Finite Group Databasebeta

← C3×#82014 #11899718

Subgroups — C3×#82014 #11899718

Order 1152.

762 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5753C211yesmaximal, characteristic, complemented
2C3×#5752C211yesmaximal, characteristic, complemented
3C32×#504C211yesmaximal, characteristic, complemented
4C3×#4909C213yesmaximal, complemented
5C3×#4908C213yesmaximal, complemented
6C3×#4904C213yesmaximal, complemented
7C2×C3×#800C213yesmaximal, complemented
8C3×#3416—31nomaximal
9#82014C311yesmaximal, characteristic, direct product factor
10C3×#1010—21no—
11C3×#1009—21no—
12C32×#131C2211yescharacteristic
13C3×#909—23no—
14C3×#908—23no—
15C2×C3×Dic12C2213yescomplemented
16C3×#904C2213yes—
17C3×#904C2213yescomplemented
18C2×C3×D24C2213yes—
19C32×M32C2213yescomplemented
20C2×C32×C16C2213yescomplemented
21C3×Dic24C2213yescomplemented
22C3×#800C2213yescomplemented
23C3×#800C2216yescomplemented
24C3×#800C2211yescharacteristic, complemented
25C3×D48C2213yescomplemented
26C3×#578—31no—
27C3×#576—31no—
28#5753C2×C311yescharacteristic, complemented
29#5752C2×C311yescharacteristic, complemented
30C3×#510—33no—
31C3×#509—33no—
32C3×#508—33no—
33C2×C3×SD32—33no—
34C3×#504—21no—
35C3×#504D311yescharacteristic, complemented
36C3×#504C2×C311yescharacteristic, complemented
37#4909C2×C313yescomplemented
38#4908C2×C313yescomplemented
39#4904C2×C313yescomplemented
40C2×#800C2×C313yescomplemented
41C32×#41D411yescharacteristic
42C3×#239—41no—
43C3×D3×Q8—41no—
44C3×#237—43no—
45C3×D3×D4—43no—
46C3×#235—23no—
47C3×#235—23no—
48C2×C3×D12—23no—
49C2×C3×Dic6—23no—
50C32×M16D413yes—
51C2×C32×C8C2313yes—
52C3×Dic12—23no—
53C3×Dic12C2311yescharacteristic
54C3×Dic12C2313yes—
55C3×D24—23no—
56C3×D24C2313yes—
57C3×D24C2311yescharacteristic
58C3×#204—23no—
59C3×#204—23no—
60C32×C16C2313yes—
61C32×C16C2311yescharacteristic
62#3416—31noSylow
63C3×2−1+4—61no—
64C3×H(5,𝔽2)—61no—
65#1010—21no—
66#1009—21no—
67C3×#137—63no—
68C3×AGL(1,ℤ/8)—63no—
69C3×#135—33no—
70C3×#135—33no—
71C2×C3×Q16—33no—
72C2×C3×D8—33no—
73C3×#131—21no—
74C3×#131C2×D311yescharacteristic
75C3×#131C22×C311yescharacteristic
76#909—23no—
77#908—23no—
78C2×Dic12C22×C313yescomplemented
79#904C22×C313yes—
80#904C22×C313yescomplemented
81C2×D24C22×C313yes—
82C3×Q32—33no—
83C3×SD32—33no—
84C3×SD32—36no—
85C3×SD32—31no—
86C3×D16—33no—
87C3×M32—23no—
88C3×M32C2×D313yescomplemented
89C3×M32C22×C313yescomplemented
90C2×C3×C16—23no—
91C2×C3×C16C2×D313yescomplemented
92C2×C3×C16C22×C313yescomplemented
93Dic24C22×C313yescomplemented
94#800C22×C313yescomplemented
95#800C22×C316yescomplemented
96#800C22×C311yescharacteristic, complemented
97D48C22×C313yescomplemented
98C22×C3×D3—43no—
99C32×Q8D811yescharacteristic
100C32×D4D813yes—
101C2×C32×C4C2×D413yes—
102C3×#67—43no—
103C3×#67—43no—
104C2×C3×Dic3—43no—
105C3×D12—21no—
106C3×D12—23no—
107C3×D12—43no—
108C3×C4×D3—43no—
109C3×C4×D3—43no—
110C3×Dic6—21no—
111C3×Dic6—23no—
112C3×Dic6—43no—
113C32×C8C2×D413yes—
114C32×C8C2411yescharacteristic
115#578—31no—
116#576—31no—
117#510—33no—
118#509—33no—
119#508—33no—
120C2×SD32—33no—
121#504C3×D311yescharacteristic, complemented
122C3×#41—21no—
123C3×#41—63no—
124C3×#41—63no—
125C3×#41—121no—
126C3×#41—123no—
127C3×#41D1211yescharacteristic
128C3×#41C3×D411yescharacteristic
129C2×C3×Q8—63no—
130C2×C3×Q8—121no—
131C2×C3×D4—63no—
132C2×C3×D4—123no—
133#239—41no—
134D3×Q8—41no—
135#237—43no—
136D3×D4—43no—
137#235—23no—
138#235—23no—
139C2×D12—23no—
140C2×Dic6—23no—
141C3×Q16—33no—
142C3×Q16—31no—
143C3×Q16—63no—
144C3×SD16—63no—
145C3×SD16—63no—
146C3×D8—31no—
147C3×D8—33no—
148C3×D8—63no—
149C3×M16—23no—
150C3×M16D1213yes—
151C3×M16C3×D413yes—
152C2×C3×C8—23no—
153C2×C3×C8C22×D313yes—
154C2×C3×C8C23×C313yes—
155Dic12—23no—
156Dic12C23×C313yes—
157Dic12C23×C311yescharacteristic
158D24—23no—
159D24C23×C311yescharacteristic
160D24C23×C313yes—
161#204—23no—
162#204—23no—
163C3×C16—21no—
164C3×C16—23no—
165C3×C16C22×D311yescharacteristic
166C3×C16C22×D313yes—
167C3×C16C23×C313yes—
168C3×C16C23×C311yescharacteristic
169C22×C32C2×D813yes—
170C2×C3×D3—41no—
171C2×C3×D3—43no—
172C2×C3×D3—163no—
173C32×C4C2×D813yes—
174C32×C4C22×D411yescharacteristic
175C3×Dic3—43no—
176C3×Dic3—41no—
1772−1+4—61no—
178H(5,𝔽2)—61no—
179#137—63no—
180AGL(1,ℤ/8)—63no—
181#135—33no—
182#135—33no—
183C2×Q16—33no—
184C2×D8—33no—
185#131C2×C3×D311yescharacteristic
186Q32—33no—
187SD32—33no—
188SD32—36no—
189SD32—31no—
190D16—33no—
191M32C2×C3×D313yescomplemented
192C2×C16C2×C3×D313yescomplemented
193C23×C3—123no—
194C22×D3—43no—
195C3×Q8—21no—
196C3×Q8—61no—
197C3×Q8—63no—
198C3×Q8—123no—
199C3×Q8D2411yescharacteristic
200C3×Q8C3×D811yescharacteristic
201C3×D4—23no—
202C3×D4—63no—
203C3×D4—61no—
204C3×D4—123no—
205C3×D4—123no—
206C3×D4—123no—
207C3×D4D2413yes—
208C3×D4C3×D813yes—
209C2×C3×C4—23no—
210C2×C3×C4—123no—
211C2×C3×C4—123no—
212C2×C3×C4—123no—
213C2×C3×C4C2×D1213yes—
214C2×C3×C4C2×C3×D413yes—
215#67—43no—
216#67—43no—
217C2×Dic3—43no—
218D12—21no—
219D12—23no—
220D12—43no—
221C4×D3—43no—
222C4×D3—43no—
223Dic6—21no—
224Dic6—23no—
225Dic6—43no—
226C3×C8—21no—
227C3×C8—23no—
228C3×C8C2×D1213yes—
229C3×C8C2×C3×D413yes—
230C3×C8C23×D311yescharacteristic
231C3×C8C24×C311yescharacteristic
232C2×C32—23no—
233C2×C32C22×D811yescharacteristic
234C3×D3—81no—
235C3×D3—83no—
236#41—63no—
237#41—63no—
238#41—121no—
239#41—123no—
240#41C3×D1211yescharacteristic
241C2×Q8—63no—
242C2×Q8—121no—
243C2×D4—63no—
244C2×D4—123no—
245Q16—33no—
246Q16—31no—
247Q16—63no—
248SD16—63no—
249SD16—63no—
250D8—31no—
251D8—33no—
252D8—63no—
253M16C3×D1213yes—
254C2×C8C22×C3×D313yes—
255C16C22×C3×D311yescharacteristic
256C16C22×C3×D313yes—
257C22×C3—23no—
258C22×C3—121no—
259C22×C3—123no—
260C22×C3—483no—
261C22×C3C2×D2413yes—
262C22×C3C2×C3×D813yes—
263C2×D3—41no—
264C2×D3—43no—
265C2×D3—163no—
266C3×C4—23no—
267C3×C4—21no—
268C3×C4—121no—
269C3×C4—123no—
270C3×C4C2×D2413yes—
271C3×C4C2×C3×D813yes—
272C3×C4C22×D1211yescharacteristic
273C3×C4C22×C3×D411yescharacteristic
274Dic3—43no—
275Dic3—41no—
276C32#341611yescharacteristic, complemented, Sylow
277C23—123no—
278Q8—61no—
279Q8—63no—
280Q8—123no—
281Q8C3×D2411yescharacteristic
282D4—63no—
283D4—61no—
284D4—123no—
285D4—123no—
286D4—123no—
287D4C3×D2413yes—
288C2×C4—123no—
289C2×C4—123no—
290C2×C4—123no—
291C2×C4C2×C3×D1213yes—
292C8C2×C3×D1213yes—
293C8C23×C3×D311yescharacteristic
294C2×C3—23no—
295C2×C3—23no—
296C2×C3—21no—
297C2×C3—43no—
298C2×C3—243no—
299C2×C3—241no—
300C2×C3C22×D2411yescharacteristic
301C2×C3C22×C3×D811yescharacteristic
302D3—83no—
303D3—81no—
304C22—121no—
305C22—123no—
306C22—483no—
307C22C2×C3×D2413yes—
308C4—121no—
309C4—123no—
310C4C2×C3×D2413yes—
311C4C22×C3×D1211yescharacteristic
312C3—21no—
313C3#8201411yescharacteristic, direct product factor
314C3C3×#341611yescharacteristic, complemented
315C2—23no—
316C2—243no—
317C2—241no—
318C2C22×C3×D2411yescharacteristic
subgroupG/H
centerC2×C3C22×D24
commutator, cosocleC3×C8C24×C3
FrattiniC8C23×C3×D3
FittingC32×#504C2
socleC2×C32C22×D8
hypercenterC3×#504D3
hypocenterC3C3×#3416
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×#131 ◁ C3×#504

Complements

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

NH ≅ G/Ntype
1. C3×#5753C2semidirect
2. C3×#5752C2semidirect
3. C32×#504C2semidirect
4. C3×#4909C2semidirect
5. C3×#4908C2semidirect
6. C3×#4904C2semidirect
7. C2×C3×#800C2semidirect
9. #82014C3direct (N × H)
15. C2×C3×Dic12C22semidirect
17. C3×#904C22semidirect
19. C32×M32C22semidirect
20. C2×C32×C16C22semidirect
21. C3×Dic24C22semidirect
22. C3×#800C22semidirect
23. C3×#800C22semidirect
24. C3×#800C22semidirect
25. C3×D48C22semidirect
28. #5753C2×C3semidirect
29. #5752C2×C3semidirect
35. C3×#504D3semidirect
36. C3×#504C2×C3semidirect
37. #4909C2×C3semidirect
38. #4908C2×C3semidirect
39. #4904C2×C3semidirect
40. C2×#800C2×C3semidirect
78. C2×Dic12C22×C3semidirect
80. #904C22×C3semidirect
88. C3×M32C2×D3semidirect
89. C3×M32C22×C3semidirect
91. C2×C3×C16C2×D3semidirect
92. C2×C3×C16C22×C3semidirect
93. Dic24C22×C3semidirect
94. #800C22×C3semidirect
95. #800C22×C3semidirect
96. #800C22×C3semidirect
97. D48C22×C3semidirect
121. #504C3×D3semidirect
191. M32C2×C3×D3semidirect
192. C2×C16C2×C3×D3semidirect
276. C32#3416semidirect
313. C3#82014direct (N × H)
314. C3C3×#3416semidirect

Maximal subgroups

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

subgroupindex
C3×#57532
C3×#57522
C32×#5042
C3×#49092
C3×#49082
C3×#49042
C2×C3×#8002
C3×#34163
#820143