Finite Group Databasebeta

← C3×#10589853 #13153401

Subgroups — C3×#10589853 #13153401

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63716C211yesmaximal, characteristic, complemented
2C3×#63715C211yesmaximal, characteristic, complemented
3C3×C9×#131C211yesmaximal, characteristic, complemented
4C3×#63471C213yesmaximal, complemented
5C3×#63470C213yesmaximal, complemented
6C3×#63467C213yesmaximal, complemented
7C2×C3×#3664C213yesmaximal, complemented
8C3×#5751—31nomaximal
9#10589853C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C3×#3701—21no—
12C3×Q8×D9—21no—
13C3×#3699—23no—
14C3×D4×D9—23no—
15C3×#3697C2213yes—
16C3×#3697C2213yescomplemented
17C2×C3×D36C2213yes—
18C2×C3×Dic18C2213yescomplemented
19C3×C9×M16C2213yescomplemented
20C2×C3×C8×C9C2213yescomplemented
21C3×D72C2213yescomplemented
22C3×#3664C2213yescomplemented
23C3×#3664C2216yescomplemented
24C3×#3664C2211yescharacteristic, complemented
25C3×Dic36C2213yescomplemented
26C3×#1010—31no—
27C3×#1009—31no—
28C32×#131D311yescharacteristic
29C3×#909—33no—
30C3×#908—33no—
31C3×#904—33no—
32C2×C3×#204—33no—
33#63716C2×C311yescharacteristic, complemented
34#63715C2×C311yescharacteristic, complemented
35C9×#131—21no—
36C9×#131C2×C311yescharacteristic, complemented
37#63471C2×C313yescomplemented
38#63470C2×C313yescomplemented
39#63467C2×C313yescomplemented
40C2×#3664C2×C313yescomplemented
41C22×C3×D9—23no—
42C3×Q8×C9D411yescharacteristic
43C3×D4×C9D413yes—
44C2×C3×C4×C9C2313yes—
45C3×#611—23no—
46C3×#611—23no—
47C2×C3×Dic9—23no—
48C3×D36—23no—
49C3×D36C2313yes—
50C3×D36C2311yescharacteristic
51C3×C4×D9—23no—
52C3×C4×D9—23no—
53C3×Dic18—23no—
54C3×Dic18C2313yes—
55C3×Dic18C2311yescharacteristic
56C3×C8×C9C2313yes—
57C3×C8×C9C2311yescharacteristic
58C3×#577—91no—
59#5751—31no—
60C32×#41C2×D311yescharacteristic
61C3×#239—61no—
62C3×D3×Q8—61no—
63C3×#237—63no—
64C3×D3×D4—63no—
65C3×#235—33no—
66C3×#235—33no—
67C2×C3×D12—33no—
68C2×C3×Dic6—33no—
69C32×M16C2×D313yes—
70C2×C32×C8C2×D313yes—
71C3×Dic12—33no—
72C3×D24—33no—
73C3×#204—33no—
74C3×#204—36no—
75C3×#204—31no—
76C9×#41—21no—
77C9×#41C22×C311yescharacteristic
78#3701—21no—
79Q8×D9—21no—
80#3699—23no—
81D4×D9—23no—
82#3697C22×C313yes—
83#3697C22×C313yescomplemented
84C2×D36C22×C313yes—
85C2×Dic18C22×C313yescomplemented
86C9×M16—23no—
87C9×M16C22×C313yescomplemented
88C2×C8×C9—23no—
89C2×C8×C9C22×C313yescomplemented
90D72C22×C313yescomplemented
91#3664C22×C313yescomplemented
92#3664C22×C316yescomplemented
93#3664C22×C311yescharacteristic, complemented
94Dic36C22×C313yescomplemented
95C22×C3×C9C2×D413yes—
96C2×C3×D9—21no—
97C2×C3×D9—23no—
98C2×C3×D9—83no—
99C3×C4×C9C2×D413yes—
100C3×C4×C9C2411yescharacteristic
101C3×Dic9—23no—
102C3×Dic9—21no—
103C3×2−1+4—91no—
104C3×H(5,𝔽2)—91no—
105#1010—31no—
106#1009—31no—
107C3×#137—93no—
108C3×AGL(1,ℤ/8)—93no—
109C3×#135—93no—
110C2×C3×SD16—93no—
111C3×#131—21no—
112C3×#131D911yescharacteristic, complemented
113C3×#131C3×D311yescharacteristic
114#909—33no—
115#908—33no—
116#904—33no—
117C2×#204—33no—
118C22×C3×D3—63no—
119C32×Q8D1211yescharacteristic
120C32×D4D1213yes—
121C2×C32×C4C22×D313yes—
122C3×#67—63no—
123C3×#67—63no—
124C2×C3×Dic3—63no—
125C3×D12—33no—
126C3×D12—31no—
127C3×D12—63no—
128C3×C4×D3—63no—
129C3×C4×D3—63no—
130C3×Dic6—33no—
131C3×Dic6—31no—
132C3×Dic6—63no—
133C22×D9—23no—
134C32×C8C22×D311yescharacteristic
135C32×C8C22×D313yes—
136Q8×C9—21no—
137Q8×C9C3×D411yescharacteristic
138D4×C9—23no—
139D4×C9C3×D413yes—
140C2×C4×C9—23no—
141C2×C4×C9C23×C313yes—
142#611—23no—
143#611—23no—
144C2×Dic9—23no—
145D36—23no—
146D36C23×C313yes—
147D36C23×C311yescharacteristic
148C4×D9—23no—
149C4×D9—23no—
150Dic18—23no—
151Dic18C23×C311yescharacteristic
152Dic18C23×C313yes—
153C8×C9—21no—
154C8×C9—23no—
155C8×C9C23×C313yes—
156C8×C9C23×C311yescharacteristic
157#577—91noSylow
158C2×C3×C9—23no—
159C2×C3×C9C22×D411yescharacteristic
160C3×D9—41no—
161C3×D9—43no—
162C3×#41—21no—
163C3×#41—93no—
164C3×#41—93no—
165C3×#41—181no—
166C3×#41—183no—
167C3×#41C2×D911yescharacteristic
168C3×#41C2×C3×D311yescharacteristic
169C2×C3×Q8—93no—
170C2×C3×Q8—181no—
171C2×C3×D4—93no—
172C2×C3×D4—183no—
173#239—61no—
174D3×Q8—61no—
175#237—63no—
176D3×D4—63no—
177#235—33no—
178#235—33no—
179C2×D12—33no—
180C2×Dic6—33no—
181C3×Q16—93no—
182C3×SD16—93no—
183C3×SD16—96no—
184C3×SD16—91no—
185C3×D8—93no—
186C3×M16—23no—
187C3×M16C2×D913yescomplemented
188C3×M16C2×C3×D313yes—
189C2×C3×C8—23no—
190C2×C3×C8C2×D913yescomplemented
191C2×C3×C8C2×C3×D313yes—
192Dic12—33no—
193D24—33no—
194#204—33no—
195#204—36no—
196#204—31no—
197C22×C32C2×D1213yes—
198C2×C3×D3—61no—
199C2×C3×D3—63no—
200C2×C3×D3—243no—
201C32×C4C2×D1213yes—
202C32×C4C23×D311yescharacteristic
203C3×Dic3—63no—
204C3×Dic3—61no—
205C22×C9—23no—
206C22×C9C2×C3×D413yes—
207C2×D9—21no—
208C2×D9—23no—
209C2×D9—83no—
210C4×C9—21no—
211C4×C9—23no—
212C4×C9C2×C3×D413yes—
213C4×C9C24×C311yescharacteristic
214Dic9—23no—
215Dic9—21no—
2162−1+4—91no—
217H(5,𝔽2)—91no—
218#137—93no—
219AGL(1,ℤ/8)—93no—
220#135—93no—
221C2×SD16—93no—
222#131C3×D911yescharacteristic, complemented
223C3×C9#57711yescharacteristic, complemented, Sylow
224C23×C3—183no—
225C22×D3—63no—
226C3×Q8—21no—
227C3×Q8—93no—
228C3×Q8—91no—
229C3×Q8—183no—
230C3×Q8D3611yescharacteristic
231C3×Q8C3×D1211yescharacteristic
232C3×D4—23no—
233C3×D4—91no—
234C3×D4—93no—
235C3×D4—183no—
236C3×D4—183no—
237C3×D4—183no—
238C3×D4D3613yes—
239C3×D4C3×D1213yes—
240C2×C3×C4—23no—
241C2×C3×C4—183no—
242C2×C3×C4—183no—
243C2×C3×C4—183no—
244C2×C3×C4C22×D913yes—
245C2×C3×C4C22×C3×D313yes—
246#67—63no—
247#67—63no—
248C2×Dic3—63no—
249D12—33no—
250D12—31no—
251D12—63no—
252C4×D3—63no—
253C4×D3—63no—
254Dic6—33no—
255Dic6—31no—
256Dic6—63no—
257C3×C8—21no—
258C3×C8—23no—
259C3×C8C22×D911yescharacteristic
260C3×C8C22×D913yes—
261C3×C8C22×C3×D311yescharacteristic
262C3×C8C22×C3×D313yes—
263C2×C32—23no—
264C2×C32C22×D1211yescharacteristic
265C3×D3—121no—
266C3×D3—123no—
267C2×C9—23no—
268C2×C9—21no—
269C2×C9—43no—
270C2×C9C22×C3×D411yescharacteristic
271D9—41no—
272D9—43no—
273#41—93no—
274#41—93no—
275#41—181no—
276#41—183no—
277#41C2×C3×D911yescharacteristic
278C2×Q8—93no—
279C2×Q8—181no—
280C2×D4—93no—
281C2×D4—183no—
282Q16—93no—
283SD16—93no—
284SD16—96no—
285SD16—91no—
286D8—93no—
287M16C2×C3×D913yescomplemented
288C2×C8C2×C3×D913yescomplemented
289C22×C3—23no—
290C22×C3—181no—
291C22×C3—183no—
292C22×C3—723no—
293C22×C3C2×D3613yes—
294C22×C3C2×C3×D1213yes—
295C2×D3—61no—
296C2×D3—63no—
297C2×D3—243no—
298C3×C4—21no—
299C3×C4—23no—
300C3×C4—181no—
301C3×C4—183no—
302C3×C4C2×D3613yes—
303C3×C4C23×D911yescharacteristic
304C3×C4C2×C3×D1213yes—
305C3×C4C23×C3×D311yescharacteristic
306Dic3—63no—
307Dic3—61no—
308C32#575111yescharacteristic
309C9—21no—
310C9C3×#57711yescharacteristic, complemented
311C23—183no—
312Q8—93no—
313Q8—91no—
314Q8—183no—
315Q8C3×D3611yescharacteristic
316D4—91no—
317D4—93no—
318D4—183no—
319D4—183no—
320D4—183no—
321D4C3×D3613yes—
322C2×C4—183no—
323C2×C4—183no—
324C2×C4—183no—
325C2×C4C22×C3×D913yes—
326C8C22×C3×D911yescharacteristic
327C8C22×C3×D913yes—
328C2×C3—23no—
329C2×C3—23no—
330C2×C3—21no—
331C2×C3—43no—
332C2×C3—363no—
333C2×C3—361no—
334C2×C3C22×D3611yescharacteristic
335C2×C3C22×C3×D1211yescharacteristic
336D3—123no—
337D3—121no—
338C22—181no—
339C22—183no—
340C22—723no—
341C22C2×C3×D3613yes—
342C4—181no—
343C4—183no—
344C4C2×C3×D3613yes—
345C4C23×C3×D911yescharacteristic
346C3—21no—
347C3#1058985311yescharacteristic, direct product factor
348C3C3×#575111yescharacteristic
349C2—23no—
350C2—363no—
351C2—361no—
352C2C22×C3×D3611yescharacteristic
subgroupG/H
centerC2×C3C22×D36
commutator, cosocleC4×C9C24×C3
FrattiniC3×C4C23×C3×D3
FittingC3×C9×#131C2
socleC2×C32C22×D12
hypercenterC3×#131D9
hypocenterC9C3×#577
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
1. C3×#63716C2semidirect
2. C3×#63715C2semidirect
3. C3×C9×#131C2semidirect
4. C3×#63471C2semidirect
5. C3×#63470C2semidirect
6. C3×#63467C2semidirect
7. C2×C3×#3664C2semidirect
9. #10589853C3direct (N × H)
16. C3×#3697C22semidirect
18. C2×C3×Dic18C22semidirect
19. C3×C9×M16C22semidirect
20. C2×C3×C8×C9C22semidirect
21. C3×D72C22semidirect
22. C3×#3664C22semidirect
23. C3×#3664C22semidirect
24. C3×#3664C22semidirect
25. C3×Dic36C22semidirect
33. #63716C2×C3semidirect
34. #63715C2×C3semidirect
36. C9×#131C2×C3semidirect
37. #63471C2×C3semidirect
38. #63470C2×C3semidirect
39. #63467C2×C3semidirect
40. C2×#3664C2×C3semidirect
83. #3697C22×C3semidirect
85. C2×Dic18C22×C3semidirect
87. C9×M16C22×C3semidirect
89. C2×C8×C9C22×C3semidirect
90. D72C22×C3semidirect
91. #3664C22×C3semidirect
92. #3664C22×C3semidirect
93. #3664C22×C3semidirect
94. Dic36C22×C3semidirect
112. C3×#131D9semidirect
187. C3×M16C2×D9semidirect
190. C2×C3×C8C2×D9semidirect
222. #131C3×D9semidirect
223. C3×C9#577semidirect
287. M16C2×C3×D9semidirect
288. C2×C8C2×C3×D9semidirect
310. C9C3×#577semidirect
347. C3#10589853direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637162
C3×#637152
C3×C9×#1312
C3×#634712
C3×#634702
C3×#634672
C2×C3×#36642
C3×#57513
#105898533