Finite Group Databasebeta

← #368190

Subgroups — #368190

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2C2×#1875C211yesmaximal, characteristic
3C2×#1874C211yesmaximal, characteristic
4C2×#1852C211yesmaximal, characteristic, complemented
5#13795C211yesmaximal, characteristic, complemented
6#11293C212yesmaximal, complemented
7#11123C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#107C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2324—24no—
23#2324—24no—
24C2×#390—21no—
25C2×#390C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#386—21no—
29C2×#385—21no—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1875C418yescomplemented
33#1874—24no—
34#1852—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C22⋊#95—21no—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121C2×C414yes—
59C2×#121D414yes—
60C2×#121Q814yes—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—24no—
64C2×C4×D4—42no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—24no—
69C2×#117—42no—
70C2×#117—42no—
71C2×#117—42no—
72C2×#117C2×C414yescomplemented
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C2×#107—24no—
99C2×#107—24no—
100C2×#107—42no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—42no—
104C2×#102—42no—
105C2×#102—42no—
106C2×#102C2×C418yescomplemented
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8C2×C414yescomplemented
113#390—28no—
114#390—22no—
115#390—22no—
116#389—22no—
117#389—22no—
118#388—28no—
119#386—28no—
120#385—28no—
121#380—22no—
122#380D414yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132#340—24no—
133#340—24no—
134#340—42no—
135#340—42no—
136#340C2×C418yescomplemented
137#336—24no—
138#336—42no—
139C25—21no—
140C25C2×D411yescharacteristic
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—82no—
152C22×D4—82no—
153C22×D4C22×C412yes—
154C22×D4C2×D412yes—
155C22×D4C2×Q812yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—82no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C22×C412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×Q812yes—
184C23×C4#4112yes—
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C22×C414yes—
201#121—28no—
202#121—28no—
203#121—28no—
204#121C4⋊C4116yes—
205C4×D4—48no—
206C4×D4—48no—
207#117—48no—
208#117—48no—
209#117—48no—
210#117—48no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×Q812yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—82no—
283C2×C42#4114yes—
284#107—48no—
285#102—48no—
286#102—48no—
287C22⋊C8—28no—
288C22⋊C8—44no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—82no—
293#95—82no—
294#95—82no—
295#95—82no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C4×D412yes—
316C24#12012yes—
317C24C22⋊Q812yes—
318C24C22×D411yescharacteristic
319C2×D4—216no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—88no—
336C2×D4—88no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4C2×C4⋊C414yes—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—88no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—88no—
413C22×C4C2×C4⋊C414yes—
414C22×C4C4×D414yes—
415C22×C4C4×D412yes—
416C22×C4#12012yes—
417C22×C4C22⋊Q814yes—
418C22×C4C22⋊Q812yes—
419C22×C4#12312yes—
420C22×C4#12312yes—
421C22×C4#12314yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—84no—
465C4⋊C4C2×C4⋊C414yes—
466C22⋊C4—216no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C42—24no—
475C42—24no—
476C42—48no—
477C42—42no—
478C42—42no—
479C42—84no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23#38514yes—
526C23C2×C4×D411yescharacteristic
527C23C2×#12011yescharacteristic
528C23C2×C22⋊Q811yescharacteristic
529C23C2×#12311yescharacteristic
530C23#52511yescharacteristic
531C23#53411yescharacteristic
532C23#53511yescharacteristic
533C23#57612yes—
534C23#57612yes—
535C23#57712yes—
536C23#57812yes—
537D4—28no—
538D4—44no—
539D4—44no—
540D4—44no—
541D4—816no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—816no—
590C2×C4—816no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4#38514yes—
604C2×C4C22×C4⋊C411yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C4×#4112yes—
607C2×C4#51812yes—
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4C2×#12311yescharacteristic
610C2×C4#52912yes—
611C2×C4#53812yes—
612C2×C4#54612yes—
613C2×C4#54612yes—
614C2×C4#54612yes—
615C2×C4#54612yes—
616C2×C4#55312yes—
617C2×C4#55312yes—
618C2×C4#55412yes—
619C2×C4#55412yes—
620C8—44no—
621C8—82no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C2×#38511yescharacteristic
645C22C4×#12311yescharacteristic
646C22#232111yescharacteristic
647C22#240411yescharacteristic
648C22#240511yescharacteristic
649C22#248711yescharacteristic
650C22#257911yescharacteristic
651C22#294612yes—
652C22#295112yes—
653C22#300811yescharacteristic
654C22#300811yescharacteristic
655C22#300911yescharacteristic
656C22#301111yescharacteristic
657C22#308112yes—
658C22#308312yes—
659C22#309612yes—
660C22#309812yes—
661C22#312112yes—
662C22#312212yes—
663C22#319812yes—
664C22#320312yes—
665C22#322012yes—
666C22#322512yes—
667C4—24no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—84no—
672C4—82no—
673C4—84no—
674C4—81no—
675C4—81no—
676C4—82no—
677C4#234912yes—
678C4#236412yes—
679C4#244412yes—
680C4#245012yes—
681C2—44no—
682C2—84no—
683C2#1401611yescharacteristic
684C2#2087611yescharacteristic
685C2#2088111yescharacteristic
686C2#2159012yes—
687C2#2205512yes—
688C2#2393911yescharacteristic
689C2#2394611yescharacteristic
690C2#2401111yescharacteristic
691C2#2401311yescharacteristic
692C2#2768712yes—
693C2#2924612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
4. C2×#1852C2semidirect
5. #13795C2semidirect
6. #11293C2semidirect
7. #11123C2semidirect
8. #10907C2semidirect
9. #10788C2semidirect
10. #10731C2semidirect
11. #10713C2semidirect
32. #1875C4semidirect
72. C2×#117C2×C4semidirect
106. C2×#102C2×C4semidirect
112. C2×C22⋊C8C2×C4semidirect
136. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
C2×#18752
C2×#18742
C2×#18522
#137952
#112932
#111232
#109072
#107882
#107312
#107132