Finite Group Databasebeta

← #6635529

Subgroups — #6635529

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37297C211yesmaximal, characteristic
2#35182C211yesmaximal, characteristic, complemented
3#32903C212yesmaximal, complemented
4#31948C214yesmaximal, complemented
5#31944C214yesmaximal, complemented
6#31004C214yesmaximal, complemented
7#30988C214yesmaximal, complemented
8#29758C212yesmaximal, complemented
9#28323C212yesmaximal, complemented
10#27986C212yesmaximal, complemented
11#24015C212yesmaximal, complemented
12#23926C212yesmaximal, complemented
13#21911C211yesmaximal, characteristic, complemented
14C2×#536C2211yescharacteristic
15C2×#535C2212yes—
16C22×C4×Q8C2211yescharacteristic, complemented
17C2×#482C2214yescomplemented
18C2×#460C2214yes—
19C2×#448C2214yescomplemented
20C2×#450C2212yescomplemented
21C2×#450C2212yescomplemented
22#2856C2212yes—
23#2853—22no—
24#2733—22no—
25#2679—22no—
26#2669—22no—
27#2669—22no—
28#2623—22no—
29#2619C2212yes—
30#2500C2214yes—
31#2448—22no—
32#2446—22no—
33#2433C2214yes—
34#2423C2214yescomplemented
35#2421C2214yescomplemented
36#2317C2211yescharacteristic, complemented
37#2311C2211yescharacteristic
38#2310C2211yescharacteristic, complemented
39#2297C2211yescharacteristic
40#2296C2211yescharacteristic, complemented
41#2097C2214yescomplemented
42#2084C2212yescomplemented
43#2083C2212yescomplemented
44#2073C2214yescomplemented
45#2071C2214yescomplemented
46#2067C2214yescomplemented
47#2065C2214yescomplemented
48#2041C2212yes—
49#2041C2212yes—
50#2034C2214yes—
51#2033C2214yes—
52#2014C2214yes—
53#2000C2214yes—
54#2000C2214yes—
55#1973C2214yescomplemented
56#1968C2214yescomplemented
57#1961C2212yes—
58#1957C2212yes—
59#1924C2214yescomplemented
60#1923C2214yescomplemented
61#1880C2214yescomplemented
62#1880C2214yescomplemented
63#1875C2214yescomplemented
64#1874C2214yescomplemented
65#1873C2212yescomplemented
66#1873C2212yescomplemented
67#1854C2214yescomplemented
68#1836C2212yescomplemented
69#1835C2212yescomplemented
70#1803C2214yescomplemented
71#1797C2212yescomplemented
72#1786C2214yescomplemented
73#1780C2212yes—
74#1750C2212yescomplemented
75C22×#41C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C23×D4C2312yes—
78C22×SD16C2314yes—
79C22×SD16C2314yes—
80#536—44no—
81#535—48no—
82C2×C4⋊Q8—21no—
83C2×#127—22no—
84C2×#127C2312yes—
85C2×#127C2312yes—
86C2×#124—22no—
87C2×#124—21no—
88C2×#123—22no—
89C2×C22⋊Q8C2312yes—
90C2×C22⋊Q8C2312yes—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2314yes—
94C2×#121C2314yes—
95C2×#120—22no—
96C2×#120—22no—
97C2×#120—24no—
98C2×#120—24no—
99C2×C4×Q8—22no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2314yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110#482—48no—
111#460—48no—
112#450—44no—
113#450—44no—
114#448—48no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×#97C2312yes—
131C2×C4×C8C2312yes—
132#400—24no—
133#399—24no—
134#398—24no—
135#397—24no—
136#394—24no—
137#394—24no—
138#394—24no—
139#392—24no—
140#392—24no—
141#392—24no—
142#391—21no—
143#391—22no—
144#391D412yes—
145#391C2314yes—
146#390—21no—
147#390—22no—
148#390—22no—
149#390D412yes—
150#390C2314yes—
151#390C2314yes—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#388—24no—
156#388—24no—
157#388—24no—
158#388—24no—
159#387—24no—
160#387—24no—
161#386—22no—
162#386D412yes—
163#386D412yes—
164#385—21no—
165#385—21no—
166#385—22no—
167#384C2312yes—
168#384C2312yes—
169#382C2312yes—
170#382C2312yes—
171#381—22no—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C22⋊C4C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2312yes—
180#336C2312yes—
181#336C2314yes—
182C25—22no—
183C2×#41—24no—
184C2×#41—42no—
185C2×#41—42no—
186C22×Q8—24no—
187C22×Q8—21no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8C2×D411yescharacteristic
191C22×Q8C2412yes—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4C2×D411yescharacteristic
203C22×D4C2×D412yes—
204C22×D4C2414yes—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C2×SD16—28no—
212C2×SD16—28no—
213C2×SD16—44no—
214C2×SD16—44no—
215C2×SD16—44no—
216C2×SD16—44no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—44no—
220#127—42no—
221#127—42no—
222#127—44no—
223#127—48no—
224#124—48no—
225#124—44no—
226#123—48no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—42no—
230C22⋊Q8—42no—
231#121—28no—
232#121—28no—
233#121—44no—
234#121—44no—
235#121—84no—
236#121—84no—
237#120—48no—
238#120—48no—
239#120—88no—
240#120—88no—
241C4×Q8—28no—
242C4×Q8—44no—
243C4×Q8—48no—
244#117—24no—
245#117—42no—
246#117—44no—
247#117—42no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2411yescharacteristic
263C2×C4⋊C4C2411yescharacteristic
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D414yes—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—24no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42C2×D412yes—
287C2×C42C2412yes—
288C2×C42C2412yes—
289#107—44no—
290#106—44no—
291C4⋊C8—44no—
292#103—44no—
293#103—44no—
294#102—44no—
295#102—44no—
296#102—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300C22⋊C8—28no—
301C22⋊C8—44no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304#97—42no—
305C4×C8—42no—
306#95—22no—
307#95—22no—
308#95—24no—
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—88no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—416no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8H(5,𝔽2)14yes—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—82no—
353C2×D4—82no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—84no—
361C2×D4—88no—
362C2×D4—88no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—84no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—82no—
388C22×C4C22×D414yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4H(5,𝔽2)14yes—
394C22×C4C2511yescharacteristic
395SD16—88no—
396SD16—88no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—84no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C22⋊C4—28no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C42—24no—
436C42—22no—
437C42—22no—
438C42—42no—
439C42—48no—
440C42—44no—
441C42—42no—
442C42—42no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—88no—
460C23—168no—
461C23—168no—
462C23—164no—
463C23—164no—
464C23#53512yes—
465C23C23×D411yescharacteristic
466Q8—28no—
467Q8—48no—
468Q8—44no—
469Q8—82no—
470D4—44no—
471D4—44no—
472D4—44no—
473D4—44no—
474D4—82no—
475D4—84no—
476D4—88no—
477D4—164no—
478D4—164no—
479D4—168no—
480D4—168no—
481D4—168no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4#53512yes—
518C2×C4#57714yes—
519C2×C4C2×H(5,𝔽2)12yes—
520C2×C4C2×H(5,𝔽2)12yes—
521C2×C4#58512yes—
522C8—44no—
523C8—44no—
524C22—22no—
525C22—24no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—44no—
530C22—42no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—1616no—
542C22C2×#53511yescharacteristic
543C22#348411yescharacteristic
544C22#348511yescharacteristic
545C22C2×#57712yes—
546C22#358512yes—
547C4—22no—
548C4—24no—
549C4—44no—
550C4—42no—
551C4—44no—
552C4—42no—
553C4—82no—
554C4—84no—
555C4—81no—
556C4—81no—
557C4—82no—
558C4#353414yes—
559C2—42no—
560C2—82no—
561C2—164no—
562C2#3672111yescharacteristic
563C2#6083312yes—
564C2#6163311yescharacteristic
565C2#6165111yescharacteristic
566C2#6180512yes—
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. #35182C2semidirect
3. #32903C2semidirect
4. #31948C2semidirect
5. #31944C2semidirect
6. #31004C2semidirect
7. #30988C2semidirect
8. #29758C2semidirect
9. #28323C2semidirect
10. #27986C2semidirect
11. #24015C2semidirect
12. #23926C2semidirect
13. #21911C2semidirect
16. C22×C4×Q8C22semidirect
17. C2×#482C22semidirect
19. C2×#448C22semidirect
20. C2×#450C22semidirect
21. C2×#450C22semidirect
34. #2423C22semidirect
35. #2421C22semidirect
36. #2317C22semidirect
38. #2310C22semidirect
40. #2296C22semidirect
41. #2097C22semidirect
42. #2084C22semidirect
43. #2083C22semidirect
44. #2073C22semidirect
45. #2071C22semidirect
46. #2067C22semidirect
47. #2065C22semidirect
55. #1973C22semidirect
56. #1968C22semidirect
59. #1924C22semidirect
60. #1923C22semidirect
61. #1880C22semidirect
62. #1880C22semidirect
63. #1875C22semidirect
64. #1874C22semidirect
65. #1873C22semidirect
66. #1873C22semidirect
67. #1854C22semidirect
68. #1836C22semidirect
69. #1835C22semidirect
70. #1803C22semidirect
71. #1797C22semidirect
72. #1786C22semidirect
74. #1750C22semidirect

Maximal subgroups

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

subgroupindex
#372972
#351822
#329032
#319482
#319442
#310042
#309882
#297582
#283232
#279862
#240152
#239262
#219112