Finite Group Databasebeta

← #364269

Subgroups — #364269

Order 512.

1452 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34109C211yesmaximal, characteristic
2#20837C211yesmaximal, characteristic
3C2×#1826C211yesmaximal, characteristic
4C2×#1733C211yesmaximal, characteristic
5#13844C211yesmaximal, characteristic
6#11226C212yesmaximal, complemented
7#11209C212yesmaximal, complemented
8#10852C212yesmaximal, complemented
9#10808C212yesmaximal, complemented
10#10732C211yesmaximal, characteristic
11#10720C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#423C2212yes—
16C2×#407C2211yescharacteristic
17#2347C411yescharacteristic, complemented
18#2347C411yescharacteristic, complemented
19#2346C411yescharacteristic, complemented
20#2346C411yescharacteristic, complemented
21#2345C412yescomplemented
22#2344C412yescomplemented
23#2343—21no—
24#2343—21no—
25#2342—22no—
26#2298—22no—
27#2296C2212yes—
28#2283—21no—
29#2280C2211yescharacteristic
30C4×#117—21no—
31C4×#117C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1851—21no—
36#1851—21no—
37#1850—22no—
38#1826—24no—
39#1733—24no—
40C2×#339C2211yescharacteristic
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2212yes—
49C2×#339C2212yes—
50#1728C2212yes—
51#1444—21no—
52#1442—21no—
53#1439—22no—
54C4×#95C2212yes—
55C24×C4C2311yescharacteristic
56C22×M16—21no—
57C22×M16—22no—
58C22×M16—24no—
59C22×M16—22no—
60C22×M16—42no—
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2312yes—
64C23×C8—21no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—24no—
73C2×#117—42no—
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117D414yes—
77C2×#117Q814yes—
78C2×#117C2312yes—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2312yes—
93C22×C42C2312yes—
94#423—24no—
95#423—42no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98#407—22no—
99#407—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#385—22no—
115#385—22no—
116#383—21no—
117#383—21no—
118#382—21no—
119#382—21no—
120#381—21no—
121#381—22no—
122#380—22no—
123#380C2×C412yes—
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126#379—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2×C412yes—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4C2×C412yes—
145C4×C22⋊C4C2×C411yescharacteristic
146C4×C22⋊C4C2×C411yescharacteristic
147C4×C22⋊C4D412yes—
148C4×C22⋊C4D411yescharacteristic
149C4×C22⋊C4D411yescharacteristic
150#376—22no—
151#376—21no—
152#376—21no—
153C2×#95—22no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—21no—
157C2×#95D411yescharacteristic
158C2×#95D411yescharacteristic
159C43—22no—
160#339—24no—
161#339—22no—
162#339—22no—
163#339—22no—
164#339—22no—
165#339—24no—
166#339—24no—
167#339—24no—
168#339—42no—
169#339—42no—
170#336—22no—
171#336—22no—
172C25C2×D411yescharacteristic
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×Q812yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4C2411yescharacteristic
194C2×M16—22no—
195C2×M16—28no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—22no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—44no—
209C2×M16—48no—
210C2×M16—44no—
211C2×M16—42no—
212C2×M16#4114yes—
213C22×C8—22no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—44no—
217C22×C8—42no—
218#117—22no—
219#117—22no—
220#117—24no—
221#117—24no—
222#117—24no—
223#117—24no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—28no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—48no—
234#117—48no—
235#117—44no—
236#117C4⋊C414yescomplemented
237#117C4⋊C414yescomplemented
238#117C2×D412yes—
239#117C2×D412yes—
240#117C2×Q812yes—
241#117C2×Q812yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4C4⋊C414yescomplemented
256C2×C4⋊C4C4⋊C414yescomplemented
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4C22⋊C412yes—
280C2×C22⋊C4C22⋊C412yes—
281C2×C22⋊C4C22⋊C412yes—
282C2×C22⋊C4C22⋊C412yes—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42C22×C412yes—
312C2×C42C22×C412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42#4114yes—
316C4⋊C8—44no—
317C4⋊C8—44no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320#95—21no—
321#95—21no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95C22⋊C411yescharacteristic
336#95C22⋊C411yescharacteristic
337#95C22⋊C411yescharacteristic
338#95C22⋊C411yescharacteristic
339#95C22⋊C412yes—
340#95C22⋊C412yes—
341C24—22no—
342C24—22no—
343C24—42no—
344C24—41no—
345C24—41no—
346C24C2×C22⋊C411yescharacteristic
347C24C2×C22⋊C411yescharacteristic
348C24#12112yes—
349C24C22⋊Q812yes—
350C24C22×D411yescharacteristic
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4#11712yes—
416C22×C4#11712yes—
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4C22⋊Q812yes—
421C22×C4C22⋊Q812yes—
422C22×C4C22⋊Q812yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433M16—24no—
434M16—24no—
435M16—48no—
436M16—42no—
437M16—42no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—41no—
442M16—42no—
443M16—42no—
444C2×C8—24no—
445C2×C8—48no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—42no—
451C2×C8—42no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—44no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4C2×C4⋊C412yes—
469C4⋊C4C2×C4⋊C412yes—
470C4⋊C4C2×C4⋊C412yes—
471C4⋊C4C2×C4⋊C412yes—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—44no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—42no—
516C42—42no—
517C42—82no—
518C42—82no—
519C23—22no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23#38012yes—
542C23#38012yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×#12111yescharacteristic
545C23C2×C22⋊Q811yescharacteristic
546C23#54612yes—
547C23#54712yes—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—84no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4#38012yes—
616C2×C4#38012yes—
617C2×C4#42114yes—
618C2×C4C22×C4⋊C411yescharacteristic
619C2×C4C2×#11711yescharacteristic
620C2×C4C4×#4112yes—
621C2×C4#52012yes—
622C2×C4C2×#12111yescharacteristic
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4#52511yescharacteristic
625C2×C4#52511yescharacteristic
626C2×C4#53312yes—
627C2×C4#53511yescharacteristic
628C2×C4#53611yescharacteristic
629C2×C4#54412yes—
630C2×C4#54612yes—
631C2×C4#54712yes—
632C2×C4#55112yes—
633C2×C4#55112yes—
634C2×C4#55212yes—
635C2×C4#55212yes—
636C8—44no—
637C8—81no—
638C8—81no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—82no—
658C22—82no—
659C22—84no—
660C22C2×#38011yescharacteristic
661C22#233912yes—
662C22#241111yescharacteristic
663C22#241311yescharacteristic
664C22#243812yes—
665C22C2×#42112yes—
666C22#290111yescharacteristic
667C22#290111yescharacteristic
668C4—22no—
669C4—22no—
670C4—24no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#229611yescharacteristic
684C4#231711yescharacteristic
685C4#234312yes—
686C4#241111yescharacteristic
687C4#241311yescharacteristic
688C4#243812yes—
689C2—21no—
690C2—21no—
691C2—22no—
692C2—42no—
693C2—42no—
694C2#1396211yescharacteristic
695C2#2065411yescharacteristic
696C2#2065511yescharacteristic
697C2#2250612yes—
698C2#2250712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11226C2semidirect
7. #11209C2semidirect
8. #10852C2semidirect
9. #10808C2semidirect
17. #2347C4semidirect
18. #2347C4semidirect
19. #2346C4semidirect
20. #2346C4semidirect
21. #2345C4semidirect
22. #2344C4semidirect
236. #117C4⋊C4semidirect
237. #117C4⋊C4semidirect
255. C2×C4⋊C4C4⋊C4semidirect
256. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341092
#208372
C2×#18262
C2×#17332
#138442
#112262
#112092
#108522
#108082
#107322
#107202