Finite Group Databasebeta

← #414649

Subgroups — #414649

Order 512.

2236 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic, complemented
2C22×C4×D8C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6#13906C211yesmaximal, characteristic, complemented
7#11324C212yesmaximal, complemented
8#11123C212yesmaximal, complemented
9#11049C212yesmaximal, complemented
10#11022C212yesmaximal
11#10834C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—22no—
15C22×C4×D4C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#474—24no—
18C2×#474—24no—
19C2×#474—24no—
20C2×C4×D8—24no—
21C2×C4×D8—24no—
22C2×C4×D8—24no—
23C22×#107C2212yes—
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2212yes—
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2212yes—
30C22×C4×C8C2211yescharacteristic
31#2542—24no—
32#2542—24no—
33C2×#399—22no—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×#380—22no—
39C2×C4×C4⋊C4C2211yescharacteristic
40#1990—24no—
41#1802—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1439—22no—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×D4C2311yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C22×D8—22no—
55C22×D8—22no—
56C22×D8—22no—
57C22×D8—22no—
58C22×D8—22no—
59C22×D8—22no—
60C22×D8Q814yescomplemented
61C23×C8C2311yescharacteristic
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8D414yes—
70C2×C22⋊Q8—216no—
71C2×C22⋊Q8—48no—
72C2×C22⋊Q8—48no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4D414yes—
84C2×C4×D4D414yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—24no—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100#474—232no—
101C4×D8—232no—
102C2×#107—22no—
103C2×#107—24no—
104C2×#107—22no—
105C2×#107—22no—
106C2×#107—24no—
107C2×#107—24no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8D414yes—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—24no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—24no—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—24no—
122C2×#102—42no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8D414yes—
126#399—216no—
127#390—24no—
128#390—24no—
129#390—24no—
130#390—42no—
131#384—24no—
132#384—24no—
133#384—22no—
134#384—22no—
135#380—48no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C2×#95—24no—
139C2×#95D412yes—
140#340—24no—
141#340—24no—
142#340—24no—
143#336—24no—
144#336—42no—
145C25—21no—
146C25—21no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C2×D412yes—
175C22×D4C2×D412yes—
176C22×D4C2×Q812yes—
177C22×D4C2×Q812yes—
178C22×D4#4112yes—
179C22×D4#4112yes—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D414yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2411yescharacteristic
198C2×D8—28no—
199C2×D8—28no—
200C2×D8—28no—
201C2×D8—28no—
202C2×D8—28no—
203C2×D8—28no—
204C2×D8—28no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8C2×D412yes—
222C22×C8C2×D414yes—
223C22×C8C2×D412yes—
224C22×C8C2×Q812yes—
225C22×C8#4112yes—
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228C4⋊Q8—44no—
229C4⋊Q8—44no—
230C22⋊Q8—464no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—44no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—44no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4D818yescomplemented
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—84no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42#4112yes—
300#107—28no—
301#107—216no—
302C4⋊C8—28no—
303#102—28no—
304#102—28no—
305#102—216no—
306#102—48no—
307C4×C8—28no—
308#95—24no—
309#95—48no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24C22×D411yescharacteristic
333C2×Q8—48no—
334C2×Q8—416no—
335C2×Q8—88no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4C22⋊Q814yes—
393C2×D4C22⋊Q814yescomplemented
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4#12012yes—
433C22×C4#12012yes—
434C22×C4#12112yes—
435C22×C4#12112yes—
436C22×C4#12712yes—
437C22×C4#12712yes—
438C22×C4C2×D818yes—
439C22×C4C2×D814yes—
440C22×C4AGL(1,ℤ/8)14yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×Q811yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4H(5,𝔽2)11yescharacteristic
452C22×C42−1+411yescharacteristic
453C22×C42−1+411yescharacteristic
454C22×C42−1+411yescharacteristic
455D8—232no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—42no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8C22⋊Q814yescomplemented
477C4⋊C4—216no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—432no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C22⋊C4—432no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—816no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42#12014yes—
507C42#12014yes—
508C42#12114yes—
509C42#12114yes—
510C23—24no—
511C23—24no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23C2×#12011yescharacteristic
557C23C2×#12111yescharacteristic
558C23C2×#12711yescharacteristic
559C23#54611yescharacteristic
560C23#54711yescharacteristic
561C23#54811yescharacteristic
562C23D4×Q811yescharacteristic
563C23C22×D812yes—
564C23C2×AGL(1,ℤ/8)12yes—
565C23#57712yes—
566C23#57812yes—
567Q8—816no—
568D4—24no—
569D4—24no—
570D4—24no—
571D4—24no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—82no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—84no—
610C2×C4—816no—
611C2×C4—816no—
612C2×C4—84no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—84no—
616C2×C4—82no—
617C2×C4#39214yes—
618C2×C4#39214yes—
619C2×C4#44718yes—
620C2×C4#45918yes—
621C2×C4#49314yes—
622C2×C4#49614yes—
623C2×C4C2×#12011yescharacteristic
624C2×C4C2×#12011yescharacteristic
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×#12111yescharacteristic
627C2×C4C2×C22⋊Q811yescharacteristic
628C2×C4C2×C22⋊Q811yescharacteristic
629C2×C4C2×C22⋊Q811yescharacteristic
630C2×C4#52511yescharacteristic
631C2×C4#52511yescharacteristic
632C2×C4#52511yescharacteristic
633C2×C4#53611yescharacteristic
634C2×C4#53611yescharacteristic
635C2×C4#53611yescharacteristic
636C2×C4#53711yescharacteristic
637C2×C4#53711yescharacteristic
638C2×C4#53711yescharacteristic
639C2×C4#54611yescharacteristic
640C2×C4#54711yescharacteristic
641C2×C4#54811yescharacteristic
642C2×C4#54811yescharacteristic
643C2×C4#54811yescharacteristic
644C2×C4D4×Q811yescharacteristic
645C2×C4D4×Q811yescharacteristic
646C2×C4#55011yescharacteristic
647C2×C4#55211yescharacteristic
648C2×C4#55211yescharacteristic
649C2×C4#55411yescharacteristic
650C2×C4#55611yescharacteristic
651C8—24no—
652C8—42no—
653C8—44no—
654C22—48no—
655C22—44no—
656C22—44no—
657C22—48no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#39211yescharacteristic
677C22#240711yescharacteristic
678C22#240911yescharacteristic
679C22#241511yescharacteristic
680C22#241711yescharacteristic
681C22#254011yescharacteristic
682C22#254211yescharacteristic
683C22C2×#44712yes—
684C22#301312yes—
685C22C2×#45912yes—
686C22#304612yes—
687C22C2×#49311yescharacteristic
688C22C2×#49611yescharacteristic
689C22#315411yescharacteristic
690C22#315611yescharacteristic
691C22#329412yes—
692C22#333012yes—
693C22#333412yes—
694C22#335412yes—
695C22#335812yes—
696C22#336712yes—
697C22Q8×D812yes—
698C22#338412yes—
699C4—24no—
700C4—48no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4—162no—
705C4#201118yes—
706C4#240711yescharacteristic
707C4#240911yescharacteristic
708C4#241511yescharacteristic
709C4#241711yescharacteristic
710C4#253611yescharacteristic
711C4#253811yescharacteristic
712C4#261611yescharacteristic
713C4#284811yescharacteristic
714C2—84no—
715C2—84no—
716C2#1517711yescharacteristic
717C2C2×#201111yescharacteristic
718C2#2342711yescharacteristic
719C2#2411612yes—
720C2#2432612yes—
721C2#2467612yes—
722C2#2481112yes—
723C2#2839711yescharacteristic
724C2#2841411yescharacteristic
725C2#2852711yescharacteristic
726C2#2854111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
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
1. C22×#474C2semidirect
3. C2×#2542C2semidirect
4. C2×#1990C2semidirect
5. C2×#1802C2semidirect
6. #13906C2semidirect
7. #11324C2semidirect
8. #11123C2semidirect
9. #11049C2semidirect
11. #10834C2semidirect
60. C22×D8Q8semidirect
272. C2×C4⋊C4D8semidirect
393. C2×D4C22⋊Q8semidirect
476. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C22×C4×D82
C2×#25422
C2×#19902
C2×#18022
#139062
#113242
#111232
#110492
#110222
#108342