Finite Group Databasebeta

← #360000

Subgroups — #360000

Order 512.

1962 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3C2×#1800C212yesmaximal, complemented
4#20626C211yesmaximal, characteristic, complemented
5#11198C212yesmaximal, complemented
6#10967C212yesmaximal
7#10962C212yesmaximal, complemented
8#10913C212yesmaximal, complemented
9#10847C212yesmaximal, complemented
10C22×#121C2211yescharacteristic
11C22×C4×D4C412yes—
12C22×C4×D4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14#2920—28no—
15C2×#432—21no—
16C2×#432—21no—
17C2×#431—21no—
18C2×#431—21no—
19C2×#423C2212yes—
20C2×#423C2212yes—
21C22×#100C2214yes—
22C2×#407C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121—21no—
28C4×#121—21no—
29C4×#121—21no—
30C4×#121—21no—
31C4×#121—21no—
32C4×#121—21no—
33C2×#390C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#388C412yes—
36C2×#386C412yes—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C412yes—
40#1800—28no—
41#1797—21no—
42#1797—21no—
43#1797—21no—
44#1797—21no—
45C2×#343C2214yes—
46C2×#343C2214yes—
47C2×#343C2214yes—
48#1728C2212yes—
49C23×D4C2×C412yes—
50C23×D4C2311yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C412yes—
53C24×C4C2×C411yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16C2314yes—
56C22×M16C2314yes—
57C2×#121—24no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—24no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—24no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—24no—
73C2×C4×D4—48no—
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C412yes—
83C22×C42—24no—
84C22×C42—24no—
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C412yes—
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#432—28no—
99#432—28no—
100#431—28no—
101#431—28no—
102#423—24no—
103#423—24no—
104#423—24no—
105#423D418yescomplemented
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×#100—28no—
111C2×#100—28no—
112C2×#100—44no—
113#407—28no—
114#407—44no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—22no—
119C4×M16—28no—
120C2×#97—22no—
121C2×C4×C8—22no—
122#390—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#388—22no—
131#388—22no—
132#388—22no—
133#388—22no—
134#386—22no—
135#386—22no—
136#386—22no—
137#386—22no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C2×#95C2×C412yes—
151C2×#95C2×C412yes—
152C2×#95C2×C412yes—
153#343—28no—
154#343—28no—
155#343—44no—
156#336—22no—
157#336—22no—
158C25C22×C411yescharacteristic
159C25C22×C412yes—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4#4114yes—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—28no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C412yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2411yescharacteristic
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—24no—
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—24no—
222C2×M16—24no—
223C2×M16C2×D418yes—
224C22×C8—24no—
225C22×C8—24no—
226#121—28no—
227#121—28no—
228#121—28no—
229#121—28no—
230C4×D4—48no—
231C4×D4—48no—
232C4×D4—48no—
233C4×D4—48no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—42no—
239C4×D4—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4#4114yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D414yes—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42#4114yes—
308C4⋊C8—28no—
309C4⋊C8—28no—
310C4⋊C8—44no—
311C4⋊C8—44no—
312#100—416no—
313C22⋊C8—28no—
314C22⋊C8—28no—
315C22⋊C8—44no—
316C22⋊C8—44no—
317#97—28no—
318C4×C8—28no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—41no—
336C24—41no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24C2×C22⋊C411yescharacteristic
342C24C2×C22⋊C411yescharacteristic
343C24#11714yes—
344C24C23×C411yescharacteristic
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—44no—
354C2×D4—416no—
355C2×D4—44no—
356C2×D4—416no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
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—416no—
429C22×C4—416no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—816no—
451C22×C4#10014yescomplemented
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4#11714yes—
463C22×C4C4×D414yes—
464C22×C4C4×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473M16—28no—
474M16—28no—
475M16—44no—
476M16—44no—
477M16—44no—
478M16—44no—
479M16—44no—
480M16—44no—
481C2×C8—28no—
482C2×C8—44no—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—44no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—44no—
490C4⋊C4—41no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C42—21no—
508C42—21no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—44no—
522C42—44no—
523C42#12012yes—
524C42#12012yes—
525C42#12114yes—
526C42#12114yes—
527C42#12312yes—
528C42#12312yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—24no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—24no—
548C23—28no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23C2×#10012yes—
573C23#41312yes—
574C23C22×C22⋊C411yescharacteristic
575C23C2×#11711yescharacteristic
576C23C2×#11711yescharacteristic
577C23C2×C4×D412yes—
578C23#52012yes—
579D4—416no—
580D4—416no—
581D4—44no—
582D4—44no—
583D4—44no—
584D4—44no—
585C2×C4—28no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—24no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—88no—
641C2×C4—88no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4#38612yes—
645C2×C4#38612yes—
646C2×C4#38612yes—
647C2×C4#38612yes—
648C2×C4C2×#10012yes—
649C2×C4C2×#10012yes—
650C2×C4#41312yes—
651C2×C4#41312yes—
652C2×C4C2×#12011yescharacteristic
653C2×C4C2×#12112yes—
654C2×C4C2×#12311yescharacteristic
655C2×C4#52512yes—
656C2×C4#52512yes—
657C2×C4#53511yescharacteristic
658C2×C4#53611yescharacteristic
659C2×C4#53712yes—
660C2×C4#54612yes—
661C2×C4#54712yes—
662C2×C4#54812yes—
663C2×C4#55012yes—
664C2×C4#55212yes—
665C2×C4#55312yes—
666C2×C4#55512yes—
667C2×C4#55612yes—
668C8—48no—
669C8—84no—
670C22—22no—
671C22—22no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—48no—
685C22—48no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—88no—
698C22—88no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22#228011yescharacteristic
704C22C2×#38611yescharacteristic
705C22#231111yescharacteristic
706C22#235212yes—
707C22#235412yes—
708C22C22×#10011yescharacteristic
709C22C2×#41311yescharacteristic
710C22C2×#41311yescharacteristic
711C22C2×#41311yescharacteristic
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—41no—
718C4—41no—
719C4—44no—
720C4—41no—
721C4—41no—
722C4—82no—
723C4—82no—
724C4#190014yes—
725C4#240712yes—
726C4#241412yes—
727C4#241512yes—
728C4#248512yes—
729C2—22no—
730C2—22no—
731C2—42no—
732C2—42no—
733C2—84no—
734C2#1387111yescharacteristic
735C2#2072611yescharacteristic
736C2#2072811yescharacteristic
737C2C2×#190011yescharacteristic
738C2#2098911yescharacteristic
739C2#2187811yescharacteristic
740C2#2188011yescharacteristic
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
1. C2×#2920C2semidirect
3. C2×#1800C2semidirect
4. #20626C2semidirect
5. #11198C2semidirect
7. #10962C2semidirect
8. #10913C2semidirect
9. #10847C2semidirect
105. #423D4semidirect
451. C22×C4#100semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×C4×#1212
C2×#18002
#206262
#111982
#109672
#109622
#109132
#108472