Finite Group Databasebeta

← #6506583

Subgroups — #6506583

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37297C211yesmaximal, characteristic
2#35639C211yesmaximal, characteristic, complemented
3#29340C212yesmaximal, complemented
4#29337C212yesmaximal, complemented
5#27986C212yesmaximal, complemented
6#27718C214yesmaximal, complemented
7#27710C214yesmaximal, complemented
8#27279C214yesmaximal, complemented
9#27264C214yesmaximal, complemented
10#25913C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#20914C212yesmaximal, complemented
13#20532C211yesmaximal, characteristic, complemented
14C2×#535C2212yes—
15C22×#125C2211yescharacteristic, complemented
16C2×#520C2211yescharacteristic
17C2×#490C2214yes—
18C2×#418C2212yescomplemented
19C22×#102C2212yescomplemented
20C2×#408C2212yes—
21#2856C2212yes—
22#2733—22no—
23#2679—22no—
24#2669—22no—
25#2669—22no—
26#2619C2212yes—
27#2579—22no—
28#2500C2214yes—
29#2487—22no—
30#2454—22no—
31#2453—22no—
32#2442C2214yescomplemented
33#2441C2214yescomplemented
34#2433C2214yes—
35#2317C2211yescharacteristic, complemented
36#2316C2211yescharacteristic, complemented
37#2311C2211yescharacteristic
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic, complemented
40#2101C2214yescomplemented
41#2099C2212yescomplemented
42#2099C2212yescomplemented
43#2073C2214yescomplemented
44#2071C2214yescomplemented
45#2041C2212yes—
46#2041C2214yes—
47#2041C2212yes—
48#2025C2214yescomplemented
49#2025C2214yescomplemented
50#1992C2214yescomplemented
51#1961C2212yes—
52#1957C2212yes—
53#1936C2214yescomplemented
54#1936C2214yescomplemented
55#1936C2214yescomplemented
56#1934C2214yescomplemented
57#1895C2214yes—
58#1893C2214yes—
59#1875C2214yes—
60#1875C2214yescomplemented
61#1875C2214yescomplemented
62#1874C2214yes—
63#1874C2214yescomplemented
64#1874C2214yescomplemented
65#1836C2212yescomplemented
66#1835C2212yescomplemented
67#1825C2214yescomplemented
68#1786C2214yescomplemented
69#1753C2214yescomplemented
70#1752C2214yescomplemented
71#1750C2212yescomplemented
72#1749C2212yescomplemented
73#1736C2212yescomplemented
74#1735C2212yescomplemented
75C22×#41C2311yescharacteristic
76C23×D4C2312yes—
77C22×M16C2312yes—
78C23×C8C2312yes—
79#535—28no—
80#535—28no—
81C2×#127—22no—
82C2×#127C2312yes—
83C2×#127C2312yes—
84C2×#125—22no—
85C2×#125—44no—
86C2×#125C2314yes—
87C2×#124—22no—
88C2×#121—22no—
89C2×#121—22no—
90C2×#121C2314yes—
91C2×#121C2314yes—
92C2×#120—22no—
93C2×#120—22no—
94C2×#120—24no—
95C2×#120—24no—
96#520—44no—
97C2×C4×Q8—21no—
98C2×C4×D4—22no—
99C2×C4×D4—21no—
100C2×#117C2312yes—
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#490—48no—
112#418—44no—
113C2×#102—44no—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122#408—28no—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2314yes—
125C2×#97C2312yes—
126C2×#97C2314yes—
127C2×C4×C8C2312yes—
128#400—24no—
129#399—24no—
130#398—24no—
131#396—24no—
132#394—24no—
133#394—24no—
134#394—24no—
135#392—24no—
136#392—24no—
137#392—24no—
138#391—21no—
139#391D412yes—
140#390—21no—
141#390—22no—
142#390—22no—
143#390D412yes—
144#390C2314yes—
145#390C2314yes—
146#389—22no—
147#389—22no—
148#389—24no—
149#389C2314yes—
150#389C2312yes—
151#389C2312yes—
152#388—24no—
153#388—24no—
154#387—24no—
155#386—22no—
156#386D412yes—
157#386D412yes—
158#385—21no—
159#385—21no—
160#385—22no—
161#385—24no—
162#384C2312yes—
163#384C2312yes—
164#382C2312yes—
165#382C2312yes—
166#380—22no—
167#380—24no—
168#380C2314yes—
169C4×C4⋊C4—22no—
170C4×C22⋊C4C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2314yes—
180#336C2312yes—
181#336C2312yes—
182C25—22no—
183C2×#41—42no—
184C2×#41—41no—
185C2×#41—41no—
186C2×#41#4118yes—
187C22×Q8C2×D411yescharacteristic
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4C2×D411yescharacteristic
201C22×D4C2×D412yes—
202C22×D4C2414yes—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—22no—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C2×M16—24no—
211C2×M16—42no—
212C2×M16—42no—
213C22×C8—24no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218#127—42no—
219#127—42no—
220#127—44no—
221#127—48no—
222#125—28no—
223#125—44no—
224#125—48no—
225#124—48no—
226#121—28no—
227#121—28no—
228#121—28no—
229#121—28no—
230#121—84no—
231#121—84no—
232#120—48no—
233#120—48no—
234#120—88no—
235#120—88no—
236C4×Q8—44no—
237C4×D4—44no—
238C4×D4—48no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—41no—
243#117—41no—
244#117—41no—
245#117—41no—
246#117—42no—
247#117—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2411yescharacteristic
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4C2×D412yes—
284C2×C22⋊C4C2×D414yes—
285C2×C22⋊C4C2×D414yes—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—42no—
291C2×C42C2×D412yes—
292C2×C42C2412yes—
293C2×C42C2412yes—
294#102—28no—
295#102—28no—
296#102—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302C22⋊C8—28no—
303C22⋊C8—28no—
304#97—42no—
305#97—44no—
306C4×C8—42no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95C2×D414yes—
314#95C2×D414yes—
315C24—24no—
316C24—44no—
317C24—82no—
318C24—82no—
319C24—84no—
320C24—84no—
321C24C22×D412yes—
322C24C22×D411yescharacteristic
323#41—48no—
324#41—48no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8#12414yes—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—44no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—82no—
354C2×D4—82no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4—84no—
360C2×D4#12414yes—
361C2×D4#12418yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—84no—
385C22×C4C22×D414yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4114yes—
391C22×C4C2×#4114yes—
392C22×C4C2×#4114yes—
393C22×C4C2511yescharacteristic
394M16—84no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—84no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4H(5,𝔽2)14yes—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—24no—
438C42—22no—
439C42—22no—
440C42—42no—
441C42—44no—
442C42—42no—
443C42—42no—
444C42—44no—
445C23—24no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—22no—
451C23—21no—
452C23—22no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—88no—
461C23—168no—
462C23—168no—
463C23—164no—
464C23—164no—
465C23C2×#12414yes—
466C23C2×#12412yes—
467C23C23×D411yescharacteristic
468Q8—44no—
469D4—48no—
470D4—44no—
471D4—44no—
472D4—44no—
473D4—44no—
474D4—44no—
475D4—88no—
476D4—164no—
477D4—164no—
478D4—168no—
479D4—168no—
480D4—168no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—164no—
519C2×C4C2×#12414yes—
520C2×C4C2×#12412yes—
521C2×C4C22×#4112yes—
522C2×C4C2×H(5,𝔽2)12yes—
523C2×C4#58511yescharacteristic
524C2×C4#58511yescharacteristic
525C8—44no—
526C8—44no—
527C22—24no—
528C22—22no—
529C22—24no—
530C22—28no—
531C22—42no—
532C22—44no—
533C22—41no—
534C22—42no—
535C22—41no—
536C22—42no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—1616no—
545C22C22×#12411yescharacteristic
546C22#348411yescharacteristic
547C22#348511yescharacteristic
548C22#358512yes—
549C22#358512yes—
550C4—24no—
551C4—22no—
552C4—44no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—82no—
557C4—81no—
558C4—84no—
559C4—81no—
560C4—82no—
561C4#350512yes—
562C4#350512yes—
563C2—42no—
564C2—44no—
565C2—164no—
566C2#3537711yescharacteristic
567C2#6062612yes—
568C2#6164811yescharacteristic
569C2#6165111yescharacteristic
570C2#6179712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37297C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35639C2semidirect
3. #29340C2semidirect
4. #29337C2semidirect
5. #27986C2semidirect
6. #27718C2semidirect
7. #27710C2semidirect
8. #27279C2semidirect
9. #27264C2semidirect
10. #25913C2semidirect
11. #21770C2semidirect
12. #20914C2semidirect
13. #20532C2semidirect
15. C22×#125C22semidirect
18. C2×#418C22semidirect
19. C22×#102C22semidirect
32. #2442C22semidirect
33. #2441C22semidirect
35. #2317C22semidirect
36. #2316C22semidirect
39. #2296C22semidirect
40. #2101C22semidirect
41. #2099C22semidirect
42. #2099C22semidirect
43. #2073C22semidirect
44. #2071C22semidirect
48. #2025C22semidirect
49. #2025C22semidirect
50. #1992C22semidirect
53. #1936C22semidirect
54. #1936C22semidirect
55. #1936C22semidirect
56. #1934C22semidirect
60. #1875C22semidirect
61. #1875C22semidirect
63. #1874C22semidirect
64. #1874C22semidirect
65. #1836C22semidirect
66. #1835C22semidirect
67. #1825C22semidirect
68. #1786C22semidirect
69. #1753C22semidirect
70. #1752C22semidirect
71. #1750C22semidirect
72. #1749C22semidirect
73. #1736C22semidirect
74. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#372972
#356392
#293402
#293372
#279862
#277182
#277102
#272792
#272642
#259132
#217702
#209142
#205322