Finite Group Databasebeta

← #360056

Subgroups — #360056

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5#13906C211yesmaximal, characteristic
6#11190C212yesmaximal, complemented
7#11023C212yesmaximal
8#10931C212yesmaximal
9#10907C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#440—24no—
16C2×#440—24no—
17C2×#440C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#2356—24no—
26#2356—24no—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#387—22no—
30C2×#384C2211yescharacteristic
31C2×#383—21no—
32C2×#382C2211yescharacteristic
33C2×#380—22no—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376—21no—
37#1804—24no—
38#1803—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1439—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16—22no—
57C22×SD16C2×C414yes—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C2×C4×Q8D414yes—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42—21no—
85C22×C42—22no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#440—232no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107C2×C414yes—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103C2×C414yes—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—42no—
106C2×#102C2×C414yes—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97C2×C414yes—
110#391—24no—
111#391—24no—
112#390—24no—
113#390—42no—
114#387—216no—
115#384—22no—
116#384—22no—
117#383—28no—
118#382—22no—
119#382D414yes—
120#380—48no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125#376—28no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—24no—
136#340—24no—
137#340—24no—
138#340—42no—
139#336—24no—
140#336—42no—
141C25—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8C22×C412yes—
154C22×Q8C2×D412yes—
155C22×Q8C2×D412yes—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C22×C412yes—
168C22×D4#4112yes—
169C22×D4#4112yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×SD16—28no—
186C2×SD16—28no—
187C2×SD16—28no—
188C2×SD16—28no—
189C2×SD16—28no—
190C2×SD16—28no—
191C2×SD16—28no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8C22×C412yes—
210C22×C8C22×C412yes—
211C22×C8#4112yes—
212C22×C8#4112yes—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8C22⋊C4116yescomplemented
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—44no—
220C4×D4—44no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4—82no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C22×C412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42#4112yes—
281C2×C42#4112yes—
282#107—28no—
283C4⋊C8—44no—
284#103—28no—
285#103—48no—
286#102—28no—
287#102—48no—
288#97—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24C22×D411yescharacteristic
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8C2×C22⋊C414yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4#11714yes—
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—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—416no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4C4×D412yes—
420C22×C4C4×D412yes—
421C22×C4#12012yes—
422C22×C4#12012yes—
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4AGL(1,ℤ/8)14yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C42−1+411yescharacteristic
439C22×C42−1+411yescharacteristic
440SD16—232no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8#11714yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—216no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—48no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4C2×C22⋊C414yes—
482C22⋊C4—48no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—216no—
486C42—416no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42C2×C22⋊C414yes—
492C42C4×D414yes—
493C42#12114yes—
494C42#12314yes—
495C23—24no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23C2×C4×D411yescharacteristic
519C23C2×#12011yescharacteristic
520C23C2×#12111yescharacteristic
521C23#54611yescharacteristic
522C23#54711yescharacteristic
523C23#54811yescharacteristic
524C23#55011yescharacteristic
525C23C2×AGL(1,ℤ/8)12yes—
526C23#57512yes—
527C23#57712yes—
528C23#57812yes—
529Q8—24no—
530Q8—24no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537D4—24no—
538D4—24no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—816no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4#38614yes—
600C2×C4#38614yes—
601C2×C4C22×C22⋊C411yescharacteristic
602C2×C4C2×#11711yescharacteristic
603C2×C4C2×#11711yescharacteristic
604C2×C4C2×C4×D411yescharacteristic
605C2×C4#52011yescharacteristic
606C2×C4#52011yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×#12311yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4#52511yescharacteristic
611C2×C4#52511yescharacteristic
612C2×C4#52511yescharacteristic
613C2×C4#53511yescharacteristic
614C2×C4#53611yescharacteristic
615C2×C4#53711yescharacteristic
616C2×C4#53711yescharacteristic
617C2×C4#54611yescharacteristic
618C2×C4#54711yescharacteristic
619C2×C4#54811yescharacteristic
620C2×C4#55011yescharacteristic
621C2×C4#55211yescharacteristic
622C2×C4#55211yescharacteristic
623C2×C4#55311yescharacteristic
624C2×C4#55311yescharacteristic
625C2×C4#55511yescharacteristic
626C2×C4#55511yescharacteristic
627C2×C4#55611yescharacteristic
628C2×C4#55611yescharacteristic
629C8—24no—
630C8—42no—
631C8—82no—
632C22—44no—
633C22—48no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—84no—
638C22—82no—
639C22—82no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22C2×#38611yescharacteristic
644C22#235211yescharacteristic
645C22#235411yescharacteristic
646C22#240711yescharacteristic
647C22#240711yescharacteristic
648C22#241411yescharacteristic
649C22#241511yescharacteristic
650C22C4×AGL(1,ℤ/8)12yes—
651C22#294712yes—
652C22#300612yes—
653C22#301312yes—
654C22#303811yescharacteristic
655C22#304011yescharacteristic
656C22#304311yescharacteristic
657C22#304411yescharacteristic
658C22#331012yes—
659C22#331512yes—
660C22#334312yes—
661C22#334812yes—
662C22#336712yes—
663C22#337612yes—
664C22#339712yes—
665C22#340212yes—
666C4—24no—
667C4—24no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—162no—
675C4#228011yescharacteristic
676C4#231111yescharacteristic
677C4#235211yescharacteristic
678C4#235411yescharacteristic
679C4#241411yescharacteristic
680C4#241511yescharacteristic
681C4#248511yescharacteristic
682C4#248511yescharacteristic
683C2—84no—
684C2#1387111yescharacteristic
685C2#2092811yescharacteristic
686C2#2093011yescharacteristic
687C2#2277812yes—
688C2#2282012yes—
689C2#2413712yes—
690C2#2420512yes—
691C2#2464411yescharacteristic
692C2#2464511yescharacteristic
693C2#2469411yescharacteristic
694C2#2469511yescharacteristic
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
2. C2×#2356C2semidirect
3. C2×#1803C2semidirect
6. #11190C2semidirect
9. #10907C2semidirect
10. #10843C2semidirect
11. #10835C2semidirect
17. C2×#440C4semidirect
216. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#23562
C2×#18032
C2×#18042
#139062
#111902
#110232
#109312
#109072
#108432
#108352