Finite Group Databasebeta

← #7623052

Subgroups — #7623052

Order 512.

1611 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60458C211yesmaximal, characteristic
2#60108C211yesmaximal, characteristic, complemented
3#33653C213yesmaximal, complemented
4#33652C213yesmaximal, complemented
5#33650C213yesmaximal, complemented
6#33650C213yesmaximal, complemented
7#33648C213yesmaximal, complemented
8#33647C213yesmaximal, complemented
9#33645C213yesmaximal, complemented
10C2×#2234C213yesmaximal, complemented
11#33411C211yesmaximal, characteristic, complemented
12#33410C211yesmaximal, characteristic, complemented
13#33381C211yesmaximal, characteristic, complemented
14#33381C211yesmaximal, characteristic, complemented
15#33334C211yesmaximal, characteristic, complemented
16C2×#576C2211yescharacteristic
17C22×#131C2211yescharacteristic
18#3488—21no—
19#3428—21no—
20C2×#504C2211yescharacteristic
21C2×#504C2211yescharacteristic
22#3062—23no—
23#3061C2213yes—
24#3058C2213yes—
25#3055—23no—
26#3050C2213yes—
27C2×#466C2213yes—
28#2913C2211yescharacteristic
29#2913C2211yescharacteristic
30#2913C2211yescharacteristic
31#2913C2213yescomplemented
32#2913C2213yescomplemented
33#2912—21no—
34#2912—23no—
35#2911—23no—
36C2×#427C2213yescomplemented
37C2×#427C2213yescomplemented
38#2897—21no—
39#2237C2211yescharacteristic, complemented
40#2237C2213yescomplemented
41#2237C2216yescomplemented
42#2236C2213yescomplemented
43#2235C2213yescomplemented
44#2234C2213yescomplemented
45#2234C2216yescomplemented
46#2234C2216yescomplemented
47#2234C2216yescomplemented
48#2234C2213yescomplemented
49#2234C2216yescomplemented
50#2233C2213yescomplemented
51#2233C2216yescomplemented
52#2232C2213yescomplemented
53#2232C2216yescomplemented
54#2159C2213yescomplemented
55#2159C2213yescomplemented
56#2159C2213yescomplemented
57#2159C2213yescomplemented
58#2158C2213yescomplemented
59#2158C2213yescomplemented
60C2×#367C2213yescomplemented
61C2×#366C2213yescomplemented
62#2140C2213yescomplemented
63#2140C2213yescomplemented
64#2139C2213yescomplemented
65#2139C2213yescomplemented
66#2138C2213yescomplemented
67#2138C2213yescomplemented
68C2×#357C2213yescomplemented
69C2×#357C2213yescomplemented
70#2113C2213yescomplemented
71#2113C2213yescomplemented
72#2112C2213yescomplemented
73C2×C22⋊C16C2213yescomplemented
74C2×H(5,𝔽2)—21no—
75C22×#41D411yescharacteristic
76#576—24no—
77#576—24no—
78C2×AGL(1,ℤ/8)—23no—
79C2×#135C2313yes—
80C22×D8C2313yes—
81C2×#131—24no—
82C2×#131—24no—
83C2×#131D411yescharacteristic
84C2×#131D413yes—
85C2×#131C2313yes—
86C2×#131C2311yescharacteristic
87C22×M16D413yes—
88C23×C8C2313yes—
89#550—41no—
90#548—41no—
91#546—43no—
92D42—43no—
93#537—23no—
94#535—23no—
95#525—23no—
96C2×#121—23no—
97#520—21no—
98#520—21no—
99#520—21no—
100#520—23no—
101#520—23no—
102#519—41no—
103#518—43no—
104C4×#41—43no—
105C4×#41—41no—
106C2×#117—23no—
107C2×#117—23no—
108#504—24no—
109#504—24no—
110C2×M32C2313yes—
111C2×M32C2313yes—
112C22×C16C2313yes—
113C22×C16C2313yes—
114#469—26no—
115#468—26no—
116#467C2313yes—
117#467C2311yescharacteristic
118#466—23no—
119#466—23no—
120#466C2316yes—
121#466C2313yes—
122#466C2313yes—
123#465—23no—
124#465—23no—
125#428—23no—
126#428—23no—
127#428—23no—
128#428—23no—
129#428—26no—
130#428—26no—
131#428—23no—
132#427—23no—
133#427—23no—
134#427—23no—
135#427—23no—
136#427C2313yes—
137#427C2313yes—
138#427C2313yes—
139#427C2311yescharacteristic
140#427C2316yes—
141#427C2313yes—
142C2×#107C2313yes—
143C2×#107C2313yes—
144C2×#107C2313yes—
145C2×#107C2313yes—
146C2×#107C2313yes—
147C2×#106—23no—
148C2×#106—23no—
149#418—23no—
150#417—23no—
151#416—23no—
152C2×#102—23no—
153#367C2313yes—
154#367C2316yes—
155#367C2311yescharacteristic
156#367C2313yes—
157#367C2313yes—
158#366C2313yes—
159#366C2316yes—
160#366C2311yescharacteristic
161#366C2313yes—
162#366C2313yes—
163#358C2311yescharacteristic
164#358C2311yescharacteristic
165#358C2313yes—
166#358C2313yes—
167#357C2313yes—
168#357C2313yes—
169#357C2316yes—
170#357C2316yes—
171#357C2313yes—
172#357C2313yes—
173C22⋊C16C2311yescharacteristic
174C22⋊C16C2313yes—
175C22⋊C16C2313yes—
176C22⋊C16C2313yes—
177C22⋊C16C2316yes—
178H(5,𝔽2)—44no—
179H(5,𝔽2)—44no—
180C2×#41—24no—
181C2×#41—24no—
182C2×#41—23no—
183C2×#41—41no—
184C2×#41D813yes—
185C2×#41D811yescharacteristic
186C2×#41C2×D413yes—
187C2×#41C2×D411yescharacteristic
188C22×Q8D811yescharacteristic
189C22×D4—23no—
190C22×D4—43no—
191C22×D4—43no—
192C22×D4D813yes—
193C23×C4C2×D413yes—
194AGL(1,ℤ/8)—412no—
195#135—26no—
196#135—26no—
197C2×Q16C2411yescharacteristic
198C2×SD16—23no—
199C2×D8—23no—
200C2×D8—26no—
201C2×D8—26no—
202C2×D8C2413yes—
203#131—24no—
204#131—212no—
205#131—26no—
206#131—26no—
207#131#4114yes—
208#131#4114yes—
209C2×M16—212no—
210C2×M16C2×D413yes—
211C2×M16C2×D416yes—
212C2×M16C2×D413yes—
213C22×C8—26no—
214C22×C8—26no—
215C22×C8C2×D413yes—
216C22×C8C2×D413yes—
217C22×C8C2×D413yes—
218C22×C8C2411yescharacteristic
219C22×C8C2413yes—
220C22×C8C2413yes—
221#127—43no—
222#124—43no—
223#123—43no—
224C22⋊Q8—21no—
225C22⋊Q8—23no—
226C22⋊Q8—43no—
227#121—23no—
228#121—26no—
229#121—23no—
230#121—46no—
231#121—43no—
232#121—43no—
233#121—43no—
234#120—43no—
235#120—46no—
236C4×Q8—43no—
237C4×Q8—43no—
238C4×Q8—41no—
239C4×Q8—41no—
240C4×Q8—41no—
241C4×D4—46no—
242C4×D4—46no—
243C4×D4—43no—
244C4×D4—43no—
245C4×D4—43no—
246C4×D4—43no—
247C4×D4—43no—
248C4×D4—43no—
249C4×D4—43no—
250#117—23no—
251#117—21no—
252#117—26no—
253#117—23no—
254#117—23no—
255#117—23no—
256#117—43no—
257#117—43no—
258#117—43no—
259#117—43no—
260C2×C4⋊C4—23no—
261C2×C4⋊C4—23no—
262C2×C4⋊C4—23no—
263C2×C4⋊C4—23no—
264C2×C4⋊C4—23no—
265C2×C22⋊C4—43no—
266C2×C22⋊C4—43no—
267C2×C22⋊C4—43no—
268C2×C42—43no—
269C2×C42—43no—
270M32—26no—
271M32—26no—
272C2×C16—26no—
273C2×C16—26no—
274C2×C16C2411yescharacteristic
275C2×C16C2413yes—
276C2×C16C2413yes—
277C2×C16C2411yescharacteristic
278#107—23no—
279#107—23no—
280#107—23no—
281#107—23no—
282#107—23no—
283#107C2413yes—
284#107C2411yescharacteristic
285#107C2413yes—
286#107C2411yescharacteristic
287#107C2413yes—
288#107C2411yescharacteristic
289#106—23no—
290#106—21no—
291#106—26no—
292#106—23no—
293#106—23no—
294#106—23no—
295#103—21no—
296#103—23no—
297#102—23no—
298#102—26no—
299#102—23no—
300C24—43no—
301C24C2×D813yes—
302#41—24no—
303#41—212no—
304#41—26no—
305#41—26no—
306#41—46no—
307#41—46no—
308#41—84no—
309#41#12314yes—
310#41#13714yes—
311C2×Q8—24no—
312C2×Q8—21no—
313C2×Q8C2×D813yes—
314C2×Q8C2×D811yescharacteristic
315C2×D4—212no—
316C2×D4—23no—
317C2×D4—43no—
318C2×D4—43no—
319C2×D4—43no—
320C2×D4—46no—
321C2×D4—46no—
322C2×D4—43no—
323C2×D4—46no—
324C2×D4—812no—
325C2×D4—166no—
326C2×D4C2×D816yes—
327C2×D4C2×D813yes—
328C2×D4C2×D813yes—
329C22×C4—26no—
330C22×C4—26no—
331C22×C4—43no—
332C22×C4—43no—
333C22×C4—43no—
334C22×C4—43no—
335C22×C4—43no—
336C22×C4—43no—
337C22×C4C2×D813yes—
338C22×C4C2×D813yes—
339C22×C4C2×D813yes—
340C22×C4C22×D413yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D413yes—
343Q16—21no—
344Q16—21no—
345SD16—46no—
346D8—23no—
347D8—23no—
348D8—46no—
349M16—26no—
350M16—212no—
351M16#123112yes—
352C2×C8—26no—
353C2×C8—26no—
354C2×C8—26no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—23no—
358C2×C8—23no—
359C2×C8C22×D411yescharacteristic
360C2×C8C22×D413yes—
361C2×C8C22×D413yes—
362C2×C8C2×#4116yes—
363C2×C8C2×#4116yes—
364C2×C8C2511yescharacteristic
365C4⋊C4—21no—
366C4⋊C4—23no—
367C4⋊C4—23no—
368C4⋊C4—21no—
369C4⋊C4—23no—
370C4⋊C4—21no—
371C4⋊C4—43no—
372C4⋊C4—43no—
373C4⋊C4—43no—
374C4⋊C4—43no—
375C4⋊C4—43no—
376C4⋊C4—41no—
377C4⋊C4—43no—
378C22⋊C4—43no—
379C22⋊C4—41no—
380C22⋊C4—46no—
381C22⋊C4—43no—
382C22⋊C4—43no—
383C22⋊C4—43no—
384C22⋊C4—41no—
385C22⋊C4—43no—
386C22⋊C4—43no—
387C22⋊C4—46no—
388C22⋊C4—43no—
389C42—43no—
390C42—41no—
391C42—46no—
392C42—43no—
393C42—43no—
394C42—43no—
395C16—21no—
396C16—21no—
397C16—23no—
398C16—23no—
399C23—26no—
400C23—43no—
401C23—43no—
402C23—43no—
403C23—86no—
404C23—163no—
405C23—163no—
406C23C22×D811yescharacteristic
407C23C22×D813yes—
408C23C22×D813yes—
409Q8—26no—
410Q8—41no—
411Q8—41no—
412Q8#48014yes—
413D4—212no—
414D4—26no—
415D4—43no—
416D4—43no—
417D4—86no—
418D4—86no—
419D4—86no—
420D4—163no—
421D4—166no—
422D4#480112yes—
423C2×C4—26no—
424C2×C4—26no—
425C2×C4—26no—
426C2×C4—21no—
427C2×C4—23no—
428C2×C4—21no—
429C2×C4—23no—
430C2×C4—43no—
431C2×C4—41no—
432C2×C4—43no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—43no—
436C2×C4—41no—
437C2×C4—86no—
438C2×C4—163no—
439C2×C4—163no—
440C2×C4—163no—
441C2×C4—163no—
442C2×C4—163no—
443C2×C4C2×#12316yes—
444C2×C4C22×D811yescharacteristic
445C2×C4C22×D813yes—
446C2×C4C22×D813yes—
447C2×C4C2×#13716yes—
448C2×C4C23×D411yescharacteristic
449C8—21no—
450C8—23no—
451C8C2×#12316yes—
452C8C22×#4111yescharacteristic
453C8C22×#4111yescharacteristic
454C22—23no—
455C22—23no—
456C22—21no—
457C22—23no—
458C22—23no—
459C22—21no—
460C22—23no—
461C22—21no—
462C22—86no—
463C22—83no—
464C22—83no—
465C22—83no—
466C22—326no—
467C22C2×#48016yes—
468C22C23×D811yescharacteristic
469C4—21no—
470C4—23no—
471C4—81no—
472C4—83no—
473C4—81no—
474C4—81no—
475C4—83no—
476C4—81no—
477C4—83no—
478C4C2×#48016yes—
479C4C22×#12311yescharacteristic
480C4C22×#13711yescharacteristic
481C2—26no—
482C2—43no—
483C2—41no—
484C2—163no—
485C2C22×#48011yescharacteristic
486C2#6263511yescharacteristic
487C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#60458C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60108C2semidirect
3. #33653C2semidirect
4. #33652C2semidirect
5. #33650C2semidirect
6. #33650C2semidirect
7. #33648C2semidirect
8. #33647C2semidirect
9. #33645C2semidirect
10. C2×#2234C2semidirect
11. #33411C2semidirect
12. #33410C2semidirect
13. #33381C2semidirect
14. #33381C2semidirect
15. #33334C2semidirect
31. #2913C22semidirect
32. #2913C22semidirect
36. C2×#427C22semidirect
37. C2×#427C22semidirect
39. #2237C22semidirect
40. #2237C22semidirect
41. #2237C22semidirect
42. #2236C22semidirect
43. #2235C22semidirect
44. #2234C22semidirect
45. #2234C22semidirect
46. #2234C22semidirect
47. #2234C22semidirect
48. #2234C22semidirect
49. #2234C22semidirect
50. #2233C22semidirect
51. #2233C22semidirect
52. #2232C22semidirect
53. #2232C22semidirect
54. #2159C22semidirect
55. #2159C22semidirect
56. #2159C22semidirect
57. #2159C22semidirect
58. #2158C22semidirect
59. #2158C22semidirect
60. C2×#367C22semidirect
61. C2×#366C22semidirect
62. #2140C22semidirect
63. #2140C22semidirect
64. #2139C22semidirect
65. #2139C22semidirect
66. #2138C22semidirect
67. #2138C22semidirect
68. C2×#357C22semidirect
69. C2×#357C22semidirect
70. #2113C22semidirect
71. #2113C22semidirect
72. #2112C22semidirect
73. C2×C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#604582
#601082
#336532
#336522
#336502
#336482
#336472
#336452
C2×#22342
#334112
#334102
#333812
#333342