Finite Group Databasebeta

← #358192

Subgroups — #358192

Order 512.

2160 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic, complemented
3C2×C4×#102C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic, complemented
5C4×#390C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#11022C212yesmaximal
10C4×#340C211yesmaximal, characteristic, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117—21no—
16C2×C4×D8—24no—
17C2×C4×D8—24no—
18C2×C4×D8C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2293—24no—
27#2293—24no—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#388—22no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4—21no—
37C2×C43C2211yescharacteristic
38C4×#102—24no—
39C4×#102—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1437—22no—
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C22×D8—22no—
53C22×D8—22no—
54C22×D8—24no—
55C22×D8—24no—
56C22×D8C2×C414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4C2×C414yes—
70C2×C4×D4D414yes—
71C2×C4×D4D414yes—
72C2×#117—28no—
73C2×#117—44no—
74C2×#117—44no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42D411yescharacteristic
89C22×C42D412yes—
90C22×C42D411yescharacteristic
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93C4×D8—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107C2×C414yes—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102C2×C414yes—
108C2×#102C2×C414yes—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8C2×C414yes—
115#390—24no—
116#390—24no—
117#390—24no—
118#390—24no—
119#389—24no—
120#389—24no—
121#388—216no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4D414yes—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—44no—
128C4×C22⋊C4—44no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C43—22no—
135C43—22no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#336—24no—
141#336—24no—
142C25—21no—
143C25—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4C22×C412yes—
160C22×D4C22×C412yes—
161C22×D4C2×D412yes—
162C22×D4C2×D412yes—
163C22×D4#4114yes—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C2×D8—216no—
188C2×D8—216no—
189C2×D8—28no—
190C2×D8—28no—
191C2×D8—28no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8C22×C412yes—
206C22×C8C22×C412yes—
207C22×C8#4114yes—
208C22×C8#4114yes—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4C22⋊C4116yescomplemented
217#117—432no—
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—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4#4114yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42D814yescomplemented
284C2×C42C22×C412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42#4112yes—
289C2×C42#4112yes—
290#107—28no—
291#102—216no—
292#102—216no—
293#102—28no—
294#102—28no—
295C4×C8—28no—
296C4×C8—44no—
297#95—28no—
298#95—24no—
299#95—44no—
300#95—42no—
301#95—44no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—41no—
308C24—41no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—41no—
313C24—41no—
314C24—42no—
315C24—42no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—42no—
323C24—42no—
324C24C22×D411yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C2×C22⋊C414yes—
360C2×D4#11714yescomplemented
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
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—22no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4C4×D412yes—
418C22×C4C4×D412yes—
419C22×C4#12012yes—
420C22×C4#12012yes—
421C22×C4#12112yes—
422C22×C4#12112yes—
423C22×C4C2×D814yes—
424C22×C4C2×D812yes—
425C22×C4#13514yes—
426C22×C4#13512yes—
427C22×C4AGL(1,ℤ/8)12yes—
428C22×C4AGL(1,ℤ/8)12yes—
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4111yescharacteristic
439D8—232no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8#11714yescomplemented
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—416no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4C2×C22⋊C414yes—
474C22⋊C4—216no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C42—28no—
481C42—48no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—48no—
487C42—44no—
488C42—84no—
489C42C2×C22⋊C414yes—
490C42C4×D414yes—
491C42#12114yes—
492C42#12314yes—
493C23—24no—
494C23—24no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—81no—
534C23—81no—
535C23—82no—
536C23—81no—
537C23—81no—
538C23—82no—
539C23C2×C4×D411yescharacteristic
540C23C2×#12011yescharacteristic
541C23C2×#12111yescharacteristic
542C23#52512yes—
543C23#53212yes—
544C23C22×D811yescharacteristic
545C23C2×#13511yescharacteristic
546C23C2×#13512yes—
547C23C2×AGL(1,ℤ/8)11yescharacteristic
548C23C2×AGL(1,ℤ/8)11yescharacteristic
549C23#57512yes—
550D4—24no—
551D4—24no—
552D4—24no—
553D4—24no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—44no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—432no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—816no—
610C2×C4#38614yes—
611C2×C4#38614yes—
612C2×C4C4×D814yes—
613C2×C4#44214yes—
614C2×C4#44714yes—
615C2×C4#44914yes—
616C2×C4#45914yes—
617C2×C4#46314yes—
618C2×C4C22×C22⋊C411yescharacteristic
619C2×C4C2×#11711yescharacteristic
620C2×C4C2×#11711yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C4×#4111yescharacteristic
623C2×C4C4×#4111yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4#52512yes—
627C2×C4#52512yes—
628C2×C4#52512yes—
629C2×C4#52512yes—
630C2×C4#52912yes—
631C2×C4#52912yes—
632C2×C4#52912yes—
633C2×C4#52912yes—
634C2×C4#53212yes—
635C2×C4#53212yes—
636C8—24no—
637C8—44no—
638C8—42no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22—84no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22—82no—
661C22C2×#38611yescharacteristic
662C22#229011yescharacteristic
663C22#229311yescharacteristic
664C22#238812yes—
665C22#239212yes—
666C22C2×C4×D811yescharacteristic
667C22C4×#13511yescharacteristic
668C22C2×#44211yescharacteristic
669C22#294311yescharacteristic
670C22C2×#44711yescharacteristic
671C22C2×#44911yescharacteristic
672C22#300212yes—
673C22C2×#45911yescharacteristic
674C22C2×#46311yescharacteristic
675C22#303512yes—
676C22#310012yes—
677C22#310212yes—
678C22#310512yes—
679C22#310612yes—
680C22#316812yes—
681C22#316912yes—
682C22#317112yes—
683C22#317112yes—
684C4—24no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—84no—
689C4—84no—
690C4—82no—
691C4#187918yes—
692C4#228011yescharacteristic
693C4#229011yescharacteristic
694C4#229311yescharacteristic
695C4#229711yescharacteristic
696C4#239212yes—
697C4#239512yes—
698C2—84no—
699C2—84no—
700C2#1383911yescharacteristic
701C2C2×#187911yescharacteristic
702C2#2092411yescharacteristic
703C2#2111311yescharacteristic
704C2#2111811yescharacteristic
705C2#2127511yescharacteristic
706C2#2127711yescharacteristic
707C2#2356712yes—
708C2#2356912yes—
709C2#2367612yes—
710C2#2368012yes—
subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2293C2semidirect
3. C2×C4×#102C2semidirect
4. C2×C4×#102C2semidirect
5. C4×#390C2semidirect
6. #11219C2semidirect
7. #11189C2semidirect
8. #11066C2semidirect
10. C4×#340C2semidirect
11. C4×#336C2semidirect
18. C2×C4×D8C4semidirect
216. C4×D4C22⋊C4semidirect
283. C2×C42D8semidirect
360. C2×D4#117semidirect
456. C2×C8#117semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#22932
C2×C4×#1022
C4×#3902
#112192
#111892
#110662
#110222
C4×#3402
C4×#3362