Finite Group Databasebeta

← #361778

Subgroups — #361778

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic, complemented
3C2×#1823C211yesmaximal, characteristic, complemented
4C2×#1798C211yesmaximal, characteristic, complemented
5C2×#1788C211yesmaximal, characteristic
6C2×#1788C211yesmaximal, characteristic
7#13777C211yesmaximal, characteristic, complemented
8#11077C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#10932C212yesmaximal, complemented
11#10931C212yesmaximal, complemented
12C23×#41C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2311—24no—
23#2311C418yescomplemented
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#386—22no—
29C2×#380—22no—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33#1824—24no—
34#1823—24no—
35#1798—24no—
36#1788—24no—
37#1788—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1437—22no—
44#1436—22no—
45C4⋊#95C2212yes—
46C22×#41—24no—
47C22×#41—24no—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41—22no—
52C22×#41—22no—
53C22×#41—42no—
54C22×#41—42no—
55C22×#41C2×C414yes—
56C23×Q8C2311yescharacteristic
57C23×D4—21no—
58C23×D4C2311yescharacteristic
59C24×C4—21no—
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C23×C8C2312yes—
63C23×C8C2312yes—
64C2×#117—42no—
65C2×#117—42no—
66C2×#117C2×C414yes—
67C2×#117D414yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C414yes—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42—22no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—24no—
89C2×C4⋊C8—42no—
90C2×C4⋊C8—42no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102—42no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109#391—22no—
110#391—22no—
111#391C2×C414yes—
112#391C2×C414yes—
113#390—22no—
114#390—22no—
115#390C2×C414yes—
116#390C2×C414yes—
117#386—216no—
118#380—22no—
119#380—22no—
120#380—216no—
121#380D414yes—
122#380D414yes—
123C4×C4⋊C4—42no—
124C4×C4⋊C4—42no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137C25—21no—
138C25C2×D411yescharacteristic
139C2×#41—216no—
140C2×#41—28no—
141C2×#41—28no—
142C2×#41—28no—
143C2×#41—28no—
144C2×#41—44no—
145C2×#41—44no—
146C2×#41—48no—
147C2×#41—48no—
148C2×#41—48no—
149C2×#41—48no—
150C2×#41—48no—
151C2×#41—48no—
152C2×#41—48no—
153C2×#41—48no—
154C2×#41—48no—
155C2×#41—44no—
156C2×#41—44no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8C22×C412yes—
166C22×Q8#4114yes—
167C22×D4—28no—
168C22×D4—24no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—44no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4C22×C412yes—
181C22×D4#4114yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—28no—
187C23×C4—21no—
188C23×C4—24no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—82no—
200C23×C4—82no—
201C23×C4C22×C412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4#4112yes—
209C23×C4#4112yes—
210C23×C4C2411yescharacteristic
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225#117—44no—
226#117—44no—
227#117—48no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
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—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C22⋊C418yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4C22⋊C418yes—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—44no—
280C2×C42—82no—
281C2×C42—82no—
282C2×C42C22×C412yes—
283C2×C42C22×C412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C4⋊C8—48no—
287#103—48no—
288#103—48no—
289#102—48no—
290#102—48no—
291C22⋊C8—44no—
292C22⋊C8—44no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—44no—
298#95—82no—
299#95—82no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—41no—
309C24—41no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C2×C22⋊C412yes—
316C24#12112yes—
317C24#12112yes—
318C24C22×D411yescharacteristic
319#41—416no—
320#41—416no—
321#41—416no—
322#41—416no—
323#41—432no—
324#41—432no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8#11714yes—
343C2×D4—216no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—48no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4#11714yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
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—24no—
400C22×C4—24no—
401C22×C4—48no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—48no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—88no—
433C22×C4C2×C22⋊C414yes—
434C22×C4C2×C22⋊C414yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C4×D414yes—
437C22×C4C4×D414yes—
438C22×C4#12112yes—
439C22×C4#12112yes—
440C22×C4#12414yes—
441C22×C4#12714yes—
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—48no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—88no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—81no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—82no—
536C23—82no—
537C23#39014yes—
538C23C22×C22⋊C411yescharacteristic
539C23C2×#12111yescharacteristic
540C23C2×#12111yescharacteristic
541C23#52512yes—
542C23#53412yes—
543C23#57612yes—
544C23#57712yes—
545C23#57712yes—
546C23#57812yes—
547Q8—24no—
548Q8—24no—
549Q8—44no—
550Q8—42no—
551Q8—42no—
552Q8—44no—
553D4—24no—
554D4—24no—
555D4—432no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—44no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—432no—
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—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—816no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4#39014yes—
616C2×C4C2×#11711yescharacteristic
617C2×C4C2×#11711yescharacteristic
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4#51811yescharacteristic
621C2×C4#51811yescharacteristic
622C2×C4#52512yes—
623C2×C4C2×#12411yescharacteristic
624C2×C4C2×#12711yescharacteristic
625C2×C4#53412yes—
626C2×C4#54612yes—
627C2×C4#54612yes—
628C2×C4#54612yes—
629C2×C4#54612yes—
630C2×C4#55112yes—
631C2×C4#55112yes—
632C2×C4#55112yes—
633C2×C4#55112yes—
634C8—82no—
635C8—82no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22#228011yescharacteristic
655C22C2×#39011yescharacteristic
656C22#230811yescharacteristic
657C22#253412yes—
658C22#253412yes—
659C22#289712yes—
660C22#289812yes—
661C22#304111yescharacteristic
662C22#304211yescharacteristic
663C22#304311yescharacteristic
664C22#304411yescharacteristic
665C22#304512yes—
666C22#304612yes—
667C22#311912yes—
668C22#312012yes—
669C22#312412yes—
670C22#312512yes—
671C22#319812yes—
672C22#320112yes—
673C22#320312yes—
674C22#320412yes—
675C4—24no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—84no—
683C4#234211yescharacteristic
684C4#234211yescharacteristic
685C4#234211yescharacteristic
686C4#234211yescharacteristic
687C4#253912yes—
688C4#256112yes—
689C2—44no—
690C2—84no—
691C2#1390311yescharacteristic
692C2#2074612yes—
693C2#2126211yescharacteristic
694C2#2126311yescharacteristic
695C2#2177212yes—
696C2#2812712yes—
697C2#2812912yes—
698C2#2813112yes—
699C2#2813312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1824C2semidirect
3. C2×#1823C2semidirect
4. C2×#1798C2semidirect
7. #13777C2semidirect
8. #11077C2semidirect
9. #11076C2semidirect
10. #10932C2semidirect
11. #10931C2semidirect
23. #2311C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#18242
C2×#18232
C2×#17982
C2×#17882
#137772
#110772
#110762
#109322
#109312