Finite Group Databasebeta

← #358123

Subgroups — #358123

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 717 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×C4×#100C212yesmaximal, complemented
4C4×#407C211yesmaximal, characteristic, complemented
5#11204C212yesmaximal, complemented
6#11198C212yesmaximal, complemented
7#11044C212yesmaximal, complemented
8#10967C212yesmaximal
9C4×#343C212yesmaximal, 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—
20C22×#100C2214yes—
21C2×#407C2212yes—
22C2×#407C2212yes—
23C2×C4×M16C2212yes—
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—
40C4×#100—28no—
41#1749—21no—
42#1749—21no—
43C4×C22⋊C8—21no—
44C4×C22⋊C8—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—28no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×#100—28no—
106C2×#100—28no—
107C2×#100—28no—
108#407—28no—
109#407—28no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C4×M16—24no—
115C4×M16—24no—
116C4×M16—24no—
117C4×M16D418yescomplemented
118C2×#97—22no—
119C2×#97—22no—
120C2×C4×C8—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—28no—
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—24no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4#4114yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—28no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
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×C412yes—
202C23×C4C22×C412yes—
203C23×C4C22×C412yes—
204C23×C4C22×C411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
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—24no—
220C2×M16C2×D418yes—
221C2×M16#4118yes—
222C22×C8—24no—
223C22×C8—24no—
224#121—28no—
225#121—28no—
226#121—28no—
227#121—28no—
228C4×D4—48no—
229C4×D4—48no—
230C4×D4—48no—
231C4×D4—48no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—42no—
237C4×D4—42no—
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—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4#4114yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—44no—
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D414yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42#4114yes—
305C4⋊C8—28no—
306C4⋊C8—28no—
307#100—232no—
308C22⋊C8—28no—
309C22⋊C8—28no—
310C22⋊C8—28no—
311C22⋊C8—28no—
312#97—28no—
313#97—44no—
314C4×C8—28no—
315C4×C8—44no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—41no—
333C24—41no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340C24#11714yes—
341C24C23×C411yescharacteristic
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—44no—
349C2×D4—416no—
350C2×D4—44no—
351C2×D4—416no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
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—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
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—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—816no—
437C22×C4#10014yescomplemented
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4#11714yes—
449C22×C4C4×D414yes—
450C22×C4C4×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4114yes—
458M16—28no—
459M16—28no—
460M16—28no—
461M16—28no—
462M16—44no—
463M16—44no—
464M16—44no—
465M16—44no—
466C2×C8—28no—
467C2×C8—28no—
468C2×C8—44no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8—44no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—44no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C42—21no—
492C42—21no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—44no—
506C42—44no—
507C42#12012yes—
508C42#12012yes—
509C42#12114yes—
510C42#12114yes—
511C42#12312yes—
512C42#12312yes—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—24no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—24no—
532C23—28no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23C2×#10012yes—
557C23#41312yes—
558C23C22×C22⋊C411yescharacteristic
559C23C2×#11711yescharacteristic
560C23C2×#11711yescharacteristic
561C23C2×C4×D412yes—
562C23C4×#4112yes—
563D4—416no—
564D4—416no—
565D4—44no—
566D4—48no—
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—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—88no—
622C2×C4—88no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4#38612yes—
626C2×C4#38612yes—
627C2×C4#38612yes—
628C2×C4#38612yes—
629C2×C4C2×#10012yes—
630C2×C4C2×#10012yes—
631C2×C4#41312yes—
632C2×C4#41312yes—
633C2×C4C2×#12011yescharacteristic
634C2×C4C2×#12112yes—
635C2×C4C2×#12311yescharacteristic
636C2×C4#52514yes—
637C2×C4#52512yes—
638C2×C4#52512yes—
639C2×C4#52512yes—
640C2×C4#52914yes—
641C2×C4#52914yes—
642C2×C4#53214yes—
643C2×C4#53212yes—
644C8—48no—
645C8—48no—
646C22—22no—
647C22—22no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—48no—
661C22—48no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—88no—
674C22—88no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22#228011yescharacteristic
680C22C2×#38611yescharacteristic
681C22#229012yes—
682C22#229312yes—
683C22#229711yescharacteristic
684C22C22×#10011yescharacteristic
685C22C2×#41311yescharacteristic
686C22C2×#41311yescharacteristic
687C22C2×#41311yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—41no—
694C4—41no—
695C4—44no—
696C4—41no—
697C4—41no—
698C4—82no—
699C4—82no—
700C4#190014yes—
701C4#238812yes—
702C4#239214yes—
703C4#239512yes—
704C2—22no—
705C2—22no—
706C2—42no—
707C2—42no—
708C2—84no—
709C2#1383911yescharacteristic
710C2#2072611yescharacteristic
711C2#2072811yescharacteristic
712C2C2×#190011yescharacteristic
713C2#2098911yescharacteristic
714C2#2142411yescharacteristic
715C2#2142611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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×C4×#100C2semidirect
4. C4×#407C2semidirect
5. #11204C2semidirect
6. #11198C2semidirect
7. #11044C2semidirect
9. C4×#343C2semidirect
117. C4×M16D4semidirect
437. 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×C4×#1002
C4×#4072
#112042
#111982
#110442
#109672
C4×#3432