Finite Group Databasebeta

← #374182

Subgroups — #374182

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic, complemented
3C2×#1893C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic, complemented
5#14137C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#481—24no—
17C2×#481—24no—
18C2×#481C418yescomplemented
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2345—24no—
26#2345—24no—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#387—22no—
30C2×#386—21no—
31C2×#386—21no—
32C2×#380—21no—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1893—24no—
38C2×#340—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1439—22no—
47C4⋊#95C2212yes—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121C2×C414yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4C2×C414yes—
78C22×C4⋊C4D414yes—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42—22no—
92C22×C42C2312yes—
93#481—232no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106C2×C414yescomplemented
99C2×#106C2×C414yescomplemented
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—42no—
107C2×#102C2×C414yescomplemented
108C2×#102C2×C414yescomplemented
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yes—
112#390—24no—
113#390—42no—
114#389—24no—
115#389—24no—
116#387—216no—
117#386—44no—
118#386—44no—
119#380—28no—
120#380—22no—
121#380D414yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D412yes—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—24no—
139#336—42no—
140C25—21no—
141C25C2×D411yescharacteristic
142C22×D4—24no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4C22×C412yes—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4C22×C412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C22×C8—28no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
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—48no—
204C22×C8—48no—
205C22×C8—44no—
206C22×C8—44no—
207C22×C8C22×C412yes—
208C22×C8C22×C412yes—
209C22×C8#4114yes—
210#121—28no—
211#121—28no—
212#121—44no—
213#121—44no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4C22⋊C4116yescomplemented
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4SD1614yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—28no—
284C2×C42—82no—
285C2×C42—82no—
286#106—28no—
287#106—28no—
288#102—28no—
289#102—28no—
290#102—48no—
291C22⋊C8—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24C4×D412yes—
314C24#12012yes—
315C24#12112yes—
316C24C2×SD1612yes—
317C24#13512yes—
318C24AGL(1,ℤ/8)12yes—
319C24#13712yes—
320C24C22×D411yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—48no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4C2×C22⋊C414yes—
417C22×C4C4×D414yes—
418C22×C4C4×D412yes—
419C22×C4#12012yes—
420C22×C4#12114yes—
421C22×C4#12112yes—
422C22×C4#12314yes—
423C22×C4C2×SD1614yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4111yescharacteristic
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+412yes—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—416no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—216no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—48no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—88no—
473C4⋊C4C2×C22⋊C414yes—
474C4⋊C4C2×C22⋊C414yes—
475C22⋊C4—28no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—88no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—81no—
518C23—81no—
519C23—82no—
520C23—81no—
521C23—81no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23#38614yes—
526C23C4×SD1614yes—
527C23#44014yes—
528C23#44814yes—
529C23#44914yes—
530C23#46014yes—
531C23#46414yes—
532C23C2×C4×D411yescharacteristic
533C23C2×#12011yescharacteristic
534C23C2×#12111yescharacteristic
535C23#54612yes—
536C23#54712yes—
537C23C22×SD1611yescharacteristic
538C23C2×#13511yescharacteristic
539C23C2×AGL(1,ℤ/8)11yescharacteristic
540C23C2×#13711yescharacteristic
541C23#57612yes—
542C23#57712yes—
543D4—42no—
544D4—44no—
545D4—42no—
546D4—816no—
547D4—81no—
548D4—81no—
549D4—82no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—816no—
594C2×C4—816no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4#38614yes—
608C2×C4C22×C22⋊C411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C4×#4112yes—
611C2×C4#52012yes—
612C2×C4C2×#12111yescharacteristic
613C2×C4C2×#12311yescharacteristic
614C2×C4#52511yescharacteristic
615C2×C4#52511yescharacteristic
616C2×C4#53212yes—
617C2×C4#53511yescharacteristic
618C2×C4#53611yescharacteristic
619C2×C4#53712yes—
620C2×C4#54612yes—
621C2×C4#54712yes—
622C2×C4#55112yes—
623C2×C4#55112yes—
624C2×C4#55212yes—
625C2×C4#55212yes—
626C8—42no—
627C8—44no—
628C8—42no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—28no—
633C22—28no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—28no—
638C22—28no—
639C22—82no—
640C22—84no—
641C22—84no—
642C22—82no—
643C22—82no—
644C22—161no—
645C22—161no—
646C22—161no—
647C22—161no—
648C22—162no—
649C22—162no—
650C22#188018yes—
651C22C2×#38611yescharacteristic
652C22#235912yes—
653C22#241111yescharacteristic
654C22#241311yescharacteristic
655C22#243012yes—
656C22C2×C4×SD1611yescharacteristic
657C22C2×#44011yescharacteristic
658C22#294612yes—
659C22C2×#44811yescharacteristic
660C22C2×#44911yescharacteristic
661C22#301212yes—
662C22C2×#46011yescharacteristic
663C22C2×#46411yescharacteristic
664C22#304111yescharacteristic
665C22#304211yescharacteristic
666C22#329512yes—
667C22#330112yes—
668C22#331612yes—
669C22#331812yes—
670C22#333512yes—
671C22#333912yes—
672C22#334512yes—
673C22#334612yes—
674C4—24no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—84no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4#229711yescharacteristic
683C4#231111yescharacteristic
684C4#235912yes—
685C4#241111yescharacteristic
686C4#241311yescharacteristic
687C4#249812yes—
688C2—28no—
689C2—162no—
690C2#1413511yescharacteristic
691C2C2×#188011yescharacteristic
692C2#2092911yescharacteristic
693C2#2291812yes—
694C2#2293612yes—
695C2#2447811yescharacteristic
696C2#2448311yescharacteristic
697C2#2457811yescharacteristic
698C2#2458311yescharacteristic
699C2#2553712yes—
700C2#2554312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2345C2semidirect
4. C22×#340C2semidirect
6. #11220C2semidirect
7. #11066C2semidirect
8. #10931C2semidirect
9. #10908C2semidirect
10. #10730C2semidirect
11. #10714C2semidirect
18. C2×#481C4semidirect
98. C2×#106C2×C4semidirect
99. C2×#106C2×C4semidirect
107. C2×#102C2×C4semidirect
108. C2×#102C2×C4semidirect
249. C2×C4⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#23452
C2×#18932
C22×#3402
#141372
#112202
#110662
#109312
#109082
#107302
#107142