Finite Group Databasebeta

← #368151

Subgroups — #368151

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1851C211yesmaximal, characteristic, complemented
4C2×#1729C211yesmaximal, characteristic, complemented
5#13839C211yesmaximal, characteristic
6#11255C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10876C212yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10#10714C212yesmaximal, complemented
11#10713C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×C8×D4—24no—
18C2×C8×D4—24no—
19C2×C8×D4—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29#1870—24no—
30#1851—24no—
31#1729—24no—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441C411yescharacteristic
41#1441C411yescharacteristic
42#1440C412yes—
43#1439C411yescharacteristic
44#1439C411yescharacteristic
45#1438C412yes—
46C4×#95C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—42no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—24no—
62C23×C8—42no—
63C23×C8—42no—
64C23×C8D414yes—
65C23×C8C2312yes—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4D414yes—
76C2×C4×D4Q814yes—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—42no—
80C2×#117—42no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2×C411yescharacteristic
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C412yes—
90C22×C42C2312yes—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93C8×D4—232no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—42no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—42no—
111C2×C22⋊C8D414yes—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—42no—
118C4×C22⋊C4—42no—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C411yescharacteristic
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127#336—24no—
128#336—24no—
129#336—24no—
130#336—24no—
131#336—24no—
132#336—42no—
133#336—42no—
134#336—42no—
135C25C22×C411yescharacteristic
136C25C22×C411yescharacteristic
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C4⋊C414yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×Q812yes—
169C23×C4C2×Q812yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×M16—28no—
174C2×M16—48no—
175C2×M16—48no—
176C2×M16—44no—
177C2×M16—44no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—28no—
182C22×C8—28no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—28no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—44no—
210C22×C8—48no—
211C22×C8—48no—
212C22×C8—48no—
213C22×C8—48no—
214C22×C8—44no—
215C22×C8—44no—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8#4114yes—
221C22×C8#4114yes—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—28no—
226#117—44no—
227#117—44no—
228#117—48no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C4⋊C414yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4C4⋊C414yes—
253C2×C22⋊C4C4⋊C414yes—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C2×D412yes—
271C2×C42C2×Q812yes—
272C4⋊C8—28no—
273C4⋊C8—48no—
274C22⋊C8—28no—
275C22⋊C8—28no—
276C22⋊C8—48no—
277C22⋊C8—44no—
278C4×C8—28no—
279#95—22no—
280#95—22no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—42no—
285#95—42no—
286#95—42no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24C2×C4⋊C412yes—
304C24C2×C4⋊C412yes—
305C24C4×D412yes—
306C24C4×D412yes—
307C24#13112yes—
308C24#13112yes—
309C24C23×C411yescharacteristic
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—28no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
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—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C2×C4⋊C412yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4#12014yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q814yes—
399C22×C4#12314yes—
400C22×C4#13114yes—
401C22×C4#13112yes—
402C22×C4#13112yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×Q811yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)12yes—
413M16—416no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—416no—
429C2×C8—416no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8#12314yes—
450C4⋊C4—28no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23#38514yes—
516C23#42414yes—
517C23C22×C4⋊C411yescharacteristic
518C23C2×C4×D411yescharacteristic
519C23C2×C4×D411yescharacteristic
520C23C4×#4112yes—
521C23#51812yes—
522C23C2×#13111yescharacteristic
523C23C2×#13111yescharacteristic
524C23C2×#13111yescharacteristic
525C23C2×#13111yescharacteristic
526C23C2×#13112yes—
527C23#56812yes—
528D4—232no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4#38514yes—
590C2×C4#42414yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52912yes—
598C2×C4#53411yescharacteristic
599C2×C4#53511yescharacteristic
600C2×C4#53812yes—
601C2×C4#54612yes—
602C2×C4#54612yes—
603C2×C4#54612yes—
604C2×C4#54612yes—
605C2×C4#55312yes—
606C2×C4#55312yes—
607C2×C4#55412yes—
608C2×C4#55412yes—
609C8—24no—
610C8—42no—
611C8—44no—
612C8—42no—
613C8—42no—
614C8—82no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—28no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22C2×#38511yescharacteristic
638C22C4×#12311yescharacteristic
639C22#232111yescharacteristic
640C22#234912yes—
641C22#236412yes—
642C22C2×#42411yescharacteristic
643C22C2×#42411yescharacteristic
644C22#290612yes—
645C22#292911yescharacteristic
646C22#292911yescharacteristic
647C22#292911yescharacteristic
648C22#292911yescharacteristic
649C22#293111yescharacteristic
650C22#293211yescharacteristic
651C22#293312yes—
652C22#296612yes—
653C22#296612yes—
654C22#296812yes—
655C22#296912yes—
656C22#297612yes—
657C22#297712yes—
658C22#297912yes—
659C22#298112yes—
660C4—24no—
661C4—24no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#240411yescharacteristic
669C4#240511yescharacteristic
670C4#244412yes—
671C4#245012yes—
672C4#248711yescharacteristic
673C4#257911yescharacteristic
674C2—44no—
675C2—44no—
676C2#1401611yescharacteristic
677C2#2083411yescharacteristic
678C2#2083911yescharacteristic
679C2#2156611yescharacteristic
680C2#2156711yescharacteristic
681C2#2202911yescharacteristic
682C2#2203111yescharacteristic
683C2#2269712yes—
684C2#2270112yes—
685C2#2312212yes—
686C2#2312312yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1851C2semidirect
4. C2×#1729C2semidirect
6. #11255C2semidirect
7. #11042C2semidirect
8. #10876C2semidirect
9. #10714C2semidirect
10. #10714C2semidirect
11. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#18702
C2×#18512
C2×#17292
#138392
#112552
#110422
#108762
#107142
#107132