Finite Group Databasebeta

← #6594137

Subgroups — #6594137

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36192C211yesmaximal, characteristic, complemented
2#36182C211yesmaximal, characteristic
3#33295C211yesmaximal, characteristic, complemented
4#33058C211yesmaximal, characteristic, complemented
5#33052C211yesmaximal, characteristic, complemented
6#29227C211yesmaximal, characteristic, complemented
7#29149C211yesmaximal, characteristic, complemented
8#27981C211yesmaximal, characteristic, complemented
9#27981C211yesmaximal, characteristic, complemented
10#27967C211yesmaximal, characteristic, complemented
11#27966C211yesmaximal, characteristic, complemented
12#27739C214yesmaximal, complemented
13#27725C214yesmaximal, complemented
14#27702C214yesmaximal, complemented
15#22498C214yesmaximal, complemented
16#21656C211yesmaximal, characteristic, complemented
17#20764C211yesmaximal, characteristic, complemented
18#20762C211yesmaximal, characteristic, complemented
19#20491C211yesmaximal, characteristic, complemented
20C2×#553C2212yes—
21C2×#490C2211yescharacteristic
22C2×#435C2212yes—
23C2×C4×M16C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2856C2211yescharacteristic
28#2856C2211yescharacteristic
29#2851C2211yescharacteristic
30#2851C2211yescharacteristic
31#2566C2212yes—
32#2552C2212yes—
33#2522—22no—
34#2522—22no—
35#2512—22no—
36#2504C2212yes—
37#2500C2212yes—
38#2454—22no—
39#2441C2212yes—
40#2441C2212yes—
41#2422—22no—
42#2416—22no—
43#2370—22no—
44#2361—22no—
45#2354C2212yes—
46C4×#127C2211yescharacteristic
47C4×#125C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2071C2214yes—
57#2071C2214yes—
58#2025C2214yes—
59#1992C2214yes—
60#1992C2211yescharacteristic
61#1992C2211yescharacteristic
62#1992C2211yescharacteristic
63#1991C2214yes—
64#1989C2214yes—
65#1961C2211yescharacteristic
66#1961C2211yescharacteristic
67#1961C2211yescharacteristic
68#1936C2214yes—
69#1934C2214yes—
70#1934C2214yes—
71#1895C2214yes—
72#1893C2214yes—
73#1875C2214yes—
74#1875C2214yes—
75#1836C2211yescharacteristic
76#1836C2211yescharacteristic
77#1836C2211yescharacteristic
78#1836C2211yescharacteristic
79#1836C2211yescharacteristic
80#1836C2211yescharacteristic
81#1836C2211yescharacteristic
82#1836C2214yes—
83#1836C2211yescharacteristic
84#1836C2211yescharacteristic
85#1834C2214yes—
86#1825C2214yes—
87#1823C2214yes—
88#1802C2214yes—
89#1771C2212yes—
90#1767C2211yescharacteristic
91#1762C2214yes—
92#1753C2212yes—
93C8×C22⋊C4C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1736C2214yes—
97#1735C2214yes—
98C4×#97C2211yescharacteristic
99C22×M16C2312yes—
100C23×C8C2312yes—
101#553—216no—
102C2×#127C2311yescharacteristic
103C2×#126—22no—
104C2×#125—22no—
105C2×#125C2311yescharacteristic
106C2×#125C2312yes—
107C2×#125C2312yes—
108C2×#123—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121C2312yes—
113C2×#121C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4D412yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#490—24no—
127#435—28no—
128C2×C4⋊C8C2312yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C4×M16—24no—
136C2×#97—24no—
137C2×#97C2312yes—
138C2×#97C2311yescharacteristic
139C2×#97C2311yescharacteristic
140C2×#97C2311yescharacteristic
141C2×#97C2312yes—
142C2×#97C2311yescharacteristic
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#399—22no—
146#399—22no—
147#398—22no—
148#398—22no—
149#397—24no—
150#396—24no—
151#396—22no—
152#396—22no—
153#394—24no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#386—24no—
180#385—22no—
181#385—22no—
182#385—24no—
183#384—22no—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382C2311yescharacteristic
192#382C2312yes—
193#382C2311yescharacteristic
194#382C2311yescharacteristic
195#382C2312yes—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2311yescharacteristic
212C4×C22⋊C4C2311yescharacteristic
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2312yes—
227#336C2312yes—
228#336C2312yes—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4C2×D411yescharacteristic
233C22×D4C2412yes—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—42no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—42no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250#127—22no—
251#127—22no—
252#126—48no—
253#125—28no—
254#125—22no—
255#125—22no—
256#125—44no—
257#125—48no—
258#123—48no—
259#121—28no—
260#121—28no—
261#121—48no—
262#121—48no—
263#121—84no—
264C4×D4—28no—
265C4×D4—44no—
266C4×D4—48no—
267C4×D4—84no—
268C4×D4#4118yes—
269#117—28no—
270#117—44no—
271#117—44no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
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—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—84no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D411yescharacteristic
316C2×C22⋊C4C2×D411yescharacteristic
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C4⋊C8—24no—
335#102—28no—
336#102—44no—
337#102—44no—
338#102—44no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341#97—24no—
342#97—22no—
343#97—22no—
344#97—24no—
345#97—22no—
346#97#4114yes—
347C4×C8—24no—
348C4×C8—42no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24—22no—
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—84no—
378C2×D4#12414yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4114yes—
412C22×C4C2511yescharacteristic
413M16—48no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—48no—
426C2×C8C2×#4114yes—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
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—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4#12414yes—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4#12414yes—
470C22⋊C4#12414yes—
471C42—24no—
472C42—24no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—42no—
479C42—42no—
480C42—44no—
481C42—44no—
482C42—82no—
483C42—82no—
484C42C2×#4114yes—
485C42C2×#4112yes—
486C422−1+412yes—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23C2×#12412yes—
501C23C2×#12412yes—
502C23C23×D411yescharacteristic
503D4—44no—
504D4—44no—
505D4—44no—
506D4—88no—
507D4—88no—
508D4—164no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—164no—
550C2×C4—164no—
551C2×C4C2×#12412yes—
552C2×C4C2×#12412yes—
553C2×C4C2×#12412yes—
554C2×C4C2×#12412yes—
555C2×C4#54714yes—
556C2×C4#57514yes—
557C2×C4C22×#4111yescharacteristic
558C2×C4C22×#4112yes—
559C2×C4C2×2−1+411yescharacteristic
560C2×C4#58512yes—
561C8—24no—
562C8—42no—
563C8—42no—
564C8—42no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—82no—
576C22—84no—
577C22—84no—
578C22—82no—
579C22—82no—
580C22C22×#12411yescharacteristic
581C22C2×#54711yescharacteristic
582C22#350011yescharacteristic
583C22C2×#57512yes—
584C22#358512yes—
585C4—22no—
586C4—22no—
587C4—24no—
588C4—24no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4#347512yes—
601C4#351512yes—
602C2—42no—
603C2—42no—
604C2—162no—
605C2#3616411yescharacteristic
606C2#6061811yescharacteristic
607C2#6062611yescharacteristic
608C2#6125512yes—
609C2#6258612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36182C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36192C2semidirect
3. #33295C2semidirect
4. #33058C2semidirect
5. #33052C2semidirect
6. #29227C2semidirect
7. #29149C2semidirect
8. #27981C2semidirect
9. #27981C2semidirect
10. #27967C2semidirect
11. #27966C2semidirect
12. #27739C2semidirect
13. #27725C2semidirect
14. #27702C2semidirect
15. #22498C2semidirect
16. #21656C2semidirect
17. #20764C2semidirect
18. #20762C2semidirect
19. #20491C2semidirect

Maximal subgroups

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

subgroupindex
#361922
#361822
#332952
#330582
#330522
#292272
#291492
#279812
#279672
#279662
#277392
#277252
#277022
#224982
#216562
#207642
#207622
#204912