Finite Group Databasebeta

← #495702

Subgroups — #495702

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2948C211yesmaximal, characteristic, complemented
2C2×#2184C211yesmaximal, characteristic
3C2×#2177C211yesmaximal, characteristic, complemented
4C2×C4×D16C211yesmaximal, characteristic, complemented
5#21524C211yesmaximal, characteristic, complemented
6#13012C211yesmaximal, characteristic, complemented
7#13012C211yesmaximal, characteristic, complemented
8#12992C211yesmaximal, characteristic, complemented
9#12937C211yesmaximal, characteristic, complemented
10#12935C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12731C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C2×#518—21no—
18C22×D16C2211yescharacteristic
19C22×D16C2211yescharacteristic
20C2×#466C2211yescharacteristic
21#2948—24no—
22#2948—24no—
23C2×#442—21no—
24C2×#442—21no—
25C2×C4×D8C2211yescharacteristic
26C2×C4×D8C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C4×D8C2211yescharacteristic
29C2×C4×D8C2211yescharacteristic, complemented
30C2×#427C2211yescharacteristic
31C2×#418—21no—
32C2×#405C2211yescharacteristic
33#2324—21no—
34#2184C418yescomplemented
35#2177—24no—
36C4×D16—24no—
37C2×#367C2211yescharacteristic, complemented
38C2×#366C2211yescharacteristic, complemented
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×#357C2211yescharacteristic
45C2×#357C2211yescharacteristic, complemented
46C2×#357C2211yescharacteristic, complemented
47C2×C22⋊C16C2211yescharacteristic
48C2×C22⋊C16C2211yescharacteristic
49C2×#346C2211yescharacteristic, complemented
50C2×C4×C16C2211yescharacteristic, complemented
51#1970C2211yescharacteristic, complemented
52#1942C2211yescharacteristic, complemented
53#1936—21no—
54#1895—21no—
55#1808C2211yescharacteristic
56C4×#107C2211yescharacteristic, complemented
57#1762—21no—
58#1380C2211yescharacteristic, complemented
59#1380C2211yescharacteristic, complemented
60#1380C2211yescharacteristic, complemented
61#1380C2211yescharacteristic, complemented
62#1374C2211yescharacteristic, complemented
63#1374C2211yescharacteristic, complemented
64C23×D4—21no—
65C22×D8—22no—
66C22×D8—22no—
67C22×D8—22no—
68C22×D8—22no—
69C22×D8—21no—
70C22×D8—22no—
71C22×D8C2×C414yescomplemented
72C22×D8C2311yescharacteristic
73C22×D8C2311yescharacteristic
74C22×D8C2311yescharacteristic
75C22×M16D411yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#121—21no—
78#518—28no—
79#518—44no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—41no—
86C2×C4×D4—41no—
87C2×#117—21no—
88C2×#117D411yescharacteristic
89C22×C22⋊C4—41no—
90C22×C22⋊C4—41no—
91C2×D16—22no—
92C2×D16—22no—
93C2×D16—22no—
94C2×D16—22no—
95C2×D16C2×C414yescomplemented
96C2×D16C2×C414yescomplemented
97C22×C16C2311yescharacteristic
98C22×C16C2311yescharacteristic
99C22×C16C2311yescharacteristic
100C22×C16C2311yescharacteristic
101#466—22no—
102#466C2×C414yescomplemented
103#442—28no—
104#442—28no—
105C4×D8—22no—
106C4×D8—22no—
107C4×D8—22no—
108C4×D8—22no—
109C4×D8—22no—
110C4×D8—22no—
111C4×D8—22no—
112C4×D8—22no—
113C4×D8D414yescomplemented
114C4×D8D414yescomplemented
115#427—22no—
116#427—22no—
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#106—21no—
123#418—28no—
124C2×#102—21no—
125C2×#102—21no—
126C2×#102—21no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×#102—21no—
131#405—22no—
132#405—22no—
133C2×#97D411yescharacteristic
134C2×#97D411yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#390—21no—
138#390—41no—
139#389—21no—
140#388—41no—
141#386—41no—
142#385—41no—
143#380—21no—
144C4×C4⋊C4—21no—
145#367—22no—
146#366—22no—
147#357—22no—
148#357—22no—
149#357—22no—
150#357—22no—
151#357—22no—
152#357—22no—
153#357C2×C414yescomplemented
154#357C2×C414yescomplemented
155C22⋊C16—22no—
156C22⋊C16C2×C414yescomplemented
157#346—22no—
158C4×C16—22no—
159#340—21no—
160#340—21no—
161#340—21no—
162C25—41no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—42no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4C2×D411yescharacteristic
180C2×D8—24no—
181C2×D8—28no—
182C2×D8—28no—
183C2×D8—24no—
184C2×D8—28no—
185C2×D8—28no—
186C2×D8—24no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8—21no—
193C2×D8—21no—
194C2×D8—21no—
195C2×D8—21no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8C22×C412yes—
199C2×D8C22×C412yes—
200C2×D8C22×C412yes—
201C2×D8C2×D412yes—
202C2×D8C2×D412yes—
203C2×D8#4112yes—
204C2×D8#4112yes—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—41no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8C22×C412yes—
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8#4112yes—
217C22×C8C2411yescharacteristic
218#121—24no—
219#121—42no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—48no—
227C4×D4—48no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—42no—
232#117—22no—
233#117—22no—
234#117—24no—
235#117—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4D811yescharacteristic
243C2×C4⋊C4D811yescharacteristic
244C2×C22⋊C4—48no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4D811yescharacteristic
257C2×C22⋊C4D811yescharacteristic
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262D16—28no—
263D16—28no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—21no—
268C2×C16—21no—
269C2×C16—21no—
270C2×C16—21no—
271C2×C16—21no—
272C2×C16—21no—
273C2×C16—21no—
274C2×C16C22×C412yes—
275C2×C16C22×C412yes—
276#107—21no—
277#107—21no—
278#107—21no—
279#107—21no—
280#107—21no—
281#107—21no—
282#107—21no—
283#107C22×C412yes—
284#107C2×D412yes—
285#107C2×D412yes—
286#106—24no—
287#102—24no—
288#102—24no—
289#102—24no—
290#102—24no—
291#102—24no—
292#102—42no—
293#102—42no—
294#97—22no—
295#97—22no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8C2×D412yes—
299C4×C8C2×D412yes—
300#95—41no—
301#95—41no—
302#95—41no—
303C24—44no—
304C24—44no—
305C24—41no—
306C24—41no—
307C24—44no—
308C24—41no—
309C24—161no—
310C24—161no—
311C24—161no—
312C24—161no—
313C24C2×D811yescharacteristic
314C2×D4—216no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—44no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—161no—
341C2×D4—161no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—82no—
356C22×C4—161no—
357C22×C4—161no—
358C22×C4—161no—
359C22×C4—161no—
360C22×C4—161no—
361C22×C4—161no—
362C22×C4—161no—
363C22×C4—161no—
364C22×C4—161no—
365C22×C4—161no—
366C22×C4C4×D412yes—
367C22×C4C2×D811yescharacteristic
368C22×C4C2×D811yescharacteristic
369C22×C4C2×D811yescharacteristic
370C22×C4C2×D811yescharacteristic
371C22×C4C2×D811yescharacteristic
372C22×C4#13512yes—
373C22×C4C22×D411yescharacteristic
374D8—216no—
375D8—22no—
376D8—22no—
377D8—22no—
378D8—22no—
379D8—22no—
380D8—22no—
381D8—22no—
382D8—22no—
383D8—41no—
384D8—41no—
385D8C4×D414yes—
386D8C4×D414yes—
387M16—44no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8C4×D412yes—
400C2×C8#12012yes—
401C2×C8C23×C411yescharacteristic
402C2×C8C22×D411yescharacteristic
403C2×C8C22×D411yescharacteristic
404C2×C8C2×#4111yescharacteristic
405C2×C8C2×#4111yescharacteristic
406C2×C8C2×#4111yescharacteristic
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—161no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—82no—
429C22⋊C4—161no—
430C22⋊C4—161no—
431C22⋊C4—161no—
432C42—21no—
433C42—21no—
434C42—44no—
435C42—161no—
436C42#12012yes—
437C42AGL(1,ℤ/8)12yes—
438C16—22no—
439C16—22no—
440C16—41no—
441C16—41no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—416no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—164no—
471C23—164no—
472C23—164no—
473C23—164no—
474C23—164no—
475C23—164no—
476C23C4×D812yes—
477C23C22×D811yescharacteristic
478D4—24no—
479D4—24no—
480D4—416no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—81no—
490D4—81no—
491D4—164no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—164no—
524C2×C4—164no—
525C2×C4—164no—
526C2×C4—164no—
527C2×C4—164no—
528C2×C4C4×D812yes—
529C2×C4#44712yes—
530C2×C4#44712yes—
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C2×#12011yescharacteristic
533C2×C4#52511yescharacteristic
534C2×C4C2×#13511yescharacteristic
535C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
536C2×C4#57511yescharacteristic
537C8—22no—
538C8—22no—
539C8—41no—
540C8C2×C4×D411yescharacteristic
541C8C2×C4×D411yescharacteristic
542C8#52511yescharacteristic
543C8#52511yescharacteristic
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—44no—
552C22—44no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—1616no—
569C22—161no—
570C22—161no—
571C22—161no—
572C22—161no—
573C22C2×C4×D811yescharacteristic
574C22C2×#44711yescharacteristic
575C22#310011yescharacteristic
576C22#341511yescharacteristic
577C22#341511yescharacteristic
578C22#341511yescharacteristic
579C22#341611yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—41no—
584C4—41no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—161no—
589C4—161no—
590C4C4×#12011yescharacteristic
591C4C4×AGL(1,ℤ/8)11yescharacteristic
592C4#300211yescharacteristic
593C4#310511yescharacteristic
594C2—42no—
595C2—84no—
596C2—84no—
597C2—162no—
598C2C4×#44711yescharacteristic
599C2#3345411yescharacteristic
600C2#3345611yescharacteristic
601C2#3352211yescharacteristic
602C2#3352611yescharacteristic
603C2#3367811yescharacteristic
604C2#3368211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2184C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2948C2semidirect
3. C2×#2177C2semidirect
4. C2×C4×D16C2semidirect
5. #21524C2semidirect
6. #13012C2semidirect
7. #13012C2semidirect
8. #12992C2semidirect
9. #12937C2semidirect
10. #12935C2semidirect
11. #12921C2semidirect
12. #12731C2semidirect
13. #12637C2semidirect
14. C4×#357C2semidirect
15. #12600C2semidirect
29. C2×C4×D8C22semidirect
34. #2184C4semidirect
37. C2×#367C22semidirect
38. C2×#366C22semidirect
45. C2×#357C22semidirect
46. C2×#357C22semidirect
49. C2×#346C22semidirect
50. C2×C4×C16C22semidirect
51. #1970C22semidirect
52. #1942C22semidirect
56. C4×#107C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1380C22semidirect
61. #1380C22semidirect
62. #1374C22semidirect
63. #1374C22semidirect
71. C22×D8C2×C4semidirect
95. C2×D16C2×C4semidirect
96. C2×D16C2×C4semidirect
102. #466C2×C4semidirect
113. C4×D8D4semidirect
114. C4×D8D4semidirect
153. #357C2×C4semidirect
154. #357C2×C4semidirect
156. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29482
C2×#21842
C2×#21772
C2×C4×D162
#215242
#130122
#129922
#129372
#129352
#129212
#127312
#126372
C4×#3572
#126002