Finite Group Databasebeta

← #6461789

Subgroups — #6461789

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#29527C214yesmaximal, complemented
4#29524C214yesmaximal, complemented
5#29523C214yesmaximal, complemented
6#29516C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic
9#23551C211yesmaximal, characteristic, complemented
10#23454C211yesmaximal, characteristic
11#21156C211yesmaximal, characteristic, complemented
12#21155C211yesmaximal, characteristic, complemented
13#21091C211yesmaximal, characteristic
14#21091C211yesmaximal, characteristic
15#21065C211yesmaximal, characteristic
16#21065C211yesmaximal, characteristic
17#21038C211yesmaximal, characteristic, complemented
18#21037C211yesmaximal, characteristic, complemented
19#20368C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2860C2214yes—
32#2644C2214yes—
33#2513—22no—
34#2512—22no—
35#2480—24no—
36#2368—24no—
37#2355—22no—
38#2355—22no—
39#2331C2214yescomplemented
40#2317C2211yescharacteristic
41#2317C2211yescharacteristic
42#2317C2211yescharacteristic
43#2316C2211yescharacteristic
44#2309C2211yescharacteristic
45#2309C2211yescharacteristic
46C4×#125C2214yescomplemented
47#2296C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#382C2211yescharacteristic
50#2274C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2092C2212yes—
56#2092C2212yes—
57#2092C2212yes—
58#2092C2212yes—
59#2059C2212yescomplemented
60#2059C2212yescomplemented
61#2058C2212yescomplemented
62#2057C2212yescomplemented
63#2025C2214yes—
64#2023C2214yes—
65#1944C2214yescomplemented
66#1943C2214yescomplemented
67#1942C2214yescomplemented
68#1941C2214yescomplemented
69#1936C2214yes—
70#1936C2214yes—
71#1934C2214yes—
72#1934C2214yes—
73#1926C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1925C2214yes—
77#1921C2214yescomplemented
78#1920C2214yescomplemented
79#1919C2214yescomplemented
80#1918C2214yescomplemented
81#1746C2212yescomplemented
82#1746C2212yescomplemented
83#1745C2212yescomplemented
84#1744C2212yescomplemented
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×D4C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102#533—24no—
103#533—24no—
104C2×C4⋊Q8C2312yes—
105C2×#126—24no—
106C2×#125C2312yes—
107C2×C22⋊Q8—22no—
108C2×C4×Q8—22no—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—22no—
119C22×C42C2311yescharacteristic
120#427—24no—
121#427—24no—
122C2×#107—24no—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106—24no—
128C2×#106C2312yes—
129C2×#106C2312yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132#418—24no—
133#418—24no—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142#405—24no—
143#405—24no—
144C2×#97C2312yes—
145C2×#97C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#397—24no—
149#397—24no—
150#397—24no—
151#396—24no—
152#391—24no—
153#391—24no—
154#390—21no—
155#390—22no—
156#390—21no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#388—24no—
170#388—24no—
171#388—24no—
172#387—24no—
173#387—24no—
174#387—24no—
175#387—24no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384—21no—
183#382—21no—
184#382—21no—
185#382—21no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#380—22no—
190#379—22no—
191#379—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—24no—
195C4×C4⋊C4—24no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2314yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4D414yes—
204#376C2312yes—
205C2×#95—22no—
206C43C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2311yescharacteristic
218#336C2311yescharacteristic
219#336C2311yescharacteristic
220#336C2311yescharacteristic
221#336C2312yes—
222#336C2312yes—
223#336C2312yes—
224#336C2312yes—
225#336C2312yes—
226#336C2312yes—
227C25—21no—
228C22×Q8—22no—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D412yes—
241C23×C4C2411yescharacteristic
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8#4114yes—
253C22×C8#4114yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C22×C8C2411yescharacteristic
259C22×C8C2411yescharacteristic
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262#126—88no—
263#125—24no—
264#125—24no—
265C22⋊Q8—216no—
266C4×Q8—216no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117#4114yes—
278#117#4114yes—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—84no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42C2×D412yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—24no—
336#107—24no—
337#107—24no—
338#107—24no—
339#106—24no—
340#106—24no—
341#106—24no—
342#106—24no—
343#102—24no—
344#102—24no—
345#102—24no—
346#102—24no—
347#102—28no—
348#102—28no—
349#97—24no—
350#97—24no—
351C4×C8—24no—
352C4×C8—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—82no—
370C24—82no—
371C24C22×D411yescharacteristic
372C2×Q8—28no—
373C2×Q8—82no—
374C2×Q8—82no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C42−1+412yes—
415C22×C42−1+412yes—
416C22×C4C2511yescharacteristic
417M16—44no—
418M16—44no—
419C2×C8—28no—
420C2×C8—28no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8C2×#4112yes—
436C2×C8C2×#4112yes—
437C2×C8C2×#4112yes—
438C2×C8C2×#4112yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4C2×#4114yes—
467C4⋊C4C2×#4114yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—48no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—22no—
490C42—22no—
491C42—44no—
492C42—44no—
493C42—42no—
494C42—42no—
495C42—84no—
496C42—84no—
497C23—22no—
498C23—22no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—88no—
505C23—84no—
506C23—162no—
507C23—162no—
508C23—164no—
509C23#52512yes—
510C23#53212yes—
511C23#53612yes—
512C23C23×D411yescharacteristic
513Q8—88no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—84no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—162no—
552C2×C4—162no—
553C2×C4—162no—
554C2×C4#52512yes—
555C2×C4#53212yes—
556C2×C4#53612yes—
557C2×C4#55214yes—
558C2×C4#55214yes—
559C2×C4#55614yes—
560C2×C4#55614yes—
561C2×C4C22×#4111yescharacteristic
562C2×C4C22×#4111yescharacteristic
563C2×C4C22×#4111yescharacteristic
564C2×C4C22×#4111yescharacteristic
565C2×C4C2×2−1+411yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C8—22no—
568C8—22no—
569C8—22no—
570C8—22no—
571C8—42no—
572C8—42no—
573C22—22no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—84no—
581C22—84no—
582C22—168no—
583C22#260414yes—
584C22C2×#52511yescharacteristic
585C22C2×#53211yescharacteristic
586C22C2×#53611yescharacteristic
587C22#358412yes—
588C22#358512yes—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—44no—
594C4—44no—
595C4—44no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4#260414yes—
602C4C2×#55211yescharacteristic
603C4C2×#55211yescharacteristic
604C4C2×#55611yescharacteristic
605C4C2×#55611yescharacteristic
606C2—24no—
607C2—162no—
608C2C2×#260411yescharacteristic
609C2#6058712yes—
610C2#6071311yescharacteristic
611C2#6071911yescharacteristic
612C2#6089012yes—
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. #34215C2semidirect
3. #29527C2semidirect
4. #29524C2semidirect
5. #29523C2semidirect
6. #29516C2semidirect
7. #23746C2semidirect
9. #23551C2semidirect
11. #21156C2semidirect
12. #21155C2semidirect
17. #21038C2semidirect
18. #21037C2semidirect
19. #20368C2semidirect
39. #2331C22semidirect
46. C4×#125C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
59. #2059C22semidirect
60. #2059C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
65. #1944C22semidirect
66. #1943C22semidirect
67. #1942C22semidirect
68. #1941C22semidirect
77. #1921C22semidirect
78. #1920C22semidirect
79. #1919C22semidirect
80. #1918C22semidirect
81. #1746C22semidirect
82. #1746C22semidirect
83. #1745C22semidirect
84. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342152
#295272
#295242
#295232
#295162
#237462
C2×#20922
#235512
#234542
#211562
#211552
#210912
#210652
#210382
#210372
#203682