Finite Group Databasebeta

← #6506566

Subgroups — #6506566

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37297C211yesmaximal, characteristic
2#35574C211yesmaximal, characteristic, complemented
3#29340C212yesmaximal, complemented
4#29331C212yesmaximal, complemented
5#27986C212yesmaximal, complemented
6#27720C214yesmaximal, complemented
7#27696C214yesmaximal, complemented
8#27273C214yesmaximal, complemented
9#27254C214yesmaximal, complemented
10#25903C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#20914C212yesmaximal, complemented
13#20530C211yesmaximal, characteristic, complemented
14C2×#535C2212yes—
15C22×C4⋊Q8C2211yescharacteristic, complemented
16C2×#520C2211yescharacteristic
17C2×#486C2214yes—
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#2452—22no—
30#2451—22no—
31#2440C2214yescomplemented
32#2439C2214yescomplemented
33#2433C2214yes—
34#2409—22no—
35#2317C2211yescharacteristic, complemented
36#2316C2211yescharacteristic, complemented
37#2311C2211yescharacteristic
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic, complemented
40#2101C2214yescomplemented
41#2100C2212yescomplemented
42#2100C2212yescomplemented
43#2073C2214yescomplemented
44#2071C2214yescomplemented
45#2041C2212yes—
46#2041C2214yes—
47#2041C2212yes—
48#2023C2214yescomplemented
49#2023C2214yescomplemented
50#1990C2214yescomplemented
51#1961C2212yes—
52#1957C2212yes—
53#1936C2214yescomplemented
54#1934C2214yescomplemented
55#1934C2214yescomplemented
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
70C8×C22⋊C4C2214yescomplemented
71#1750C2212yescomplemented
72C4×C22⋊C8C2212yescomplemented
73#1736C2212yescomplemented
74#1735C2212yescomplemented
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C23×D4C2312yes—
78C22×M16C2312yes—
79C23×C8C2312yes—
80#535—28no—
81#535—28no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—44no—
84C2×C4⋊Q8C2314yes—
85C2×#127—22no—
86C2×#127C2312yes—
87C2×#127C2312yes—
88C2×#124—22no—
89C2×C22⋊Q8—24no—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121C2314yes—
93C2×#121C2314yes—
94C2×#120—22no—
95C2×#120—22no—
96C2×#120—24no—
97C2×#120—24no—
98#520—44no—
99C2×C4×Q8—21no—
100C2×C4×D4—22no—
101C2×C4×D4—21no—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#486—48no—
113#418—44no—
114C2×#102—44no—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123#408—28no—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128C2×C4×C8C2314yes—
129#398—24no—
130#396—24no—
131#395—24no—
132#394—24no—
133#394—24no—
134#394—24no—
135#393—24no—
136#392—24no—
137#392—24no—
138#392—24no—
139#391—21no—
140#391—24no—
141#391D412yes—
142#390—21no—
143#390—22no—
144#390—22no—
145#390D412yes—
146#390C2314yes—
147#390C2314yes—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2312yes—
152#389C2312yes—
153#388—24no—
154#388—24no—
155#387—24no—
156#386—22no—
157#386D412yes—
158#386D412yes—
159#385—21no—
160#385—21no—
161#385—22no—
162#385—24no—
163#384C2312yes—
164#384C2312yes—
165#382C2312yes—
166#382C2312yes—
167#380—22no—
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×Q8—24no—
188C22×Q8—44no—
189C22×Q8—84no—
190C22×Q8C2×D411yescharacteristic
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4C2×D411yescharacteristic
204C22×D4C2×D412yes—
205C22×D4C2414yes—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C2×M16—24no—
214C2×M16—42no—
215C2×M16—42no—
216C22×C8—24no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8C2414yes—
220C22×C8C2414yes—
221C4⋊Q8—28no—
222C4⋊Q8—44no—
223C4⋊Q8—48no—
224#127—42no—
225#127—42no—
226#127—44no—
227#127—48no—
228#124—48no—
229C22⋊Q8—88no—
230#121—28no—
231#121—28no—
232#121—28no—
233#121—28no—
234#121—84no—
235#121—84no—
236#120—48no—
237#120—48no—
238#120—88no—
239#120—88no—
240C4×Q8—44no—
241C4×D4—44no—
242C4×D4—48no—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—41no—
247#117—41no—
248#117—41no—
249#117—41no—
250#117—42no—
251#117—42no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2411yescharacteristic
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4C2×D414yes—
287C2×C22⋊C4C2×D412yes—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—42no—
293C2×C42C2×D412yes—
294C2×C42C2412yes—
295C2×C42C2412yes—
296#102—28no—
297#102—28no—
298#102—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304C22⋊C8—28no—
305C22⋊C8—28no—
306#97—42no—
307C4×C8—42no—
308C4×C8—44no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95C2×D414yes—
316#95C2×D414yes—
317C24—24no—
318C24—44no—
319C24—82no—
320C24—82no—
321C24—84no—
322C24—84no—
323C24C22×D412yes—
324C24C22×D411yescharacteristic
325#41—48no—
326#41—48no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—84no—
332C2×Q8—816no—
333C2×Q8#12414yes—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—82no—
354C2×D4—82no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4#12414yes—
367C2×D4#12418yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—84no—
391C22×C4C22×D414yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C2×#4114yes—
397C22×C4C2×#4114yes—
398C22×C4C2×#4114yes—
399C22×C4C2511yescharacteristic
400M16—84no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—84no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4H(5,𝔽2)14yes—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—42no—
446C42—44no—
447C42—42no—
448C42—42no—
449C42—44no—
450C23—21no—
451C23—22no—
452C23—21no—
453C23—22no—
454C23—24no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—88no—
466C23—168no—
467C23—168no—
468C23—164no—
469C23—164no—
470C23C2×#12414yes—
471C23C2×#12412yes—
472C23C23×D411yescharacteristic
473Q8—44no—
474Q8—816no—
475D4—48no—
476D4—44no—
477D4—44no—
478D4—44no—
479D4—44no—
480D4—44no—
481D4—88no—
482D4—164no—
483D4—164no—
484D4—168no—
485D4—168no—
486D4—168no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4—164no—
523C2×C4C2×#12414yes—
524C2×C4C2×#12412yes—
525C2×C4#57614yes—
526C2×C4#57714yes—
527C2×C4C22×#4112yes—
528C2×C4C2×H(5,𝔽2)12yes—
529C2×C4#58511yescharacteristic
530C2×C4#58511yescharacteristic
531C8—44no—
532C8—44no—
533C22—24no—
534C22—22no—
535C22—24no—
536C22—28no—
537C22—42no—
538C22—44no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—42no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—1616no—
551C22C22×#12411yescharacteristic
552C22#348411yescharacteristic
553C22#348511yescharacteristic
554C22C2×#57612yes—
555C22C2×#57712yes—
556C4—24no—
557C4—22no—
558C4—24no—
559C4—24no—
560C4—44no—
561C4—44no—
562C4—82no—
563C4—82no—
564C4—81no—
565C4—81no—
566C4—84no—
567C4#350512yes—
568C4#350512yes—
569C2—42no—
570C2—44no—
571C2—164no—
572C2#3537711yescharacteristic
573C2#6062212yes—
574C2#6163011yescharacteristic
575C2#6163311yescharacteristic
576C2#6178812yes—
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. #35574C2semidirect
3. #29340C2semidirect
4. #29331C2semidirect
5. #27986C2semidirect
6. #27720C2semidirect
7. #27696C2semidirect
8. #27273C2semidirect
9. #27254C2semidirect
10. #25903C2semidirect
11. #21770C2semidirect
12. #20914C2semidirect
13. #20530C2semidirect
15. C22×C4⋊Q8C22semidirect
18. C2×#418C22semidirect
19. C22×#102C22semidirect
31. #2440C22semidirect
32. #2439C22semidirect
35. #2317C22semidirect
36. #2316C22semidirect
39. #2296C22semidirect
40. #2101C22semidirect
41. #2100C22semidirect
42. #2100C22semidirect
43. #2073C22semidirect
44. #2071C22semidirect
48. #2023C22semidirect
49. #2023C22semidirect
50. #1990C22semidirect
53. #1936C22semidirect
54. #1934C22semidirect
55. #1934C22semidirect
56. #1934C22semidirect
60. #1875C22semidirect
61. #1875C22semidirect
63. #1874C22semidirect
64. #1874C22semidirect
65. #1836C22semidirect
66. #1835C22semidirect
67. #1825C22semidirect
68. #1786C22semidirect
69. #1753C22semidirect
70. C8×C22⋊C4C22semidirect
71. #1750C22semidirect
72. C4×C22⋊C8C22semidirect
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
#355742
#293402
#293312
#279862
#277202
#276962
#272732
#272542
#259032
#217702
#209142
#205302