Finite Group Databasebeta

← C32×#4749 #13162185

Subgroups — C32×#4749 #13162185

Order 1728.

832 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#932C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3C32×#908C211yesmaximal, characteristic, complemented
4C32×#907C211yesmaximal, characteristic, complemented
5C33×#100C211yesmaximal, characteristic, complemented
6C32×#823C211yesmaximal, characteristic, complemented
7C32×#822C211yesmaximal, characteristic, complemented
8C32×#472—31nomaximal
9C3×#4749C314yesmaximal, direct product factor
10C2×C33×D4C2211yescharacteristic
11C32×#235C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C33×M16C2211yescharacteristic
14C33×M16C2211yescharacteristic
15C32×#214—21no—
16C32×#213—21no—
17C32×#213—21no—
18C32×#213—21no—
19C32×#208C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C32×D24—21no—
22C32×#204—21no—
23C32×ΓL(1,𝔽25)—21no—
24C32×D3×C8—21no—
25C32×AGL(1,ℤ/8)—31no—
26C32×AGL(1,ℤ/8)—31no—
27C2×C32×D8—31no—
28C32×#131—31no—
29C3×#932C2×C314yescomplemented
30C2×C3×#213C2×C314yescomplemented
31C3×#908C2×C314yescomplemented
32C3×#907C2×C314yescomplemented
33C32×#108—31no—
34C32×#100—24no—
35C32×#100—31no—
36C32×#100D311yescharacteristic, complemented
37C32×#100C2×C314yescomplemented
38C3×#823C2×C314yescomplemented
39C3×#822C2×C314yescomplemented
40C23×C33—21no—
41C22×C32×D3—21no—
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4C2311yescharacteristic
45C32×#67—21no—
46C32×D12—21no—
47C32×D12—21no—
48C32×D12D411yescharacteristic
49C32×C4×D3—21no—
50C32×Dic6D411yescharacteristic
51C33×C8—21no—
52C33×C8—21no—
53C32×C3⋊C8—21no—
54C32×C3⋊C8D411yescharacteristic
55C32×C3⋊C8D411yescharacteristic
56C3×#472—34no—
57#4749C3211yescharacteristic, direct product factor
58C32×#41—31no—
59C2×C32×D4—24no—
60C2×C32×D4—31no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C314yes—
63C3×#235C22×C314yes—
64C2×C3×D12C22×C314yes—
65C32×SD16—61no—
66C32×SD16—61no—
67C32×D8—61no—
68C32×D8—61no—
69C32×D8—61no—
70C32×D8—61no—
71C32×M16—24no—
72C32×M16—24no—
73C32×M16—31no—
74C32×M16—61no—
75C32×M16C2×D311yescharacteristic
76C32×M16C2×D311yescharacteristic
77C32×M16C22×C314yes—
78C32×M16C22×C314yes—
79C2×C32×C8—31no—
80C2×C32×C8—61no—
81C3×#214—24no—
82C3×#213—24no—
83C3×#213—24no—
84C3×#213—24no—
85C3×#208C22×C314yes—
86C2×C3×C3⋊C8C22×C314yes—
87C3×D24—24no—
88C3×#204—24no—
89C3×ΓL(1,𝔽25)—24no—
90C3×D3×C8—24no—
91C22×C33—41no—
92C22×C33—81no—
93C22×C33#4111yescharacteristic
94C2×C32×D3—21no—
95C2×C32×D3—41no—
96C2×C32×D3—81no—
97C33×C4C2×D411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C32×Dic3—21no—
100C3×AGL(1,ℤ/8)—34no—
101C3×AGL(1,ℤ/8)—34no—
102C2×C3×D8—34no—
103C3×#131—34no—
104#932C2×C3211yescharacteristic, complemented
105C2×#213C2×C3211yescharacteristic, complemented
106#908C2×C3211yescharacteristic, complemented
107#907C2×C3211yescharacteristic, complemented
108C3×#108—34no—
109C3×#100—24no—
110C3×#100—34no—
111C3×#100C3×D314yescomplemented
112C3×#100C2×C3211yescharacteristic, complemented
113#823C2×C3211yescharacteristic, complemented
114#822C2×C3211yescharacteristic, complemented
115C23×C32—28no—
116C23×C32—21no—
117C23×C32—24no—
118C23×C32—61no—
119C22×C3×D3—24no—
120C32×Q8—31no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—24no—
125C32×D4—31no—
126C32×D4—44no—
127C32×D4—44no—
128C32×D4—61no—
129C32×D4—61no—
130C32×D4—61no—
131C2×C32×C4—24no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C23×C314yes—
135C3×#67—24no—
136C3×D12—24no—
137C3×D12—24no—
138C3×D12C3×D414yes—
139C3×C4×D3—24no—
140C3×Dic6C3×D414yes—
141C32×C8—21no—
142C32×C8—21no—
143C32×C8—24no—
144C32×C8—24no—
145C32×C8—31no—
146C32×C8—31no—
147C32×C8—44no—
148C32×C8—44no—
149C32×C8—61no—
150C3×C3⋊C8—24no—
151C3×C3⋊C8C3×D414yes—
152C3×C3⋊C8C3×D414yes—
153#472—31noSylow
154C2×C33—21no—
155C2×C33—81no—
156C2×C33#12111yescharacteristic
157C32×D3—41no—
158C32×D3—81no—
159C3×#41—34no—
160C2×C3×D4—24no—
161C2×C3×D4—34no—
162C2×C3×D4C2×C3×D314yes—
163C2×C3×D4C22×C3211yescharacteristic
164#235C22×C3211yescharacteristic
165C2×D12C22×C3211yescharacteristic
166C3×SD16—64no—
167C3×SD16—64no—
168C3×D8—64no—
169C3×D8—64no—
170C3×D8—64no—
171C3×D8—64no—
172C3×M16—24no—
173C3×M16—24no—
174C3×M16—34no—
175C3×M16—64no—
176C3×M16C2×C3×D314yes—
177C3×M16C2×C3×D314yes—
178C3×M16C22×C3211yescharacteristic
179C3×M16C22×C3211yescharacteristic
180C2×C3×C8—34no—
181C2×C3×C8—64no—
182#214—21no—
183#213—21no—
184#213—21no—
185#213—21no—
186#208C22×C3211yescharacteristic
187C2×C3⋊C8C22×C3211yescharacteristic
188D24—21no—
189#204—21no—
190ΓL(1,𝔽25)—21no—
191D3×C8—21no—
192C22×C32—24no—
193C22×C32—48no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—61no—
197C22×C32—81no—
198C22×C32—84no—
199C22×C32—88no—
200C22×C32—121no—
201C22×C32—241no—
202C22×C32#23511yescharacteristic
203C22×C32C3×#4114yes—
204C2×C3×D3—24no—
205C2×C3×D3—44no—
206C2×C3×D3—84no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—61no—
210C32×C4D3×D411yescharacteristic
211C32×C4D3×D411yescharacteristic
212C32×C4C2×C3×D414yes—
213C32×C4C2×C3×D414yes—
214C3×Dic3—24no—
215AGL(1,ℤ/8)—31no—
216AGL(1,ℤ/8)—31no—
217C2×D8—31no—
218#131—31no—
219#108—31no—
220#100—31no—
221#100C32×D311yescharacteristic, complemented
222C33#47211yescharacteristic, complemented, Sylow
223C23×C3—28no—
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—64no—
227C22×D3—21no—
228C3×Q8—34no—
229C3×D4—24no—
230C3×D4—24no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—34no—
234C3×D4—44no—
235C3×D4—44no—
236C3×D4—64no—
237C3×D4—64no—
238C3×D4—64no—
239C2×C3×C4—24no—
240C2×C3×C4—64no—
241C2×C3×C4C22×C3×D314yes—
242C2×C3×C4C23×C3211yescharacteristic
243#67—21no—
244D12—21no—
245D12—21no—
246D12C32×D411yescharacteristic
247C4×D3—21no—
248Dic6C32×D411yescharacteristic
249C3×C8—24no—
250C3×C8—24no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8—34no—
254C3×C8—34no—
255C3×C8—44no—
256C3×C8—44no—
257C3×C8—64no—
258C3⋊C8—21no—
259C3⋊C8C32×D411yescharacteristic
260C3⋊C8C32×D411yescharacteristic
261C2×C32—21no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—44no—
265C2×C32—88no—
266C2×C32—81no—
267C2×C32—84no—
268C2×C32—121no—
269C2×C32—241no—
270C2×C32#88411yescharacteristic
271C2×C32C3×#12114yes—
272C3×D3—44no—
273C3×D3—84no—
274#41—31no—
275C2×D4—31no—
276C2×D4C2×C32×D311yescharacteristic
277SD16—61no—
278SD16—61no—
279D8—61no—
280D8—61no—
281D8—61no—
282D8—61no—
283M16—31no—
284M16—61no—
285M16C2×C32×D311yescharacteristic
286M16C2×C32×D311yescharacteristic
287C2×C8—31no—
288C2×C8—61no—
289C22×C3—24no—
290C22×C3—48no—
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—64no—
294C22×C3—81no—
295C22×C3—84no—
296C22×C3—88no—
297C22×C3—124no—
298C22×C3—244no—
299C22×C3C3×#23514yes—
300C22×C3C32×#4111yescharacteristic
301C2×D3—21no—
302C2×D3—41no—
303C2×D3—81no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—64no—
307C3×C4C3×D3×D414yes—
308C3×C4C3×D3×D414yes—
309C3×C4C2×C32×D411yescharacteristic
310C3×C4C2×C32×D411yescharacteristic
311Dic3—21no—
312C32—24no—
313C32#474911yescharacteristic, direct product factor
314C32C3×#47214yescomplemented
315C23—21no—
316C23—61no—
317Q8—31no—
318D4—21no—
319D4—21no—
320D4—31no—
321D4—61no—
322D4—61no—
323D4—61no—
324C2×C4—61no—
325C2×C4C22×C32×D311yescharacteristic
326C8—21no—
327C8—21no—
328C8—31no—
329C8—31no—
330C8—61no—
331C2×C3—21no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—44no—
335C2×C3—88no—
336C2×C3—81no—
337C2×C3—84no—
338C2×C3—124no—
339C2×C3—244no—
340C2×C3C3×#88414yes—
341C2×C3C32×#12111yescharacteristic
342D3—41no—
343D3—81no—
344C22—41no—
345C22—61no—
346C22—81no—
347C22—121no—
348C22—241no—
349C22C32×#23511yescharacteristic
350C4—61no—
351C4C32×D3×D411yescharacteristic
352C4C32×D3×D411yescharacteristic
353C3—24no—
354C3C3×#474914yesdirect product factor
355C3C32×#47211yescharacteristic, complemented
356C2—21no—
357C2—81no—
358C2—121no—
359C2—241no—
360C2C32×#88411yescharacteristic
subgroupG/H
centerC2×C32#884
commutator, cosocleC2×C3×C4C23×C32
FrattiniC2×C4C22×C32×D3
FittingC33×#100C2
socleC2×C33#121
hypercenterC32×#100D3
hypocenterC3C32×#472
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C32×#100

Complements

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

NH ≅ G/Ntype
1. C32×#932C2semidirect
2. C2×C32×#213C2semidirect
3. C32×#908C2semidirect
4. C32×#907C2semidirect
5. C33×#100C2semidirect
6. C32×#823C2semidirect
7. C32×#822C2semidirect
9. C3×#4749C3direct (N × H)
29. C3×#932C2×C3semidirect
30. C2×C3×#213C2×C3semidirect
31. C3×#908C2×C3semidirect
32. C3×#907C2×C3semidirect
36. C32×#100D3semidirect
37. C32×#100C2×C3semidirect
38. C3×#823C2×C3semidirect
39. C3×#822C2×C3semidirect
57. #4749C32direct (N × H)
104. #932C2×C32semidirect
105. C2×#213C2×C32semidirect
106. #908C2×C32semidirect
107. #907C2×C32semidirect
111. C3×#100C3×D3semidirect
112. C3×#100C2×C32semidirect
113. #823C2×C32semidirect
114. #822C2×C32semidirect
221. #100C32×D3semidirect
222. C33#472semidirect
313. C32#4749direct (N × H)
314. C32C3×#472semidirect
354. C3C3×#4749direct (N × H)
355. C3C32×#472semidirect

Maximal subgroups

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

subgroupindex
C32×#9322
C2×C32×#2132
C32×#9082
C32×#9072
C33×#1002
C32×#8232
C32×#8222
C32×#4723
C3×#47493