Finite Group Databasebeta

← #368154

Subgroups — #368154

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic, complemented
4C22×#336C211yesmaximal, characteristic, complemented
5C4×#386C211yesmaximal, characteristic
6#11255C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10877C212yesmaximal, complemented
9#10715C212yesmaximal, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#435—24no—
18C2×#435—24no—
19C2×#435—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#386C411yescharacteristic
28C2×#386C411yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31#1869—24no—
32#1850—24no—
33C2×#336—24no—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1437C412yes—
43#1436C412yes—
44C22⋊#95C411yescharacteristic
45C22⋊#95C411yescharacteristic
46C4×#95C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—42no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—24no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—42no—
65C23×C8—42no—
66C23×C8D414yes—
67C23×C8C2312yes—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2312yes—
72C23×C8C2311yescharacteristic
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4D414yes—
78C2×C4×D4Q814yes—
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C411yescharacteristic
85C22×C22⋊C4C2×C412yes—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—42no—
89C22×C42—42no—
90C22×C42C2×C412yes—
91C22×C42C2312yes—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94#435—232no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—42no—
112C2×C22⋊C8D414yes—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116#386—24no—
117#386—24no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—42no—
121C4×C22⋊C4—42no—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129#336—24no—
130#336—24no—
131#336—24no—
132#336—24no—
133#336—24no—
134#336—42no—
135#336—42no—
136#336—42no—
137C25C22×C411yescharacteristic
138C25C22×C411yescharacteristic
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4C4⋊C414yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4M1614yescomplemented
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C22×C412yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×Q812yes—
181C23×C4C2×Q812yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×M16—28no—
186C2×M16—48no—
187C2×M16—48no—
188C2×M16—44no—
189C2×M16—44no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—28no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—44no—
222C22×C8—48no—
223C22×C8—48no—
224C22×C8—48no—
225C22×C8—48no—
226C22×C8—44no—
227C22×C8—44no—
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230C22×C8C2×D412yes—
231C22×C8C2×D412yes—
232C22×C8#4114yes—
233C22×C8#4114yes—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—28no—
237C4×D4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C4⋊C414yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4C4⋊C414yes—
257C2×C22⋊C4C4⋊C414yes—
258C2×C22⋊C4M1614yescomplemented
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—48no—
281C2×C42C2×D412yes—
282C2×C42C2×Q812yes—
283C4⋊C8—28no—
284C4⋊C8—48no—
285C22⋊C8—28no—
286C22⋊C8—28no—
287C22⋊C8—48no—
288C22⋊C8—44no—
289#97—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—42no—
295#95—42no—
296#95—42no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—21no—
302C24—21no—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24C2×C4⋊C412yes—
312C24C2×C4⋊C412yes—
313C24C4×D412yes—
314C24C4×D412yes—
315C24C2×M1612yes—
316C24C2×M1612yes—
317C24#13112yes—
318C24#13112yes—
319C24C23×C411yescharacteristic
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
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—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C2×C4⋊C412yes—
411C22×C4C2×C4⋊C412yes—
412C22×C4C2×C4⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C4×D412yes—
415C22×C4C4×D412yes—
416C22×C4#12014yes—
417C22×C4C22⋊Q814yes—
418C22×C4C22⋊Q814yes—
419C22×C4#12314yes—
420C22×C4C2×M1614yes—
421C22×C4C2×M1614yes—
422C22×C4#13114yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4H(5,𝔽2)12yes—
433M16—416no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—416no—
449C2×C8—416no—
450C2×C8—42no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
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—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8#12314yes—
470C4⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—48no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—42no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23#38514yes—
530C23#42314yes—
531C23#42414yes—
532C23#43514yes—
533C23#43514yes—
534C23#43614yes—
535C23#43614yes—
536C23C22×C4⋊C411yescharacteristic
537C23C2×C4×D411yescharacteristic
538C23C2×C4×D411yescharacteristic
539C23C4×#4112yes—
540C23#51812yes—
541C23C22×M1611yescharacteristic
542C23C22×M1611yescharacteristic
543C23C2×#13111yescharacteristic
544C23C2×#13111yescharacteristic
545C23C2×#13112yes—
546C23#56812yes—
547D4—416no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4#38514yes—
612C2×C4C2×#12011yescharacteristic
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4C2×C22⋊Q811yescharacteristic
615C2×C4C2×#12311yescharacteristic
616C2×C4C2×#12311yescharacteristic
617C2×C4#52511yescharacteristic
618C2×C4#52912yes—
619C2×C4#53411yescharacteristic
620C2×C4#53511yescharacteristic
621C2×C4#53812yes—
622C2×C4#54612yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4#55312yes—
627C2×C4#55312yes—
628C2×C4#55412yes—
629C2×C4#55412yes—
630C8—24no—
631C8—42no—
632C8—44no—
633C8—42no—
634C8—42no—
635C8—82no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—82no—
657C22#185118yes—
658C22C2×#38511yescharacteristic
659C22C4×#12311yescharacteristic
660C22#232111yescharacteristic
661C22#234912yes—
662C22#236412yes—
663C22C2×#42311yescharacteristic
664C22C2×#42411yescharacteristic
665C22#290612yes—
666C22C2×#43511yescharacteristic
667C22C2×#43511yescharacteristic
668C22C2×#43611yescharacteristic
669C22C2×#43611yescharacteristic
670C22#292911yescharacteristic
671C22#293011yescharacteristic
672C22#293512yes—
673C22#296512yes—
674C22#296612yes—
675C22#296812yes—
676C22#297012yes—
677C22#297312yes—
678C22#297412yes—
679C22#297612yes—
680C22#297712yes—
681C4—24no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4#240411yescharacteristic
690C4#240511yescharacteristic
691C4#244412yes—
692C4#245012yes—
693C4#248711yescharacteristic
694C4#257911yescharacteristic
695C2—28no—
696C2—82no—
697C2#1401611yescharacteristic
698C2C2×#185111yescharacteristic
699C2#2083811yescharacteristic
700C2#2156311yescharacteristic
701C2#2156411yescharacteristic
702C2#2203211yescharacteristic
703C2#2203411yescharacteristic
704C2#2269712yes—
705C2#2269812yes—
706C2#2311712yes—
707C2#2312012yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1850C2semidirect
4. C22×#336C2semidirect
6. #11255C2semidirect
7. #11042C2semidirect
8. #10877C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10712C2semidirect
167. C23×C4M16semidirect
258. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#18692
C2×#18502
C22×#3362
C4×#3862
#112552
#110422
#108772
#107152
#107142
#107122