Finite Group Databasebeta

← #392739

Subgroups — #392739

Order 512.

2068 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C4⋊C4C211yesmaximal, characteristic
2C2×#1921C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic
4#14697C211yesmaximal, characteristic
5#10907C212yesmaximal, complemented
6#10885C212yesmaximal, complemented
7#10807C212yesmaximal, complemented
8#10730C211yesmaximal, characteristic
9#10730C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42—21no—
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#385—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4—28no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2212yes—
29C22×#95—21no—
30#1921—28no—
31#1852C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C22⋊#95—21no—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4Q814yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42—28no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—44no—
81C22×C42—44no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—24no—
88C2×#107C2×C418yescomplemented
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2×C414yes—
95#389—24no—
96#389—24no—
97#385—44no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380C2×C414yes—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4D418yes—
108C2×#95—28no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340C2×C418yescomplemented
126#336—24no—
127#336—24no—
128C25C2×D411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4Q1614yes—
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×Q812yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C22×C414yes—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—216no—
196C2×C4⋊C4—216no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—216no—
208C2×C4⋊C4—216no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×Q814yes—
235C2×C4⋊C4#4114yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C42—216no—
244C2×C42—216no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—28no—
258C2×C42—28no—
259C2×C42—432no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C2×D414yes—
265C2×C42#4114yes—
266#107—216no—
267C22⋊C8—28no—
268C22⋊C8—44no—
269#95—28no—
270#95—24no—
271#95—28no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—42no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24C4×D412yes—
288C24C22⋊Q812yes—
289C24#12312yes—
290C24C2×Q1612yes—
291C24#13512yes—
292C24AGL(1,ℤ/8)12yes—
293C24AGL(1,ℤ/8)12yes—
294C24C22×D411yescharacteristic
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4#11714yes—
402C22×C4C4×D412yes—
403C22×C4C4×Q814yes—
404C22×C4#12012yes—
405C22×C4#12012yes—
406C22×C4C22⋊Q812yes—
407C22×C4#12312yes—
408C22×C4#12312yes—
409C22×C4#12312yes—
410C22×C4#12514yes—
411C22×C4#12614yes—
412C22×C4C2×Q1612yes—
413C22×C4C2×Q1612yes—
414C22×C4AGL(1,ℤ/8)12yes—
415C22×C4#13712yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×Q811yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4112yes—
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4111yescharacteristic
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
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—
439C2×C8—42no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4C4×D418yes—
459C4⋊C4C22⋊Q818yes—
460C22⋊C4—48no—
461C22⋊C4—84no—
462C42—28no—
463C42—28no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—24no—
468C42—416no—
469C42—416no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23#38714yes—
499C23C4×Q1614yes—
500C23#44214yes—
501C23#47714yes—
502C23#47814yes—
503C23#48214yes—
504C23#48414yes—
505C23C2×C4×D411yescharacteristic
506C23C2×#12011yescharacteristic
507C23C2×C22⋊Q811yescharacteristic
508C23C2×#12311yescharacteristic
509C23C2×#12311yescharacteristic
510C23#52511yescharacteristic
511C23#52511yescharacteristic
512C23C22×Q1611yescharacteristic
513C23C2×#13511yescharacteristic
514C23C2×AGL(1,ℤ/8)11yescharacteristic
515C23C2×AGL(1,ℤ/8)11yescharacteristic
516C23C2×AGL(1,ℤ/8)11yescharacteristic
517C23C2×#13711yescharacteristic
518C23#57511yescharacteristic
519C23#57511yescharacteristic
520C2×C4—232no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—432no—
552C2×C4—432no—
553C2×C4—416no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4#38714yes—
603C2×C4#45114yes—
604C2×C4#48414yes—
605C2×C4C2×#11711yescharacteristic
606C2×C4C2×C4×D412yes—
607C2×C4C2×C4×Q811yescharacteristic
608C2×C4C4×#4112yes—
609C2×C4C2×C22⋊Q812yes—
610C2×C4#52512yes—
611C2×C4#52512yes—
612C2×C4#52512yes—
613C2×C4#52512yes—
614C2×C4C2×#12511yescharacteristic
615C2×C4C2×#12611yescharacteristic
616C2×C4#52912yes—
617C2×C4#52912yes—
618C2×C4#52912yes—
619C2×C4#52912yes—
620C2×C4#53312yes—
621C8—44no—
622C8—82no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22#192118yescomplemented
638C22C2×#38711yescharacteristic
639C22C4×#12011yescharacteristic
640C22C4×#12311yescharacteristic
641C22#284811yescharacteristic
642C22#284911yescharacteristic
643C22#285311yescharacteristic
644C22#285511yescharacteristic
645C22C2×C4×Q1611yescharacteristic
646C22C2×#44211yescharacteristic
647C22C4×AGL(1,ℤ/8)11yescharacteristic
648C22C4×#13711yescharacteristic
649C22C2×#45111yescharacteristic
650C22#300211yescharacteristic
651C22#300311yescharacteristic
652C22#300611yescharacteristic
653C22C2×#47811yescharacteristic
654C22C2×#47711yescharacteristic
655C22#307711yescharacteristic
656C22#307811yescharacteristic
657C22C2×#48211yescharacteristic
658C22C2×#48411yescharacteristic
659C22C2×#48411yescharacteristic
660C22#309111yescharacteristic
661C22#309211yescharacteristic
662C22#309211yescharacteristic
663C22#309311yescharacteristic
664C22#309411yescharacteristic
665C22#310411yescharacteristic
666C22#310411yescharacteristic
667C22#310511yescharacteristic
668C22#310811yescharacteristic
669C22#310911yescharacteristic
670C22#311211yescharacteristic
671C22#311411yescharacteristic
672C22#311411yescharacteristic
673C4—24no—
674C4—24no—
675C4—24no—
676C4—48no—
677C4—48no—
678C4—42no—
679C4—42no—
680C4—82no—
681C4—82no—
682C4—161no—
683C4—161no—
684C4#229012yes—
685C4C4×C22⋊Q812yes—
686C4#285112yes—
687C4#285212yes—
688C2—28no—
689C2#1469511yescharacteristic
690C2C2×#192111yescharacteristic
691C2#2103311yescharacteristic
692C2C4×#45111yescharacteristic
693C2#2145211yescharacteristic
694C2C4×#48411yescharacteristic
695C2#2157811yescharacteristic
696C2#3295711yescharacteristic
697C2#3295911yescharacteristic
698C2#3298111yescharacteristic
699C2#3298311yescharacteristic
700C2#3309611yescharacteristic
701C2#3309911yescharacteristic
702C2#3314911yescharacteristic
703C2#3315111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1921C2semidirect
5. #10907C2semidirect
6. #10885C2semidirect
7. #10807C2semidirect
31. #1852C4semidirect
88. C2×#107C2×C4semidirect
125. #340C2×C4semidirect
637. C22#1921semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C4⋊C42
C2×#19212
C2×#18522
#146972
#109072
#108852
#108072
#107302
#107122