Finite Group Databasebeta

← #6629791

Subgroups — #6629791

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic
3#33154C214yesmaximal, complemented
4#33153C214yesmaximal, complemented
5#33149C212yesmaximal, complemented
6#33148C212yesmaximal, complemented
7#33147C212yesmaximal, complemented
8#33146C212yesmaximal, complemented
9#21313C212yesmaximal, complemented
10#21171C212yesmaximal, complemented
11#21034C214yesmaximal, complemented
12#20846C212yesmaximal
13#20702C212yesmaximal
14#20626C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—21no—
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#428C2214yes—
21C2×#423C2212yes—
22C2×#407C2212yes—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2212yes—
31C2×#391—21no—
32C2×#389—21no—
33C2×#389C2212yes—
34C2×#384C2211yescharacteristic
35C2×#380—21no—
36C2×#380C2212yes—
37C4×#117—21no—
38C4×#117—21no—
39C4×#117—21no—
40C4×#117—21no—
41C4×#117C2212yes—
42C4×#117C2212yes—
43C4×#117C2212yes—
44C4×#117C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50C2×C4×C22⋊C4—21no—
51C2×#376—21no—
52C2×C43—21no—
53#1988C2214yescomplemented
54#1988C2214yescomplemented
55#1987C2214yescomplemented
56#1987C2214yescomplemented
57#1948C2214yescomplemented
58#1948C2214yescomplemented
59#1947C2214yescomplemented
60#1947C2214yescomplemented
61#1921C2218yescomplemented
62#1920C2218yescomplemented
63#1919C2218yescomplemented
64#1918C2218yescomplemented
65#1854C2218yes—
66#1853C2214yes—
67#1852C2214yes—
68#1825C2218yes—
69#1824C2214yes—
70#1823C2214yes—
71#1797C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1797C2212yes—
75#1736C2214yes—
76#1736C2214yes—
77#1736C2214yes—
78#1728C2212yes—
79C23×Q8—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—48no—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—21no—
97C2×#117—21no—
98C2×#117—24no—
99C2×#117—28no—
100C2×#117Q818yescomplemented
101C2×#117C2312yes—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—28no—
121C22×C42—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#428—216no—
138C2×#107C2318yes—
139C2×#106C2318yes—
140#423—28no—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143#407—28no—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#384—21no—
159#384—21no—
160#384C2312yes—
161#384C2312yes—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380C2314yes—
169#380C2314yes—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194#376—22no—
195#376—22no—
196#376—22no—
197#376—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—22no—
203C2×#95—22no—
204C43—22no—
205C43—22no—
206C43—22no—
207C43—22no—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×Q8—44no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×Q814yes—
252C23×C4#4114yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×M16—28no—
257C2×M16—28no—
258C2×M16—28no—
259C22×C8C2418yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268#117—28no—
269#117—28no—
270#117—28no—
271#117—28no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—24no—
277#117C2×Q818yes—
278#117C2×Q818yescomplemented
279#117#4118yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×Q814yescomplemented
305C2×C4⋊C4C2×Q814yescomplemented
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—48no—
321C2×C42—28no—
322C2×C42—28no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42#4118yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—216no—
368#106—216no—
369C4⋊C8—28no—
370C4⋊C8—28no—
371C22⋊C8—28no—
372C22⋊C8—28no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—41no—
390C24—41no—
391C24C22⋊Q814yes—
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—816no—
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—216no—
411C22×C4—216no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4C22⋊Q814yes—
473C22×C4C22⋊Q814yes—
474C22×C4C22⋊Q814yes—
475C22×C4#12314yes—
476C22×C4#12314yes—
477C22×C4AGL(1,ℤ/8)14yes—
478C22×C4#13714yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×Q812yes—
491C22×C4C22×Q812yes—
492C22×C4C22×Q812yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2511yescharacteristic
499M16—416no—
500C2×C8—28no—
501C2×C8—28no—
502C2×C8—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4C22×Q814yes—
524C4⋊C4C22×Q814yes—
525C4⋊C4C2×#4114yes—
526C4⋊C4C2×#4114yes—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C42—28no—
534C42—28no—
535C42—28no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—28no—
547C42—22no—
548C42—22no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—44no—
557C42#12314yes—
558C42#12314yes—
559C42C2×#4114yes—
560C42C2×#4114yes—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—28no—
570C23—28no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23C2×C22⋊Q812yes—
583C23C2×C22⋊Q812yes—
584C23C2×C22⋊Q812yes—
585C23C2×#12312yes—
586C23#52512yes—
587C23C2×AGL(1,ℤ/8)12yes—
588C23C2×#13712yes—
589C23C23×D411yescharacteristic
590Q8—88no—
591Q8—88no—
592C2×C4—28no—
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—21no—
604C2×C4—21no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—48no—
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—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—88no—
652C2×C4—88no—
653C2×C4C2×C22⋊Q812yes—
654C2×C4C2×C22⋊Q812yes—
655C2×C4C2×C22⋊Q812yes—
656C2×C4C2×#12312yes—
657C2×C4C2×#12312yes—
658C2×C4C2×#12312yes—
659C2×C4C2×#12312yes—
660C2×C4C2×#12312yes—
661C2×C4#52512yes—
662C2×C4#52918yes—
663C2×C4#53318yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C2×#13712yes—
667C2×C4C2×#13712yes—
668C2×C4#57512yes—
669C2×C4#57512yes—
670C2×C4C23×Q811yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4112yes—
673C2×C4C22×#4112yes—
674C8—48no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—416no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22#285514yes—
696C22#307914yes—
697C22C22×C22⋊Q811yescharacteristic
698C22C22×#12311yescharacteristic
699C22C2×#52511yescharacteristic
700C22C22×AGL(1,ℤ/8)11yescharacteristic
701C22C22×#13711yescharacteristic
702C22C2×#57511yescharacteristic
703C22C2×#57511yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—44no—
711C4—44no—
712C4—41no—
713C4—41no—
714C4—44no—
715C4—44no—
716C4—84no—
717C4—84no—
718C4#285514yes—
719C4#309414yes—
720C4#309514yes—
721C4C2×#52912yes—
722C4C2×#53312yes—
723C2—24no—
724C2—24no—
725C2—42no—
726C2C2×#285511yescharacteristic
727C2C2×#307911yescharacteristic
728C2#6048411yescharacteristic
729C2C2×#309411yescharacteristic
730C2C2×#309511yescharacteristic
731C2#6056111yescharacteristic
732C2#6056211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33154C2semidirect
4. #33153C2semidirect
5. #33149C2semidirect
6. #33148C2semidirect
7. #33147C2semidirect
8. #33146C2semidirect
9. #21313C2semidirect
10. #21171C2semidirect
11. #21034C2semidirect
53. #1988C22semidirect
54. #1988C22semidirect
55. #1987C22semidirect
56. #1987C22semidirect
57. #1948C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1947C22semidirect
61. #1921C22semidirect
62. #1920C22semidirect
63. #1919C22semidirect
64. #1918C22semidirect
100. C2×#117Q8semidirect
278. #117C2×Q8semidirect
304. C2×C4⋊C4C2×Q8semidirect
305. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×C4×#1172
#331542
#331532
#331492
#331482
#331472
#331462
#213132
#211712
#210342
#208462
#207022
#206262