Finite Group Databasebeta

← #374319

Subgroups — #374319

Order 512.

2162 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#424C211yesmaximal, characteristic, complemented
2C2×#2392C211yesmaximal, characteristic
3C2×#1955C211yesmaximal, characteristic, complemented
4C2×#1954C211yesmaximal, characteristic, complemented
5C2×#1870C212yesmaximal, complemented
6C2×#1869C212yesmaximal, complemented
7C2×#1750C211yesmaximal, characteristic, complemented
8#11042C212yesmaximal, complemented
9#11042C212yesmaximal, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#424—24no—
18C2×#424—24no—
19C2×#424—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26C22×C22⋊C8C2212yes—
27C22×C22⋊C8C2212yes—
28C22×C22⋊C8C2212yes—
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2392—24no—
32#2392—24no—
33C2×#398C411yescharacteristic
34C2×#397C411yescharacteristic
35C2×#396C412yes—
36C2×#394C411yescharacteristic
37C2×#394C412yes—
38C2×#392C411yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41#1955—24no—
42#1954—24no—
43#1870—28no—
44#1869—28no—
45#1750—24no—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C23×D4C2×C412yes—
52C23×D4C2×C411yescharacteristic
53C24×C4C2312yes—
54C24×C4C2311yescharacteristic
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—42no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—24no—
64C23×C8—42no—
65C23×C8C2312yes—
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117D414yes—
88C2×#117D414yes—
89C22×C4⋊C4C2×C412yes—
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C4⋊C4C2×C412yes—
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C22⋊C4C2×C412yes—
95C22×C22⋊C4C2×C412yes—
96C22×C22⋊C4C2×C412yes—
97C22×C22⋊C4C2×C412yes—
98C22×C22⋊C4C2×C412yes—
99C22×C22⋊C4C2×C411yescharacteristic
100C22×C22⋊C4C2×C411yescharacteristic
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#424—232no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8D414yescomplemented
116C2×C4⋊C8D414yescomplemented
117C2×C22⋊C8—24no—
118C2×C22⋊C8—24no—
119C2×C22⋊C8—24no—
120C2×C22⋊C8—24no—
121C2×C22⋊C8—24no—
122C2×C22⋊C8—24no—
123C2×C22⋊C8—24no—
124C2×C22⋊C8—24no—
125C2×C22⋊C8—24no—
126C2×C22⋊C8—24no—
127C2×C22⋊C8—42no—
128C2×C22⋊C8—42no—
129C2×C22⋊C8—42no—
130C2×C22⋊C8—42no—
131C2×C22⋊C8D418yescomplemented
132C2×#97—22no—
133C2×#97—22no—
134C2×#97—22no—
135C2×C4×C8—22no—
136C2×C4×C8—22no—
137C2×C4×C8—22no—
138#398—24no—
139#397—24no—
140#396—28no—
141#394—24no—
142#394—28no—
143#392—24no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148C2×#95C2×C411yescharacteristic
149C2×#95C2×C411yescharacteristic
150C2×#95C2×C412yes—
151C2×#95C2×C412yes—
152#336—24no—
153#336—24no—
154#336—24no—
155#336—24no—
156#336—24no—
157C25C22×C412yes—
158C25C22×C411yescharacteristic
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
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—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4C22×C412yes—
192C23×C4C22×C412yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C412yes—
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D414yes—
202C23×C4#4114yes—
203C23×C4C2411yescharacteristic
204C2×M16—28no—
205C2×M16—48no—
206C2×M16—48no—
207C2×M16—44no—
208C2×M16—44no—
209C22×C8—28no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—48no—
239C22×C8—48no—
240C22×C8—44no—
241C22×C8—44no—
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C22×C8C2×D412yes—
246C22×C8C2×D414yes—
247C22×C8C2×D414yes—
248C4×D4—48no—
249C4×D4—48no—
250C4×D4—48no—
251C4×D4—48no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4—44no—
256#117—28no—
257#117—28no—
258#117—28no—
259#117—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4C22⋊C414yes—
274C2×C4⋊C4C22⋊C414yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4C22⋊C414yes—
295C2×C22⋊C4C22⋊C414yes—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42#4112yes—
313C2×C42#4112yes—
314C4⋊C8—28no—
315C4⋊C8—28no—
316C4⋊C8—28no—
317C4⋊C8—28no—
318C22⋊C8—216no—
319C22⋊C8—48no—
320C22⋊C8—48no—
321C22⋊C8—48no—
322C22⋊C8—48no—
323#97—44no—
324C4×C8—44no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24C2×C22⋊C412yes—
351C24C4×D414yes—
352C24#13114yes—
353C24C23×C411yescharacteristic
354C2×D4—28no—
355C2×D4—216no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—44no—
366C22×C4—21no—
367C22×C4—21no—
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—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
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—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—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
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—42no—
443C22×C4—42no—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C412yes—
448C22×C4C2×C22⋊C412yes—
449C22×C4C4×D412yes—
450C22×C4C4×D412yes—
451C22×C4C4×D412yes—
452C22×C4C4×D412yes—
453C22×C4C4×D414yes—
454C22×C4#12118yes—
455C22×C4#12414yes—
456C22×C4#12714yes—
457C22×C4#13114yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468M16—416no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—416no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—42no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42#12014yes—
525C42#12114yes—
526C42#12114yes—
527C42#12314yes—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23#39014yes—
584C23C22×C22⋊C411yescharacteristic
585C23C2×C4×D412yes—
586C23C2×C4×D411yescharacteristic
587C23C2×C4×D411yescharacteristic
588C23#52012yes—
589C23C2×#13112yes—
590C23C2×#13112yes—
591C23#56812yes—
592C23#56812yes—
593D4—416no—
594D4—432no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4#38614yes—
653C2×C4#38614yes—
654C2×C4#39014yes—
655C2×C4C2×#12011yescharacteristic
656C2×C4C2×#12111yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×#12112yes—
659C2×C4C2×#12311yescharacteristic
660C2×C4C2×#12411yescharacteristic
661C2×C4C2×#12711yescharacteristic
662C2×C4#53511yescharacteristic
663C2×C4#53611yescharacteristic
664C2×C4#53712yes—
665C2×C4D4212yes—
666C2×C4D4212yes—
667C2×C4#54712yes—
668C2×C4#54712yes—
669C2×C4#54812yes—
670C2×C4#54812yes—
671C2×C4#55312yes—
672C2×C4#55312yes—
673C8—44no—
674C8—44no—
675C8—42no—
676C8—42no—
677C8—42no—
678C8—42no—
679C22—24no—
680C22—28no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22C2×#38611yescharacteristic
704C22C2×#39011yescharacteristic
705C22#231111yescharacteristic
706C22#236112yes—
707C22#236112yes—
708C22#288012yes—
709C22#288012yes—
710C22#292911yescharacteristic
711C22#292912yes—
712C22#293011yescharacteristic
713C22#293012yes—
714C22#293111yescharacteristic
715C22#293211yescharacteristic
716C22#293312yes—
717C22#293312yes—
718C22#293312yes—
719C22#293512yes—
720C22#299212yes—
721C22#299312yes—
722C22#299412yes—
723C22#299512yes—
724C4—24no—
725C4—24no—
726C4—24no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—82no—
731C4—82no—
732C4#241612yes—
733C4#241612yes—
734C4#243311yescharacteristic
735C4#243611yescharacteristic
736C4#250111yescharacteristic
737C4#250411yescharacteristic
738C2—44no—
739C2—84no—
740C2#1413711yescharacteristic
741C2#2089911yescharacteristic
742C2#2090011yescharacteristic
743C2#2119112yes—
744C2#2185512yes—
745C2#2299112yes—
746C2#2299212yes—
747C2#2299312yes—
748C2#2299412yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
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
1. C22×#424C2semidirect
3. C2×#1955C2semidirect
4. C2×#1954C2semidirect
5. C2×#1870C2semidirect
6. C2×#1869C2semidirect
7. C2×#1750C2semidirect
8. #11042C2semidirect
9. #11042C2semidirect
10. #10835C2semidirect
11. #10834C2semidirect
115. C2×C4⋊C8D4semidirect
116. C2×C4⋊C8D4semidirect
131. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#23922
C2×#19552
C2×#19542
C2×#18702
C2×#18692
C2×#17502
#110422
#108352
#108342