Finite Group Databasebeta

← #6432487

Subgroups — #6432487

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#23765C214yesmaximal, complemented
4#23764C214yesmaximal, complemented
5#23759C214yesmaximal, complemented
6#23758C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23733C211yesmaximal, characteristic, complemented
9#23659C211yesmaximal, characteristic, complemented
10#23642C212yesmaximal, complemented
11#23631C211yesmaximal, characteristic, complemented
12#23482C212yesmaximal
13C4×#428C212yesmaximal, complemented
14C4×#417C212yesmaximal, complemented
15C4×#407C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25C4×C22⋊Q8C2212yes—
26C4×C22⋊Q8C2212yes—
27C4×#123—21no—
28C4×#123—21no—
29C4×#121C2212yes—
30C4×#121C2212yes—
31C4×#120—21no—
32C4×#120—21no—
33#2297C2211yescharacteristic
34#2296C2211yescharacteristic
35#2290—22no—
36#2283—22no—
37#2283—22no—
38#2283C2214yes—
39#2283C2214yes—
40C2×#383—21no—
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43#2280C2211yescharacteristic
44C2×#381—21no—
45C2×#380C2212yes—
46C42×D4—22no—
47C4×#117C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4—21no—
50C2×#376—21no—
51#2101C2214yescomplemented
52#2101C2214yescomplemented
53#2097C2214yescomplemented
54#2092C2214yescomplemented
55#2085C2214yescomplemented
56#2085C2214yescomplemented
57#2077C2214yes—
58#2077C2214yes—
59#2076C2214yes—
60#2075C2214yes—
61#2074C2214yescomplemented
62#2073C2214yescomplemented
63#2036C2214yes—
64#2035C2214yes—
65#2034C2214yes—
66#2033C2214yes—
67#1776C2214yescomplemented
68#1776C2214yescomplemented
69C4×#107C2214yescomplemented
70C4×#106C2214yescomplemented
71#1763C2214yescomplemented
72#1762C2214yescomplemented
73C4×#103C2214yescomplemented
74C4×#102C2214yescomplemented
75#1749C2214yes—
76C4×C22⋊C8C2214yes—
77#1736C2212yes—
78#1736C2212yes—
79#1736C2212yes—
80#1736C2212yes—
81#1728C2211yescharacteristic
82#1728C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C22×M16C2312yes—
88C22×M16C2312yes—
89#525—24no—
90#525—24no—
91C2×#123—21no—
92C2×C22⋊Q8C2312yes—
93C2×#121C2312yes—
94C2×#120—21no—
95C4×#41—28no—
96C2×C4×Q8C2311yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98C2×C4×D4—22no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×#117—21no—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C42—28no—
113C22×C42—22no—
114C22×C42—21no—
115C22×C42D411yescharacteristic
116C22×C42D412yes—
117C22×C42D411yescharacteristic
118C22×C42D412yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#428—28no—
124#428—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129#417—28no—
130#417—28no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135#407—28no—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C4×M16—28no—
139C2×#97C2314yes—
140C2×C4×C8C2314yes—
141#391C2312yes—
142#390C2312yes—
143#389C2312yes—
144#389C2312yes—
145#388—22no—
146#388—22no—
147#387—22no—
148#387—22no—
149#386—22no—
150#386—22no—
151#386—21no—
152#386—21no—
153#385—22no—
154#385—22no—
155#385—21no—
156#385—21no—
157#383—24no—
158#383—24no—
159#382—22no—
160#382—22no—
161#382C2314yes—
162#382C2314yes—
163#381—24no—
164#381—24no—
165#381—21no—
166#380—22no—
167#380—22no—
168#380C2312yes—
169#380C2314yes—
170#380C2312yes—
171#380C2314yes—
172#379—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4—24no—
183C4×C22⋊C4—24no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192#376—24no—
193#376—24no—
194#376—21no—
195C2×#95—22no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—22no—
201C43—21no—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#336C2314yes—
211#336C2314yes—
212C25C2×D411yescharacteristic
213C2×#41—22no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41#4114yes—
217C22×Q8C2411yescharacteristic
218C22×D4—22no—
219C22×D4—21no—
220C22×D4C2411yescharacteristic
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C2×M16—24no—
250C2×M16—24no—
251C2×M16#4118yes—
252C22×C8C2414yes—
253C22×C8C2414yes—
254#123—28no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257#121—24no—
258#121—24no—
259#120—28no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×D4—216no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—48no—
266C4×D4—48no—
267C4×D4—44no—
268C4×D4—44no—
269#117—22no—
270#117—24no—
271#117—24no—
272#117—44no—
273#117#4114yes—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—48no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—48no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42#4118yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—28no—
337#107—28no—
338#106—28no—
339#106—28no—
340#103—28no—
341#103—28no—
342#102—28no—
343#102—28no—
344C22⋊C8—28no—
345C22⋊C8—28no—
346#97—28no—
347C4×C8—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—42no—
370C24C22×D412yes—
371C24C22×D411yescharacteristic
372#41—216no—
373#41—48no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8C2×#4112yes—
378C2×D4—28no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4C2×#4112yes—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—416no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4#12314yes—
443C22×C4#12314yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4114yes—
456C22×C4C2511yescharacteristic
457M16—216no—
458M16—48no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8C2×#4118yes—
465C4⋊C4—28no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4C2×#4114yes—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—24no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C42—24no—
496C42—24no—
497C42—24no—
498C42—24no—
499C42—24no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—48no—
504C42—44no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—48no—
510C42—44no—
511C42#12314yes—
512C42#12314yes—
513C42C2×#4114yes—
514C42C2×#4114yes—
515C23—22no—
516C23—22no—
517C23—28no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—48no—
526C23—48no—
527C23—44no—
528C23—44no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23C2×#12312yes—
534C23#52512yes—
535C23#53212yes—
536C23#57514yes—
537C23C23×D411yescharacteristic
538Q8—24no—
539Q8—42no—
540D4—24no—
541D4—416no—
542D4—48no—
543D4—42no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4C2×#12312yes—
595C2×C4C2×#12312yes—
596C2×C4C2×#12312yes—
597C2×C4C2×#12312yes—
598C2×C4C2×#12312yes—
599C2×C4#52512yes—
600C2×C4#52918yes—
601C2×C4#52918yes—
602C2×C4#53212yes—
603C2×C4#57514yes—
604C2×C4C22×#4112yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4112yes—
608C8—28no—
609C8—44no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—416no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22#239514yes—
628C22C22×#12311yescharacteristic
629C22C2×#52511yescharacteristic
630C22C2×#53211yescharacteristic
631C22C2×#57512yes—
632C22C2×#57512yes—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—84no—
643C4—81no—
644C4—81no—
645C4—84no—
646C4#239514yes—
647C4#309314yes—
648C4C2×#52912yes—
649C4C2×#52912yes—
650C2—24no—
651C2—44no—
652C2—82no—
653C2C2×#239511yescharacteristic
654C2C2×#309311yescharacteristic
655C2#6051111yescharacteristic
656C2#6056312yes—
657C2#6070212yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #23765C2semidirect
4. #23764C2semidirect
5. #23759C2semidirect
6. #23758C2semidirect
7. #23746C2semidirect
8. #23733C2semidirect
9. #23659C2semidirect
10. #23642C2semidirect
11. #23631C2semidirect
13. C4×#428C2semidirect
14. C4×#417C2semidirect
15. C4×#407C2semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2097C22semidirect
54. #2092C22semidirect
55. #2085C22semidirect
56. #2085C22semidirect
61. #2074C22semidirect
62. #2073C22semidirect
67. #1776C22semidirect
68. #1776C22semidirect
69. C4×#107C22semidirect
70. C4×#106C22semidirect
71. #1763C22semidirect
72. #1762C22semidirect
73. C4×#103C22semidirect
74. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#22832
#237652
#237642
#237592
#237582
#237462
#237332
#236592
#236422
#236312
#234822
C4×#4282
C4×#4172
C4×#4072