Finite Group Databasebeta

← #6479548

Subgroups — #6479548

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35262C211yesmaximal, characteristic
2#34048C211yesmaximal, characteristic, complemented
3#33097C214yesmaximal, complemented
4#29910C214yesmaximal, complemented
5#27874C214yesmaximal, complemented
6#27869C214yesmaximal, complemented
7#27619C214yesmaximal, complemented
8#27591C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic, complemented
10#21939C211yesmaximal, characteristic, complemented
11#21767C211yesmaximal, characteristic
12#21761C211yesmaximal, characteristic
13#20685C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2854C2212yes—
19#2854C2212yes—
20#2624C2214yes—
21#2504—21no—
22#2501—21no—
23#2500—22no—
24#2482—22no—
25#2405—21no—
26#2405—21no—
27#2317C2212yes—
28#2314C2214yes—
29#2314C2214yescomplemented
30#2313—21no—
31#2313—21no—
32#2312—21no—
33#2312—21no—
34#2309C2212yes—
35#2298—22no—
36#2298—22no—
37#2285C2212yescomplemented
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41C4×#117C2212yescomplemented
42#2101C2214yescomplemented
43#2101C2214yescomplemented
44#2092C2214yescomplemented
45#2092C2214yescomplemented
46#2084C2214yescomplemented
47#2083C2214yescomplemented
48#2071C2214yescomplemented
49#2071C2214yescomplemented
50#1926C2214yescomplemented
51#1926C2214yescomplemented
52#1925C2214yescomplemented
53#1925C2214yescomplemented
54#1921C2214yescomplemented
55#1920C2214yescomplemented
56#1919C2214yescomplemented
57#1918C2214yescomplemented
58#1895C2214yes—
59#1893C2212yes—
60#1893C2212yes—
61#1875C2214yes—
62#1875C2214yes—
63#1854C2214yes—
64#1854C2214yes—
65#1854C2214yescomplemented
66#1854C2214yescomplemented
67#1824C2212yes—
68#1823C2212yes—
69#1818C2214yes—
70#1817C2212yescomplemented
71#1817C2212yescomplemented
72#1809C2214yescomplemented
73#1808C2212yescomplemented
74#1807C2212yescomplemented
75#1786C2212yes—
76#1786C2212yes—
77#1729C2212yescomplemented
78#1729C2212yescomplemented
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×M16C2312yes—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#123—22no—
86C2×#123—22no—
87C2×#121—21no—
88C2×#121—21no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121C2312yes—
92C2×#121C2312yes—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117D412yes—
96C2×#117D412yes—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2312yes—
101C2×#117C2312yes—
102C22×C4⋊C4D411yescharacteristic
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118#407—24no—
119C2×C22⋊C8—24no—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2312yes—
124#399—24no—
125#397—24no—
126#394—24no—
127#394—24no—
128#390—22no—
129#390—22no—
130#390C2314yes—
131#390C2314yes—
132#389C2314yes—
133#389C2314yes—
134#388—24no—
135#388—24no—
136#388—24no—
137#388—24no—
138#388—24no—
139#388—24no—
140#387—24no—
141#387—24no—
142#387—24no—
143#387—24no—
144#386—22no—
145#386—22no—
146#385—22no—
147#385—22no—
148#384—22no—
149#384C2314yes—
150#382—22no—
151#382C2314yes—
152#381—21no—
153#381—21no—
154#381—21no—
155#380—22no—
156#380—22no—
157#380C2312yes—
158#380C2312yes—
159#380C2312yes—
160#380C2312yes—
161#380C2312yes—
162#380C2312yes—
163#379—21no—
164#379—21no—
165#379—21no—
166#379—22no—
167#379—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C4×C22⋊C4—22no—
173C4×C22⋊C4—22no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180#376—22no—
181C43—22no—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C25C2×D411yescharacteristic
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—84no—
205C22×D4C2412yes—
206C23×C4—21no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C2×M16—24no—
222C2×M16—42no—
223C2×M16—42no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2414yes—
231#123—84no—
232#123—84no—
233#121—24no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121—44no—
238#121—44no—
239#117—28no—
240#117—28no—
241#117—24no—
242#117—24no—
243#117—24no—
244#117—24no—
245#117—42no—
246#117—42no—
247#117—42no—
248#117—42no—
249#117—42no—
250#117—42no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2412yes—
302C2×C42C2412yes—
303#107—44no—
304#107—44no—
305#106—44no—
306#106—44no—
307#102—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327C24—22no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—44no—
332C24—42no—
333C24—82no—
334C24—82no—
335C24#12314yes—
336C24C22×D411yescharacteristic
337C24C22×D411yescharacteristic
338C24C22×D411yescharacteristic
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—84no—
352C2×D4—82no—
353C2×D4—82no—
354C2×D4—816no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—82no—
381C22×C4#12314yes—
382C22×C4#12314yes—
383C22×C4#12314yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2511yescharacteristic
398M16—84no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—82no—
407C2×C8—82no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C42—24no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—84no—
454C42—84no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—88no—
476C23—168no—
477C23C2×#12312yes—
478C23C2×#12312yes—
479C23C2×#12312yes—
480C23C23×D411yescharacteristic
481D4—42no—
482D4—42no—
483D4—44no—
484D4—84no—
485D4—88no—
486D4—88no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4C2×#12312yes—
528C2×C4C2×#12312yes—
529C2×C4C2×#12312yes—
530C2×C4C22×#4111yescharacteristic
531C2×C4C22×#4111yescharacteristic
532C2×C4#58512yes—
533C2×C4#58512yes—
534C8—42no—
535C8—42no—
536C8—44no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—88no—
550C22—82no—
551C22—84no—
552C22—84no—
553C22—82no—
554C22—82no—
555C22—168no—
556C22C22×#12311yescharacteristic
557C22#344912yes—
558C22#358412yes—
559C22#358512yes—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4—82no—
571C4#349111yescharacteristic
572C4#349111yescharacteristic
573C4#349112yes—
574C2—42no—
575C2—42no—
576C2—42no—
577C2—162no—
578C2#3509711yescharacteristic
579C2#6051611yescharacteristic
580C2#6051811yescharacteristic
581C2#6098514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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. #34048C2semidirect
3. #33097C2semidirect
4. #29910C2semidirect
5. #27874C2semidirect
6. #27869C2semidirect
7. #27619C2semidirect
8. #27591C2semidirect
9. #21943C2semidirect
10. #21939C2semidirect
13. #20685C2semidirect
29. #2314C22semidirect
37. #2285C22semidirect
41. C4×#117C22semidirect
42. #2101C22semidirect
43. #2101C22semidirect
44. #2092C22semidirect
45. #2092C22semidirect
46. #2084C22semidirect
47. #2083C22semidirect
48. #2071C22semidirect
49. #2071C22semidirect
50. #1926C22semidirect
51. #1926C22semidirect
52. #1925C22semidirect
53. #1925C22semidirect
54. #1921C22semidirect
55. #1920C22semidirect
56. #1919C22semidirect
57. #1918C22semidirect
65. #1854C22semidirect
66. #1854C22semidirect
70. #1817C22semidirect
71. #1817C22semidirect
72. #1809C22semidirect
73. #1808C22semidirect
74. #1807C22semidirect
77. #1729C22semidirect
78. #1729C22semidirect

Maximal subgroups

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

subgroupindex
#352622
#340482
#330972
#299102
#278742
#278692
#276192
#275912
#219432
#219392
#217672
#217612
#206852