Finite Group Databasebeta

← C4×#3416 #7622616

Subgroups — C4×#3416 #7622616

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3416C211yesmaximal, characteristic
2C4×#578C211yesmaximal, characteristic, complemented
3C4×#576C211yesmaximal, characteristic, complemented
4#33460C213yesmaximal, complemented
5#33457C213yesmaximal, complemented
6#33455C213yesmaximal, complemented
7#33454C213yesmaximal, complemented
8C4×#510C213yesmaximal, complemented
9C4×#509C213yesmaximal, complemented
10C4×#508C213yesmaximal, complemented
11C2×C4×SD32C213yesmaximal, complemented
12#33411C211yesmaximal, characteristic, complemented
13#33382C211yesmaximal, characteristic, complemented
14#33381C211yesmaximal, characteristic, complemented
15C4×#504C211yesmaximal, characteristic, complemented
16C2×#578C2211yescharacteristic
17C2×#576C2211yescharacteristic
18C4×2−1+4—21no—
19C4×H(5,𝔽2)—21no—
20#3416C4116yesdirect product factor
21C2×#510C2213yes—
22C2×#509C2213yes—
23C2×#508C2213yes—
24C22×SD32C2213yes—
25C2×#504C2211yescharacteristic
26C2×#504C2211yescharacteristic
27#2951C2213yes—
28#2951C2213yescomplemented
29#2950C2213yescomplemented
30#2948C2213yes—
31#2947—23no—
32#2946—23no—
33C4×#137—23no—
34C4×AGL(1,ℤ/8)—23no—
35C4×#135C2213yes—
36C4×#135C2213yescomplemented
37C2×C4×Q16C2213yescomplemented
38C2×C4×D8C2213yes—
39#2913C2211yescharacteristic
40#2913C2211yescharacteristic
41#2898—21no—
42#2897—21no—
43C4×#131C2211yescharacteristic
44#2177C2216yescomplemented
45#2176C2216yescomplemented
46#2175C2216yescomplemented
47#2175C2216yescomplemented
48C4×Q32C2213yescomplemented
49C4×Q32C2213yescomplemented
50C4×SD32C2213yescomplemented
51C4×SD32C2216yescomplemented
52C4×SD32C2216yescomplemented
53C4×SD32C2216yescomplemented
54C4×SD32C2211yescharacteristic, complemented
55C4×SD32C2213yescomplemented
56C4×SD32C2213yescomplemented
57C4×D16C2213yescomplemented
58C4×D16C2213yescomplemented
59#2159C2213yescomplemented
60#2159C2213yescomplemented
61#2158C2213yescomplemented
62C2×#367C2213yescomplemented
63#2141C2213yescomplemented
64#2140C2213yes—
65#2139C2213yes—
66#2139C2213yescomplemented
67#2138C2213yes—
68#2138C2213yescomplemented
69C2×#358C2213yescomplemented
70C2×#357C2213yes—
71#2108C2213yescomplemented
72#2108C2213yescomplemented
73C4×M32C2213yescomplemented
74C2×C4×C16C2213yescomplemented
75C2×2−1+4—21no—
76C2×H(5,𝔽2)—21no—
77#578—24no—
78#578C2×C418yescomplemented
79#576—24no—
80#576C2×C418yescomplemented
81C2×#137—23no—
82C2×AGL(1,ℤ/8)—23no—
83C2×#135C2313yes—
84C2×#135C2313yes—
85C22×Q16C2313yes—
86C22×D8C2313yes—
87C2×#131D411yescharacteristic
88C2×#131C2311yescharacteristic
89#520—21no—
90#520—21no—
91#520—23no—
92#520—23no—
93#520—41no—
94#520—43no—
95#519—23no—
96#519—41no—
97#518—23no—
98#518—43no—
99C4×#41—23no—
100C4×#41—23no—
101C4×#41—41no—
102C4×#41—43no—
103C4×#41D411yescharacteristic
104C2×C4×Q8—23no—
105C2×C4×Q8—41no—
106C2×C4×D4—23no—
107C2×C4×D4—43no—
108#510C2×C4124yescomplemented
109#509C2×C4124yescomplemented
110#508C2×C4124yescomplemented
111C2×Q32C2313yes—
112C2×SD32C2×C4124yescomplemented
113C2×SD32C2313yes—
114C2×SD32C2316yes—
115C2×SD32C2311yescharacteristic
116C2×D16C2313yes—
117#504—24no—
118#504C2×C418yescomplemented
119C2×M32C2313yes—
120C2×M32C2313yes—
121C22×C16C2313yes—
122C22×C16C2313yes—
123#442—26no—
124#441—26no—
125#440—26no—
126#440—26no—
127C4×Q16—23no—
128C4×Q16—23no—
129C4×Q16C2313yes—
130C4×Q16C2316yes—
131C4×Q16C2311yescharacteristic
132C4×Q16C2313yes—
133C4×Q16C2313yes—
134C4×SD16—23no—
135C4×SD16—23no—
136C4×SD16—23no—
137C4×SD16—23no—
138C4×D8—23no—
139C4×D8—23no—
140C4×D8C2313yes—
141C4×D8C2316yes—
142C4×D8C2313yes—
143C4×D8C2311yescharacteristic
144C4×D8C2313yes—
145#428—23no—
146#428—23no—
147#427C2313yes—
148#427C2313yes—
149C2×#107C2313yes—
150C2×#107C2313yes—
151#419—23no—
152#418—23no—
153#417—23no—
154#417—23no—
155#416—23no—
156#416—23no—
157C2×#103—23no—
158C2×#102—23no—
159#405D413yes—
160#405C2313yes—
161C4×M16D413yes—
162C2×C4×C8C2313yes—
163#367C2313yes—
164#367C2316yes—
165#367C2311yescharacteristic
166#366C2313yes—
167#366C2313yes—
168#358C2313yes—
169#358C2316yes—
170#358C2311yescharacteristic
171#358C2313yes—
172#358C2313yes—
173#357C2313yes—
174#357C2316yes—
175#357C2313yes—
176#357C2311yescharacteristic
177#357C2313yes—
178#346C2316yes—
179C4×C16C2311yescharacteristic
180C4×C16C2313yes—
181C4×C16C2313yes—
182C4×C16C2313yes—
1832−1+4—28no—
1842−1+4—44no—
185H(5,𝔽2)—28no—
186H(5,𝔽2)—44no—
187C2×#41—23no—
188C2×#41—23no—
189C2×#41—41no—
190C2×#41—43no—
191C2×#41C2×D411yescharacteristic
192C22×Q8—23no—
193C22×Q8—41no—
194C22×D4—23no—
195C22×D4—43no—
196C23×C4—43no—
197#137—224no—
198AGL(1,ℤ/8)—224no—
199#135—26no—
200#135—26no—
201#135C22×C4112yes—
202#135C22×C4112yescomplemented
203C2×Q16—23no—
204C2×Q16—26no—
205C2×Q16C22×C4112yescomplemented
206C2×Q16C2413yes—
207C2×Q16C2411yescharacteristic
208C2×SD16—23no—
209C2×SD16—23no—
210C2×D8—23no—
211C2×D8—26no—
212C2×D8C22×C4112yes—
213C2×D8C2413yes—
214C2×D8C2411yescharacteristic
215#131—24no—
216#131C22×C414yes—
217#131#4114yes—
218C2×M16C2×D413yes—
219C2×M16C2×D413yes—
220C22×C8C2×D413yes—
221C22×C8C2413yes—
222C4×Q8—23no—
223C4×Q8—26no—
224C4×Q8—21no—
225C4×Q8—23no—
226C4×Q8—23no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—43no—
230C4×Q8—43no—
231C4×Q8—43no—
232C4×Q8—43no—
233C4×Q8D811yescharacteristic
234C4×Q8D811yescharacteristic
235C4×D4—23no—
236C4×D4—26no—
237C4×D4—21no—
238C4×D4—23no—
239C4×D4—23no—
240C4×D4—43no—
241C4×D4—43no—
242C4×D4—43no—
243C4×D4—43no—
244C4×D4—43no—
245C4×D4—43no—
246C4×D4—43no—
247C4×D4—43no—
248C4×D4—46no—
249C4×D4—46no—
250C4×D4—43no—
251C4×D4—43no—
252C4×D4D813yes—
253C4×D4D813yes—
254#117—23no—
255#117—23no—
256#117—43no—
257#117—43no—
258#117—46no—
259#117—43no—
260#117—43no—
261#117C2×D413yes—
262C2×C4⋊C4—23no—
263C2×C4⋊C4—23no—
264C2×C4⋊C4—43no—
265C2×C4⋊C4—43no—
266C2×C4⋊C4—43no—
267C2×C22⋊C4—43no—
268C2×C22⋊C4—46no—
269C2×C42—43no—
270C2×C42—43no—
271C2×C42—43no—
272C2×C42C2×D413yes—
273Q32C22×C4112yescomplemented
274SD32C22×C4112yescomplemented
275SD32C22×C4124yescomplemented
276SD32C22×C414yescomplemented
277D16C22×C4112yescomplemented
278M32—26no—
279M32C22×C4112yescomplemented
280C2×C16—26no—
281C2×C16C22×C4112yescomplemented
282C2×C16C2411yescharacteristic
283C2×C16C2413yes—
284C2×C16C2411yescharacteristic
285C2×C16C2413yes—
286#107—23no—
287#107—23no—
288#107C2411yescharacteristic
289#107C2413yes—
290#107C2411yescharacteristic
291#107C2413yes—
292#106—23no—
293#106—23no—
294#103—23no—
295#103—26no—
296#103—21no—
297#103—23no—
298#103—23no—
299#102—23no—
300#102—26no—
301#102—21no—
302#102—23no—
303#102—23no—
304#97C2×D416yes—
305C4×C8C2×D413yes—
306C4×C8C2×D413yes—
307C4×C8C2411yescharacteristic
308C4×C8C2413yes—
309C24—43no—
310#41—212no—
311#41—212no—
312#41—48no—
313#41—424no—
314#41—46no—
315#41—46no—
316#41C4×D414yes—
317#41#13514yes—
318C2×Q8—212no—
319C2×Q8—21no—
320C2×Q8—23no—
321C2×Q8—48no—
322C2×Q8—43no—
323C2×Q8—46no—
324C2×Q8C2×D811yescharacteristic
325C2×D4—212no—
326C2×D4—23no—
327C2×D4—21no—
328C2×D4—43no—
329C2×D4—43no—
330C2×D4—424no—
331C2×D4—43no—
332C2×D4—46no—
333C2×D4C2×D813yes—
334C22×C4—43no—
335C22×C4—43no—
336C22×C4—43no—
337C22×C4—43no—
338C22×C4—43no—
339C22×C4—41no—
340C22×C4—43no—
341C22×C4—43no—
342C22×C4—43no—
343C22×C4—46no—
344C22×C4—166no—
345C22×C4C2×D813yes—
346C22×C4C2×D813yes—
347C22×C4C22×D413yes—
348Q16—212no—
349Q16—23no—
350Q16—21no—
351Q16C23×C416yes—
352Q16C23×C412yes—
353SD16—212no—
354SD16—212no—
355D8—212no—
356D8—23no—
357D8—21no—
358D8C23×C412yes—
359D8C23×C416yes—
360M16—26no—
361M16C4×D4112yes—
362C2×C8—26no—
363C2×C8C23×C416yes—
364C2×C8C22×D411yescharacteristic
365C2×C8C22×D413yes—
366C2×C8C22×D413yes—
367C2×C8C2×#4116yes—
368C2×C8C2511yescharacteristic
369C4⋊C4—21no—
370C4⋊C4—23no—
371C4⋊C4—21no—
372C4⋊C4—23no—
373C4⋊C4—43no—
374C4⋊C4—43no—
375C4⋊C4—43no—
376C4⋊C4—46no—
377C4⋊C4—43no—
378C4⋊C4—41no—
379C4⋊C4—43no—
380C4⋊C4C2×D813yes—
381C4⋊C4C2×D813yes—
382C22⋊C4—43no—
383C22⋊C4—43no—
384C22⋊C4—43no—
385C22⋊C4—46no—
386C22⋊C4—43no—
387C22⋊C4—41no—
388C22⋊C4—43no—
389C22⋊C4C2×D816yes—
390C42—43no—
391C42—43no—
392C42—43no—
393C42—43no—
394C42—46no—
395C42—41no—
396C42—43no—
397C42C2×D813yes—
398C42C2×D813yes—
399C42C22×D413yes—
400C42C22×D411yescharacteristic
401C16—21no—
402C16—23no—
403C16C23×C412yes—
404C16C23×C416yes—
405C23—41no—
406C23—43no—
407C23—412no—
408C23—163no—
409C23—163no—
410C23C22×D813yes—
411Q8—22no—
412Q8—26no—
413Q8—412no—
414Q8—43no—
415Q8—41no—
416Q8C4×D814yes—
417D4—26no—
418D4—22no—
419D4—412no—
420D4—412no—
421D4—412no—
422D4—43no—
423D4—41no—
424D4C4×D8112yes—
425C2×C4—26no—
426C2×C4—43no—
427C2×C4—412no—
428C2×C4—41no—
429C2×C4—43no—
430C2×C4—412no—
431C2×C4—412no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—43no—
435C2×C4—43no—
436C2×C4—82no—
437C2×C4—86no—
438C2×C4—163no—
439C2×C4—163no—
440C2×C4—166no—
441C2×C4—163no—
442C2×C4—163no—
443C2×C4C2×C4×D416yes—
444C2×C4C22×D813yes—
445C2×C4C22×D813yes—
446C2×C4C22×D811yescharacteristic
447C2×C4C22×D813yes—
448C2×C4C22×D811yescharacteristic
449C2×C4C2×#13516yes—
450C2×C4C23×D411yescharacteristic
451C8—21no—
452C8—23no—
453C8C2×C4×D416yes—
454C8C24×C411yescharacteristic
455C8C22×#4111yescharacteristic
456C22—23no—
457C22—23no—
458C22—42no—
459C22—46no—
460C22—83no—
461C22—81no—
462C22—81no—
463C22—83no—
464C22—1612no—
465C22C2×C4×D816yes—
466C22C23×D811yescharacteristic
467C4—23no—
468C4—21no—
469C4—23no—
470C4—42no—
471C4—46no—
472C4—81no—
473C4—83no—
474C4—81no—
475C4—83no—
476C4C2×C4×D816yes—
477C4#341612yesdirect product factor
478C4C22×C4×D411yescharacteristic
479C4C22×#13511yescharacteristic
480C2—26no—
481C2—82no—
482C2—86no—
483C2C22×C4×D811yescharacteristic
484C2C2×#341611yescharacteristic
485C2#6263511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC8C24×C4
Frattini, cosocleC2×C8C25
socleC22C23×D8
prime radicalC2×#3416C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#578C2semidirect
3. C4×#576C2semidirect
4. #33460C2semidirect
5. #33457C2semidirect
6. #33455C2semidirect
7. #33454C2semidirect
8. C4×#510C2semidirect
9. C4×#509C2semidirect
10. C4×#508C2semidirect
11. C2×C4×SD32C2semidirect
12. #33411C2semidirect
13. #33382C2semidirect
14. #33381C2semidirect
15. C4×#504C2semidirect
20. #3416C4direct (N × H)
28. #2951C22semidirect
29. #2950C22semidirect
36. C4×#135C22semidirect
37. C2×C4×Q16C22semidirect
44. #2177C22semidirect
45. #2176C22semidirect
46. #2175C22semidirect
47. #2175C22semidirect
48. C4×Q32C22semidirect
49. C4×Q32C22semidirect
50. C4×SD32C22semidirect
51. C4×SD32C22semidirect
52. C4×SD32C22semidirect
53. C4×SD32C22semidirect
54. C4×SD32C22semidirect
55. C4×SD32C22semidirect
56. C4×SD32C22semidirect
57. C4×D16C22semidirect
58. C4×D16C22semidirect
59. #2159C22semidirect
60. #2159C22semidirect
61. #2158C22semidirect
62. C2×#367C22semidirect
63. #2141C22semidirect
66. #2139C22semidirect
68. #2138C22semidirect
69. C2×#358C22semidirect
71. #2108C22semidirect
72. #2108C22semidirect
73. C4×M32C22semidirect
74. C2×C4×C16C22semidirect
78. #578C2×C4semidirect
80. #576C2×C4semidirect
108. #510C2×C4semidirect
109. #509C2×C4semidirect
110. #508C2×C4semidirect
112. C2×SD32C2×C4semidirect
118. #504C2×C4semidirect
202. #135C22×C4semidirect
205. C2×Q16C22×C4semidirect
273. Q32C22×C4semidirect
274. SD32C22×C4semidirect
275. SD32C22×C4semidirect
276. SD32C22×C4semidirect
277. D16C22×C4semidirect
279. M32C22×C4semidirect
281. C2×C16C22×C4semidirect
477. C4#3416direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#34162
C4×#5782
C4×#5762
#334602
#334572
#334552
#334542
C4×#5102
C4×#5092
C4×#5082
C2×C4×SD322
#334112
#333822
#333812
C4×#5042