C23#54611yescharacteristic
496C23#54811yescharacteristic
497C23#55011yescharacteristic
498C23C22×Q1611yescharacteristic
499C23C2×#13511yescharacteristic
500C23C2×#13711yescharacteristic
501C23C2×#13711yescharacteristic
502C23#57711yescharacteristic
503C23#57711yescharacteristic
504C23#57811yescharacteristic
505C23#57811yescharacteristic
506Q8—24no—
507Q8—24no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—44no—
518Q8—82no—
519Q8—82no—
520Q8—82no—
521Q8—82no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4#39414yes—
585C2×C4#39414yes—
586C2×C4#46214yes—
587C2×C4#46214yes—
588C2×C4#46414yes—
589C2×C4#46414yes—
590C2×C4#46714yes—
591C2×C4#47014yes—
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12111yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4C2×#12411yescharacteristic
598C2×C4#52911yescharacteristic
599C2×C4#52911yescharacteristic
600C2×C4#53211yescharacteristic
601C2×C4#53211yescharacteristic
602C2×C4#53511yescharacteristic
603C2×C4#53711yescharacteristic
604C2×C4#54011yescharacteristic
605C2×C4#54011yescharacteristic
606C2×C4#54211yescharacteristic
607C2×C4#54211yescharacteristic
608C2×C4D4211yescharacteristic
609C2×C4D4211yescharacteristic
610C2×C4#54611yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54711yescharacteristic
614C2×C4#54711yescharacteristic
615C2×C4#54811yescharacteristic
616C2×C4#55011yescharacteristic
617C2×C4#55311yescharacteristic
618C2×C4#55311yescharacteristic
619C2×C4#55511yescharacteristic
620C8—42no—
621C8—42no—
622C8—44no—
623C22C2×#39411yescharacteristic
624C22#241411yescharacteristic
625C22#241611yescharacteristic
626C22#242611yescharacteristic
627C22#242711yescharacteristic
628C22#242811yescharacteristic
629C22#242811yescharacteristic
630C22C2×#46211yescharacteristic
631C22C2×#46211yescharacteristic
632C22C2×#46411yescharacteristic
633C22C2×#46411yescharacteristic
634C22#304311yescharacteristic
635C22#304411yescharacteristic
636C22#304611yescharacteristic
637C22#304611yescharacteristic
638C22C2×#46711yescharacteristic
639C22C2×#47011yescharacteristic
640C22#306311yescharacteristic
641C22#306311yescharacteristic
642C22#327611yescharacteristic
643C22#327811yescharacteristic
644C22#328411yescharacteristic
645C22D4×Q1611yescharacteristic
646C22#329911yescharacteristic
647C22#330211yescharacteristic
648C22#332811yescharacteristic
649C22#333111yescharacteristic
650C22#336311yescharacteristic
651C22#336411yescharacteristic
652C22#336911yescharacteristic
653C22#337411yescharacteristic
654C22#339411yescharacteristic
655C22#339511yescharacteristic
656C22#339811yescharacteristic
657C22#340011yescharacteristic
658C4—24no—
659C4—44no—
660C4—44no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—162no—
667C4#205618yescomplemented
668C4#240711yescharacteristic
669C4#240811yescharacteristic
670C4#242011yescharacteristic
671C4#242211yescharacteristic
672C4#249011yescharacteristic
673C4#249111yescharacteristic
674C4#262411yescharacteristic
675C4#285411yescharacteristic
676C2#1512611yescharacteristic
677C2C2×#205611yescharacteristic
678C2#2354511yescharacteristic
679C2#2462911yescharacteristic
680C2#2463211yescharacteristic
681C2#2472911yescharacteristic
682C2#2473211yescharacteristic
683C2#2529411yescharacteristic
684C2#2530211yescharacteristic
685C2#2532311yescharacteristic
686C2#2532411yescharacteristic
687C2#2536911yescharacteristic
688C2#2537011yescharacteristic
689C2#2538611yescharacteristic
690C2#2539411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
255. C2×C4⋊C4Q16semidirect
667. C4#2056semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#20562
C2×#19562
C2×#19392
C2×C4×#1032
#141552
#139262
#111902
#110772
#110242
#108492