Finite Group Databasebeta

← #368018

Subgroups — #368018

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic, complemented
2C2×#2363C211yesmaximal, characteristic
3C2×#1895C211yesmaximal, characteristic
4C2×#1874C211yesmaximal, characteristic
5C2×#1864C212yesmaximal, complemented
6#14130C211yesmaximal, characteristic, complemented
7#11328C212yesmaximal, complemented
8#11123C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10715C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#428—28no—
18C2×#428—24no—
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24#2363—24no—
25#2363—24no—
26C2×#390—21no—
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388—21no—
31C2×#386—21no—
32C2×#385—21no—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1895C418yescomplemented
38#1874—24no—
39#1864—28no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1444—21no—
47#1442—21no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4C2311yescharacteristic
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—24no—
58C22×M16—24no—
59C22×M16C2×C414yescomplemented
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121C2×C414yes—
66C2×#121D414yes—
67C2×#121Q814yes—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—24no—
71C2×C4×D4—42no—
72C2×#117—22no—
73C2×#117—24no—
74C2×#117—42no—
75C2×#117D414yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42—21no—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#428—232no—
101C2×#107—24no—
102C2×#107—24no—
103C2×#107—24no—
104C2×#106—24no—
105C2×#106—24no—
106C2×#106D418yescomplemented
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—42no—
112C2×#102C2×C418yescomplemented
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116#390—28no—
117#390—22no—
118#390—22no—
119#389—22no—
120#389—22no—
121#389—24no—
122#389—42no—
123#388—28no—
124#386—28no—
125#385—44no—
126#380—22no—
127#380C2×C414yescomplemented
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141#340—24no—
142#340—24no—
143#340—42no—
144#340C2×C418yescomplemented
145#336—24no—
146#336—24no—
147C25—21no—
148C25C2×D411yescharacteristic
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—82no—
160C22×D4—82no—
161C22×D4C22×C412yes—
162C22×D4C2×D412yes—
163C22×D4C2×Q812yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4—82no—
183C23×C4C22×C412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×Q812yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C2×M16—216no—
196C2×M16—28no—
197C2×M16—28no—
198C2×M16—28no—
199C2×M16—216no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8C22×C414yes—
212C22×C8C2×D414yes—
213C22×C8#4114yes—
214#121—28no—
215#121—28no—
216#121—28no—
217#121C4⋊C4116yescomplemented
218C4×D4—48no—
219C4×D4—44no—
220C4×D4—44no—
221#117—28no—
222#117—44no—
223#117—48no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
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—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4C22×C414yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×Q812yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—82no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—44no—
292C2×C42—82no—
293#107—216no—
294#106—216no—
295#102—216no—
296#102—48no—
297C22⋊C8—44no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—44no—
304#95—44no—
305#95—82no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—42no—
318C24—42no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—42no—
324C24—42no—
325C24C4×D412yes—
326C24#12012yes—
327C24C22⋊Q812yes—
328C24C22×D411yescharacteristic
329C2×D4—216no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4C2×C4⋊C414yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4C2×C4⋊C414yes—
426C22×C4C4×D414yes—
427C22×C4C4×D412yes—
428C22×C4#12012yes—
429C22×C4C22⋊Q814yes—
430C22×C4C22⋊Q812yes—
431C22×C4#12314yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4#13712yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×Q811yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4111yescharacteristic
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444M16—232no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8C4×D418yes—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—84no—
477C4⋊C4—88no—
478C4⋊C4C2×C4⋊C414yes—
479C4⋊C4#12014yes—
480C22⋊C4—216no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C42—48no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—42no—
495C42—84no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23#38514yes—
542C23C2×C4×D411yescharacteristic
543C23C2×#12011yescharacteristic
544C23C2×C22⋊Q811yescharacteristic
545C23D4212yes—
546C23#54712yes—
547C23C2×AGL(1,ℤ/8)11yescharacteristic
548C23C2×#13711yescharacteristic
549C23#57511yescharacteristic
550C23#57511yescharacteristic
551C23#57612yes—
552C23#57712yes—
553D4—28no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—44no—
558D4—42no—
559D4—816no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4—816no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4#38514yes—
622C2×C4C22×C4⋊C411yescharacteristic
623C2×C4C2×C4×D411yescharacteristic
624C2×C4C2×C4×D412yes—
625C2×C4#52012yes—
626C2×C4C2×#12011yescharacteristic
627C2×C4C2×C22⋊Q811yescharacteristic
628C2×C4C2×#12311yescharacteristic
629C2×C4#52511yescharacteristic
630C2×C4#52512yes—
631C2×C4#53411yescharacteristic
632C2×C4#53611yescharacteristic
633C2×C4#53912yes—
634C2×C4D4212yes—
635C2×C4#54712yes—
636C2×C4#54812yes—
637C2×C4D4×Q812yes—
638C2×C4#55312yes—
639C2×C4#55412yes—
640C8—28no—
641C8—82no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22C2×#38511yescharacteristic
665C22#236112yes—
666C22#240311yescharacteristic
667C22#240611yescharacteristic
668C22#244912yes—
669C22C4×AGL(1,ℤ/8)11yescharacteristic
670C22C4×#13711yescharacteristic
671C22#294612yes—
672C22#300611yescharacteristic
673C22#300711yescharacteristic
674C22#300811yescharacteristic
675C22#300911yescharacteristic
676C22#307911yescharacteristic
677C22#308011yescharacteristic
678C22#308112yes—
679C22#326612yes—
680C22#326812yes—
681C22#327412yes—
682C22#327512yes—
683C22#334512yes—
684C22#334712yes—
685C22#335112yes—
686C22#335712yes—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—81no—
693C4—81no—
694C4—82no—
695C4—84no—
696C4—84no—
697C4C4×#12011yescharacteristic
698C4#232211yescharacteristic
699C4D4×C4⋊C412yes—
700C4#240911yescharacteristic
701C4#244312yes—
702C4#257911yescharacteristic
703C2—44no—
704C2—84no—
705C2#1401511yescharacteristic
706C2#2087511yescharacteristic
707C2#2087611yescharacteristic
708C2#2303512yes—
709C2#2303912yes—
710C2#2386211yescharacteristic
711C2#2386911yescharacteristic
712C2#2410111yescharacteristic
713C2#2410511yescharacteristic
714C2#2635912yes—
715C2#2636112yes—
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
1. C22×#428C2semidirect
5. C2×#1864C2semidirect
6. #14130C2semidirect
7. #11328C2semidirect
8. #11123C2semidirect
9. #10908C2semidirect
10. #10731C2semidirect
11. #10715C2semidirect
37. #1895C4semidirect
59. C22×M16C2×C4semidirect
106. C2×#106D4semidirect
112. C2×#102C2×C4semidirect
127. #380C2×C4semidirect
144. #340C2×C4semidirect
217. #121C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#23632
C2×#18952
C2×#18742
C2×#18642
#141302
#113282
#111232
#109082
#107312
#107152