Finite Group Databasebeta

← #381842

Subgroups — #381842

Order 512.

1658 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×#1954C211yesmaximal, characteristic, complemented
2C2×#1870C212yesmaximal, complemented
3C2×#1753C211yesmaximal, characteristic, complemented
4C4×#392C211yesmaximal, characteristic
5#11255C211yesmaximal, characteristic, complemented
6#11147C212yesmaximal, complemented
7#11042C211yesmaximal, characteristic, complemented
8#10964C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic, complemented
10#10922C212yesmaximal, complemented
11#10797C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#392C411yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27#1954—24no—
28#1870—28no—
29#1753—24no—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1444C411yescharacteristic
39#1439C411yescharacteristic
40#1439C412yes—
41#1438C412yes—
42#1438C411yescharacteristic
43C4×#95C2211yescharacteristic
44C23×D4C2×C412yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2312yes—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16D414yescomplemented
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C2×C4×D4—42no—
66C2×C4×D4—42no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C412yes—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8—42no—
101C2×#97—22no—
102C2×#97—42no—
103C2×C4×C8—42no—
104C2×C4×C8D414yescomplemented
105#392—24no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—42no—
109C4×C22⋊C4—42no—
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C412yes—
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117#336—24no—
118#336—24no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—42no—
123#336—42no—
124#336—42no—
125C25C22×C411yescharacteristic
126C25C22×C412yes—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4M1614yescomplemented
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C2×M16—28no—
173C2×M16—48no—
174C2×M16—44no—
175C2×M16—44no—
176C2×M16—48no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
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×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8#4112yes—
210C22×C8#4112yes—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—48no—
214C4×D4—48no—
215C4×D4—48no—
216C4×D4—48no—
217C4×D4—48no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C2×D412yes—
257C2×C42#4112yes—
258C4⋊C8—44no—
259C22⋊C8—28no—
260C22⋊C8—48no—
261C22⋊C8—48no—
262C22⋊C8—48no—
263C22⋊C8—48no—
264C22⋊C8—48no—
265#97—44no—
266C4×C8—44no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—24no—
271#95—22no—
272#95—42no—
273#95—42no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—21no—
292C24—22no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C4×D412yes—
299C24C2×M1614yes—
300C24C23×C411yescharacteristic
301C2×D4—28no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—48no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4C2×M1612yes—
403C22×C4#13114yes—
404C22×C4#13112yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415M16—416no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
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—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8#12114yes—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C42—22no—
460C42—22no—
461C42—24no—
462C42—42no—
463C42—41no—
464C42—41no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23C2×C4×D411yescharacteristic
528C23C2×C4×D411yescharacteristic
529C23C4×#4111yescharacteristic
530C23#51812yes—
531C23#51812yes—
532C23C22×M1611yescharacteristic
533C23C2×#13111yescharacteristic
534C23C2×#13112yes—
535C23#56812yes—
536C23#56811yescharacteristic
537C23#56811yescharacteristic
538D4—416no—
539D4—432no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
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#43514yes—
600C2×C4C2×#12111yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4#52911yescharacteristic
603C2×C4#53211yescharacteristic
604C2×C4#53411yescharacteristic
605C2×C4#53412yes—
606C2×C4#53511yescharacteristic
607C2×C4#53511yescharacteristic
608C2×C4#53511yescharacteristic
609C2×C4#53512yes—
610C2×C4#53812yes—
611C2×C4#53812yes—
612C2×C4#53812yes—
613C2×C4#53812yes—
614C2×C4#53812yes—
615C2×C4#54012yes—
616C2×C4#54012yes—
617C2×C4#54012yes—
618C8—24no—
619C8—42no—
620C8—42no—
621C8—42no—
622C8—82no—
623C8—82no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—82no—
650C22C4×#12111yescharacteristic
651C22#232111yescharacteristic
652C22#232112yes—
653C22#232811yescharacteristic
654C22#233312yes—
655C22C2×#43511yescharacteristic
656C22#293011yescharacteristic
657C22#293011yescharacteristic
658C22#293111yescharacteristic
659C22#293312yes—
660C22D4×M1611yescharacteristic
661C22#293511yescharacteristic
662C22#296911yescharacteristic
663C22#296912yes—
664C22#297011yescharacteristic
665C22#297312yes—
666C22#297412yes—
667C22#297612yes—
668C22#297612yes—
669C22#297812yes—
670C22#297812yes—
671C22#297912yes—
672C22#298112yes—
673C4—24no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—82no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4#261711yescharacteristic
682C4#261911yescharacteristic
683C4#263111yescharacteristic
684C4#263511yescharacteristic
685C4#264812yes—
686C4#266112yes—
687C2—84no—
688C2—82no—
689C2#1431111yescharacteristic
690C2#2148611yescharacteristic
691C2#2149411yescharacteristic
692C2#2203311yescharacteristic
693C2#2203411yescharacteristic
694C2#2203612yes—
695C2#2203912yes—
696C2#2226211yescharacteristic
697C2#2226511yescharacteristic
698C2#2233212yes—
699C2#2233512yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
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
1. C2×#1954C2semidirect
2. C2×#1870C2semidirect
3. C2×#1753C2semidirect
5. #11255C2semidirect
6. #11147C2semidirect
7. #11042C2semidirect
8. #10964C2semidirect
9. #10964C2semidirect
10. #10922C2semidirect
11. #10797C2semidirect
52. C22×M16D4semidirect
104. C2×C4×C8D4semidirect
138. C22×D4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18702
C2×#17532
C4×#3922
#112552
#111472
#110422
#109642
#109222
#107972