Finite Group Databasebeta

← #196845

Subgroups — #196845

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic, complemented
3#21738C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic, complemented
5C22×#327C211yesmaximal, characteristic, complemented
6#9156C212yesmaximal, complemented
7#9155C212yesmaximal, complemented
8#9148C212yesmaximal, complemented
9#9147C212yesmaximal, complemented
10#8023C211yesmaximal, characteristic, complemented
11C2×#1294C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C22×#121C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2854C2212yes—
19#2847C2212yes—
20#2618—21no—
21#2617—21no—
22#2543—22no—
23#2534—22no—
24#2486C412yescomplemented
25#2416C414yescomplemented
26#2408C412yescomplemented
27#2349—24no—
28#2349—24no—
29#2308—21no—
30C2×#389—21no—
31C2×#385—21no—
32C2×#385—21no—
33C2×#382C2211yescharacteristic, complemented
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic, complemented
37C2×C4×C4⋊C4C2211yescharacteristic, complemented
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic, complemented
40#1869—22no—
41#1869—22no—
42#1817C2212yescomplemented
43#1779C2212yescomplemented
44#1519—24no—
45C2×#327—24no—
46C2×#327C2212yes—
47C2×#327C2212yes—
48C2×#327C2212yes—
49C2×#327C2212yes—
50#1294C2212yescomplemented
51#1294C2212yescomplemented
52#1294C2212yescomplemented
53#1294C2212yescomplemented
54C23×D4—21no—
55C23×D4C2×C411yescharacteristic, complemented
56C23×D4D411yescharacteristic
57C24×C4—21no—
58C24×C4—21no—
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C23×C8—21no—
62C23×C8—21no—
63#534—44no—
64#534—44no—
65C2×#123—22no—
66C2×#123—22no—
67C2×#121—24no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121C2×C412yescomplemented
75C2×#121C2×C414yescomplemented
76C2×#121C2312yes—
77C2×#120—22no—
78C2×#120—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4—28no—
84C2×C4×D4—44no—
85C2×C4×D4—44no—
86C2×C4×D4C2×C412yes—
87C2×C4×D4C2×C412yes—
88C2×C4×D4D412yes—
89C2×C4×D4D412yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2×C411yescharacteristic, complemented
95C22×C4⋊C4C2×C412yescomplemented
96C22×C4⋊C4D411yescharacteristic
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D411yescharacteristic
102C22×C22⋊C4D411yescharacteristic
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8C2312yes—
113C2×C22⋊C8C2312yes—
114#398—22no—
115#397—22no—
116#396—24no—
117#394—22no—
118#394—24no—
119#392—22no—
120#390C2×C414yescomplemented
121#389—28no—
122#389C2×C414yescomplemented
123#388—24no—
124#386—22no—
125#386—22no—
126#386—24no—
127#386—24no—
128#385—28no—
129#385—28no—
130#382—22no—
131#382—22no—
132#381—22no—
133#380—22no—
134#380—21no—
135#380—21no—
136#380—22no—
137#380—22no—
138#380C2×C414yescomplemented
139#380C2×C412yescomplemented
140#380D414yescomplemented
141#380C2312yes—
142#379—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95D414yescomplemented
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157#336—24no—
158#327—24no—
159#327—24no—
160#327—24no—
161#327—24no—
162C25—21no—
163C25C2×D411yescharacteristic
164C25C2×D411yescharacteristic
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—82no—
182C22×D4C22⋊C412yes—
183C22×D4C22⋊C412yes—
184C22×D4D814yes—
185C22×D4SD1614yes—
186C22×D4C22×C412yes—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—82no—
206C23×C4—82no—
207C23×C4—82no—
208C23×C4—82no—
209C23×C4C22×C412yes—
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4#4112yes—
222C23×C4#4112yes—
223C23×C4C2411yescharacteristic
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—82no—
229C22×C8—82no—
230#123—48no—
231#123—44no—
232#123—44no—
233#121—24no—
234#121—24no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—416no—
240#121—42no—
241#121—42no—
242#121—42no—
243#121—42no—
244#121—84no—
245#120—44no—
246#120—44no—
247#120—48no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—432no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4—82no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4C22⋊C412yes—
275C2×C4⋊C4C22⋊C412yes—
276C2×C4⋊C4C22×C412yes—
277C2×C4⋊C4C22×C414yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4C22⋊C412yes—
301C2×C22⋊C4C22⋊C412yes—
302C2×C22⋊C4C22⋊C412yes—
303C2×C22⋊C4C22⋊C412yes—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C22×C412yes—
312C2×C42C2×D412yes—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—84no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—22no—
330#95—21no—
331#95C2×D412yes—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—82no—
348C24—82no—
349C24C2×C22⋊C412yes—
350C24C2×C22⋊C411yescharacteristic
351C24C2×C22⋊C411yescharacteristic
352C24#12012yes—
353C24#12312yes—
354C24C2×D812yes—
355C24C2×D812yes—
356C24C2×SD1612yes—
357C24C2×SD1612yes—
358C24AGL(1,ℤ/8)12yes—
359C24#13712yes—
360C24C22×D411yescharacteristic
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—42no—
383C2×D4—82no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4#10218yes—
393C2×D4#10218yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4C2×C22⋊C412yes—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C412yes—
467C22×C4C4×D412yes—
468C22×C4C4×D412yes—
469C22×C4C4×D412yes—
470C22×C4C4×D412yes—
471C22×C4#12012yes—
472C22×C4#12112yes—
473C22×C4#12112yes—
474C22×C4#12112yes—
475C22×C4#12112yes—
476C22×C4#12312yes—
477C22×C4C2×D812yes—
478C22×C4C2×SD1612yes—
479C22×C4C23×C411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C2×C8—44no—
486C2×C8—44no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—84no—
490C2×C8—84no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—48no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—48no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—164no—
563C23—164no—
564C23#38612yes—
565C23#38612yes—
566C23C2×#10214yes—
567C23C2×#10212yes—
568C23C2×#10212yes—
569C23#41712yes—
570C23#41712yes—
571C23#44714yes—
572C23#44814yes—
573C23#48014yes—
574C23#48114yes—
575C23C22×C22⋊C411yescharacteristic
576C23C2×#12011yescharacteristic
577C23C2×#12311yescharacteristic
578C23C22×D811yescharacteristic
579C23C22×SD1611yescharacteristic
580C23C2×AGL(1,ℤ/8)11yescharacteristic
581C23C2×#13711yescharacteristic
582D4—216no—
583D4—216no—
584D4—82no—
585D4—82no—
586D4—84no—
587D4—82no—
588D4—82no—
589D4—88no—
590D4—82no—
591D4—82no—
592D4—168no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—816no—
650C2×C4—816no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—84no—
658C2×C4—164no—
659C2×C4#38612yes—
660C2×C4#38612yes—
661C2×C4C2×#10214yes—
662C2×C4C2×C4×D411yescharacteristic
663C2×C4C2×C4×D411yescharacteristic
664C2×C4C2×#12111yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C8—82no—
667C8—82no—
668C22—22no—
669C22—24no—
670C22—22no—
671C22—24no—
672C22—22no—
673C22—22no—
674C22—24no—
675C22—24no—
676C22—21no—
677C22—21no—
678C22—22no—
679C22—22no—
680C22—21no—
681C22—21no—
682C22—22no—
683C22—22no—
684C22—24no—
685C22—28no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—82no—
698C22—84no—
699C22—84no—
700C22—82no—
701C22—82no—
702C22—168no—
703C22#187414yes—
704C22#187414yes—
705C22C2×#38611yescharacteristic
706C22#288312yes—
707C22C22×#10211yescharacteristic
708C22C2×#41711yescharacteristic
709C22#290112yes—
710C22C2×#44711yescharacteristic
711C22C2×#44811yescharacteristic
712C22C2×#48011yescharacteristic
713C22C2×#48111yescharacteristic
714C4—24no—
715C4—21no—
716C4—21no—
717C4—21no—
718C4—21no—
719C4—22no—
720C4—22no—
721C4—44no—
722C4—42no—
723C4—42no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—84no—
729C2—21no—
730C2—21no—
731C2—21no—
732C2—21no—
733C2—24no—
734C2—24no—
735C2—44no—
736C2—162no—
737C2#1761312yes—
738C2#1762212yes—
739C2C2×#187411yescharacteristic
740C2#2098011yescharacteristic
741C2#2098411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2349C2semidirect
3. #21738C2semidirect
4. C2×#1519C2semidirect
5. C22×#327C2semidirect
6. #9156C2semidirect
7. #9155C2semidirect
8. #9148C2semidirect
9. #9147C2semidirect
10. #8023C2semidirect
11. C2×#1294C2semidirect
24. #2486C4semidirect
25. #2416C4semidirect
26. #2408C4semidirect
33. C2×#382C22semidirect
36. C2×#380C22semidirect
37. C2×C4×C4⋊C4C22semidirect
39. C22×#95C22semidirect
42. #1817C22semidirect
43. #1779C22semidirect
50. #1294C22semidirect
51. #1294C22semidirect
52. #1294C22semidirect
53. #1294C22semidirect
55. C23×D4C2×C4semidirect
74. C2×#121C2×C4semidirect
75. C2×#121C2×C4semidirect
94. C22×C4⋊C4C2×C4semidirect
95. C22×C4⋊C4C2×C4semidirect
120. #390C2×C4semidirect
122. #389C2×C4semidirect
138. #380C2×C4semidirect
139. #380C2×C4semidirect
140. #380D4semidirect
152. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
#350002
C2×#23492
#217382
C2×#15192
C22×#3272
#91562
#91552
#91482
#91472
#80232
C2×#12942