Finite Group Databasebeta

← #364134

Subgroups — #364134

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1900C212yesmaximal
3#20626C211yesmaximal, characteristic, complemented
4C2×#1728C211yesmaximal, characteristic, complemented
5#11204C212yesmaximal, complemented
6#11044C212yesmaximal, complemented
7#10802C212yesmaximal, complemented
8#10762C212yesmaximal, complemented
9#10739C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×#121C2211yescharacteristic
12C22×C4×D4C412yes—
13C22×C4×D4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#423C2212yes—
16C22×#100C2214yes—
17C2×#407C2212yes—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C4×#121—21no—
22C4×#121—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C2×#390C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#388C412yes—
29C2×#386C412yes—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C412yes—
33#1900—28no—
34#1797—21no—
35#1797—21no—
36#1797—21no—
37#1797—21no—
38C2×#343C2214yes—
39C2×#343C2214yes—
40C2×#343C2214yes—
41#1728—21no—
42#1728—21no—
43#1728C2212yes—
44#1728C2212yes—
45#1728C2212yes—
46C2×#336—21no—
47C2×#336—21no—
48C23×D4C2×C412yes—
49C23×D4C2311yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C412yes—
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—28no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16—44no—
59C22×M16C2314yes—
60C22×M16C2314yes—
61C23×C8—22no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121D414yes—
69C2×#121Q814yes—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—44no—
79C2×C4×D4—48no—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C412yes—
89C22×C42—44no—
90C22×C42C2×C412yes—
91C22×C42C2×C411yescharacteristic
92C22×C42C2×C411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42D411yescharacteristic
98C22×C42D411yescharacteristic
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#423—44no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×#100—28no—
107C2×#100—28no—
108C2×#100—44no—
109#407—24no—
110#407—24no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113#390—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144#343—28no—
145#343—28no—
146#343—44no—
147#336—22no—
148#336—22no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153C25C22×C411yescharacteristic
154C25C22×C412yes—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—24no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4C4⋊C418yes—
167C22×D4C2×D412yes—
168C22×D4C2×Q812yes—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C412yes—
195C23×C4C22×C412yes—
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D412yes—
203C23×C4C2×Q812yes—
204C23×C4#4112yes—
205C23×C4#4112yes—
206C23×C4C2411yescharacteristic
207C2×M16—28no—
208C2×M16—24no—
209C2×M16—24no—
210C2×M16—24no—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—24no—
219C2×M16—416no—
220C2×M16—416no—
221C2×M16—44no—
222C2×M16—44no—
223C2×M16#4118yes—
224C22×C8—28no—
225C22×C8—24no—
226C22×C8—24no—
227C22×C8—44no—
228C22×C8—44no—
229#121—24no—
230#121—24no—
231#121—24no—
232#121—24no—
233#121—24no—
234#121—24no—
235#121—24no—
236#121—24no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—48no—
242C4×D4—48no—
243C4×D4—48no—
244C4×D4—48no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×Q812yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C4⋊C418yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C4⋊C8—44no—
308C4⋊C8—44no—
309#100—416no—
310C22⋊C8—44no—
311C22⋊C8—44no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—41no—
327C24—41no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C2×C4⋊C412yes—
335C24C2×C4⋊C414yes—
336C24#11712yes—
337C24C23×C411yescharacteristic
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—44no—
347C2×D4—416no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—416no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—416no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—416no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—816no—
428C22×C4#10014yescomplemented
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C4⋊C412yes—
440C22×C4C2×C4⋊C414yes—
441C22×C4#11712yes—
442C22×C4#12112yes—
443C22×C4#12112yes—
444C22×C4#12112yes—
445C22×C4#12112yes—
446C22×C4C22⋊Q812yes—
447C22×C4C22⋊Q812yes—
448C22×C4C22⋊Q812yes—
449C22×C4C22⋊Q812yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×Q811yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458M16—28no—
459M16—28no—
460M16—416no—
461M16—44no—
462M16—44no—
463M16—44no—
464M16—44no—
465M16—44no—
466M16—44no—
467C2×C8—28no—
468C2×C8—416no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—44no—
511C42—44no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—28no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23#38012yes—
554C23#38012yes—
555C23C2×#10012yes—
556C23#41312yes—
557C23C22×C22⋊C411yescharacteristic
558C23C22×C4⋊C411yescharacteristic
559C23C2×#11711yescharacteristic
560C23C4×#4112yes—
561C23#51812yes—
562D4—416no—
563D4—416no—
564D4—44no—
565D4—44no—
566D4—44no—
567D4—44no—
568C2×C4—28no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4#38012yes—
630C2×C4#38012yes—
631C2×C4C2×#10012yes—
632C2×C4C2×#10012yes—
633C2×C4#41312yes—
634C2×C4#41312yes—
635C2×C4C2×#12111yescharacteristic
636C2×C4C2×#12111yescharacteristic
637C2×C4C2×C22⋊Q811yescharacteristic
638C2×C4C2×C22⋊Q811yescharacteristic
639C2×C4#52512yes—
640C2×C4#53212yes—
641C2×C4#53412yes—
642C2×C4#53412yes—
643C2×C4#54614yes—
644C2×C4#55114yes—
645C2×C4#55314yes—
646C2×C4#55414yes—
647C8—48no—
648C8—84no—
649C22—22no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—48no—
667C22—48no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—88no—
676C22—88no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22#180014yes—
682C22C2×#38011yescharacteristic
683C22#229711yescharacteristic
684C22#230811yescharacteristic
685C22#234912yes—
686C22#235312yes—
687C22C22×#10011yescharacteristic
688C22C2×#41311yescharacteristic
689C22C2×#41311yescharacteristic
690C22C2×#41311yescharacteristic
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—44no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—82no—
703C4—82no—
704C4#253412yes—
705C4#254312yes—
706C4#256412yes—
707C4#257912yes—
708C2—22no—
709C2—22no—
710C2—24no—
711C2—42no—
712C2—84no—
713C2#1396011yescharacteristic
714C2C2×#180011yescharacteristic
715C2#2063711yescharacteristic
716C2#2142411yescharacteristic
717C2#2142611yescharacteristic
718C2#2175211yescharacteristic
719C2#2175411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20626C2semidirect
4. C2×#1728C2semidirect
5. #11204C2semidirect
6. #11044C2semidirect
7. #10802C2semidirect
8. #10762C2semidirect
9. #10739C2semidirect
428. C22×C4#100semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#19002
#206262
C2×#17282
#112042
#110442
#108022
#107622
#107392