Finite Group Databasebeta

← #405343

Subgroups — #405343

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1874C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#14130C211yesmaximal, characteristic, complemented
7#11267C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11189C212yesmaximal, complemented
10#11066C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#123—21no—
14C22×#123—21no—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21#2501—24no—
22#2501—24no—
23C2×#399—21no—
24C2×#399—21no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1893—24no—
34#1874—24no—
35#1823—24no—
36C2×#340—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1442—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×#123—28no—
62C2×#123—28no—
63C2×#123—44no—
64C2×#123—44no—
65C2×#123—44no—
66C2×#123—44no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121D414yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4D414yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D411yescharacteristic
102C22×C22⋊C4D411yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—42no—
110C2×#102—42no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114#399—28no—
115#399—28no—
116#390—22no—
117#390—22no—
118#390—22no—
119#390—22no—
120#389—24no—
121#389—42no—
122#385—44no—
123#384—22no—
124#384—22no—
125#384—22no—
126#384—22no—
127#380—22no—
128#380D414yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#336—24no—
144#336—42no—
145C25—21no—
146C25C2×D411yescharacteristic
147C22×D4—24no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4C2×D412yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—82no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C22×C8—28no—
198C22×C8—22no—
199C22×C8—22no—
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×C8—48no—
212C22×C8—48no—
213C22×C8—44no—
214C22×C8—44no—
215C22×C8#4114yes—
216#123—432no—
217#123—432no—
218#121—28no—
219#121—28no—
220#121—44no—
221#121—44no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4#4114yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
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⋊C4—42no—
301C2×C22⋊C4D814yes—
302C2×C22⋊C4SD1614yes—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311#102—48no—
312#102—48no—
313C22⋊C8—44no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—44no—
318#95—44no—
319#95—82no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—41no—
326C24—41no—
327C24—42no—
328C24—42no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24—41no—
333C24—42no—
334C24—42no—
335C24#12012yes—
336C24#12112yes—
337C24#12412yes—
338C24C2×D812yes—
339C24C2×SD1612yes—
340C24AGL(1,ℤ/8)12yes—
341C24AGL(1,ℤ/8)12yes—
342C24C22×D411yescharacteristic
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—48no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4#12012yes—
448C22×C4#12114yes—
449C22×C4#12114yes—
450C22×C4#12112yes—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4#12412yes—
454C22×C4C2×D812yes—
455C22×C4C2×D812yes—
456C22×C4C2×SD1612yes—
457C22×C4C2×SD1612yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—416no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—48no—
509C4⋊C4—88no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—42no—
523C42—42no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—81no—
552C23—81no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—81no—
557C23—81no—
558C23—82no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23#39414yes—
564C23#44714yes—
565C23#45014yes—
566C23#45914yes—
567C23#46014yes—
568C23#48614yes—
569C23#49014yes—
570C23C2×#12011yescharacteristic
571C23C2×#12111yescharacteristic
572C23C2×#12411yescharacteristic
573C23#54612yes—
574C23#54712yes—
575C23C22×D811yescharacteristic
576C23C22×SD1611yescharacteristic
577C23C2×AGL(1,ℤ/8)11yescharacteristic
578C23C2×AGL(1,ℤ/8)11yescharacteristic
579C23#57712yes—
580C23#57812yes—
581D4—44no—
582D4—44no—
583D4—816no—
584D4—81no—
585D4—81no—
586D4—81no—
587D4—81no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—816no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—816no—
655C2×C4#39414yes—
656C2×C4C2×#12111yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×#12311yescharacteristic
659C2×C4C2×#12311yescharacteristic
660C2×C4#52511yescharacteristic
661C2×C4#52511yescharacteristic
662C2×C4#52912yes—
663C2×C4#52912yes—
664C2×C4#53511yescharacteristic
665C2×C4#53611yescharacteristic
666C2×C4#53912yes—
667C2×C4#53912yes—
668C2×C4#54612yes—
669C2×C4#54612yes—
670C2×C4#54712yes—
671C2×C4#54712yes—
672C2×C4#55112yes—
673C2×C4#55212yes—
674C8—44no—
675C8—82no—
676C22—24no—
677C22—24no—
678C22—28no—
679C22—24no—
680C22—28no—
681C22—28no—
682C22—28no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—162no—
694C22—162no—
695C22—162no—
696C22—162no—
697C22#204118yes—
698C22C2×#39411yescharacteristic
699C22#240411yescharacteristic
700C22#240611yescharacteristic
701C22#242512yes—
702C22#248812yes—
703C22C2×#44711yescharacteristic
704C22C2×#45011yescharacteristic
705C22#301011yescharacteristic
706C22#301111yescharacteristic
707C22C2×#45911yescharacteristic
708C22C2×#46011yescharacteristic
709C22#304612yes—
710C22C2×#48611yescharacteristic
711C22C2×#49011yescharacteristic
712C22#313712yes—
713C22#329412yes—
714C22#329812yes—
715C22#330612yes—
716C22#330812yes—
717C22#333412yes—
718C22#333512yes—
719C22#334312yes—
720C22#334512yes—
721C4—24no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—84no—
727C4—84no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4#241111yescharacteristic
732C4#241311yescharacteristic
733C4#241912yes—
734C4#242512yes—
735C4#262311yescharacteristic
736C4#285311yescharacteristic
737C2—28no—
738C2—162no—
739C2#1491211yescharacteristic
740C2C2×#204111yescharacteristic
741C2#2351011yescharacteristic
742C2#2396011yescharacteristic
743C2#2396111yescharacteristic
744C2#2408811yescharacteristic
745C2#2408911yescharacteristic
746C2#2520012yes—
747C2#2520412yes—
748C2#2769512yes—
749C2#2771712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×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
4. C2×#1823C2semidirect
5. C22×#340C2semidirect
6. #14130C2semidirect
7. #11267C2semidirect
8. #11207C2semidirect
9. #11189C2semidirect
10. #11066C2semidirect
11. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#18932
C2×#18742
C2×#18232
C22×#3402
#141302
#112672
#112072
#111892
#110662
#107142