Finite Group Databasebeta

← #6461977

Subgroups — #6461977

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#29590C214yesmaximal, complemented
4#29586C214yesmaximal, complemented
5#29580C212yesmaximal, complemented
6#29578C212yesmaximal, complemented
7#29574C212yesmaximal, complemented
8#29572C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23731C211yesmaximal, characteristic
11#23658C211yesmaximal, characteristic, complemented
12C2×#2071C211yesmaximal, characteristic
13#21160C212yesmaximal, complemented
14#21089C212yesmaximal
15#21035C212yesmaximal, complemented
16#20876C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#518C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#418C2212yes—
22C22×#102C2212yes—
23C2×#405C2212yes—
24#2859C2212yescomplemented
25#2859C2212yescomplemented
26#2647C2214yescomplemented
27#2571—21no—
28#2571—21no—
29#2552—21no—
30#2552—21no—
31#2442—22no—
32#2442—22no—
33#2357—21no—
34#2357—21no—
35#2354—21no—
36#2354—21no—
37#2342—22no—
38#2342—22no—
39#2319C2212yes—
40#2319C2212yes—
41#2317C2212yes—
42#2317C2211yescharacteristic
43#2316C2212yes—
44#2309C2211yescharacteristic
45#2296C2211yescharacteristic
46#2293C2212yes—
47#2293C2212yes—
48C2×#390C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2101C2212yescomplemented
54#2101C2212yescomplemented
55#2100C2211yescharacteristic, complemented
56#2100C2211yescharacteristic, complemented
57#2099C2211yescharacteristic, complemented
58#2099C2211yescharacteristic, complemented
59#2094C2212yes—
60#2093C2212yes—
61#2092C2214yes—
62#2084C2212yescomplemented
63#2084C2212yescomplemented
64#2083C2212yescomplemented
65#2083C2212yescomplemented
66#2071C2212yes—
67#2071C2212yes—
68#2071C2214yes—
69#1944C2214yescomplemented
70#1943C2214yescomplemented
71#1942C2214yescomplemented
72#1941C2214yescomplemented
73#1936C2214yes—
74#1936C2214yes—
75#1934C2214yes—
76#1934C2214yes—
77#1921C2214yescomplemented
78#1920C2214yescomplemented
79#1919C2214yescomplemented
80#1918C2214yescomplemented
81#1864C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1746C2214yescomplemented
86#1745C2212yescomplemented
87#1744C2212yescomplemented
88#1736C2212yes—
89#1735C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×#125—22no—
99C2×#125—22no—
100#518—24no—
101#518—24no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2312yes—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—22no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C42C2311yescharacteristic
121#428—28no—
122#428—28no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127#418—28no—
128C2×#102—28no—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133#405—28no—
134C2×#97C2314yes—
135C2×C4×C8C2314yes—
136#400—24no—
137#400—24no—
138#398—24no—
139#398—24no—
140#390—21no—
141#390—21no—
142#390—21no—
143#390—21no—
144#390C2312yes—
145#390C2312yes—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#389C2311yescharacteristic
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2311yescharacteristic
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#385—21no—
173#385—21no—
174#385—22no—
175#385—24no—
176#384—21no—
177#384—22no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—22no—
185#382—21no—
186#382—22no—
187#382—22no—
188#382C2312yes—
189#381—22no—
190#381—22no—
191#380—21no—
192#380—21no—
193#380—24no—
194#380—22no—
195#380—22no—
196#379—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214#376C2312yes—
215C43C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235C25—21no—
236C22×D4—21no—
237C22×D4—24no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4#4114yes—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×M16—24no—
250C2×M16—24no—
251C2×M16#4118yes—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2414yes—
258#125—84no—
259#125—84no—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—24no—
273#117—84no—
274#117—84no—
275#117#4114yes—
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—22no—
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—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—216no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#107—28no—
349#106—28no—
350#106—28no—
351#102—28no—
352#102—28no—
353#102—28no—
354#102—28no—
355#97—28no—
356C4×C8—28no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—22no—
376C24—24no—
377C24—42no—
378C24—42no—
379C24—81no—
380C24—81no—
381C24—81no—
382C24—81no—
383C24C22×D411yescharacteristic
384C2×D4—24no—
385C2×D4—216no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—48no—
392C2×D4C2×#4114yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4114yes—
431C22×C42−1+414yes—
432C22×C4C2511yescharacteristic
433M16—216no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—48no—
443C2×C8C2×#4118yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4C2×#4112yes—
480C4⋊C4C2×#4112yes—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—42no—
507C42—42no—
508C42—44no—
509C42—42no—
510C42—82no—
511C42—82no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—24no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—48no—
524C23—84no—
525C23—84no—
526C23—162no—
527C23—162no—
528C23—162no—
529C23—162no—
530C23#52512yes—
531C23#53212yes—
532C23#53612yes—
533C23C23×D411yescharacteristic
534D4—28no—
535D4—48no—
536D4—44no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—88no—
586C2×C4—88no—
587C2×C4—162no—
588C2×C4—164no—
589C2×C4#52512yes—
590C2×C4#53212yes—
591C2×C4#53612yes—
592C2×C4#55218yes—
593C2×C4#55218yes—
594C2×C4#57612yes—
595C2×C4#57712yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4112yes—
599C2×C4C2×2−1+412yes—
600C8—28no—
601C8—42no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—168no—
617C22#261114yes—
618C22C2×#52511yescharacteristic
619C22C2×#53211yescharacteristic
620C22C2×#53611yescharacteristic
621C22C2×#57611yescharacteristic
622C22C2×#57711yescharacteristic
623C22#358511yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—84no—
639C4—81no—
640C4—81no—
641C4#261114yes—
642C4C2×#55212yes—
643C4C2×#55212yes—
644C2—24no—
645C2—84no—
646C2C2×#261111yescharacteristic
647C2#6059011yescharacteristic
648C2#6059111yescharacteristic
649C2#6072211yescharacteristic
650C2#6072711yescharacteristic
651C2#6087111yescharacteristic
652C2#6087311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #29590C2semidirect
4. #29586C2semidirect
5. #29580C2semidirect
6. #29578C2semidirect
7. #29574C2semidirect
8. #29572C2semidirect
9. #23744C2semidirect
11. #23658C2semidirect
13. #21160C2semidirect
15. #21035C2semidirect
17. #20368C2semidirect
24. #2859C22semidirect
25. #2859C22semidirect
26. #2647C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
58. #2099C22semidirect
62. #2084C22semidirect
63. #2084C22semidirect
64. #2083C22semidirect
65. #2083C22semidirect
69. #1944C22semidirect
70. #1943C22semidirect
71. #1942C22semidirect
72. #1941C22semidirect
77. #1921C22semidirect
78. #1920C22semidirect
79. #1919C22semidirect
80. #1918C22semidirect
85. #1746C22semidirect
86. #1745C22semidirect
87. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352862
#342552
#295902
#295862
#295802
#295782
#295742
#295722
#237442
#237312
#236582
C2×#20712
#211602
#210892
#210352
#208762
#203682