Finite Group Databasebeta

← #6432489

Subgroups — #6432489

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C211yesmaximal, characteristic
2#34048C211yesmaximal, characteristic
3#23773C214yesmaximal, complemented
4#23772C214yesmaximal, complemented
5#23769C214yesmaximal, complemented
6#23768C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic, complemented
8#23732C211yesmaximal, characteristic, complemented
9#23657C211yesmaximal, characteristic, complemented
10#23642C212yesmaximal
11#23629C211yesmaximal, characteristic, complemented
12#23481C212yesmaximal
13C4×#428C212yesmaximal, complemented
14C4×#419C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#419C2212yes—
22C2×#419C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2323—22no—
26#2323—22no—
27#2323C2214yes—
28#2323C2214yes—
29#2314C2212yes—
30#2314C2212yes—
31#2314C2212yes—
32#2314C2212yes—
33#2313—21no—
34#2313—21no—
35#2312—21no—
36#2312—21no—
37#2298—22no—
38#2298—22no—
39C2×#391C2211yescharacteristic
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#387—21no—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47#2280C2212yes—
48C2×#380C2211yescharacteristic
49C4×#117C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yescomplemented
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2097C2214yescomplemented
55#2096C2212yescomplemented
56#2095C2212yescomplemented
57#2085C2214yescomplemented
58#2084C2212yescomplemented
59#2083C2212yescomplemented
60#2077C2214yes—
61#2077C2214yes—
62#2076C2214yes—
63#2075C2214yes—
64#2074C2214yescomplemented
65#2072C2212yescomplemented
66#2072C2212yescomplemented
67#2036C2214yes—
68#2036C2214yes—
69#2035C2214yes—
70#2035C2214yes—
71#1776C2214yescomplemented
72#1776C2214yescomplemented
73C4×#107C2214yescomplemented
74C4×#106C2214yescomplemented
75#1763C2214yescomplemented
76#1763C2214yescomplemented
77C4×#103C2214yescomplemented
78C4×#103C2214yescomplemented
79#1750C2214yes—
80#1750C2214yes—
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×M16C2312yes—
90C22×M16C2312yes—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2314yes—
95C2×C4×Q8—22no—
96C2×C4×Q8—28no—
97C2×C4×Q8C2314yes—
98C2×#117—28no—
99C2×#117D412yes—
100C2×#117D412yes—
101C2×#117C2312yes—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#428—28no—
122#428—28no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127#419—28no—
128#419—28no—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133#407—28no—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C4×M16—28no—
137C2×#97C2314yes—
138C2×C4×C8C2314yes—
139#391—21no—
140#391—21no—
141#391C2312yes—
142#391C2312yes—
143#389—24no—
144#389—24no—
145#389—21no—
146#389—21no—
147#389C2312yes—
148#389C2312yes—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—24no—
158#388—24no—
159#388—24no—
160#388—24no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—24no—
170#387—24no—
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#382C2312yes—
176#382C2312yes—
177#381—22no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380—22no—
183#380C2312yes—
184#380C2312yes—
185#380C2312yes—
186#380C2312yes—
187#380C2314yes—
188#379—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2314yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203#376—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C43—22no—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2312yes—
221#336C2312yes—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—24no—
226C22×Q8—24no—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C2×M16—24no—
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C2×M16—24no—
250C2×M16#4118yes—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C22⋊Q8—216no—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256C4×Q8—28no—
257C4×Q8—28no—
258C4×Q8—48no—
259#117—28no—
260#117—28no—
261#117—24no—
262#117—24no—
263#117—24no—
264#117—24no—
265#117—24no—
266#117—24no—
267#117—24no—
268#117—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4#4114yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42—82no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42#4118yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2412yes—
338#107—28no—
339#107—28no—
340#106—28no—
341#106—28no—
342#103—28no—
343#103—28no—
344#103—28no—
345#103—28no—
346C22⋊C8—28no—
347C22⋊C8—28no—
348#97—28no—
349C4×C8—28no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—216no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8—42no—
387C2×Q8C2×#4114yes—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
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—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4#12314yes—
433C22×C4#12314yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4114yes—
446C22×C4C2511yescharacteristic
447M16—216no—
448M16—48no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8C2×#4118yes—
455C4⋊C4—216no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4C2×#4114yes—
486C22⋊C4—216no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—84no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—48no—
503C42—44no—
504C42—44no—
505C42—82no—
506C42—82no—
507C42—82no—
508C42—82no—
509C42—84no—
510C42#12318yes—
511C42C2×#4114yes—
512C42C2×#4114yes—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23C2×#12312yes—
532C23#52512yes—
533C23#53212yes—
534C23C23×D411yescharacteristic
535Q8—28no—
536Q8—48no—
537Q8—44no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
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—42no—
590C2×C4—42no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4C2×#12314yes—
596C2×C4C2×#12314yes—
597C2×C4C2×#12312yes—
598C2×C4#52512yes—
599C2×C4#52918yes—
600C2×C4#52918yes—
601C2×C4#53212yes—
602C2×C4C22×#4112yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4112yes—
606C8—28no—
607C8—44no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—88no—
622C22#239514yes—
623C22C22×#12311yescharacteristic
624C22C2×#52511yescharacteristic
625C22C2×#53211yescharacteristic
626C22#358511yescharacteristic
627C22#358611yescharacteristic
628C22#358611yescharacteristic
629C22#358611yescharacteristic
630C4—24no—
631C4—24no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#239514yes—
646C4C2×#52912yes—
647C4C2×#52912yes—
648C2—24no—
649C2—82no—
650C2C2×#239511yescharacteristic
651C2#6051911yescharacteristic
652C2#6052011yescharacteristic
653C2#6056511yescharacteristic
654C2#6056611yescharacteristic
655C2#6070311yescharacteristic
656C2#6070511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23773C2semidirect
4. #23772C2semidirect
5. #23769C2semidirect
6. #23768C2semidirect
7. #23747C2semidirect
8. #23732C2semidirect
9. #23657C2semidirect
11. #23629C2semidirect
13. C4×#428C2semidirect
14. C4×#419C2semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
54. #2097C22semidirect
55. #2096C22semidirect
56. #2095C22semidirect
57. #2085C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
64. #2074C22semidirect
65. #2072C22semidirect
66. #2072C22semidirect
71. #1776C22semidirect
72. #1776C22semidirect
73. C4×#107C22semidirect
74. C4×#106C22semidirect
75. #1763C22semidirect
76. #1763C22semidirect
77. C4×#103C22semidirect
78. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
#340482
#237732
#237722
#237692
#237682
#237472
#237322
#236572
#236422
#236292
#234812
C4×#4282
C4×#4192
#203772