Finite Group Databasebeta

← #360247

Subgroups — #360247

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#416C211yesmaximal, characteristic
2C2×#1927C212yesmaximal, complemented
3C2×#1807C211yesmaximal, characteristic, complemented
4C2×#1788C211yesmaximal, characteristic
5#13937C211yesmaximal, characteristic
6#13801C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11121C212yesmaximal, complemented
9#10931C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#106C2212yes—
16C2×#416—28no—
17C2×#416C418yescomplemented
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#390C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1927—28no—
32#1807—24no—
33#1788—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1438—22no—
40#1437—22no—
41C4⋊#95C2212yes—
42#1434—21no—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C4×#95—21no—
46C22×#41—22no—
47C22×#41—24no—
48C22×#41—24no—
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×C4C2311yescharacteristic
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—24no—
64C23×C8—42no—
65C23×C8C2×C414yes—
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117C2×C414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42C2312yes—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89#416—232no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103C2×C414yes—
95C2×#103C2×C414yes—
96C2×#103D418yes—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102C2×C414yes—
103C2×#102C2×C414yes—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107#391—24no—
108#391—42no—
109#390—24no—
110#390—42no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380D414yes—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—42no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—42no—
131#336—24no—
132#336—42no—
133C25—21no—
134C25C2×D411yescharacteristic
135C2×#41—216no—
136C2×#41—216no—
137C2×#41—28no—
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—28no—
141C2×#41—44no—
142C2×#41—44no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—48no—
147C2×#41—48no—
148C2×#41—48no—
149C2×#41—48no—
150C2×#41—48no—
151C2×#41—44no—
152C2×#41—44no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8C22×C412yes—
162C22×Q8C2×D414yes—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4C22×C412yes—
177C22×D4#4114yes—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C22×C412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4#4112yes—
201C23×C4C2411yescharacteristic
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—28no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—28no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—44no—
219C22×C8—48no—
220C22×C8—48no—
221C22×C8C22×C412yes—
222C22×C8C22×C412yes—
223C22×C8C2×D414yes—
224#117—28no—
225#117—44no—
226#117—48no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
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—22no—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—81no—
261C2×C22⋊C4—81no—
262C2×C22⋊C4—81no—
263C2×C22⋊C4—81no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—82no—
275#106—48no—
276#103—28no—
277#103—48no—
278#103C22⋊C4116yescomplemented
279#102—28no—
280#102—28no—
281#102—48no—
282C22⋊C8—44no—
283#95—24no—
284#95—22no—
285#95—22no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—82no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—24no—
302C24—41no—
303C24—41no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24C4×D412yes—
312C24#12112yes—
313C24#12312yes—
314C24C22×D411yescharacteristic
315#41—232no—
316#41—416no—
317#41—416no—
318#41—416no—
319#41—432no—
320#41—432no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8C2×C22⋊C414yes—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4—48no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4#11714yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4#11714yes—
426C22×C4C4×D414yes—
427C22×C4C4×D412yes—
428C22×C4#12112yes—
429C22×C4#12312yes—
430C22×C4#12414yes—
431C22×C4#12614yes—
432C22×C4#13514yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4112yes—
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—416no—
451C2×C8—42no—
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×C8C2×C22⋊C414yes—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4C2×C22⋊C414yes—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—88no—
478C22⋊C4—84no—
479C22⋊C4—88no—
480C22⋊C4—84no—
481C42—48no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—88no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—82no—
523C23—81no—
524C23—81no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—81no—
531C23—81no—
532C23#38814yes—
533C23C2×C4×D411yescharacteristic
534C23C2×#12111yescharacteristic
535C23C2×#12311yescharacteristic
536C23D4212yes—
537C23#54612yes—
538C23C2×#13512yes—
539C23#57512yes—
540C23#57711yescharacteristic
541C23#57711yescharacteristic
542C23#57711yescharacteristic
543C23#57711yescharacteristic
544Q8—24no—
545Q8—24no—
546Q8—42no—
547Q8—42no—
548Q8—44no—
549Q8—44no—
550D4—24no—
551D4—24no—
552D4—432no—
553D4—42no—
554D4—42no—
555D4—44no—
556D4—44no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—432no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
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—82no—
610C2×C4—82no—
611C2×C4#38814yes—
612C2×C4C22×C22⋊C411yescharacteristic
613C2×C4C2×#11711yescharacteristic
614C2×C4C2×#11711yescharacteristic
615C2×C4C2×C4×D411yescharacteristic
616C2×C4#51811yescharacteristic
617C2×C4#51811yescharacteristic
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4C2×#12411yescharacteristic
621C2×C4C2×#12611yescharacteristic
622C2×C4#53812yes—
623C2×C4#53812yes—
624C2×C4D4212yes—
625C2×C4#54612yes—
626C2×C4#54612yes—
627C2×C4#54612yes—
628C2×C4#55112yes—
629C2×C4#55112yes—
630C8—24no—
631C8—42no—
632C8—82no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22C2×#38811yescharacteristic
656C22D4×C22⋊C411yescharacteristic
657C22#234211yescharacteristic
658C22#241812yes—
659C22#242412yes—
660C22C4×#13511yescharacteristic
661C22#294311yescharacteristic
662C22#295211yescharacteristic
663C22#295211yescharacteristic
664C22#303512yes—
665C22#304211yescharacteristic
666C22#304311yescharacteristic
667C22#309112yes—
668C22#309911yescharacteristic
669C22#309911yescharacteristic
670C22#327112yes—
671C22#327112yes—
672C22#328212yes—
673C22#328412yes—
674C22#329612yes—
675C22#330312yes—
676C22#330712yes—
677C22#331112yes—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—84no—
685C4—162no—
686C4#228011yescharacteristic
687C4#231211yescharacteristic
688C4D4×C22⋊C411yescharacteristic
689C4#234211yescharacteristic
690C4#247212yes—
691C4#248012yes—
692C2—44no—
693C2—84no—
694C2#1387311yescharacteristic
695C2#2105811yescharacteristic
696C2#2106811yescharacteristic
697C2#2244111yescharacteristic
698C2#2244911yescharacteristic
699C2#2256811yescharacteristic
700C2#2257211yescharacteristic
701C2#2486812yes—
702C2#2487612yes—
703C2#2508312yes—
704C2#2509112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1927C2semidirect
3. C2×#1807C2semidirect
6. #13801C2semidirect
7. #11190C2semidirect
8. #11121C2semidirect
9. #10931C2semidirect
10. #10731C2semidirect
11. #10714C2semidirect
17. C2×#416C4semidirect
278. #103C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4162
C2×#19272
C2×#18072
C2×#17882
#139372
#138012
#111902
#111212
#109312
#107312
#107142