Finite Group Databasebeta

← #6661108

Subgroups — #6661108

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35261C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic, complemented
3#32754C214yesmaximal, complemented
4#30301C212yesmaximal, complemented
5#30194C212yesmaximal, complemented
6#29972C212yesmaximal, complemented
7#29948C212yesmaximal, complemented
8#29754C212yesmaximal, complemented
9#29700C212yesmaximal, complemented
10#27644C212yesmaximal, complemented
11#24848C214yesmaximal, complemented
12#24558C212yesmaximal, complemented
13#24550C212yesmaximal, complemented
14#24305C212yesmaximal, complemented
15#20628C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#480C2212yes—
20C2×#480C2212yes—
21C2×#478C2214yescomplemented
22C2×#424C2212yes—
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2856C2211yescharacteristic
26#2851C2211yescharacteristic
27#2847C2212yes—
28#2629—22no—
29#2629—22no—
30#2629C2212yes—
31#2629C2212yes—
32#2629C2212yes—
33#2629C2212yes—
34#2629C2211yescharacteristic
35#2624C2212yes—
36#2619C2211yescharacteristic
37#2476—22no—
38#2471—22no—
39#2443—22no—
40#2424—22no—
41C2×#398—22no—
42C2×#396—22no—
43#2324C2212yes—
44#2314C2211yescharacteristic
45C4×#121C2212yes—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48C2×#380C2212yes—
49#2101C2214yescomplemented
50#2088C2214yescomplemented
51#2087C2214yescomplemented
52#2083C2214yescomplemented
53#2083C2214yescomplemented
54#2076C2214yes—
55#2075C2214yes—
56#2071C2214yescomplemented
57#2067C2214yescomplemented
58#2067C2214yescomplemented
59#2059C2214yescomplemented
60#2059C2214yescomplemented
61#2041C2214yescomplemented
62#2034C2214yes—
63#2033C2214yes—
64#2025C2214yescomplemented
65#1989C2214yescomplemented
66#1947C2214yescomplemented
67#1895C2214yes—
68#1893C2214yes—
69#1874C2212yes—
70#1874C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1809C2214yes—
76#1808C2214yes—
77#1802C2214yescomplemented
78#1798C2212yes—
79#1798C2212yes—
80#1797C2212yes—
81#1797C2212yes—
82#1729C2212yes—
83C24×C4C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#540—44no—
87C2×#127C2311yescharacteristic
88C2×#125—22no—
89C2×#125—44no—
90C2×#125C2312yes—
91C2×#125C2311yescharacteristic
92C2×#125C2312yes—
93C2×#123—22no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#120—22no—
100C2×C4×D4—22no—
101C2×C4×D4C2312yes—
102C2×#117—44no—
103C2×#117D412yes—
104C2×#117D412yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4D412yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—22no—
116C22×C22⋊C4D412yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#480—44no—
120#480—44no—
121#478—48no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126#424—44no—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133#407—42no—
134C2×C22⋊C8—42no—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139#399—22no—
140#398—24no—
141#398—24no—
142#398—24no—
143#398—24no—
144#398—22no—
145#397—22no—
146#396—24no—
147#396—24no—
148#396—24no—
149#396—24no—
150#396—22no—
151#394—22no—
152#394—22no—
153#394—22no—
154#392—22no—
155#390—22no—
156#390—22no—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2314yes—
163#389C2314yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#387—22no—
169#387—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#385—22no—
176#385—22no—
177#384—22no—
178#384C2312yes—
179#384C2311yescharacteristic
180#384C2312yes—
181#382C2311yescharacteristic
182#380—22no—
183#380—22no—
184#380C2312yes—
185#380C2312yes—
186#380C2314yes—
187#380C2314yes—
188#380C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C2×#95—22no—
192C2×#95—22no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25C2×D411yescharacteristic
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—42no—
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C2×M16—24no—
225C2×M16—42no—
226C2×M16—42no—
227C22×C8—24no—
228C22×C8—42no—
229C22×C8—42no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232#127—42no—
233#125—28no—
234#125—42no—
235#125—42no—
236#125—42no—
237#125—48no—
238#123—48no—
239#121—24no—
240#121—28no—
241#121—24no—
242#121—48no—
243#121—44no—
244#121—84no—
245#120—84no—
246C4×D4—28no—
247C4×D4—84no—
248#117—24no—
249#117—24no—
250#117—42no—
251#117—42no—
252#117—44no—
253#117—44no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—84no—
272C2×C4⋊C4—84no—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—84no—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—42no—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310#107—44no—
311#107—44no—
312#107—44no—
313#106—44no—
314C4⋊C8—28no—
315C4⋊C8—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319#102—44no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322C22⋊C8—42no—
323C22⋊C8—42no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—41no—
340C24—42no—
341C24C22×D411yescharacteristic
342C24C22×D411yescharacteristic
343C24C22×D411yescharacteristic
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+414yes—
409C22×C42−1+412yes—
410C22×C4C2511yescharacteristic
411M16—84no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—84no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—82no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—42no—
465C42—82no—
466C42H(5,𝔽2)14yes—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—44no—
480C23—48no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—84no—
490C23—82no—
491C23#53612yes—
492C23#53712yes—
493C23#57614yes—
494C23C23×D411yescharacteristic
495D4—44no—
496D4—44no—
497D4—88no—
498D4—164no—
499D4—164no—
500D4—164no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—84no—
545C2×C4—88no—
546C2×C4—88no—
547C2×C4—164no—
548C2×C4#53414yes—
549C2×C4#53612yes—
550C2×C4#53712yes—
551C2×C4C2×H(5,𝔽2)11yescharacteristic
552C2×C4C2×H(5,𝔽2)12yes—
553C2×C4C2×2−1+412yes—
554C2×C4C2×2−1+411yescharacteristic
555C8—44no—
556C8—44no—
557C22—22no—
558C22—24no—
559C22—24no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—42no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—41no—
570C22—41no—
571C22—88no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22#325914yes—
580C22C2×#53411yescharacteristic
581C22C2×#53611yescharacteristic
582C22C2×#53711yescharacteristic
583C22C2×#57612yes—
584C22#358612yes—
585C4—22no—
586C4—22no—
587C4—24no—
588C4—41no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—84no—
596C4—82no—
597C4#353612yes—
598C4#353812yes—
599C2—24no—
600C2—42no—
601C2—81no—
602C2—162no—
603C2#3744211yescharacteristic
604C2#6076212yes—
605C2#6087112yes—
606C2C2×#325911yescharacteristic
607C2#6092111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35261C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2629C2semidirect
3. #32754C2semidirect
4. #30301C2semidirect
5. #30194C2semidirect
6. #29972C2semidirect
7. #29948C2semidirect
8. #29754C2semidirect
9. #29700C2semidirect
10. #27644C2semidirect
11. #24848C2semidirect
12. #24558C2semidirect
13. #24550C2semidirect
14. #24305C2semidirect
15. #20628C2semidirect
21. C2×#478C22semidirect
49. #2101C22semidirect
50. #2088C22semidirect
51. #2087C22semidirect
52. #2083C22semidirect
53. #2083C22semidirect
56. #2071C22semidirect
57. #2067C22semidirect
58. #2067C22semidirect
59. #2059C22semidirect
60. #2059C22semidirect
61. #2041C22semidirect
64. #2025C22semidirect
65. #1989C22semidirect
66. #1947C22semidirect
77. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#352612
C2×#26292
#327542
#303012
#301942
#299722
#299482
#297542
#297002
#276442
#248482
#245582
#245502
#243052
#206282