Finite Group Databasebeta

← #6476670

Subgroups — #6476670

Order 512.

2184 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#33109C212yesmaximal, complemented
4#32906C212yesmaximal, complemented
5#30109C214yesmaximal, complemented
6#29329C212yesmaximal, complemented
7#28304C212yesmaximal, complemented
8#27681C212yesmaximal, complemented
9#24652C214yesmaximal, complemented
10#24212C212yesmaximal, complemented
11#24006C212yesmaximal
12#23921C212yesmaximal
13#22044C214yesmaximal, complemented
14#20714C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#484C2214yes—
21C2×#462C2214yescomplemented
22C2×#451C2214yes—
23C2×#451C2214yes—
24C2×#431C2212yes—
25#2857—21no—
26#2857—21no—
27#2857—21no—
28#2857—21no—
29#2857C2212yes—
30#2857C2212yes—
31#2857C2212yes—
32#2857C2212yes—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8C2212yes—
38C4×C22⋊Q8C2212yes—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#391C2212yes—
42C2×#391C2211yescharacteristic
43C2×#389—21no—
44C2×#389—21no—
45C2×#389C2211yescharacteristic
46C2×#387—21no—
47C2×#385—21no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×#380—21no—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55#2100C2214yescomplemented
56#2095C2214yescomplemented
57#2043C2214yescomplemented
58#2036C2218yes—
59#2016C2214yescomplemented
60#2001C2218yes—
61#1981C2214yescomplemented
62#1969C2218yescomplemented
63#1947C2214yescomplemented
64#1933C2218yescomplemented
65#1928C2218yescomplemented
66#1878C2218yescomplemented
67#1872C2214yes—
68#1872C2214yes—
69#1872C2214yes—
70#1852C2214yes—
71#1817C2212yes—
72#1803C2214yescomplemented
73#1797C2212yes—
74#1797C2212yes—
75#1787C2214yes—
76#1779C2212yes—
77C4×C22⋊C8C2212yes—
78C4×C22⋊C8C2212yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×Q16C2318yes—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—48no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42—44no—
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#484—48no—
136#462—48no—
137#451—216no—
138#451—48no—
139#431—44no—
140C2×#107C2318yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2318yes—
143C2×#103C2318yes—
144C2×#103C2318yes—
145C2×#103C2318yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2314yes—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391C2314yes—
155#391C2314yes—
156#391C2312yes—
157#391C2312yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2312yes—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—21no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—22no—
212C2×#95—22no—
213#340C2318yes—
214#340C2318yes—
215#336C2314yes—
216#336C2314yes—
217C25C2×D411yescharacteristic
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—28no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8C2414yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4Q1618yescomplemented
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×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×Q16—216no—
262C2×Q16—48no—
263C2×Q16—48no—
264C22×C8C2418yes—
265C4⋊Q8—28no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8#4118yes—
281C4×Q8—28no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—48no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—42no—
346C2×C42—44no—
347C2×C42—44no—
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×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—48no—
365C4⋊C8—28no—
366#103—216no—
367#103—48no—
368#103—48no—
369#103—48no—
370C22⋊C8—28no—
371C22⋊C8—28no—
372C22⋊C8—44no—
373C4×C8—44no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—41no—
393C24—41no—
394C24C2×Q1614yes—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—28no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—28no—
405C2×Q8—24no—
406C2×Q8—416no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—816no—
413C2×Q8C2×#4114yes—
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—22no—
427C22×C4—24no—
428C22×C4—24no—
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—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—416no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4C2×Q1614yes—
487C22×C4C2×Q1614yes—
488C22×C4C2×Q1614yes—
489C22×C4C2×Q1614yes—
490C22×C4C2×Q1614yes—
491C22×C4C2×Q1614yes—
492C22×C4#13714yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4114yes—
505C22×C4H(5,𝔽2)14yes—
506C22×C4C2511yescharacteristic
507Q16—816no—
508C2×C8—28no—
509C2×C8—28no—
510C2×C8—28no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4C2×#4114yes—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—44no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—44no—
559C42—44no—
560C42—44no—
561C42—44no—
562C42H(5,𝔽2)14yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—28no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—28no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23#52512yes—
585C23#53512yes—
586C23C22×Q1612yes—
587C23C22×Q1612yes—
588C23C22×Q1612yes—
589C23C2×#13712yes—
590C23#57512yes—
591C23C23×D411yescharacteristic
592Q8—28no—
593Q8—48no—
594Q8—44no—
595Q8—48no—
596Q8—48no—
597Q8—88no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—84no—
659C2×C4—88no—
660C2×C4—84no—
661C2×C4—88no—
662C2×C4#52512yes—
663C2×C4#53412yes—
664C2×C4#53412yes—
665C2×C4#53512yes—
666C2×C4C22×Q1612yes—
667C2×C4C22×Q1612yes—
668C2×C4C22×Q1612yes—
669C2×C4C22×Q1612yes—
670C2×C4C2×#13712yes—
671C2×C4C2×#13712yes—
672C2×C4#57512yes—
673C2×C4C22×#4112yes—
674C2×C4C2×H(5,𝔽2)12yes—
675C2×C4C2×H(5,𝔽2)12yes—
676C8—48no—
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—22no—
691C22—22no—
692C22—416no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22#310414yes—
698C22#320914yes—
699C22C2×#52511yescharacteristic
700C22C2×#53411yescharacteristic
701C22C2×#53511yescharacteristic
702C22C23×Q1611yescharacteristic
703C22C22×#13711yescharacteristic
704C22C2×#57511yescharacteristic
705C22C2×#57511yescharacteristic
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—41no—
715C4—44no—
716C4—41no—
717C4—44no—
718C4—84no—
719C4—82no—
720C4—84no—
721C4—82no—
722C4#318914yes—
723C4#351914yes—
724C2—24no—
725C2—24no—
726C2—42no—
727C2#3507611yescharacteristic
728C2C2×#310411yescharacteristic
729C2#6056211yescharacteristic
730C2C2×#318911yescharacteristic
731C2#6075111yescharacteristic
732C2C2×#320911yescharacteristic
733C2#6080511yescharacteristic
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. #33109C2semidirect
4. #32906C2semidirect
5. #30109C2semidirect
6. #29329C2semidirect
7. #28304C2semidirect
8. #27681C2semidirect
9. #24652C2semidirect
10. #24212C2semidirect
13. #22044C2semidirect
21. C2×#462C22semidirect
55. #2100C22semidirect
56. #2095C22semidirect
57. #2043C22semidirect
59. #2016C22semidirect
61. #1981C22semidirect
62. #1969C22semidirect
63. #1947C22semidirect
64. #1933C22semidirect
65. #1928C22semidirect
66. #1878C22semidirect
72. #1803C22semidirect
242. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×C4×C22⋊Q82
#331092
#329062
#301092
#293292
#283042
#276812
#246522
#242122
#240062
#239212
#220442
#207142