Finite Group Databasebeta

← #6495689

Subgroups — #6495689

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#33119C212yesmaximal, complemented
4#32914C212yesmaximal
5#29987C212yesmaximal, complemented
6#29798C212yesmaximal, complemented
7#29568C214yesmaximal, complemented
8#29369C212yesmaximal, complemented
9#29368C212yesmaximal, complemented
10#29338C211yesmaximal, characteristic, complemented
11#29332C211yesmaximal, characteristic, complemented
12#28311C211yesmaximal, characteristic
13#28302C211yesmaximal, characteristic
14#24071C212yesmaximal, complemented
15#23942C212yesmaximal
16#22162C214yesmaximal, complemented
17#20784C211yesmaximal, characteristic, complemented
18C2×#541C2212yes—
19C2×#518C2211yescharacteristic
20C2×#484C2212yes—
21C2×#483C2212yes—
22C2×#461C2214yes—
23C2×#450C2212yes—
24C2×#450C2212yes—
25#2868C2211yescharacteristic, complemented
26#2859C2211yescharacteristic, complemented
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2642C2212yescomplemented
30#2621C2212yes—
31#2506—22no—
32#2506—22no—
33#2496—22no—
34#2496—22no—
35#2357—22no—
36#2354—22no—
37#2342—22no—
38#2342—22no—
39#2327C2212yescomplemented
40#2319C2212yes—
41#2319C2212yes—
42#2317C2211yescharacteristic
43#2309C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic, complemented
45C4×C4⋊Q8C2211yescharacteristic, complemented
46#2296C2211yescharacteristic
47#2293C2212yes—
48#2293C2212yes—
49C2×#390C2211yescharacteristic
50#2274C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2094C2212yes—
55#2093C2212yes—
56#2059C2214yescomplemented
57#2043C2214yescomplemented
58#2043C2212yescomplemented
59#2043C2212yescomplemented
60#2025C2214yes—
61#2013C2214yes—
62#2013C2212yes—
63#2013C2212yes—
64#1981C2214yescomplemented
65#1973C2214yes—
66#1968C2214yes—
67#1949C2214yescomplemented
68#1935C2214yescomplemented
69#1933C2214yescomplemented
70#1928C2214yescomplemented
71#1927C2214yescomplemented
72#1880C2214yes—
73#1880C2214yes—
74#1877C2214yes—
75#1877C2214yes—
76#1873C2212yes—
77#1873C2212yes—
78#1873C2212yes—
79#1873C2212yes—
80#1865C2214yescomplemented
81#1863C2214yescomplemented
82#1839C2212yescomplemented
83#1831C2212yescomplemented
84#1818C2212yes—
85#1804C2214yes—
86#1798C2212yes—
87#1768C2212yescomplemented
88#1749C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1746C2212yescomplemented
91C23×D4C2311yescharacteristic
92C22×SD16C2314yes—
93C22×SD16C2314yes—
94#541—44no—
95#541—44no—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#126—22no—
101C2×#126—22no—
102C2×#125—22no—
103C2×#125—22no—
104C2×#125C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107#518—28no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C42C2311yescharacteristic
123#484—44no—
124#483—44no—
125#461—216no—
126#450—44no—
127#450—44no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143#400—24no—
144#400—24no—
145#398—24no—
146#397—24no—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#390—21no—
152#390—21no—
153#390—22no—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—24no—
177#387—24no—
178#385—22no—
179#385—21no—
180#385—21no—
181#384—21no—
182#384—22no—
183#384—22no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—21no—
187#382C2312yes—
188#381—22no—
189#381—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#379—21no—
195#379—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4D412yes—
209#376C2311yescharacteristic
210#376C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2312yes—
222C25—21no—
223C22×Q8C2412yes—
224C22×Q8C2412yes—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—24no—
228C22×D4—42no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—21no—
233C23×C4C2×D412yes—
234C23×C4C2411yescharacteristic
235C2×SD16—28no—
236C2×SD16—28no—
237C2×SD16—44no—
238C2×SD16—44no—
239C2×SD16—44no—
240C2×SD16—44no—
241C22×C8C2414yes—
242C22×C8C2414yes—
243C4⋊Q8—28no—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248#126—48no—
249#126—48no—
250#125—42no—
251#125—42no—
252#125—48no—
253#125—48no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C4×Q8—42no—
259C4×Q8—42no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×D4—216no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4#4118yes—
267#117—24no—
268#117—24no—
269#117—24no—
270#117—42no—
271#117—84no—
272#117—84no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
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—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—216no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—84no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D411yescharacteristic
318C2×C22⋊C4C2×D411yescharacteristic
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—42no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—44no—
338#106—44no—
339C4⋊C8—28no—
340#103—28no—
341#103—28no—
342#103—44no—
343#103—44no—
344#103—44no—
345#103—44no—
346#102—44no—
347#102—44no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350#97—42no—
351C4×C8—42no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C24—24no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—82no—
372C24—82no—
373C24C22×D411yescharacteristic
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—416no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4C2×#4114yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C4C2511yescharacteristic
421SD16—88no—
422SD16—88no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4C2×#4114yes—
458C22⋊C4—24no—
459C22⋊C4—28no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—24no—
463C22⋊C4—28no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C42—24no—
476C42—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—44no—
488C42—44no—
489C42—82no—
490C42—82no—
491C42C2×#4114yes—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—24no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—88no—
505C23—162no—
506C23—162no—
507C23—164no—
508C23#53512yes—
509C23C23×D411yescharacteristic
510Q8—44no—
511Q8—44no—
512Q8—44no—
513Q8—44no—
514D4—28no—
515D4—48no—
516D4—44no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
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—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—84no—
558C2×C4—88no—
559C2×C4—88no—
560C2×C4—162no—
561C2×C4#52514yes—
562C2×C4#53512yes—
563C2×C4#57714yes—
564C2×C4C22×#4112yes—
565C2×C4C2×H(5,𝔽2)12yes—
566C2×C4#58512yes—
567C8—44no—
568C8—44no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—168no—
582C22C2×#52511yescharacteristic
583C22C2×#53511yescharacteristic
584C22#344911yescharacteristic
585C22C2×#57712yes—
586C22#358612yes—
587C4—24no—
588C4—24no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4#349414yes—
604C2—42no—
605C2—84no—
606C2#3530111yescharacteristic
607C2#6059112yes—
608C2#6082011yescharacteristic
609C2#6082811yescharacteristic
610C2#6098212yes—
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. #35319C2semidirect
3. #33119C2semidirect
5. #29987C2semidirect
6. #29798C2semidirect
7. #29568C2semidirect
8. #29369C2semidirect
9. #29368C2semidirect
10. #29338C2semidirect
11. #29332C2semidirect
14. #24071C2semidirect
16. #22162C2semidirect
17. #20784C2semidirect
25. #2868C22semidirect
26. #2859C22semidirect
29. #2642C22semidirect
39. #2327C22semidirect
44. C4×C4⋊Q8C22semidirect
45. C4×C4⋊Q8C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
56. #2059C22semidirect
57. #2043C22semidirect
58. #2043C22semidirect
59. #2043C22semidirect
64. #1981C22semidirect
67. #1949C22semidirect
68. #1935C22semidirect
69. #1933C22semidirect
70. #1928C22semidirect
71. #1927C22semidirect
80. #1865C22semidirect
81. #1863C22semidirect
82. #1839C22semidirect
83. #1831C22semidirect
87. #1768C22semidirect
90. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#353192
#342552
#331192
#329142
#299872
#297982
#295682
#293692
#293682
#293382
#293322
#283112
#283022
#240712
#239422
#221622
#207842