Finite Group Databasebeta

← #360145

Subgroups — #360145

Order 512.

2280 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#442C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1879C212yesmaximal
4C2×#1804C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6#13941C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10843C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4—21no—
17C2×#442—24no—
18C2×#442—24no—
19C2×#442C418yescomplemented
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#97C2211yescharacteristic
27D4×C4⋊C4—24no—
28D4×C4⋊C4—24no—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#385—22no—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1879—28no—
37#1804—24no—
38#1802—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1439—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C22⋊#95—22no—
49C23×D4C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4—22no—
53C24×C4—21no—
54C22×D8—22no—
55C22×D8—22no—
56C22×D8—22no—
57C22×D8—22no—
58C22×D8C2×C414yes—
59C22×D8D414yes—
60C22×D8Q814yes—
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—42no—
77C2×C4×D4—42no—
78C2×C4×D4C2×C414yes—
79C2×C4×D4C2×C414yescomplemented
80C2×C4×D4D414yes—
81C2×C4×D4D414yes—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—48no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100#442—232no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106C2×C414yescomplemented
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—24no—
112C2×#102—24no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—42no—
117C2×#102C2×C414yes—
118C2×#102C2×C414yescomplemented
119C2×#97—22no—
120C2×#97—22no—
121C2×#97C2×C414yescomplemented
122#390—24no—
123#390—42no—
124#389—24no—
125#389—24no—
126#385—216no—
127#384—22no—
128#384D414yes—
129#382—22no—
130#382—22no—
131#380—28no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95D412yes—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145#336—42no—
146C25—21no—
147C25—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
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×D4C22×C412yes—
167C22×D4C22×C412yes—
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C22×D4C2×D412yes—
171C22×D4C2×D412yes—
172C22×D4C2×Q812yes—
173C22×D4C2×Q812yes—
174C22×D4#4112yes—
175C22×D4#4112yes—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—82no—
194C23×C4—82no—
195C23×C4—82no—
196C23×C4—82no—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2411yescharacteristic
202C2×D8—28no—
203C2×D8—28no—
204C2×D8—28no—
205C2×D8—28no—
206C2×D8—28no—
207C2×D8—28no—
208C2×D8C4⋊C4116yescomplemented
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8C22×C412yes—
227C22×C8C22×C412yes—
228C22×C8C2×D412yes—
229C22×C8C2×Q812yes—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—48no—
236C4×D4—48no—
237C4×D4—48no—
238C4×D4—48no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4C22⋊C4116yescomplemented
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—432no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—82no—
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—82no—
292C2×C22⋊C4—82no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C22×C412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42#4112yes—
304C2×C42#4112yes—
305#106—28no—
306C4⋊C8—44no—
307#102—28no—
308#102—28no—
309#102—48no—
310#102—48no—
311#97—28no—
312#95—28no—
313#95—24no—
314#95—42no—
315#95—42no—
316#95—82no—
317#95—82no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—41no—
333C24—41no—
334C24—41no—
335C24—41no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24C22×D411yescharacteristic
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—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4C2×C22⋊C414yes—
390C2×D4C2×C4⋊C414yes—
391C2×D4C2×C4⋊C414yescomplemented
392C2×D4#11714yes—
393C2×D4#12114yes—
394C2×D4#12114yes—
395C2×D4C22⋊Q814yes—
396C2×D4C22⋊Q814yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—416no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4C4×D412yes—
456C22×C4C4×D412yes—
457C22×C4#12012yes—
458C22×C4#12012yes—
459C22×C4#12112yes—
460C22×C4#12112yes—
461C22×C4AGL(1,ℤ/8)14yes—
462C22×C4C23×C411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×Q811yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4H(5,𝔽2)11yescharacteristic
473C22×C4H(5,𝔽2)11yescharacteristic
474C22×C42−1+411yescharacteristic
475C22×C42−1+411yescharacteristic
476D8—232no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8C2×C4⋊C414yescomplemented
498C4⋊C4—28no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—42no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—84no—
518C4⋊C4C2×C22⋊C414yes—
519C22⋊C4—216no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42C2×C22⋊C414yes—
529C42C4×D414yes—
530C42#12114yes—
531C42#12314yes—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—82no—
569C23—82no—
570C23—81no—
571C23—81no—
572C23—81no—
573C23—81no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—82no—
578C23C2×C4×D411yescharacteristic
579C23C2×#12011yescharacteristic
580C23C2×#12111yescharacteristic
581C23D4211yescharacteristic
582C23#54611yescharacteristic
583C23#54811yescharacteristic
584C23D4×Q811yescharacteristic
585C23C2×AGL(1,ℤ/8)12yes—
586C23#57512yes—
587C23#57612yes—
588C23#57712yes—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—24no—
593D4—24no—
594D4—24no—
595D4—24no—
596D4—24no—
597D4—24no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4#38018yes—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—432no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—816no—
639C2×C4—832no—
640C2×C4—81no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—81no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4#38614yes—
655C2×C4#38614yes—
656C2×C4C22×C22⋊C411yescharacteristic
657C2×C4C22×C4⋊C411yescharacteristic
658C2×C4C2×#11711yescharacteristic
659C2×C4C2×C4×D411yescharacteristic
660C2×C4#52011yescharacteristic
661C2×C4#52011yescharacteristic
662C2×C4C2×#12111yescharacteristic
663C2×C4C2×#12111yescharacteristic
664C2×C4C2×#12111yescharacteristic
665C2×C4C2×C22⋊Q811yescharacteristic
666C2×C4C2×C22⋊Q811yescharacteristic
667C2×C4C2×#12311yescharacteristic
668C2×C4#53511yescharacteristic
669C2×C4#53611yescharacteristic
670C2×C4#53711yescharacteristic
671C2×C4#53711yescharacteristic
672C2×C4D4211yescharacteristic
673C2×C4#54611yescharacteristic
674C2×C4#54711yescharacteristic
675C2×C4#54711yescharacteristic
676C2×C4#54811yescharacteristic
677C2×C4D4×Q811yescharacteristic
678C2×C4#55211yescharacteristic
679C2×C4#55211yescharacteristic
680C2×C4#55311yescharacteristic
681C2×C4#55311yescharacteristic
682C2×C4#55411yescharacteristic
683C2×C4#55411yescharacteristic
684C8—24no—
685C8—42no—
686C8—82no—
687C22—28no—
688C22—44no—
689C22—44no—
690C22—48no—
691C22—48no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—48no—
697C22—48no—
698C22—44no—
699C22—48no—
700C22—82no—
701C22—82no—
702C22—84no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—84no—
708C22—84no—
709C22C2×#38611yescharacteristic
710C22D4×C4⋊C411yescharacteristic
711C22#235211yescharacteristic
712C22#240811yescharacteristic
713C22#240911yescharacteristic
714C22#241611yescharacteristic
715C22#241711yescharacteristic
716C22C4×AGL(1,ℤ/8)12yes—
717C22#294612yes—
718C22#300311yescharacteristic
719C22#300511yescharacteristic
720C22#300811yescharacteristic
721C22#301011yescharacteristic
722C22#303612yes—
723C22#304512yes—
724C22#326412yes—
725C22#326712yes—
726C22#330712yes—
727C22#331812yes—
728C22#337012yes—
729C22#337312yes—
730C22#338412yes—
731C22#338912yes—
732C4—24no—
733C4—24no—
734C4—44no—
735C4—44no—
736C4—84no—
737C4—82no—
738C4—82no—
739C4—162no—
740C4C2×#38011yescharacteristic
741C4#231111yescharacteristic
742C4D4×C4⋊C411yescharacteristic
743C4#235211yescharacteristic
744C4#241611yescharacteristic
745C4#241711yescharacteristic
746C4#248611yescharacteristic
747C4#248711yescharacteristic
748C2—48no—
749C2—84no—
750C2#1387211yescharacteristic
751C2#2092511yescharacteristic
752C2#2093211yescharacteristic
753C2#2267612yes—
754C2#2277412yes—
755C2#2424811yescharacteristic
756C2#2424911yescharacteristic
757C2#2433811yescharacteristic
758C2#2433911yescharacteristic
759C2#2473812yes—
760C2#2482312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
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×D4×C4⋊C4C2semidirect
4. C2×#1804C2semidirect
5. C2×#1802C2semidirect
6. #13941C2semidirect
7. #11220C2semidirect
8. #11066C2semidirect
9. #10843C2semidirect
10. #10835C2semidirect
11. #10788C2semidirect
19. C2×#442C4semidirect
79. C2×C4×D4C2×C4semidirect
103. C2×#106C2×C4semidirect
118. C2×#102C2×C4semidirect
121. C2×#97C2×C4semidirect
208. C2×D8C4⋊C4semidirect
241. C4×D4C22⋊C4semidirect
391. C2×D4C2×C4⋊C4semidirect
497. C2×C8C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4422
C2×D4×C4⋊C42
C2×#18792
C2×#18042
C2×#18022
#139412
#112202
#110662
#108432
#108352
#107882