Finite Group Databasebeta

← #412715

Subgroups — #412715

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C211yesmaximal, characteristic, complemented
2C2×#2065C211yesmaximal, characteristic
3C2×#1973C211yesmaximal, characteristic
4C2×#1973C211yesmaximal, characteristic
5C2×#1968C211yesmaximal, characteristic
6C2×#1943C211yesmaximal, characteristic, complemented
7C2×#1846C211yesmaximal, characteristic, complemented
8C2×#1771C211yesmaximal, characteristic, complemented
9#15237C211yesmaximal, characteristic, complemented
10#15222C211yesmaximal, characteristic
11#11122C211yesmaximal, characteristic, complemented
12#11121C211yesmaximal, characteristic
13#11052C211yesmaximal, characteristic, complemented
14#11051C211yesmaximal, characteristic
15#10863C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43#2066—24no—
44#2065—24no—
45#1973—24no—
46#1973—24no—
47#1968—24no—
48#1943—24no—
49#1846—24no—
50#1771—24no—
51C2×#340C2211yescharacteristic
52C2×#340C2211yescharacteristic
53C2×#340C2211yescharacteristic
54C2×#340C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57#1443—22no—
58#1441—22no—
59#1439—22no—
60#1439—22no—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C23×Q8C2311yescharacteristic
64C23×D4C2311yescharacteristic
65C22×SD16—22no—
66C22×SD16—22no—
67C22×SD16—22no—
68C22×SD16—22no—
69C22×SD16—22no—
70C22×SD16—22no—
71C22×SD16Q814yes—
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D412yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107D414yes—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—22no—
104C2×#106—22no—
105C2×#106—41no—
106C2×#106—41no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8D414yes—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—41no—
117C2×#103—41no—
118C2×#103—41no—
119C2×#102—22no—
120C2×#102—22no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—41no—
126C2×#102—41no—
127C2×#102—41no—
128C2×#97—22no—
129C2×#97—22no—
130C2×#97—22no—
131C2×#97—22no—
132C2×#97—22no—
133C2×#97D414yes—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136C2×C4×C8D414yes—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—41no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390—41no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#382—22no—
154#382—22no—
155#382—41no—
156#382—41no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C2×#95—22no—
160C2×#95—22no—
161C2×#95—21no—
162C2×#95—21no—
163C2×#95—22no—
164C2×#95—22no—
165C2×#95—22no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—22no—
169C2×#95D412yes—
170#340—22no—
171#340—22no—
172#340—22no—
173#340—22no—
174#336—22no—
175#336—22no—
176C25—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8C2×Q812yes—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C2×Q812yes—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C2×SD16—28no—
218C2×SD16—28no—
219C2×SD16—28no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C22×C8C2×D412yes—
246C22×C8C2×D412yes—
247C22×C8C2×Q812yes—
248C22×C8#4112yes—
249C22×C8#4112yes—
250C22×C8#4112yes—
251C22×C8#4112yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—82no—
302C2×C22⋊C4—82no—
303C2×C22⋊C4—82no—
304C2×C22⋊C4—82no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42#4112yes—
323#107—28no—
324#107—44no—
325#106—44no—
326#106—44no—
327#106—44no—
328C4⋊C8—28no—
329#103—44no—
330#103—44no—
331#103—44no—
332#102—44no—
333#102—44no—
334#102—44no—
335#97—28no—
336#97—44no—
337C4×C8—28no—
338#95—24no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—82no—
349C24—22no—
350C24—41no—
351C24—41no—
352C24—41no—
353C24—41no—
354C24—41no—
355C24—41no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24C22×D411yescharacteristic
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4#12114yes—
476C22×C4#12712yes—
477C22×C4#12712yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×Q811yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4111yescharacteristic
487C22×C4C2×#4111yescharacteristic
488C22×C4H(5,𝔽2)11yescharacteristic
489C22×C4H(5,𝔽2)11yescharacteristic
490C22×C42−1+411yescharacteristic
491C22×C42−1+411yescharacteristic
492SD16—416no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8#12114yes—
514C2×C8#12114yes—
515C2×C8#12114yes—
516C2×C8#12114yes—
517C2×C8C22⋊Q814yes—
518C2×C8C22⋊Q814yes—
519C2×C8C22⋊Q814yes—
520C2×C8#12614yes—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—48no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—84no—
547C4⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—82no—
570C23—82no—
571C23—81no—
572C23—82no—
573C23—82no—
574C23—81no—
575C23—81no—
576C23—81no—
577C23—81no—
578C23—82no—
579C23—82no—
580C23—81no—
581C23—82no—
582C23—82no—
583C23—81no—
584C23—81no—
585C23—82no—
586C23—82no—
587C23C2×#12111yescharacteristic
588C23C2×#12711yescharacteristic
589C23#53211yescharacteristic
590C23D4211yescharacteristic
591C23#54711yescharacteristic
592C23D4×Q811yescharacteristic
593C23#55011yescharacteristic
594C23#57512yes—
595C23#57512yes—
596C23#57712yes—
597C23#57712yes—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—81no—
603Q8—81no—
604Q8—81no—
605Q8—81no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—81no—
611D4—81no—
612D4—81no—
613D4—81no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
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—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—81no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—82no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—82no—
674C2×C4—82no—
675C2×C4C2×#12111yescharacteristic
676C2×C4C2×#12111yescharacteristic
677C2×C4C2×#12111yescharacteristic
678C2×C4C2×#12111yescharacteristic
679C2×C4C2×C22⋊Q811yescharacteristic
680C2×C4C2×C22⋊Q811yescharacteristic
681C2×C4C2×C22⋊Q811yescharacteristic
682C2×C4C2×#12611yescharacteristic
683C2×C4#53211yescharacteristic
684C2×C4#53211yescharacteristic
685C2×C4#53511yescharacteristic
686C2×C4#53611yescharacteristic
687C2×C4#53711yescharacteristic
688C2×C4#53711yescharacteristic
689C2×C4#54211yescharacteristic
690C2×C4#54211yescharacteristic
691C2×C4D4211yescharacteristic
692C2×C4#54711yescharacteristic
693C2×C4#54811yescharacteristic
694C2×C4#54811yescharacteristic
695C2×C4D4×Q811yescharacteristic
696C2×C4#55011yescharacteristic
697C2×C4#55311yescharacteristic
698C2×C4#55311yescharacteristic
699C2×C4#55411yescharacteristic
700C2×C4#55411yescharacteristic
701C2×C4#55611yescharacteristic
702C2×C4#55611yescharacteristic
703C8—24no—
704C8—42no—
705C8—42no—
706C8#39618yes—
707C22—82no—
708C22—84no—
709C22—84no—
710C22—82no—
711C22—82no—
712C22—161no—
713C22—161no—
714C22—162no—
715C22—161no—
716C22—161no—
717C22—162no—
718C22#241611yescharacteristic
719C22#245211yescharacteristic
720C22#251211yescharacteristic
721C22#253811yescharacteristic
722C22#254111yescharacteristic
723C22#254211yescharacteristic
724C22#285611yescharacteristic
725C22#305412yes—
726C22#305712yes—
727C22#315311yescharacteristic
728C22#315311yescharacteristic
729C22#315411yescharacteristic
730C22#315411yescharacteristic
731C22#317412yes—
732C22#318312yes—
733C22#326712yes—
734C22#327112yes—
735C22#334812yes—
736C22#335012yes—
737C22#338812yes—
738C22#338812yes—
739C22#340112yes—
740C22#340112yes—
741C4—42no—
742C4—42no—
743C4—44no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—161no—
749C4—161no—
750C4—161no—
751C4—161no—
752C4C2×#39611yescharacteristic
753C4#241611yescharacteristic
754C4#245211yescharacteristic
755C4#251211yescharacteristic
756C4#253811yescharacteristic
757C4#254511yescharacteristic
758C4#256611yescharacteristic
759C4#262011yescharacteristic
760C2—162no—
761C2#1513411yescharacteristic
762C2#2478612yes—
763C2#2649412yes—
764C2#2802412yes—
765C2#2827012yes—
766C2#2847911yescharacteristic
767C2#2848111yescharacteristic
768C2#2854711yescharacteristic
769C2#2854911yescharacteristic
770C2#3317711yescharacteristic
771C2#3317811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
1. C2×#2066C2semidirect
6. C2×#1943C2semidirect
7. C2×#1846C2semidirect
8. C2×#1771C2semidirect
9. #15237C2semidirect
11. #11122C2semidirect
13. #11052C2semidirect
15. #10863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#20652
C2×#19732
C2×#19682
C2×#19432
C2×#18462
C2×#17712
#152372
#152222
#111222
#111212
#110522
#110512
#108632