Finite Group Databasebeta

← #414873

Subgroups — #414873

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1939C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic, complemented
6#13906C211yesmaximal, characteristic
7#11257C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11189C212yesmaximal
10#11076C212yesmaximal
11#10849C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#476—24no—
15C2×#476—24no—
16C2×#476—24no—
17C22×#106C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#390C2212yes—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1990—24no—
31#1939—24no—
32C4×#102—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1446—22no—
41#1439—22no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#476—232no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—42no—
90C2×C4⋊C8D414yes—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—24no—
94C2×#102—22no—
95C2×#102—24no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—22no—
99C2×#102Q814yescomplemented
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103#390—24no—
104#390—42no—
105#384—22no—
106#384—22no—
107#380—44no—
108#380—44no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D412yes—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—42no—
127#336—24no—
128C25—21no—
129C22×Q8—24no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—81no—
133C22×Q8—81no—
134C22×Q8—81no—
135C22×Q8—81no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C2×D412yes—
148C22×D4C2×Q812yes—
149C22×D4#4112yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C2×D414yes—
186C22×C8C2×Q812yes—
187C22×C8#4112yes—
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8—44no—
191C4⋊Q8—44no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—44no—
195C4×D4—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4SD1614yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×Q812yes—
240C2×C4⋊C4#4112yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265#106—216no—
266C4⋊C8—28no—
267C4⋊C8—48no—
268#102—28no—
269#102—216no—
270#102—28no—
271C4×C8—44no—
272#95—24no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—82no—
277#95—82no—
278#95—82no—
279#95—82no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24C22×D411yescharacteristic
296C2×Q8—48no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4C22⋊Q814yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
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—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4#12112yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C2×SD1614yes—
397C22×C4C2×SD1612yes—
398C22×C4#13512yes—
399C22×C4AGL(1,ℤ/8)12yes—
400C22×C4#13712yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×Q811yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413C22×C42−1+411yescharacteristic
414C22×C42−1+411yescharacteristic
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C4⋊C4—28no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C22⋊C4—48no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—84no—
469C42#12014yes—
470C42#12314yes—
471C42#12314yes—
472C42#12414yes—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
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—
508C23C2×#12111yescharacteristic
509C23C2×#12111yescharacteristic
510C23C2×#12111yescharacteristic
511C23#54611yescharacteristic
512C23#54711yescharacteristic
513C23#54811yescharacteristic
514C23D4×Q811yescharacteristic
515C23C22×SD1611yescharacteristic
516C23C2×#13511yescharacteristic
517C23C2×AGL(1,ℤ/8)11yescharacteristic
518C23C2×#13711yescharacteristic
519C23#57611yescharacteristic
520C23#57611yescharacteristic
521C23#57711yescharacteristic
522C23#57711yescharacteristic
523Q8—816no—
524D4—24no—
525D4—24no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—816no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4#39414yes—
589C2×C4#39414yes—
590C2×C4#46014yes—
591C2×C4#46114yes—
592C2×C4#46314yes—
593C2×C4#46414yes—
594C2×C4#46514yes—
595C2×C4#46814yes—
596C2×C4C2×#12011yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×#12311yescharacteristic
599C2×C4C2×#12311yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4C2×#12411yescharacteristic
602C2×C4#52911yescharacteristic
603C2×C4#52911yescharacteristic
604C2×C4#53311yescharacteristic
605C2×C4#53311yescharacteristic
606C2×C4#53611yescharacteristic
607C2×C4#53711yescharacteristic
608C2×C4#54111yescharacteristic
609C2×C4#54111yescharacteristic
610C2×C4#54211yescharacteristic
611C2×C4#54211yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4#54711yescharacteristic
614C2×C4#54811yescharacteristic
615C2×C4#54811yescharacteristic
616C2×C4#54811yescharacteristic
617C2×C4D4×Q811yescharacteristic
618C2×C4D4×Q811yescharacteristic
619C2×C4#55011yescharacteristic
620C2×C4#55211yescharacteristic
621C2×C4#55211yescharacteristic
622C2×C4#55411yescharacteristic
623C2×C4#55611yescharacteristic
624C8—44no—
625C8—42no—
626C8—42no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C2×#39411yescharacteristic
643C22#241511yescharacteristic
644C22#241711yescharacteristic
645C22#242611yescharacteristic
646C22#242711yescharacteristic
647C22#242911yescharacteristic
648C22#242911yescharacteristic
649C22C2×#46111yescharacteristic
650C22C2×#46311yescharacteristic
651C22C2×#46011yescharacteristic
652C22C2×#46411yescharacteristic
653C22#304111yescharacteristic
654C22#304211yescharacteristic
655C22#304511yescharacteristic
656C22#304511yescharacteristic
657C22C2×#46511yescharacteristic
658C22C2×#46811yescharacteristic
659C22#306211yescharacteristic
660C22#306211yescharacteristic
661C22#329511yescharacteristic
662C22#330311yescharacteristic
663C22#332511yescharacteristic
664C22#333111yescharacteristic
665C22#333511yescharacteristic
666C22#334111yescharacteristic
667C22#335411yescharacteristic
668C22#335711yescharacteristic
669C22#336011yescharacteristic
670C22#336211yescharacteristic
671C22#336611yescharacteristic
672C22#336911yescharacteristic
673C22Q8×SD1611yescharacteristic
674C22#338011yescharacteristic
675C22#338411yescharacteristic
676C22#338511yescharacteristic
677C4—24no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4—162no—
685C4#205518yes—
686C4#240711yescharacteristic
687C4#240911yescharacteristic
688C4#242111yescharacteristic
689C4#242311yescharacteristic
690C4#248911yescharacteristic
691C4#249211yescharacteristic
692C4#262511yescharacteristic
693C4#285511yescharacteristic
694C2—84no—
695C2#1517911yescharacteristic
696C2C2×#205511yescharacteristic
697C2#2354411yescharacteristic
698C2#2467711yescharacteristic
699C2#2468011yescharacteristic
700C2#2481211yescharacteristic
701C2#2481511yescharacteristic
702C2#2527311yescharacteristic
703C2#2527411yescharacteristic
704C2#2533911yescharacteristic
705C2#2534611yescharacteristic
706C2#2545011yescharacteristic
707C2#2545111yescharacteristic
708C2#2547011yescharacteristic
709C2#2547711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#1990C2semidirect
3. C2×#1939C2semidirect
5. #14161C2semidirect
7. #11257C2semidirect
8. #11231C2semidirect
11. #10849C2semidirect
99. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#19902
C2×#19392
C2×C4×#1022
#141612
#139062
#112572
#112312
#111892
#110762
#108492