Finite Group Databasebeta

← #6461796

Subgroups — #6461796

Order 512.

1648 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#29527C214yesmaximal, complemented
4#29524C214yesmaximal, complemented
5#29521C212yesmaximal, complemented
6#29518C212yesmaximal, complemented
7#29514C212yesmaximal, complemented
8#29513C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23731C211yesmaximal, characteristic
11#23552C211yesmaximal, characteristic, complemented
12C2×#2023C211yesmaximal, characteristic
13#21160C212yesmaximal, complemented
14#21089C212yesmaximal
15#21063C212yesmaximal
16#21036C212yesmaximal, complemented
17#20368C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#418C2212yes—
22C22×#102C2212yes—
23C2×#405C2212yes—
24#2860C2212yes—
25#2860C2212yes—
26#2644C2212yes—
27#2644C2212yes—
28#2513—21no—
29#2513—21no—
30#2512—21no—
31#2512—21no—
32#2480—22no—
33#2480—22no—
34#2369—21no—
35#2369—21no—
36#2365—21no—
37#2365—21no—
38#2345—22no—
39Q8×C22⋊C4—22no—
40#2331C2214yescomplemented
41#2317C2211yescharacteristic
42#2316C2211yescharacteristic
43#2316C2212yes—
44#2309C2211yescharacteristic
45#2309C2211yescharacteristic
46C4×C4⋊Q8C2212yescomplemented
47C4×C4⋊Q8C2212yescomplemented
48#2296C2211yescharacteristic
49C2×#390C2211yescharacteristic
50#2285C2211yescharacteristic
51#2274C2212yes—
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#2058C2212yescomplemented
63#2058C2212yescomplemented
64#2057C2212yescomplemented
65#2057C2212yescomplemented
66#2023C2212yes—
67#2023C2212yes—
68#2023C2214yes—
69#1944C2214yescomplemented
70#1943C2214yescomplemented
71#1942C2214yescomplemented
72#1941C2214yescomplemented
73#1936C2214yes—
74#1936C2214yes—
75#1934C2214yes—
76#1934C2214yes—
77#1926C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1921C2214yescomplemented
82#1920C2214yescomplemented
83#1919C2214yescomplemented
84#1918C2214yescomplemented
85#1746C2214yescomplemented
86#1745C2212yescomplemented
87#1744C2212yescomplemented
88#1736C2212yes—
89#1735C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#126—22no—
105C2×#126—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42C2311yescharacteristic
119#428—28no—
120#428—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125#418—28no—
126C2×#102—28no—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131#405—28no—
132C2×#97C2314yes—
133C2×C4×C8C2314yes—
134#397—22no—
135#397—22no—
136#397—22no—
137#397—22no—
138#397—22no—
139#397—22no—
140#396—22no—
141#396—22no—
142#391—22no—
143#391—22no—
144#391—24no—
145#391—24no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—24no—
181#387—24no—
182#385—21no—
183#385—21no—
184#385—22no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#382—21no—
189#382C2312yes—
190#380—21no—
191#380—21no—
192#380—24no—
193#380—22no—
194#379—22no—
195#379—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C22⋊C4—24no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D412yes—
213#376C2312yes—
214C43C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234C25—21no—
235C22×Q8—28no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—82no—
239C22×Q8—82no—
240C22×D4—21no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16#4118yes—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264#126—84no—
265#126—84no—
266C4×Q8—84no—
267C4×Q8—84no—
268#117—24no—
269#117—22no—
270#117—22no—
271#117—24no—
272#117—24no—
273#117—24no—
274#117—22no—
275#117—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
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—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4#4118yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
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×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—28no—
346#107—28no—
347#106—28no—
348#106—28no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—28no—
353#97—28no—
354C4×C8—28no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—81no—
378C24—81no—
379C24—81no—
380C24—81no—
381C24C22×D411yescharacteristic
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8—816no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—48no—
397C2×D4—48no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C42−1+412yes—
430C22×C42−1+412yes—
431C22×C4C2511yescharacteristic
432M16—216no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—48no—
442C2×C8C2×#4118yes—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C2×#4118yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C42—22no—
496C42—22no—
497C42—24no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—84no—
510C42—84no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—162no—
526C23—162no—
527C23—162no—
528C23—162no—
529C23#52512yes—
530C23#53212yes—
531C23#53612yes—
532C23C23×D411yescharacteristic
533Q8—416no—
534Q8—88no—
535D4—44no—
536D4—44no—
537D4—84no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
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—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—88no—
587C2×C4—162no—
588C2×C4#52512yes—
589C2×C4#53212yes—
590C2×C4#53612yes—
591C2×C4#55218yes—
592C2×C4#55618yes—
593C2×C4#57612yes—
594C2×C4#57712yes—
595C2×C4C22×#4112yes—
596C2×C4C22×#4112yes—
597C2×C4C2×2−1+411yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C8—28no—
600C8—42no—
601C8—42no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—168no—
616C22#260414yes—
617C22C2×#52511yescharacteristic
618C22C2×#53211yescharacteristic
619C22C2×#53611yescharacteristic
620C22C2×#57611yescharacteristic
621C22C2×#57711yescharacteristic
622C22#358511yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—48no—
630C4—41no—
631C4—42no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—81no—
640C4#260414yes—
641C4C2×#55212yes—
642C4C2×#55612yes—
643C2—24no—
644C2—162no—
645C2C2×#260411yescharacteristic
646C2#6057811yescharacteristic
647C2#6058211yescharacteristic
648C2#6072211yescharacteristic
649C2#6072711yescharacteristic
650C2#6088811yescharacteristic
651C2#6089211yescharacteristic
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
2. #34216C2semidirect
3. #29527C2semidirect
4. #29524C2semidirect
5. #29521C2semidirect
6. #29518C2semidirect
7. #29514C2semidirect
8. #29513C2semidirect
9. #23744C2semidirect
11. #23552C2semidirect
13. #21160C2semidirect
16. #21036C2semidirect
17. #20368C2semidirect
40. #2331C22semidirect
46. C4×C4⋊Q8C22semidirect
47. C4×C4⋊Q8C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
58. #2099C22semidirect
62. #2058C22semidirect
63. #2058C22semidirect
64. #2057C22semidirect
65. #2057C22semidirect
69. #1944C22semidirect
70. #1943C22semidirect
71. #1942C22semidirect
72. #1941C22semidirect
81. #1921C22semidirect
82. #1920C22semidirect
83. #1919C22semidirect
84. #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
#352662
#342162
#295272
#295242
#295212
#295182
#295142
#295132
#237442
#237312
#235522
C2×#20232
#211602
#210892
#210632
#210362
#203682