Finite Group Databasebeta

← #369341

Subgroups — #369341

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C2×#2367C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1983C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#10835C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11#10703C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#489—24no—
17C2×#489—24no—
18C2×#489C418yescomplemented
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#97C2211yescharacteristic
25#2367—24no—
26#2367—24no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#388—21no—
30C2×#388—21no—
31C2×#385—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36#1988—24no—
37#1983—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1446—21no—
45#1446—21no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4—21no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yescomplemented
62C2×#124—22no—
63C2×#124—22no—
64C2×#124—22no—
65C2×#124C2×C414yes—
66C2×#124D414yes—
67C2×#124D414yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—24no—
71C2×C4×Q8—42no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—24no—
75C2×C4×D4—42no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#489—232no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103C2×C418yescomplemented
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—42no—
110C2×#102C2×C418yes—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97C2×C414yescomplemented
114#391—22no—
115#391—22no—
116#390—22no—
117#390—22no—
118#388—28no—
119#388—28no—
120#385—28no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4D414yes—
127C4×C22⋊C4—28no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95D411yescharacteristic
137C2×#95D411yescharacteristic
138#340—24no—
139#340—24no—
140#340—24no—
141#340—24no—
142#336—42no—
143#336—42no—
144C25—21no—
145C22×Q8—24no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—82no—
156C22×Q8—82no—
157C22×Q8C22×C412yes—
158C22×Q8C2×D412yes—
159C22×Q8C2×D412yes—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C22×C412yes—
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—82no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
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×C8C22×C414yes—
201C4⋊Q8—28no—
202C4⋊Q8—28no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205#124—28no—
206#124—28no—
207#124—28no—
208#124C22⋊C4116yes—
209C4×Q8—48no—
210C4×Q8—44no—
211C4×Q8—44no—
212C4×D4—48no—
213C4×D4—44no—
214C4×D4—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4#4114yes—
245C2×C4⋊C4#4114yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—82no—
259C2×C42—24no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—44no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—82no—
274C2×C42C22×C412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280C4⋊C8—44no—
281#103—216no—
282#103—48no—
283#102—216no—
284#102—48no—
285#97—28no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—42no—
292#95—82no—
293#95—82no—
294#95—82no—
295#95—82no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—42no—
300C24—42no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—48no—
326C2×Q8—88no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8C2×C22⋊C414yes—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4C2×C22⋊C414yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—88no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4#12012yes—
410C22×C4#12012yes—
411C22×C4#12112yes—
412C22×C4#12112yes—
413C22×C4#13712yes—
414C22×C4#13712yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+412yes—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
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—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—48no—
456C4⋊C4#12414yes—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—84no—
462C42—48no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—42no—
468C42—42no—
469C42—84no—
470C42—84no—
471C42C2×C22⋊C414yes—
472C42C4×D414yes—
473C42#12114yes—
474C42#12314yes—
475C23—24no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23C2×C4×D411yescharacteristic
499C23C2×#12011yescharacteristic
500C23C2×#12111yescharacteristic
501C23#54612yes—
502C23#54812yes—
503C23C2×#13711yescharacteristic
504C23C2×#13711yescharacteristic
505C23#57511yescharacteristic
506C23#57511yescharacteristic
507C23#57612yes—
508C23#57712yes—
509Q8—28no—
510Q8—42no—
511Q8—44no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—816no—
516D4—28no—
517D4—42no—
518D4—44no—
519D4—42no—
520D4—42no—
521D4—42no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—816no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#38614yes—
583C2×C4#38614yes—
584C2×C4C22×C22⋊C411yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C4×#4112yes—
587C2×C4#52012yes—
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12311yescharacteristic
590C2×C4C2×#12411yescharacteristic
591C2×C4#52912yes—
592C2×C4#53211yescharacteristic
593C2×C4#53511yescharacteristic
594C2×C4#53611yescharacteristic
595C2×C4#54212yes—
596C2×C4#54612yes—
597C2×C4#54612yes—
598C2×C4#54712yes—
599C2×C4#54712yes—
600C2×C4#55612yes—
601C2×C4#55912yes—
602C8—44no—
603C8—82no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22C2×#38611yescharacteristic
616C22#236712yes—
617C22#242712yes—
618C22#243211yescharacteristic
619C22#243511yescharacteristic
620C22C4×#13711yescharacteristic
621C22C4×#13711yescharacteristic
622C22#294612yes—
623C22#300711yescharacteristic
624C22#300711yescharacteristic
625C22#300811yescharacteristic
626C22#301011yescharacteristic
627C22#303911yescharacteristic
628C22#304011yescharacteristic
629C22#304512yes—
630C22#331412yes—
631C22#331612yes—
632C22#331912yes—
633C22#332112yes—
634C22#336912yes—
635C22#337012yes—
636C22#337512yes—
637C22#337612yes—
638C4—24no—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—81no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—84no—
649C4C4×#12411yescharacteristic
650C4#232611yescharacteristic
651C4#233912yes—
652C4#244712yes—
653C4#254411yescharacteristic
654C4#254711yescharacteristic
655C2—84no—
656C2#1402711yescharacteristic
657C2#2092711yescharacteristic
658C2#2093211yescharacteristic
659C2#2323012yes—
660C2#2323212yes—
661C2#2533512yes—
662C2#2533612yes—
663C2#2566011yescharacteristic
664C2#2566211yescharacteristic
665C2#2589511yescharacteristic
666C2#2589711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1988C2semidirect
5. #14161C2semidirect
6. #11190C2semidirect
8. #10835C2semidirect
9. #10789C2semidirect
11. #10703C2semidirect
18. C2×#489C4semidirect
61. C2×C4⋊Q8C2×C4semidirect
104. C2×#103C2×C4semidirect
113. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×#23672
C2×#19882
C2×#19832
#141612
#111902
#111892
#108352
#107892
#107882
#107032