Finite Group Databasebeta

← #380258

Subgroups — #380258

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic, complemented
3C2×#1788C211yesmaximal, characteristic
4C2×#1788C211yesmaximal, characteristic
5#14264C211yesmaximal, characteristic, complemented
6#11124C212yesmaximal, complemented
7#11123C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2311—24no—
25#2311—24no—
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#386—22no—
30C2×#380—21no—
31C2×#380—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34#1852—24no—
35#1788—24no—
36#1788C418yescomplemented
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1442—21no—
44#1439—22no—
45C4⋊#95C2212yes—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41—42no—
50C22×#41—42no—
51C22×#41—42no—
52C22×#41—42no—
53C22×#41C2×C414yes—
54C22×#41D414yes—
55C22×#41Q814yes—
56C23×Q8C2311yescharacteristic
57C23×D4—21no—
58C23×D4C2311yescharacteristic
59C24×C4—21no—
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2312yes—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—42no—
68C2×#117—42no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D414yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C22×C42C2312yes—
87C2×#107—24no—
88C2×#107—42no—
89C2×#107—42no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103C2×C418yescomplemented
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—42no—
100C2×#102—42no—
101C2×#102—42no—
102C2×#102—42no—
103C2×#102C2×C418yescomplemented
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8C2×C414yes—
110#391—24no—
111#391—24no—
112#390—24no—
113#390—24no—
114#389—42no—
115#389—42no—
116#386—216no—
117#380—28no—
118#380—28no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380C2×C414yes—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130#340—24no—
131#340—42no—
132#340—42no—
133#336—42no—
134#336—42no—
135C25—21no—
136C25C2×D411yescharacteristic
137C2×#41—28no—
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—48no—
141C2×#41—48no—
142C2×#41—44no—
143C2×#41—44no—
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×#41C4⋊C4116yescomplemented
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8C22×C412yes—
163C22×Q8C2×D412yes—
164C22×Q8C2×Q812yes—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4C22×C412yes—
178C22×D4C2×D412yes—
179C22×D4C2×Q812yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—82no—
197C23×C4—81no—
198C23×C4—81no—
199C23×C4C22×C412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×Q812yes—
206C23×C4#4112yes—
207C23×C4#4112yes—
208C23×C4C2411yescharacteristic
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8C22×C414yes—
226#117—44no—
227#117—44no—
228#117—48no—
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—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
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×C22⋊C4—82no—
272C2×C22⋊C4—81no—
273C2×C22⋊C4—81no—
274C2×C22⋊C4—82no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281#107—48no—
282#103—48no—
283#103—44no—
284#103—44no—
285#102—48no—
286#102—44no—
287#102—44no—
288C22⋊C8—28no—
289C22⋊C8—44no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—44no—
294#95—42no—
295#95—82no—
296#95—82no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—22no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—42no—
309C24—42no—
310C24—41no—
311C24—42no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—42no—
316C24C4×D412yes—
317C24#12012yes—
318C24C22⋊Q812yes—
319C24C22×D411yescharacteristic
320#41—416no—
321#41—416no—
322#41—432no—
323#41—416no—
324#41—416no—
325#41—432no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8C2×C4⋊C414yes—
344C2×D4—216no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4C2×C4⋊C414yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
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—416no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—416no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
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—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4C2×C4⋊C414yes—
437C22×C4C4×D414yes—
438C22×C4C4×D412yes—
439C22×C4#12012yes—
440C22×C4C22⋊Q814yes—
441C22×C4C22⋊Q812yes—
442C22×C4#12314yes—
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×Q811yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4H(5,𝔽2)12yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—88no—
488C22⋊C4—48no—
489C22⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—84no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—42no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23#38514yes—
543C23C2×C4×D411yescharacteristic
544C23C2×#12011yescharacteristic
545C23C2×C22⋊Q811yescharacteristic
546C23#53412yes—
547C23#53512yes—
548C23#57612yes—
549C23#57611yescharacteristic
550C23#57711yescharacteristic
551C23#57711yescharacteristic
552C23#57812yes—
553C23#57811yescharacteristic
554Q8—24no—
555Q8—24no—
556Q8—44no—
557Q8—42no—
558Q8—42no—
559Q8—44no—
560D4—24no—
561D4—24no—
562D4—432no—
563D4—44no—
564D4—42no—
565D4—42no—
566D4—44no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—432no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—816no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4#38514yes—
623C2×C4C22×C4⋊C411yescharacteristic
624C2×C4C2×C4×D411yescharacteristic
625C2×C4#51812yes—
626C2×C4#51812yes—
627C2×C4C2×C22⋊Q811yescharacteristic
628C2×C4C2×#12311yescharacteristic
629C2×C4#53411yescharacteristic
630C2×C4#53411yescharacteristic
631C2×C4#53411yescharacteristic
632C2×C4#53411yescharacteristic
633C2×C4#53412yes—
634C2×C4#53512yes—
635C2×C4#53812yes—
636C2×C4#53812yes—
637C2×C4#53812yes—
638C2×C4#53812yes—
639C2×C4#54312yes—
640C2×C4#54312yes—
641C8—44no—
642C8—81no—
643C8—81no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22C2×#38511yescharacteristic
667C22#232112yes—
668C22#261311yescharacteristic
669C22#261311yescharacteristic
670C22#262612yes—
671C22#294611yescharacteristic
672C22#294711yescharacteristic
673C22#295112yes—
674C22#300811yescharacteristic
675C22#301111yescharacteristic
676C22#301211yescharacteristic
677C22#301311yescharacteristic
678C22#308312yes—
679C22#308311yescharacteristic
680C22#308411yescharacteristic
681C22#319812yes—
682C22#320112yes—
683C22#320312yes—
684C22#320312yes—
685C22#322012yes—
686C22#322312yes—
687C22#322412yes—
688C22#322512yes—
689C4—24no—
690C4—44no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4—84no—
695C4—161no—
696C4—161no—
697C4#231511yescharacteristic
698C4#231511yescharacteristic
699C4#232112yes—
700C4#262211yescharacteristic
701C4#262211yescharacteristic
702C4#262612yes—
703C2—44no—
704C2—84no—
705C2#1426211yescharacteristic
706C2#2087811yescharacteristic
707C2#2088111yescharacteristic
708C2#2205512yes—
709C2#2205712yes—
710C2#2960711yescharacteristic
711C2#2960811yescharacteristic
712C2#2963011yescharacteristic
713C2#2963311yescharacteristic
714C2#2996112yes—
715C2#2996412yes—
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×#1852C2semidirect
5. #14264C2semidirect
6. #11124C2semidirect
7. #11123C2semidirect
8. #11067C2semidirect
9. #11066C2semidirect
10. #10730C2semidirect
11. #10713C2semidirect
36. #1788C4semidirect
96. C2×#103C2×C4semidirect
103. C2×#102C2×C4semidirect
154. C2×#41C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#18522
C2×#17882
#142642
#111242
#111232
#110672
#110662
#107302
#107132