Finite Group Databasebeta

← #7228769

Subgroups — #7228769

Order 512.

1655 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36289C211yesmaximal, characteristic
3#33072C212yesmaximal, complemented
4#32818C212yesmaximal, complemented
5#32647C212yesmaximal, complemented
6#32261C212yesmaximal, complemented
7#32209C214yesmaximal, complemented
8#30799C214yesmaximal, complemented
9#27541C214yesmaximal, complemented
10#25950C214yesmaximal, complemented
11#25850C212yesmaximal, complemented
12#24033C212yesmaximal
13#23352C211yesmaximal, characteristic, complemented
14C2×#554C2211yescharacteristic
15C2×D42C2211yescharacteristic
16C2×#501C2212yescomplemented
17C2×#481C2212yes—
18C2×#474C2212yes—
19C22×#102C2212yes—
20#2859C2212yescomplemented
21#2858C2211yescharacteristic
22#2850—21no—
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2567C2212yescomplemented
28#2563C2211yescharacteristic
29#2548C2211yescharacteristic
30#2504—21no—
31#2497C2214yescomplemented
32#2490C2214yes—
33#2487C2212yes—
34#2472—24no—
35#2440C2214yescomplemented
36#2435—21no—
37#2379—21no—
38D4×C22⋊C4C2214yes—
39C4×#124C2211yescharacteristic
40C2×#390—21no—
41C42×D4C2211yescharacteristic
42#2093C2214yescomplemented
43#2076C2214yescomplemented
44#2076C2214yescomplemented
45#2071C2214yescomplemented
46#2067C2214yescomplemented
47#2067C2214yescomplemented
48#2067C2214yescomplemented
49#2059C2214yescomplemented
50#2057C2214yescomplemented
51#2041C2214yes—
52#2034C2214yes—
53#2023C2214yescomplemented
54#1989C2212yescomplemented
55#1985C2212yescomplemented
56#1948C2212yescomplemented
57#1947C2212yescomplemented
58#1936C2214yescomplemented
59#1925C2214yescomplemented
60#1924C2214yescomplemented
61#1921C2214yescomplemented
62#1919C2214yescomplemented
63#1893C2212yes—
64#1869C2212yes—
65#1864C2214yes—
66#1862C2214yes—
67#1862C2214yes—
68#1852C2214yescomplemented
69#1850C2214yescomplemented
70#1831C2212yescomplemented
71#1808C2214yescomplemented
72#1802C2212yes—
73#1802C2212yes—
74#1786C2214yes—
75#1786C2212yes—
76#1762C2212yes—
77C23×D4—21no—
78C23×D4C2312yes—
79C23×C8C2312yes—
80#554—42no—
81#554—42no—
82D42—42no—
83D42—42no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#125—22no—
88C2×#125C2312yes—
89C2×#124—21no—
90C2×#124—22no—
91C2×#124D411yescharacteristic
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×#121—21no—
96C2×#121C2312yes—
97C2×#120—22no—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4C2311yescharacteristic
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—24no—
106C22×C42—21no—
107#501—44no—
108#481—44no—
109#474—44no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×#102—44no—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2312yes—
127#400—22no—
128#400—24no—
129#399—22no—
130#398—24no—
131#397—24no—
132#396—24no—
133#395—22no—
134#395—22no—
135#395—24no—
136#394—22no—
137#394—24no—
138#393—22no—
139#391—22no—
140#391D411yescharacteristic
141#390—21no—
142#390—21no—
143#390—24no—
144#390—21no—
145#390—21no—
146#390D411yescharacteristic
147#390C2314yes—
148#390C2314yes—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#388D414yes—
157#387—24no—
158#386—22no—
159#386—24no—
160#386—24no—
161#385—24no—
162#385—22no—
163#384—22no—
164#384C2312yes—
165#384C2312yes—
166#382—21no—
167#382—22no—
168#382D411yescharacteristic
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172#379—24no—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—22no—
178C4×C22⋊C4C2314yes—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25—22no—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8C2×D411yescharacteristic
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—82no—
204C22×D4C2×D411yescharacteristic
205C22×D4C2412yes—
206C22×D4C2411yescharacteristic
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4C2412yes—
212C22×C8—44no—
213C22×C8—42no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—41no—
217C4⋊Q8—41no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220#127—41no—
221#127—41no—
222#125—24no—
223#125—42no—
224#125—84no—
225#124—24no—
226#124—84no—
227#124—82no—
228C22⋊Q8—48no—
229C22⋊Q8—44no—
230C22⋊Q8—88no—
231#121—24no—
232#121—42no—
233#121—44no—
234#120—48no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—41no—
244C4×D4—41no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—88no—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D414yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—82no—
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2412yes—
292C2×C42C2412yes—
293#107—44no—
294#107—44no—
295#107—44no—
296#106—28no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300#102—28no—
301#102—28no—
302#102—44no—
303#102—44no—
304#102—44no—
305#102—44no—
306C22⋊C8—44no—
307C22⋊C8—42no—
308#97—42no—
309#95—21no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95C2×D414yes—
319#95C2×D414yes—
320C24—21no—
321C24—22no—
322C24—44no—
323C24—48no—
324C24—44no—
325C24—84no—
326C24—82no—
327C24C22×D412yes—
328C2×Q8—22no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—42no—
332C2×Q8—84no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—42no—
352C2×D4—82no—
353C2×D4—84no—
354C2×D4—88no—
355C2×D4—816no—
356C2×D4—82no—
357C2×D4#13714yes—
358C2×D4H(5,𝔽2)12yes—
359C2×D42−1+414yes—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—82no—
376C22×C4—84no—
377C22×C4—82no—
378C22×C4—84no—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C2511yescharacteristic
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—84no—
394C4⋊C4—21no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—84no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—84no—
417C4⋊C4—82no—
418C4⋊C4H(5,𝔽2)12yes—
419C22⋊C4—24no—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—42no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4—82no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—22no—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42—22no—
440C42—22no—
441C42—41no—
442C42—42no—
443C42—82no—
444C42—82no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—88no—
460C23—88no—
461C23—816no—
462C23—168no—
463C23—164no—
464C23C2×#13714yes—
465C23C23×D411yescharacteristic
466Q8—82no—
467Q8—162no—
468Q8—164no—
469D4—22no—
470D4—24no—
471D4—44no—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—84no—
476D4—88no—
477D4—88no—
478D4—162no—
479D4#56314yes—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—48no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—88no—
515C2×C4—164no—
516C2×C4—164no—
517C2×C4C2×#13712yes—
518C2×C4#57614yes—
519C2×C4C2×H(5,𝔽2)11yescharacteristic
520C2×C4C2×H(5,𝔽2)11yescharacteristic
521C2×C4C2×2−1+412yes—
522C2×C4#58512yes—
523C8—44no—
524C8—44no—
525C22—24no—
526C22—24no—
527C22—28no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—82no—
536C22—82no—
537C22—84no—
538C22—84no—
539C22—82no—
540C22—1616no—
541C22#348411yescharacteristic
542C22#348912yes—
543C22C22×#13711yescharacteristic
544C22C2×#57612yes—
545C22#358511yescharacteristic
546C4—22no—
547C4—24no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—84no—
555C4—82no—
556C4—82no—
557C4—84no—
558C4C2×#56311yescharacteristic
559C4#354511yescharacteristic
560C4#356112yes—
561C2—44no—
562C2—44no—
563C2—162no—
564C2#4613611yescharacteristic
565C2#6157011yescharacteristic
566C2#6166111yescharacteristic
567C2#6212712yes—
568C2#6216312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46152C2semidirect
3. #33072C2semidirect
4. #32818C2semidirect
5. #32647C2semidirect
6. #32261C2semidirect
7. #32209C2semidirect
8. #30799C2semidirect
9. #27541C2semidirect
10. #25950C2semidirect
11. #25850C2semidirect
13. #23352C2semidirect
16. C2×#501C22semidirect
20. #2859C22semidirect
27. #2567C22semidirect
31. #2497C22semidirect
35. #2440C22semidirect
42. #2093C22semidirect
43. #2076C22semidirect
44. #2076C22semidirect
45. #2071C22semidirect
46. #2067C22semidirect
47. #2067C22semidirect
48. #2067C22semidirect
49. #2059C22semidirect
50. #2057C22semidirect
53. #2023C22semidirect
54. #1989C22semidirect
55. #1985C22semidirect
56. #1948C22semidirect
57. #1947C22semidirect
58. #1936C22semidirect
59. #1925C22semidirect
60. #1924C22semidirect
61. #1921C22semidirect
62. #1919C22semidirect
68. #1852C22semidirect
69. #1850C22semidirect
70. #1831C22semidirect
71. #1808C22semidirect

Maximal subgroups

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

subgroupindex
#461522
#362892
#330722
#328182
#326472
#322612
#322092
#307992
#275412
#259502
#258502
#240332
#233522