Finite Group Databasebeta

← #6493795

Subgroups — #6493795

Order 512.

1677 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35261C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic, complemented
3#33150C212yesmaximal, complemented
4#33107C212yesmaximal, complemented
5#29949C212yesmaximal, complemented
6#29929C212yesmaximal, complemented
7#27869C214yesmaximal, complemented
8#27684C212yesmaximal, complemented
9#27645C212yesmaximal, complemented
10#27641C214yesmaximal, complemented
11#24557C212yesmaximal, complemented
12#24553C212yesmaximal, complemented
13#21967C212yesmaximal, complemented
14#21897C212yesmaximal, complemented
15#20628C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#481C2212yes—
19C2×#481C2212yes—
20C2×#424C2212yes—
21C2×#407C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2856C2211yescharacteristic
24#2851C2211yescharacteristic
25#2847C2212yes—
26#2629C2211yescharacteristic
27#2624C2212yes—
28#2619C2211yescharacteristic
29#2476—22no—
30#2471—22no—
31#2443—22no—
32#2424—22no—
33#2324C2212yes—
34#2314—22no—
35#2314—22no—
36#2314C2212yes—
37#2314C2211yescharacteristic
38#2314C2212yes—
39#2314C2212yes—
40#2314C2212yes—
41C4×#121C2212yes—
42C2×#388—22no—
43C2×#387—22no—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48#2101C2214yescomplemented
49#2092C2214yescomplemented
50#2084C2214yescomplemented
51#2077C2214yes—
52#2077C2214yes—
53#2071C2214yescomplemented
54#2034C2214yes—
55#2033C2214yes—
56#1989C2214yescomplemented
57#1982C2214yescomplemented
58#1949C2214yescomplemented
59#1949C2214yescomplemented
60#1948C2214yescomplemented
61#1948C2214yescomplemented
62#1926C2214yescomplemented
63#1926C2214yescomplemented
64#1920C2214yescomplemented
65#1918C2214yescomplemented
66#1895C2214yes—
67#1893C2214yes—
68#1874C2212yes—
69#1874C2212yes—
70#1853C2212yes—
71#1852C2212yes—
72#1824C2212yes—
73#1824C2212yes—
74#1809C2214yes—
75#1807C2214yes—
76#1804C2214yescomplemented
77#1802C2214yescomplemented
78#1798C2212yes—
79#1798C2212yes—
80#1797C2212yes—
81#1797C2212yes—
82#1729C2212yes—
83C24×C4C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#540—28no—
87C2×#127C2311yescharacteristic
88C2×#125C2311yescharacteristic
89C2×#123—22no—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121C2312yes—
93C2×#121C2312yes—
94C2×#121C2312yes—
95C2×#120—22no—
96C2×C4×D4—22no—
97C2×C4×D4C2312yes—
98C2×#117—28no—
99C2×#117D412yes—
100C2×#117D412yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—22no—
106C22×C4⋊C4D412yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—22no—
110C22×C22⋊C4D412yes—
111C22×C42—22no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#481—44no—
115#481—44no—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120#424—28no—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127#407—42no—
128C2×C22⋊C8—42no—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133#399—22no—
134#398—22no—
135#397—22no—
136#396—22no—
137#394—22no—
138#394—22no—
139#394—22no—
140#392—22no—
141#390—22no—
142#390—22no—
143#390C2312yes—
144#390C2312yes—
145#389C2312yes—
146#389C2312yes—
147#388—22no—
148#388—24no—
149#388—24no—
150#388—24no—
151#388—24no—
152#388—22no—
153#388—22no—
154#387—24no—
155#387—24no—
156#387—24no—
157#387—24no—
158#387—22no—
159#387—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#386—22no—
164#386—22no—
165#385—22no—
166#385—22no—
167#384—22no—
168#384C2312yes—
169#384C2312yes—
170#384C2311yescharacteristic
171#382—22no—
172#382C2312yes—
173#382C2312yes—
174#382C2311yescharacteristic
175#380—22no—
176#380—22no—
177#380C2314yes—
178#380C2314yes—
179#380C2312yes—
180#380C2312yes—
181#380C2312yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—22no—
188C2×#95—22no—
189C2×#95—22no—
190C2×#95—22no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25C2×D411yescharacteristic
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4C2412yes—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—42no—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C2×M16—24no—
223C2×M16—42no—
224C2×M16—42no—
225C22×C8—24no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230#127—24no—
231#125—24no—
232#123—48no—
233#121—28no—
234#121—24no—
235#121—24no—
236#121—44no—
237#121—48no—
238#121—84no—
239#120—84no—
240C4×D4—28no—
241C4×D4—84no—
242#117—24no—
243#117—28no—
244#117—28no—
245#117—24no—
246#117—24no—
247#117#4118yes—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—84no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—84no—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—84no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—44no—
302#106—44no—
303#106—44no—
304#106—44no—
305C4⋊C8—28no—
306C4⋊C8—28no—
307#102—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—42no—
333C24—41no—
334C24C22×D411yescharacteristic
335C24C22×D411yescharacteristic
336C24C22×D411yescharacteristic
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4114yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+412yes—
403C22×C4C2511yescharacteristic
404M16—84no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—84no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—82no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4#12318yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4#12318yes—
449C42—24no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—84no—
454C42—84no—
455C42—82no—
456C42C2×#4114yes—
457C42C2×#4114yes—
458C23—22no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—44no—
471C23—48no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—84no—
481C23—82no—
482C23C2×#12312yes—
483C23#53712yes—
484C23#57714yes—
485C23C23×D411yescharacteristic
486D4—44no—
487D4—44no—
488D4—88no—
489D4—164no—
490D4—164no—
491D4—164no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—88no—
536C2×C4—88no—
537C2×C4—164no—
538C2×C4C2×#12314yes—
539C2×C4C2×#12314yes—
540C2×C4C2×#12312yes—
541C2×C4#53712yes—
542C2×C4C22×#4112yes—
543C2×C4C2×H(5,𝔽2)11yescharacteristic
544C2×C4C2×2−1+411yescharacteristic
545C2×C4#58512yes—
546C8—44no—
547C8—44no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—22no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—88no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22#326014yes—
571C22C22×#12311yescharacteristic
572C22C2×#53711yescharacteristic
573C22#344911yescharacteristic
574C22C2×#57712yes—
575C22#358512yes—
576C4—24no—
577C4—24no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—41no—
583C4—84no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—84no—
588C4#349412yes—
589C4#349512yes—
590C2—24no—
591C2—42no—
592C2—81no—
593C2—162no—
594C2#3526311yescharacteristic
595C2#6052512yes—
596C2C2×#326011yescharacteristic
597C2#6092011yescharacteristic
598C2#6098012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35261C2
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. C2×#2314C2semidirect
3. #33150C2semidirect
4. #33107C2semidirect
5. #29949C2semidirect
6. #29929C2semidirect
7. #27869C2semidirect
8. #27684C2semidirect
9. #27645C2semidirect
10. #27641C2semidirect
11. #24557C2semidirect
12. #24553C2semidirect
13. #21967C2semidirect
14. #21897C2semidirect
15. #20628C2semidirect
48. #2101C22semidirect
49. #2092C22semidirect
50. #2084C22semidirect
53. #2071C22semidirect
56. #1989C22semidirect
57. #1982C22semidirect
58. #1949C22semidirect
59. #1949C22semidirect
60. #1948C22semidirect
61. #1948C22semidirect
62. #1926C22semidirect
63. #1926C22semidirect
64. #1920C22semidirect
65. #1918C22semidirect
76. #1804C22semidirect
77. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#352612
C2×#23142
#331502
#331072
#299492
#299292
#278692
#276842
#276452
#276412
#245572
#245532
#219672
#218972
#206282