Finite Group Databasebeta

← #6629789

Subgroups — #6629789

Order 512.

2442 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#33151C214yesmaximal, complemented
4#33150C214yesmaximal, complemented
5#33149C212yesmaximal, complemented
6#33148C212yesmaximal, complemented
7#33147C212yesmaximal, complemented
8#33146C212yesmaximal, complemented
9#21312C212yesmaximal, complemented
10#21170C212yesmaximal, complemented
11#21033C214yesmaximal, complemented
12#20844C212yesmaximal
13#20700C212yesmaximal
14#20626C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42—21no—
19C23×C42C2211yescharacteristic
20C2×#428C2214yes—
21C2×#423C2212yes—
22C2×#407C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855—21no—
28#2855—21no—
29#2851C2212yes—
30#2851C2212yes—
31#2848—21no—
32#2848—21no—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37#2296C2212yes—
38#2285C2211yescharacteristic
39#2280C2212yes—
40C2×C4×C4⋊C4—21no—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4—21no—
44C2×C4×C4⋊C4—21no—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C22×#95—21no—
53#1989C2218yescomplemented
54#1988C2214yescomplemented
55#1987C2214yescomplemented
56#1949C2218yescomplemented
57#1948C2214yescomplemented
58#1947C2214yescomplemented
59#1921C2218yescomplemented
60#1920C2218yescomplemented
61#1919C2218yescomplemented
62#1918C2218yescomplemented
63#1853C2214yes—
64#1853C2214yes—
65#1852C2214yes—
66#1852C2214yes—
67#1824C2214yes—
68#1824C2214yes—
69#1823C2214yes—
70#1823C2214yes—
71#1797C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1797C2212yes—
75#1736C2214yes—
76#1736C2214yes—
77#1736C2214yes—
78#1728C2212yes—
79C24×C4—21no—
80C24×C4—21no—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16C2314yes—
85#533—28no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125C2312yes—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4Q818yescomplemented
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C42—28no—
115C22×C42—22no—
116C22×C42—22no—
117C22×C42—22no—
118C22×C42—22no—
119C22×C42—22no—
120C22×C42—22no—
121C22×C42—48no—
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#428—216no—
135C2×#107C2318yes—
136C2×#106C2318yes—
137#423—28no—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140#407—28no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#385—22no—
160#385—22no—
161#385—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#382C2312yes—
168#381—22no—
169#380—21no—
170#380—21no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—21no—
175#380C2312yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#379—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2314yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C2×#95—22no—
203C2×#95—22no—
204C2×#95—22no—
205C2×#95—22no—
206C2×#95—22no—
207C2×#95—22no—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—21no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—28no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
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×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×Q814yes—
255C23×C4#4114yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×M16—28no—
260C2×M16—28no—
261C2×M16—28no—
262C22×C8C2418yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265#125—28no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274#117—28no—
275#117—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4C2×Q818yes—
307C2×C4⋊C4C2×Q818yes—
308C2×C4⋊C4C2×Q818yes—
309C2×C4⋊C4#4118yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2414yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—48no—
340C2×C42—48no—
341C2×C42—48no—
342C2×C42—48no—
343C2×C42—48no—
344C2×C42—48no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42#4118yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—216no—
363#106—216no—
364C4⋊C8—28no—
365C4⋊C8—28no—
366C22⋊C8—28no—
367C22⋊C8—28no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—22no—
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—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24C22⋊Q814yes—
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391C24C22×D411yescharacteristic
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C22×C4—232no—
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—216no—
413C22×C4—216no—
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—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
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—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—416no—
460C22×C4—432no—
461C22×C4—432no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4C22⋊Q814yes—
471C22×C4C22⋊Q814yes—
472C22×C4C22⋊Q814yes—
473C22×C4#12314yes—
474C22×C4#12314yes—
475C22×C4AGL(1,ℤ/8)14yes—
476C22×C4#13714yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
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×Q812yes—
489C22×C4C22×Q812yes—
490C22×C4C22×Q812yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2511yescharacteristic
497M16—416no—
498C2×C8—28no—
499C2×C8—28no—
500C2×C8—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4C22×Q818yes—
526C4⋊C4C2×#4118yes—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—48no—
546C42—48no—
547C42—44no—
548C42—44no—
549C42—48no—
550C42—48no—
551C42#12314yes—
552C42#12314yes—
553C42C2×#4114yes—
554C42C2×#4114yes—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—28no—
559C23—28no—
560C23—28no—
561C23—28no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23C2×C22⋊Q812yes—
575C23C2×C22⋊Q812yes—
576C23C2×C22⋊Q812yes—
577C23C2×#12312yes—
578C23#52512yes—
579C23C2×AGL(1,ℤ/8)12yes—
580C23C2×#13712yes—
581C23#57512yes—
582C23#57512yes—
583C23C23×D411yescharacteristic
584Q8—84no—
585Q8—84no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
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—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—28no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—416no—
627C2×C4—416no—
628C2×C4—44no—
629C2×C4—416no—
630C2×C4—416no—
631C2×C4—416no—
632C2×C4—416no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4C2×C22⋊Q812yes—
650C2×C4C2×C22⋊Q812yes—
651C2×C4C2×C22⋊Q812yes—
652C2×C4C2×#12312yes—
653C2×C4C2×#12312yes—
654C2×C4C2×#12312yes—
655C2×C4C2×#12312yes—
656C2×C4C2×#12312yes—
657C2×C4#52512yes—
658C2×C4#52918yes—
659C2×C4#53318yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4C2×#13712yes—
663C2×C4C2×#13712yes—
664C2×C4C23×Q811yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4112yes—
667C2×C4C22×#4112yes—
668C8—48no—
669C22—232no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
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#285514yes—
689C22#307714yes—
690C22#307814yes—
691C22C22×C22⋊Q811yescharacteristic
692C22C22×#12311yescharacteristic
693C22C2×#52511yescharacteristic
694C22C22×AGL(1,ℤ/8)11yescharacteristic
695C22C22×#13711yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57511yescharacteristic
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—48no—
706C4—41no—
707C4—41no—
708C4—48no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#285514yes—
714C4#309214yes—
715C4C2×#52912yes—
716C4C2×#53312yes—
717C2—24no—
718C2—24no—
719C2—24no—
720C2C2×#285511yescharacteristic
721C2C2×#307711yescharacteristic
722C2C2×#307811yescharacteristic
723C2C2×#309211yescharacteristic
724C2#6051111yescharacteristic
725C2#6055811yescharacteristic
726C2#6056011yescharacteristic
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. #33151C2semidirect
4. #33150C2semidirect
5. #33149C2semidirect
6. #33148C2semidirect
7. #33147C2semidirect
8. #33146C2semidirect
9. #21312C2semidirect
10. #21170C2semidirect
11. #21033C2semidirect
53. #1989C22semidirect
54. #1988C22semidirect
55. #1987C22semidirect
56. #1949C22semidirect
57. #1948C22semidirect
58. #1947C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1919C22semidirect
62. #1918C22semidirect
107. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×C4×C4⋊C42
#331512
#331502
#331492
#331482
#331472
#331462
#213122
#211702
#210332
#208442
#207002
#206262