Finite Group Databasebeta

← #366987

Subgroups — #366987

Order 512.

2182 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#459C211yesmaximal, characteristic
2C2×#1982C211yesmaximal, characteristic, complemented
3C2×#1970C212yesmaximal, complemented
4C2×#1947C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
7#14713C211yesmaximal, characteristic, complemented
8C4×#390C211yesmaximal, characteristic, complemented
9#11123C212yesmaximal, complemented
10#11022C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C23×D8C2212yes—
13C22×#127C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#459—24no—
16C2×#459—24no—
17C2×#459C418yescomplemented
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×C4×C8C2212yes—
26C2×#390—22no—
27C2×#390—22no—
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#390C2212yes—
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—22no—
35C2×C43C2211yescharacteristic
36#1982—24no—
37#1970—28no—
38#1947—24no—
39#1802—24no—
40C4×C4⋊C8—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1437—22no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×D8—24no—
53C22×D8—24no—
54C22×D8—24no—
55C22×D8—24no—
56C22×D8—42no—
57C22×D8—42no—
58C22×D8C2×C418yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—22no—
65C2×#127—22no—
66C2×#127C2×C414yes—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C42—21no—
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#459—232no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—42no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8C2×C414yes—
97C2×C4⋊C8D414yes—
98C2×C4⋊C8D414yes—
99C2×C4⋊C8D414yes—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—42no—
109C2×#102—42no—
110C2×#102—42no—
111C2×#102C2×C418yes—
112C2×C4×C8—24no—
113C2×C4×C8—24no—
114C2×C4×C8—42no—
115#390—22no—
116#390—22no—
117#390—24no—
118#390—22no—
119#390—24no—
120#390—42no—
121#390—42no—
122#390—48no—
123#390—48no—
124#390D414yescomplemented
125#384—22no—
126#384—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C22⋊C4—48no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C43—22no—
134C43—22no—
135#340—24no—
136#340—24no—
137#336—24no—
138C25—21no—
139C25—22no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4C22×C414yes—
175C22×D4C22×C412yes—
176C22×D4C2×D412yes—
177C22×D4#4112yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×D8—216no—
193C2×D8—216no—
194C2×D8—48no—
195C2×D8—48no—
196C2×D8—48no—
197C2×D8—48no—
198C2×D8—48no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8C22×C414yes—
210C22×C8C2×D414yes—
211C22×C8C2×D414yes—
212C22×C8C2×D414yes—
213#127—28no—
214#127—28no—
215#127—44no—
216#127—44no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—44no—
220C4×D4—44no—
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—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4#4112yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—44no—
281C2×C42D818yescomplemented
282C2×C42C22×C412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42#4112yes—
288C2×C42#4112yes—
289C2×C42#4112yes—
290#107—48no—
291C4⋊C8—28no—
292C4⋊C8C22⋊C4116yescomplemented
293#102—216no—
294#102—48no—
295#102—48no—
296#102—48no—
297C4×C8—48no—
298#95—42no—
299#95—44no—
300#95—44no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—44no—
320C24—44no—
321C24—44no—
322C24—44no—
323C24C22×D411yescharacteristic
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—416no—
346C2×D4—41no—
347C2×D4—42no—
348C2×D4—41no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—41no—
379C2×D4—42no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—416no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4C4×D414yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—48no—
418C22×C4—416no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4C4×D412yes—
428C22×C4C4×D412yes—
429C22×C4#12112yes—
430C22×C4#12114yes—
431C22×C4#12112yes—
432C22×C4#12712yes—
433C22×C4#12712yes—
434C22×C4C2×D814yes—
435C22×C4C2×D814yes—
436C22×C4C2×D814yes—
437C22×C4#13514yes—
438C22×C4AGL(1,ℤ/8)14yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C4H(5,𝔽2)11yescharacteristic
452C22×C42−1+411yescharacteristic
453D8—432no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8C2×C22⋊C418yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—816no—
490C22⋊C4—816no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—216no—
496C42—48no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—84no—
502C42C2×C22⋊C414yes—
503C42C4×D414yes—
504C42#12114yes—
505C42#12414yes—
506C23—24no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—44no—
523C23—44no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—84no—
535C23—84no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—82no—
550C23—82no—
551C23—84no—
552C23C2×C4×D411yescharacteristic
553C23C2×#12111yescharacteristic
554C23C2×#12111yescharacteristic
555C23#52511yescharacteristic
556C23C2×#12711yescharacteristic
557C23#53511yescharacteristic
558C23#53711yescharacteristic
559C23C22×D812yes—
560C23C2×#13512yes—
561C23C2×AGL(1,ℤ/8)12yes—
562C23#57512yes—
563D4—24no—
564D4—24no—
565D4—44no—
566D4—42no—
567D4—44no—
568D4—44no—
569D4—42no—
570D4—44no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—816no—
576D4—82no—
577D4—82no—
578D4—816no—
579D4—82no—
580D4—82no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—816no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—82no—
619C2×C4#39014yes—
620C2×C4#39014yes—
621C2×C4C4×D818yes—
622C2×C4#45918yescomplemented
623C2×C4#46618yes—
624C2×C4#49314yes—
625C2×C4#49514yes—
626C2×C4C22×C22⋊C411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C4×#4111yescharacteristic
630C2×C4#51811yescharacteristic
631C2×C4#52011yescharacteristic
632C2×C4C2×#12111yescharacteristic
633C2×C4C2×#12411yescharacteristic
634C2×C4#52911yescharacteristic
635C2×C4#53211yescharacteristic
636C2×C4#54011yescharacteristic
637C2×C4#54211yescharacteristic
638C2×C4D4211yescharacteristic
639C2×C4D4211yescharacteristic
640C2×C4D4211yescharacteristic
641C2×C4#54611yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54711yescharacteristic
645C2×C4#54811yescharacteristic
646C2×C4#54811yescharacteristic
647C2×C4#54811yescharacteristic
648C2×C4#55011yescharacteristic
649C2×C4#55011yescharacteristic
650C2×C4#55011yescharacteristic
651C2×C4#55311yescharacteristic
652C2×C4#55311yescharacteristic
653C2×C4#55511yescharacteristic
654C8—28no—
655C8—44no—
656C22—44no—
657C22—48no—
658C22—82no—
659C22—82no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—88no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—88no—
672C22—162no—
673C22—162no—
674C22—164no—
675C22—164no—
676C22—162no—
677C22—162no—
678C22C2×#39011yescharacteristic
679C22C4×#12111yescharacteristic
680C22#232411yescharacteristic
681C22#241411yescharacteristic
682C22#241611yescharacteristic
683C22#254011yescharacteristic
684C22#254111yescharacteristic
685C22C2×C4×D812yes—
686C22C4×AGL(1,ℤ/8)12yes—
687C22C2×#45912yes—
688C22#303512yes—
689C22C2×#46612yes—
690C22#305512yes—
691C22#310012yes—
692C22#310612yes—
693C22C2×#49311yescharacteristic
694C22C2×#49511yescharacteristic
695C22#315111yescharacteristic
696C22#315311yescharacteristic
697C22#320512yes—
698C22#321212yes—
699C22#324512yes—
700C22#324712yes—
701C4—28no—
702C4—24no—
703C4—44no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—162no—
708C4#197018yescomplemented
709C4D4×C22⋊C411yescharacteristic
710C4#234111yescharacteristic
711C4#236111yescharacteristic
712C4#236311yescharacteristic
713C4#242611yescharacteristic
714C4#242811yescharacteristic
715C4#249011yescharacteristic
716C4#249111yescharacteristic
717C2—84no—
718C2—164no—
719C2#1399811yescharacteristic
720C2C2×#197011yescharacteristic
721C2#2123611yescharacteristic
722C2C4×#45912yes—
723C2#2214512yes—
724C2#2465012yes—
725C2#2475012yes—
726C2#2836911yescharacteristic
727C2#2837111yescharacteristic
728C2#2843711yescharacteristic
729C2#2843911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#459C2
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
2. C2×#1982C2semidirect
3. C2×#1970C2semidirect
4. C2×#1947C2semidirect
5. C2×#1802C2semidirect
6. C2×C4×C4⋊C8C2semidirect
7. #14713C2semidirect
8. C4×#390C2semidirect
9. #11123C2semidirect
10. #11022C2semidirect
11. #10925C2semidirect
17. C2×#459C4semidirect
124. #390D4semidirect
281. C2×C42D8semidirect
292. C4⋊C8C22⋊C4semidirect
622. C2×C4#459semidirect
708. C4#1970semidirect

Maximal subgroups

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

subgroupindex
C22×#4592
C2×#19822
C2×#19702
C2×#19472
C2×#18022
C2×C4×C4⋊C82
#147132
C4×#3902
#111232
#110222
#109252