Finite Group Databasebeta

← #369225

Subgroups — #369225

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#490C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic, complemented
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1982C211yesmaximal, characteristic
5#14154C211yesmaximal, characteristic
6#11066C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10931C212yesmaximal, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#490—24no—
15C2×#490—24no—
16C2×#490C418yescomplemented
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#97C2211yescharacteristic
23#2370—24no—
24#2370—24no—
25C2×#390—21no—
26C2×#390—21no—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389—21no—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387—21no—
33C2×#387—21no—
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1989—24no—
41#1982—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48#1438—21no—
49#1438—21no—
50C23×D4—21no—
51C23×D4—21no—
52C23×D4C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127C2×C414yes—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×#125C2×C414yes—
65C2×#125D414yes—
66C2×#125D414yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#490—232no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102—42no—
103C2×#102C2×C418yes—
104C2×#102C2×C418yes—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97C2×C414yes—
108#390—24no—
109#390—24no—
110#390—42no—
111#390—42no—
112#390—44no—
113#390—44no—
114#389—28no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#387—28no—
120#387—28no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4D414yes—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95D411yescharacteristic
139C2×#95D411yescharacteristic
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#336—42no—
145#336—42no—
146C25—22no—
147C22×D4—24no—
148C22×D4—24no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—82no—
171C22×D4—82no—
172C22×D4C22×C414yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
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×C8C22×C414yes—
199#127—28no—
200#127—28no—
201#127—44no—
202#127—44no—
203#125—28no—
204#125—28no—
205#125—28no—
206#125C22⋊C4116yescomplemented
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
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—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4#4114yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4—82no—
262C2×C42—24no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—82no—
279C2×C42—82no—
280C2×C42C22×C412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42#4112yes—
285C2×C42#4112yes—
286C4⋊C8—44no—
287#102—216no—
288#102—216no—
289#102—48no—
290#102—48no—
291#97—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—42no—
300C24—24no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C22×D411yescharacteristic
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—88no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—88no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4#12012yes—
412C22×C4#12012yes—
413C22×C4#12112yes—
414C22×C4#12112yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4AGL(1,ℤ/8)12yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
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—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—84no—
462C4⋊C4C2×C22⋊C414yes—
463C4⋊C4C2×C22⋊C414yes—
464C4⋊C4#12414yes—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—48no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—42no—
475C42—42no—
476C42—84no—
477C42—84no—
478C42—84no—
479C42C2×C22⋊C414yes—
480C42C4×D414yes—
481C42#12114yes—
482C42#12314yes—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23C2×C4×D411yescharacteristic
519C23C2×#12011yescharacteristic
520C23C2×#12111yescharacteristic
521C23#54612yes—
522C23#54812yes—
523C23C2×AGL(1,ℤ/8)11yescharacteristic
524C23C2×AGL(1,ℤ/8)11yescharacteristic
525C23#57511yescharacteristic
526C23#57511yescharacteristic
527C23#57612yes—
528C23#57712yes—
529D4—44no—
530D4—44no—
531D4—44no—
532D4—44no—
533D4—816no—
534D4—816no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—82no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4#38614yes—
599C2×C4#38614yes—
600C2×C4C22×C22⋊C411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C4×#4112yes—
603C2×C4#52012yes—
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12311yescharacteristic
606C2×C4C2×#12411yescharacteristic
607C2×C4#52912yes—
608C2×C4#53211yescharacteristic
609C2×C4#53511yescharacteristic
610C2×C4#53611yescharacteristic
611C2×C4#54212yes—
612C2×C4#54812yes—
613C2×C4#54812yes—
614C2×C4#55012yes—
615C2×C4#55012yes—
616C2×C4#55612yes—
617C2×C4#55912yes—
618C8—44no—
619C8—82no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—162no—
628C22—162no—
629C22—162no—
630C22—162no—
631C22—162no—
632C22—162no—
633C22—162no—
634C22—162no—
635C22C2×#38611yescharacteristic
636C22#236712yes—
637C22#242612yes—
638C22#243111yescharacteristic
639C22#243611yescharacteristic
640C22C4×AGL(1,ℤ/8)11yescharacteristic
641C22C4×AGL(1,ℤ/8)11yescharacteristic
642C22#294612yes—
643C22#300611yescharacteristic
644C22#300611yescharacteristic
645C22#300811yescharacteristic
646C22#301011yescharacteristic
647C22#303811yescharacteristic
648C22#304011yescharacteristic
649C22#304512yes—
650C22#330912yes—
651C22#331012yes—
652C22#331712yes—
653C22#331812yes—
654C22#336612yes—
655C22#336812yes—
656C22#337012yes—
657C22#337112yes—
658C4—24no—
659C4—24no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—84no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4C4×#12411yescharacteristic
670C4#232611yescharacteristic
671C4#234112yes—
672C4#244812yes—
673C4#254511yescharacteristic
674C4#254811yescharacteristic
675C2—164no—
676C2#1402611yescharacteristic
677C2#2092811yescharacteristic
678C2#2093211yescharacteristic
679C2#2322912yes—
680C2#2323112yes—
681C2#2528812yes—
682C2#2529112yes—
683C2#2558111yescharacteristic
684C2#2558311yescharacteristic
685C2#2594411yescharacteristic
686C2#2594611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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
2. C2×#2370C2semidirect
3. C2×#1989C2semidirect
6. #11066C2semidirect
7. #11066C2semidirect
8. #10931C2semidirect
9. #10931C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect
16. C2×#490C4semidirect
206. #125C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4902
C2×#23702
C2×#19892
C2×#19822
#141542
#110662
#109312
#108352
#107032