Finite Group Databasebeta

← #360089

Subgroups — #360089

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#424C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4C2×#1869C211yesmaximal, characteristic
5C2×#1798C212yesmaximal, complemented
6C2×#1798C211yesmaximal, characteristic, complemented
7#11202C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10832C212yesmaximal, complemented
10#10715C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
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#2293—24no—
27#2293—24no—
28C2×#390C412yes—
29C2×#389C412yes—
30C2×#388C412yes—
31C2×#388C412yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1870—24no—
37#1869—24no—
38#1798—28no—
39#1798—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—42no—
55C23×C8—22no—
56C23×C8—24no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—42no—
60C23×C8C2312yes—
61C23×C8C2312yes—
62C23×C8C2312yes—
63C23×C8C2312yes—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4D414yes—
69C2×C4×D4Q814yes—
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—24no—
75C2×#117—24no—
76C2×#117D414yes—
77C2×#117D414yes—
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4C2×C412yes—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C412yes—
83C22×C22⋊C4C2×C412yes—
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2×C411yescharacteristic
88C22×C42C2×C412yes—
89C22×C42C2×C412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#424—232no—
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⋊C8—24no—
103C2×C4⋊C8—24no—
104C2×C4⋊C8D418yescomplemented
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8—42no—
116C2×C22⋊C8—42no—
117#390—28no—
118#389—28no—
119#388—28no—
120#388—28no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C43—22no—
131C43—22no—
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—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×Q812yes—
171C23×C4C2×Q812yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×M16—28no—
176C2×M16—48no—
177C2×M16—48no—
178C2×M16—44no—
179C2×M16—44no—
180C22×C8—28no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
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×C8—22no—
206C22×C8—22no—
207C22×C8—48no—
208C22×C8—48no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8C2×D414yes—
212C22×C8C2×D414yes—
213C4×D4—28no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—28no—
217#117—28no—
218#117—216no—
219#117—28no—
220#117—44no—
221#117—48no—
222#117—44no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22⋊C418yes—
234C2×C4⋊C4C4⋊C414yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
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—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C22⋊C418yes—
251C2×C22⋊C4C4⋊C414yes—
252C2×C22⋊C4C4⋊C414yes—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D414yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×Q812yes—
280C2×C42#4114yes—
281C4⋊C8—216no—
282C4⋊C8—216no—
283C4⋊C8—48no—
284C22⋊C8—48no—
285C22⋊C8—48no—
286C22⋊C8—44no—
287C22⋊C8—44no—
288#95—24no—
289#95—24no—
290#95—24no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24C2×C22⋊C412yes—
306C24C2×C4⋊C412yes—
307C24C2×C4⋊C412yes—
308C24#11712yes—
309C24C23×C411yescharacteristic
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—48no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
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—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×C22⋊C414yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C4⋊C412yes—
387C22×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4#11712yes—
391C22×C4C4×D414yes—
392C22×C4C4×D414yes—
393C22×C4#12114yes—
394C22×C4#12114yes—
395C22×C4C22⋊Q814yes—
396C22×C4C22⋊Q814yes—
397C22×C4#13114yes—
398C22×C4#13114yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+412yes—
409M16—416no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—416no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
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—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—216no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C42—22no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—48no—
457C42—48no—
458C42—41no—
459C42—42no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—42no—
464C42#12014yes—
465C42#12118yes—
466C42#12314yes—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—42no—
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—41no—
508C23#38014yes—
509C23C22×C22⋊C411yescharacteristic
510C23C22×C4⋊C411yescharacteristic
511C23C2×#11711yescharacteristic
512C23C2×C4×D412yes—
513C23#52012yes—
514C23C2×#13112yes—
515C23C2×#13112yes—
516C23#56812yes—
517C23#56812yes—
518D4—416no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
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—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
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—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4#38014yes—
577C2×C4#38618yes—
578C2×C4#42414yescomplemented
579C2×C4C2×#12011yescharacteristic
580C2×C4C2×#12111yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12112yes—
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×#12311yescharacteristic
586C2×C4#53511yescharacteristic
587C2×C4#53611yescharacteristic
588C2×C4#53712yes—
589C2×C4D4212yes—
590C2×C4#54612yes—
591C2×C4#54712yes—
592C2×C4#54812yes—
593C2×C4D4×Q812yes—
594C2×C4#55212yes—
595C2×C4#55312yes—
596C2×C4#55412yes—
597C8—44no—
598C8—44no—
599C8—82no—
600C8—82no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#38011yescharacteristic
620C22C2×#38611yescharacteristic
621C22#231111yescharacteristic
622C22D4×C4⋊C412yes—
623C22#235212yes—
624C22#288012yes—
625C22#288012yes—
626C22C2×#42411yescharacteristic
627C22#290511yescharacteristic
628C22#290612yes—
629C22#292211yescharacteristic
630C22#292311yescharacteristic
631C22#292412yes—
632C22#293112yes—
633C22#293212yes—
634C22#293512yes—
635C22#293512yes—
636C22#299212yes—
637C22#299312yes—
638C22#299412yes—
639C22#299512yes—
640C4—24no—
641C4—24no—
642C4—28no—
643C4—42no—
644C4—44no—
645C4—42no—
646C4—44no—
647C4—82no—
648C4#240811yescharacteristic
649C4#240911yescharacteristic
650C4#241612yes—
651C4#241712yes—
652C4#248611yescharacteristic
653C4#248711yescharacteristic
654C2—44no—
655C2—82no—
656C2#1387211yescharacteristic
657C2#2062811yescharacteristic
658C2#2063011yescharacteristic
659C2#2090012yes—
660C2#2185512yes—
661C2#2265112yes—
662C2#2265212yes—
663C2#2275612yes—
664C2#2275712yes—
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×#1798C2semidirect
7. #11202C2semidirect
8. #11042C2semidirect
9. #10832C2semidirect
10. #10715C2semidirect
11. #10714C2semidirect
104. C2×C4⋊C8D4semidirect
578. C2×C4#424semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#22932
C2×#18702
C2×#18692
C2×#17982
#112022
#110422
#108322
#107152
#107142