Finite Group Databasebeta

← #360088

Subgroups — #360088

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#424C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4C2×#1869C211yesmaximal, characteristic
5C2×#1798C211yesmaximal, characteristic, complemented
6C2×#1797C212yesmaximal, complemented
7#11202C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10833C212yesmaximal, complemented
10#10715C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#424—28no—
18C2×#424—24no—
19C22×C4⋊C8C2212yes—
20C22×C4⋊C8C2212yes—
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26#2290—24no—
27#2290—24no—
28C2×#388C412yes—
29C2×#387C412yes—
30C2×#386C412yes—
31C2×#385C412yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×#376C2211yescharacteristic
36#1870—24no—
37#1869—24no—
38#1798—24no—
39#1797—28no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C412yes—
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—42no—
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—42no—
61C23×C8C2312yes—
62C23×C8C2312yes—
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4D414yes—
70C2×C4×D4Q814yes—
71C2×#117—24no—
72C2×#117—24no—
73C2×#117D414yes—
74C2×#117D414yes—
75C22×C4⋊C4C2×C412yes—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C42—24no—
84C22×C42—22no—
85C22×C42—22no—
86C22×C42—24no—
87C22×C42C2×C412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#424—232no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8D418yescomplemented
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—42no—
114C2×C22⋊C8—42no—
115#388—28no—
116#387—28no—
117#386—28no—
118#385—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125#376—22no—
126#376—22no—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132#336—24no—
133#336—24no—
134#336—24no—
135#336—24no—
136#336—24no—
137#336—42no—
138#336—42no—
139C25C22×C411yescharacteristic
140C25C22×C411yescharacteristic
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—24no—
144C22×D4—42no—
145C22×D4C4⋊C414yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×Q812yes—
178C23×C4C2×Q812yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C2×M16—28no—
183C2×M16—48no—
184C2×M16—48no—
185C2×M16—44no—
186C2×M16—44no—
187C22×C8—28no—
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×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—22no—
214C22×C8—48no—
215C22×C8—48no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8C2×D414yes—
219C22×C8C2×D414yes—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4—28no—
224#117—28no—
225#117—216no—
226#117—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22⋊C418yes—
235C2×C4⋊C4C4⋊C414yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4C22⋊C418yes—
249C2×C22⋊C4C4⋊C414yes—
250C2×C22⋊C4C4⋊C414yes—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—44no—
269C2×C42—48no—
270C2×C42—44no—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D414yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×Q812yes—
278C2×C42#4114yes—
279C4⋊C8—216no—
280C4⋊C8—216no—
281C4⋊C8—48no—
282C22⋊C8—48no—
283C22⋊C8—48no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24C2×C22⋊C412yes—
305C24C2×C4⋊C412yes—
306C24C2×C4⋊C412yes—
307C24#11712yes—
308C24#13114yes—
309C24C23×C411yescharacteristic
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—48no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—216no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
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—22no—
352C22×C4—22no—
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—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—48no—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C414yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4C2×C4⋊C412yes—
396C22×C4C2×C4⋊C412yes—
397C22×C4C2×C4⋊C412yes—
398C22×C4#11712yes—
399C22×C4C4×D414yes—
400C22×C4C4×D414yes—
401C22×C4#12114yes—
402C22×C4#12114yes—
403C22×C4C22⋊Q814yes—
404C22×C4C22⋊Q814yes—
405C22×C4#13114yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×Q811yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+412yes—
416M16—416no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—416no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
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—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—216no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C42—22no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—41no—
462C42—42no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—42no—
467C42—48no—
468C42#12014yes—
469C42#12118yes—
470C42#12314yes—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23#38014yes—
512C23#42414yescomplemented
513C23C22×C22⋊C411yescharacteristic
514C23C22×C4⋊C411yescharacteristic
515C23C2×#11711yescharacteristic
516C23C2×C4×D412yes—
517C23#52012yes—
518C23C2×#13112yes—
519C23C2×#13112yes—
520C23#56812yes—
521C23#56812yes—
522D4—416no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—432no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4#38014yes—
583C2×C4#38618yes—
584C2×C4C2×#12011yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12111yescharacteristic
587C2×C4C2×#12112yes—
588C2×C4C2×C22⋊Q811yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4#53511yescharacteristic
592C2×C4#53611yescharacteristic
593C2×C4#53712yes—
594C2×C4D4212yes—
595C2×C4#54612yes—
596C2×C4#54712yes—
597C2×C4#54812yes—
598C2×C4D4×Q812yes—
599C2×C4#55212yes—
600C2×C4#55312yes—
601C2×C4#55412yes—
602C8—44no—
603C8—44no—
604C8—82no—
605C8—82no—
606C22—24no—
607C22—28no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22C2×#38011yescharacteristic
625C22C2×#38611yescharacteristic
626C22#231111yescharacteristic
627C22D4×C4⋊C412yes—
628C22#235212yes—
629C22#288012yes—
630C22#288012yes—
631C22C2×#42411yescharacteristic
632C22#290511yescharacteristic
633C22#290612yes—
634C22#292211yescharacteristic
635C22#292311yescharacteristic
636C22#292412yes—
637C22#292912yes—
638C22#293012yes—
639C22#293312yes—
640C22#293312yes—
641C22#299212yes—
642C22#299312yes—
643C22#299412yes—
644C22#299512yes—
645C4—24no—
646C4—28no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—44no—
652C4—82no—
653C4#240811yescharacteristic
654C4#240911yescharacteristic
655C4#241612yes—
656C4#241712yes—
657C4#248611yescharacteristic
658C4#248711yescharacteristic
659C2—44no—
660C2—44no—
661C2#1387211yescharacteristic
662C2#2062811yescharacteristic
663C2#2063011yescharacteristic
664C2#2089912yes—
665C2#2185512yes—
666C2#2264712yes—
667C2#2264912yes—
668C2#2275212yes—
669C2#2275312yes—
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
1. C22×#424C2semidirect
5. C2×#1798C2semidirect
6. C2×#1797C2semidirect
7. #11202C2semidirect
8. #11042C2semidirect
9. #10833C2semidirect
10. #10715C2semidirect
11. #10714C2semidirect
102. C2×C4⋊C8D4semidirect
512. C23#424semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#22902
C2×#18702
C2×#18692
C2×#17982
C2×#17972
#112022
#110422
#108332
#107152
#107142