Finite Group Databasebeta

← #7622614

Subgroups — #7622614

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3414C211yesmaximal, characteristic
2#60167C211yesmaximal, characteristic, complemented
3C2×#2951C211yesmaximal, characteristic, complemented
4#33460C214yesmaximal, complemented
5#33459C214yesmaximal, complemented
6#33455C214yesmaximal, complemented
7#33454C214yesmaximal, complemented
8C4×#510C214yesmaximal, complemented
9C4×#509C214yesmaximal, complemented
10C2×#2159C211yesmaximal, characteristic, complemented
11#33380C211yesmaximal, characteristic, complemented
12C2×#2139C211yesmaximal, characteristic, complemented
13#33324C211yesmaximal, characteristic, complemented
14C22×#135C2211yescharacteristic
15C22×#135C2211yescharacteristic
16#3428—21no—
17C2×#520—21no—
18#3414C4116yescomplemented
19C2×#510C2212yes—
20C2×#510C2212yes—
21C2×#509C2212yes—
22C2×#509C2212yes—
23C2×#508C2214yes—
24C22×M32C2211yescharacteristic
25C22×M32C2211yescharacteristic
26#2952—22no—
27#2951—22no—
28#2951—22no—
29#2951C2214yes—
30#2951C2214yescomplemented
31#2951C2212yescomplemented
32#2951C2212yescomplemented
33#2950C2212yescomplemented
34#2949—22no—
35#2948C2212yescomplemented
36#2943—22no—
37C2×#440—22no—
38C4×#135C2214yes—
39C2×C4×Q16C2212yescomplemented
40C2×C4×D8C2212yescomplemented
41C2×#427C2211yescharacteristic
42C22×#107C2211yescharacteristic
43#2896—21no—
44C2×#417—21no—
45C2×#405C2211yescharacteristic
46#2177C2218yescomplemented
47#2176C2218yescomplemented
48#2175C2218yescomplemented
49#2175C2218yescomplemented
50C4×Q32C2218yescomplemented
51C4×SD32C2218yescomplemented
52C4×SD32C2218yescomplemented
53C4×D16C2218yescomplemented
54#2159C2214yescomplemented
55#2159C2214yescomplemented
56C2×#367C2212yescomplemented
57C2×#366C2212yescomplemented
58#2141C2212yescomplemented
59#2140C2212yescomplemented
60#2139C2214yes—
61#2139C2214yescomplemented
62#2139C2212yescomplemented
63#2139C2212yescomplemented
64#2138C2214yes—
65C2×#358C2212yescomplemented
66C2×#357C2212yescomplemented
67#2108C2214yescomplemented
68#2108C2214yescomplemented
69C4×M32C2214yescomplemented
70C22×#41—21no—
71C22×#41—21no—
72C2×#135—21no—
73C2×#135—21no—
74C2×#135—21no—
75C2×#135—21no—
76C2×#135—24no—
77C2×#135—24no—
78C2×#135C2×C418yescomplemented
79C2×#135C2×C418yescomplemented
80C2×#135C2312yes—
81C2×#135C2312yes—
82C2×#135C2312yes—
83C2×#135C2312yes—
84C22×Q16C2311yescharacteristic
85C22×Q16C2311yescharacteristic
86C22×SD16—21no—
87C22×SD16—21no—
88C22×D8C2311yescharacteristic
89C22×D8C2311yescharacteristic
90C22×M16D411yescharacteristic
91C23×C8C2311yescharacteristic
92#520—24no—
93#520—24no—
94#520—22no—
95#520—22no—
96#520—42no—
97#520—42no—
98#520—41no—
99#520—41no—
100#519—22no—
101#518—22no—
102#518—42no—
103C4×#41—24no—
104C4×#41—42no—
105C2×C4×Q8—22no—
106C2×C4×D4—22no—
107C2×C4×D4—42no—
108C2×#117—21no—
109C2×#117—42no—
110C2×#117—41no—
111C2×#117D411yescharacteristic
112C22×C4⋊C4—21no—
113C22×C4⋊C4—41no—
114#510C2×C4116yescomplemented
115#510C2×C4116yescomplemented
116#509C2×C4116yescomplemented
117#509C2×C4116yescomplemented
118#508C2×C4132yescomplemented
119C2×Q32C2314yes—
120C2×SD32C2314yes—
121C2×SD32C2314yes—
122C2×D16C2314yes—
123C2×M32—24no—
124C2×M32C2×C418yescomplemented
125C2×M32C2312yes—
126C2×M32C2312yes—
127C2×M32C2312yes—
128C2×M32C2312yes—
129C22×C16C2312yes—
130C22×C16C2312yes—
131#442—24no—
132#441—24no—
133#440—28no—
134#440—28no—
135#440—24no—
136#440—24no—
137C4×Q16—24no—
138C4×Q16C2318yes—
139C4×Q16C2314yes—
140C4×Q16C2314yes—
141C4×SD16—24no—
142C4×SD16—24no—
143C4×D8—24no—
144C4×D8C2318yes—
145C4×D8C2314yes—
146C4×D8C2314yes—
147#427—21no—
148#427—21no—
149#427C2312yes—
150#427C2312yes—
151C2×#107—21no—
152C2×#107—21no—
153C2×#107—21no—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2311yescharacteristic
158C2×#107C2311yescharacteristic
159C2×#106—21no—
160#419—22no—
161#418—22no—
162#417—24no—
163#417—24no—
164#417—22no—
165#417—22no—
166#416—24no—
167C2×#103—22no—
168C2×#102—22no—
169#405D412yes—
170#405D412yes—
171#405C2312yes—
172#405C2312yes—
173C2×#97D412yes—
174C2×C4×C8C2312yes—
175#367C2318yes—
176#366C2318yes—
177#358C2318yes—
178#358C2314yes—
179#358C2314yes—
180#357C2318yes—
181#357C2314yes—
182#357C2314yes—
183#346C2318yes—
184C4×C16C2318yes—
185C2×#41—28no—
186C2×#41—28no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—41no—
192C2×#41—41no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41—44no—
196C2×#41—44no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—41no—
202C22×D4—41no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—42no—
207C23×C4C2×D411yescharacteristic
208#135—28no—
209#135—28no—
210#135—28no—
211#135—28no—
212#135—24no—
213#135—24no—
214#135—24no—
215#135—24no—
216#135C22×C418yes—
217#135C22×C418yes—
218#135C22×C418yes—
219#135C22×C418yescomplemented
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16—22no—
223C2×Q16—22no—
224C2×Q16C22×C414yes—
225C2×Q16C22×C414yescomplemented
226C2×Q16C2412yes—
227C2×Q16C2412yes—
228C2×SD16—28no—
229C2×SD16—28no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×SD16—22no—
233C2×SD16—22no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—22no—
237C2×D8—22no—
238C2×D8C22×C414yes—
239C2×D8C22×C414yescomplemented
240C2×D8C2412yes—
241C2×D8C2412yes—
242C2×M16—24no—
243C2×M16C2×D412yes—
244C2×M16C2×D412yes—
245C22×C8C22×C414yes—
246C22×C8C2×D412yes—
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8#4114yes—
251C22×C8C2411yescharacteristic
252C22×C8C2411yescharacteristic
253C22×C8C2411yescharacteristic
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—44no—
260C4×D4—28no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—44no—
264C4×D4—44no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—48no—
268C4×D4—48no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—44no—
272#117—22no—
273#117—22no—
274#117—41no—
275#117—41no—
276#117—44no—
277#117—44no—
278#117—42no—
279#117—42no—
280#117—44no—
281#117—44no—
282#117D812yes—
283#117D812yes—
284#117C2×D412yes—
285#117C2×D412yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4D811yescharacteristic
300C2×C4⋊C4D811yescharacteristic
301C2×C22⋊C4—41no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4D812yes—
305C2×C42—41no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42C2×D412yes—
309Q32C22×C4116yescomplemented
310SD32C22×C4116yescomplemented
311SD32C22×C4116yescomplemented
312D16C22×C4116yescomplemented
313M32—24no—
314M32—24no—
315M32C22×C418yes—
316M32C22×C418yescomplemented
317C2×C16—24no—
318C2×C16C22×C418yescomplemented
319C2×C16C2414yes—
320C2×C16C2414yes—
321#107—22no—
322#107—22no—
323#107—22no—
324#107—21no—
325#107—21no—
326#107C2412yes—
327#107C2414yes—
328#107C2412yes—
329#106—22no—
330#106—22no—
331#103—28no—
332#103—24no—
333#103—24no—
334#102—28no—
335#102—24no—
336#102—24no—
337#97C2×D414yes—
338#97C2×D414yes—
339C4×C8C2×D414yes—
340C4×C8C2414yes—
341C24—41no—
342C24—41no—
343C24C2×D811yescharacteristic
344#41—28no—
345#41—28no—
346#41—28no—
347#41—28no—
348#41—44no—
349#41—44no—
350#41—44no—
351#41—44no—
352#41—48no—
353#41—48no—
354#41—48no—
355#41—48no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—164no—
395C22×C4—162no—
396C22×C4C4×D414yes—
397C22×C4C2×D812yes—
398C22×C4C2×D812yes—
399C22×C4C2×D811yescharacteristic
400C22×C4C2×D811yescharacteristic
401C22×C4C2×D811yescharacteristic
402C22×C4#13514yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406Q16—24no—
407Q16—24no—
408Q16—22no—
409Q16—22no—
410Q16C23×C414yes—
411Q16C23×C414yes—
412SD16—28no—
413SD16—28no—
414SD16—28no—
415SD16—28no—
416D8—24no—
417D8—24no—
418D8—22no—
419D8—22no—
420D8C23×C414yes—
421D8C23×C414yes—
422M16—24no—
423M16—24no—
424C2×C8—24no—
425C2×C8C4×D414yes—
426C2×C8C4×D414yes—
427C2×C8C4×D414yes—
428C2×C8C23×C412yes—
429C2×C8C23×C412yes—
430C2×C8C23×C412yes—
431C2×C8C22×D414yes—
432C2×C8C22×D411yescharacteristic
433C2×C8C22×D411yescharacteristic
434C2×C8C22×D411yescharacteristic
435C2×C8C2×#4112yes—
436C2×C8C2×#4112yes—
437C2×C8C2×#4112yes—
438C2×C8C2511yescharacteristic
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4C2×D812yes—
451C4⋊C4C2×D812yes—
452C4⋊C4C2×D812yes—
453C4⋊C4C2×D812yes—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4C2×D814yes—
460C22⋊C4C2×D814yes—
461C42—42no—
462C42—42no—
463C42—44no—
464C42—44no—
465C42—48no—
466C42C2×D814yes—
467C42C22×D414yes—
468C16—24no—
469C16C23×C418yes—
470C23—41no—
471C23—41no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—82no—
477C23—82no—
478C23—161no—
479C23—161no—
480C23C4×D814yes—
481C23C22×D811yescharacteristic
482C23C22×D811yescharacteristic
483C23C22×D811yescharacteristic
484Q8—24no—
485Q8—24no—
486Q8—44no—
487Q8—44no—
488Q8—42no—
489Q8—42no—
490D4—24no—
491D4—24no—
492D4—48no—
493D4—48no—
494D4—48no—
495D4—48no—
496D4—44no—
497D4—44no—
498D4—42no—
499D4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—84no—
514C2×C4—164no—
515C2×C4—164no—
516C2×C4—168no—
517C2×C4—161no—
518C2×C4—161no—
519C2×C4—162no—
520C2×C4—162no—
521C2×C4C4×D814yes—
522C2×C4C4×D814yes—
523C2×C4C4×D814yes—
524C2×C4C2×C4×D412yes—
525C2×C4C2×C4×D412yes—
526C2×C4C2×C4×D412yes—
527C2×C4C22×D814yes—
528C2×C4C22×D814yes—
529C2×C4C22×D811yescharacteristic
530C2×C4C22×D811yescharacteristic
531C2×C4C22×D811yescharacteristic
532C2×C4C2×#13512yes—
533C2×C4C2×#13512yes—
534C2×C4C2×#13512yes—
535C2×C4C23×D411yescharacteristic
536C8—24no—
537C8C2×C4×D412yes—
538C8C2×C4×D412yes—
539C8C2×C4×D412yes—
540C8C24×C411yescharacteristic
541C8C22×#4111yescharacteristic
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—88no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—164no—
557C22C2×C4×D812yes—
558C22C2×C4×D812yes—
559C22C2×C4×D812yes—
560C22C23×D811yescharacteristic
561C4—24no—
562C4—24no—
563C4—44no—
564C4—44no—
565C4—84no—
566C4—82no—
567C4—82no—
568C4C2×C4×D812yes—
569C4C2×C4×D812yes—
570C4C2×C4×D812yes—
571C4C22×C4×D411yescharacteristic
572C4C22×#13511yescharacteristic
573C2—22no—
574C2—22no—
575C2—22no—
576C2—84no—
577C2—84no—
578C2C22×C4×D811yescharacteristic
579C2#6263611yescharacteristic
580C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutatorC8C24×C4
Frattini, cosocleC2×C8C25
prime radicalC2×#3414C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60167C2semidirect
3. C2×#2951C2semidirect
4. #33460C2semidirect
5. #33459C2semidirect
6. #33455C2semidirect
7. #33454C2semidirect
8. C4×#510C2semidirect
9. C4×#509C2semidirect
10. C2×#2159C2semidirect
11. #33380C2semidirect
12. C2×#2139C2semidirect
13. #33324C2semidirect
18. #3414C4semidirect
30. #2951C22semidirect
31. #2951C22semidirect
32. #2951C22semidirect
33. #2950C22semidirect
35. #2948C22semidirect
39. C2×C4×Q16C22semidirect
40. C2×C4×D8C22semidirect
46. #2177C22semidirect
47. #2176C22semidirect
48. #2175C22semidirect
49. #2175C22semidirect
50. C4×Q32C22semidirect
51. C4×SD32C22semidirect
52. C4×SD32C22semidirect
53. C4×D16C22semidirect
54. #2159C22semidirect
55. #2159C22semidirect
56. C2×#367C22semidirect
57. C2×#366C22semidirect
58. #2141C22semidirect
59. #2140C22semidirect
61. #2139C22semidirect
62. #2139C22semidirect
63. #2139C22semidirect
65. C2×#358C22semidirect
66. C2×#357C22semidirect
67. #2108C22semidirect
68. #2108C22semidirect
69. C4×M32C22semidirect
78. C2×#135C2×C4semidirect
79. C2×#135C2×C4semidirect
114. #510C2×C4semidirect
115. #510C2×C4semidirect
116. #509C2×C4semidirect
117. #509C2×C4semidirect
118. #508C2×C4semidirect
124. C2×M32C2×C4semidirect
219. #135C22×C4semidirect
225. C2×Q16C22×C4semidirect
239. C2×D8C22×C4semidirect
309. Q32C22×C4semidirect
310. SD32C22×C4semidirect
311. SD32C22×C4semidirect
312. D16C22×C4semidirect
316. M32C22×C4semidirect
318. C2×C16C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#34142
#601672
C2×#29512
#334602
#334592
#334552
#334542
C4×#5102
C4×#5092
C2×#21592
#333802
C2×#21392
#333242