Finite Group Databasebeta

← #6434533

Subgroups — #6434533

Order 512.

1902 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35321C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30128C212yesmaximal, complemented
4#30127C212yesmaximal, complemented
5#30084C212yesmaximal, complemented
6#30049C212yesmaximal, complemented
7#29962C212yesmaximal, complemented
8#29915C212yesmaximal, complemented
9#29723C212yesmaximal, complemented
10#29625C212yesmaximal
11#23824C211yesmaximal, characteristic, complemented
12#23819C211yesmaximal, characteristic, complemented
13#23803C211yesmaximal, characteristic, complemented
14#23799C211yesmaximal, characteristic, complemented
15#23690C214yesmaximal, complemented
16#21459C214yesmaximal, complemented
17#20780C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518C2212yes—
20C2×#450C2214yes—
21C2×#440C2214yescomplemented
22C2×C4×SD16C2214yescomplemented
23#2863C2212yescomplemented
24#2855C2212yes—
25#2647C2211yescharacteristic, complemented
26#2642C2211yescharacteristic, complemented
27#2629C2211yescharacteristic
28#2621C2211yescharacteristic
29#2585—22no—
30#2573—22no—
31#2524—22no—
32#2515—22no—
33#2366—22no—
34#2366—22no—
35D4×C22⋊C4—22no—
36D4×C22⋊C4—22no—
37#2328C2212yes—
38#2328C2212yes—
39#2323C2212yes—
40#2320C2212yescomplemented
41#2309C2212yes—
42C4×C4⋊Q8C2211yescharacteristic, complemented
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#125C2211yescharacteristic, complemented
48#2285C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2100C2212yescomplemented
51#2099C2212yescomplemented
52#2096C2212yescomplemented
53#2095C2212yescomplemented
54#2094C2212yescomplemented
55#2093C2212yescomplemented
56#2085C2214yescomplemented
57#2074C2214yescomplemented
58#2073C2212yescomplemented
59#2073C2212yescomplemented
60#2067C2214yescomplemented
61#2065C2214yescomplemented
62#2055C2214yescomplemented
63#2055C2214yescomplemented
64#2041C2212yes—
65#2041C2212yes—
66#2041C2214yes—
67#2025C2212yescomplemented
68#2025C2212yescomplemented
69#2014C2214yes—
70#2014C2212yes—
71#2014C2212yes—
72#2013C2212yescomplemented
73#2013C2212yescomplemented
74#1877C2214yescomplemented
75#1877C2214yescomplemented
76#1873C2214yes—
77#1864C2214yescomplemented
78#1862C2214yescomplemented
79#1854C2214yes—
80#1840C2212yescomplemented
81#1831C2212yescomplemented
82#1817C2212yes—
83#1798C2212yes—
84#1786C2214yes—
85#1768C2212yescomplemented
86#1762C2214yescomplemented
87C4×#102C2214yescomplemented
88#1750C2212yes—
89#1745C2211yescharacteristic, complemented
90#1744C2211yescharacteristic, complemented
91C23×D4—22no—
92C23×D4C2312yes—
93C22×SD16C2314yes—
94C22×SD16C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127—28no—
97C2×#127—22no—
98C2×#127C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#127C2312yes—
101C2×#125C2311yescharacteristic
102C2×C22⋊Q8C2312yes—
103#518—48no—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C22⋊C4—22no—
119C22×C22⋊C4—22no—
120C22×C42C2311yescharacteristic
121#450—216no—
122#440—48no—
123C4×SD16—48no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#400—22no—
141#398—22no—
142#398—22no—
143#398—22no—
144#397—22no—
145#396—22no—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#390—24no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—24no—
177#386—24no—
178#386—24no—
179#386—24no—
180#385—22no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#383—22no—
185#383—22no—
186#382—22no—
187#382C2311yescharacteristic
188#382C2312yes—
189#381—22no—
190#380—22no—
191#380C2312yes—
192#379—24no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201#376C2311yescharacteristic
202#376C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C25—22no—
216C22×Q8C2412yes—
217C22×D4—216no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—44no—
224C22×D4—44no—
225C22×D4—84no—
226C22×D4—84no—
227C22×D4#4118yes—
228C22×D4C2412yes—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4C2×D412yes—
235C23×C4C2411yescharacteristic
236C2×SD16—28no—
237C2×SD16—28no—
238C2×SD16—44no—
239C2×SD16—44no—
240C2×SD16—44no—
241C2×SD16—44no—
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—42no—
245#127—216no—
246#127—24no—
247#127—24no—
248#127—24no—
249#127—24no—
250#125—42no—
251C22⋊Q8—28no—
252C4×Q8—28no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—44no—
256C4×D4—44no—
257C4×D4—48no—
258C4×D4—48no—
259C4×D4—84no—
260C4×D4—84no—
261C4×D4—84no—
262C4×D4—84no—
263#117—24no—
264#117—42no—
265#117—44no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4C2×D411yescharacteristic
299C2×C22⋊C4C2×D411yescharacteristic
300C2×C22⋊C4C2×D412yes—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316#107—44no—
317#106—44no—
318C4⋊C8—44no—
319#103—44no—
320#103—44no—
321#102—28no—
322#102—28no—
323#102—44no—
324#102—44no—
325#102—44no—
326#102—44no—
327C22⋊C8—28no—
328#97—24no—
329C4×C8—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24—22no—
351C24—22no—
352C24—28no—
353C24—44no—
354C24—44no—
355C24—84no—
356C24—84no—
357C24C22×D411yescharacteristic
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—216no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—416no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—416no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—832no—
383C2×D4C2×#4114yes—
384C2×D4C2×#4114yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C2×#4114yes—
414C22×C4C2511yescharacteristic
415SD16—416no—
416SD16—416no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—82no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—42no—
466C42—42no—
467C42—44no—
468C42—44no—
469C42—82no—
470C42H(5,𝔽2)14yes—
471C23—22no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—416no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—816no—
487C23—168no—
488C23—164no—
489C23—164no—
490C23#52512yes—
491C23C23×D411yescharacteristic
492Q8—44no—
493Q8—44no—
494D4—28no—
495D4—28no—
496D4—416no—
497D4—416no—
498D4—416no—
499D4—44no—
500D4—44no—
501D4—816no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—164no—
540C2×C4—164no—
541C2×C4—164no—
542C2×C4—164no—
543C2×C4#52512yes—
544C2×C4#53512yes—
545C2×C4#53512yes—
546C2×C4#57714yes—
547C2×C4C22×#4112yes—
548C2×C4C2×H(5,𝔽2)12yes—
549C2×C4#58512yes—
550C8—44no—
551C8—44no—
552C22—22no—
553C22—24no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—44no—
558C22—44no—
559C22—48no—
560C22—48no—
561C22—84no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—1616no—
568C22C2×#52511yescharacteristic
569C22C2×#53511yescharacteristic
570C22#344911yescharacteristic
571C22C2×#57712yes—
572C22#358512yes—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—24no—
577C4—42no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4#345914yes—
589C2—42no—
590C2—84no—
591C2—84no—
592C2#3459311yescharacteristic
593C2#6057912yes—
594C2#6081811yescharacteristic
595C2#6083011yescharacteristic
596C2#6099412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #35321C2semidirect
3. #30128C2semidirect
4. #30127C2semidirect
5. #30084C2semidirect
6. #30049C2semidirect
7. #29962C2semidirect
8. #29915C2semidirect
9. #29723C2semidirect
11. #23824C2semidirect
12. #23819C2semidirect
13. #23803C2semidirect
14. #23799C2semidirect
15. #23690C2semidirect
16. #21459C2semidirect
17. #20780C2semidirect
21. C2×#440C22semidirect
22. C2×C4×SD16C22semidirect
23. #2863C22semidirect
25. #2647C22semidirect
26. #2642C22semidirect
40. #2320C22semidirect
42. C4×C4⋊Q8C22semidirect
47. C4×#125C22semidirect
50. #2100C22semidirect
51. #2099C22semidirect
52. #2096C22semidirect
53. #2095C22semidirect
54. #2094C22semidirect
55. #2093C22semidirect
56. #2085C22semidirect
57. #2074C22semidirect
58. #2073C22semidirect
59. #2073C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
62. #2055C22semidirect
63. #2055C22semidirect
67. #2025C22semidirect
68. #2025C22semidirect
72. #2013C22semidirect
73. #2013C22semidirect
74. #1877C22semidirect
75. #1877C22semidirect
77. #1864C22semidirect
78. #1862C22semidirect
80. #1840C22semidirect
81. #1831C22semidirect
85. #1768C22semidirect
86. #1762C22semidirect
87. C4×#102C22semidirect
89. #1745C22semidirect
90. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353212
#342102
#301282
#301272
#300842
#300492
#299622
#299152
#297232
#296252
#238242
#238192
#238032
#237992
#236902
#214592
#207802