Finite Group Databasebeta

← C3×#10592445 #13162784

Subgroups — C3×#10592445 #13162784

Order 1728.

1096 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64361C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3C3×#64118C212yesmaximal, complemented
4C33×#100C211yesmaximal, characteristic, complemented
5C3×#63652C212yesmaximal, complemented
6#10592445C311yesmaximal, characteristic, direct product factor
7C3×#4746—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C3×#3830—22no—
11C3×D4×C3⋊D3—22no—
12C3×#3828C2212yes—
13C2×C3×#634C2211yescharacteristic, complemented
14C33×M16C2212yescomplemented
15C3×#3757—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
17C3×#3746—22no—
18C3×#3744—22no—
19#64361C2×C311yescharacteristic
20C3×#1004—34no—
21#64145C2×C311yescharacteristic, complemented
22#64118C2×C312yescomplemented
23C3×#936—34no—
24C3×#909—38no—
25C32×#100—24no—
26C32×#100D314yescomplemented
27C32×#100C2×C311yescharacteristic, complemented
28#63652C2×C312yescomplemented
29C3×#825—38no—
30C23×C33D412yes—
31C22×C3×C3⋊D3—22no—
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34C3×#638—24no—
35C3×#638—22no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3—21no—
39C3×C4⋊(C3⋊D3)D412yes—
40C3×C4×C3⋊D3—22no—
41C3×#634—22no—
42C3×#634D412yes—
43C33×C8—22no—
44C3×#455—91no—
45#4746—34no—
46C2×C32×D4—24no—
47C2×C32×D4C2×D314yes—
48C2×C32×D4C22×C311yescharacteristic
49C2×#638—22no—
50#3830—22no—
51D4×C3⋊D3—22no—
52#3828C22×C312yes—
53C2×#634C22×C311yescharacteristic, complemented
54C2×C3×#67—68no—
55C3×#237—68no—
56C3×D3×D4—68no—
57C3×#235—38no—
58C2×C3×Dic6—34no—
59C32×M16—28no—
60C32×M16C2×D318yescomplemented
61C32×M16C22×C312yescomplemented
62#3757—22no—
63C4×C3⋊Dic3C22×C311yescharacteristic, complemented
64#3746—22no—
65#3744—22no—
66C3×#217—68no—
67C3×C4×Dic3—34no—
68C3×Dic12—68no—
69C3×#204—68no—
70C22×C33—22no—
71C22×C33—42no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—22no—
74C2×C3×C3⋊D3—82no—
75C33×C4C2×D412yes—
76C3×C3⋊Dic3—22no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3—41no—
79C3×H(5,𝔽2)—91no—
80#1004—34no—
81C3×#137—92no—
82C3×#124—91no—
83#936—34no—
84#909—38no—
85C3×#104—92no—
86C3×#100—24no—
87C3×#100C3×D314yescomplemented
88C3×#100C3⋊D311yescharacteristic, complemented
89#825—38no—
90C23×C32—28no—
91C23×C32D1218yes—
92C23×C32C3×D412yes—
93C22×C3⋊D3—22no—
94C22×C3×D3—68no—
95C32×D4—22no—
96C32×D4—28no—
97C32×D4—48no—
98C2×C32×C4—24no—
99C2×C32×C4C22×D314yes—
100C2×C32×C4C23×C311yescharacteristic
101#638—24no—
102#638—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C4⋊(C3⋊D3)C3×D412yes—
107C4×C3⋊D3—22no—
108#634—22no—
109#634C3×D412yes—
110C3×#67—616no—
111C3×#67—68no—
112C2×C3×Dic3—68no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—64no—
115C3×D12—38no—
116C3×C4×D3—68no—
117C3×Dic6—38no—
118C3×Dic6—68no—
119C32×C8—22no—
120C32×C8—28no—
121C32×C8—48no—
122#455—91noSylow
123C2×C33—21no—
124C2×C33—42no—
125C2×C33#12011yescharacteristic
126C3×C3⋊D3—42no—
127C3×#41—92no—
128C3×#41—182no—
129C2×C3×Q8—91no—
130C2×C3×D4—24no—
131C2×C3×D4—182no—
132C2×C3×D4—182no—
133C2×C3×D4C2×C3×D314yes—
134C2×C3×D4C2×C3⋊D311yescharacteristic
135C2×#67—68no—
136#237—68no—
137D3×D4—68no—
138#235—38no—
139C2×Dic6—34no—
140C3×Q16—182no—
141C3×SD16—182no—
142C3×M16—28no—
143C3×M16C2×C3×D318yescomplemented
144C3×M16C2×C3⋊D312yescomplemented
145C3×C22⋊C4—182no—
146C3×C42—91no—
147#217—68no—
148C4×Dic3—34no—
149Dic12—68no—
150#204—68no—
151C22×C32—22no—
152C22×C32—24no—
153C22×C32—28no—
154C22×C32—48no—
155C22×C32—42no—
156C22×C32—48no—
157C22×C32—88no—
158C22×C32C2×D1214yes—
159C22×C32C2×C3×D411yescharacteristic
160C2×C3⋊D3—22no—
161C2×C3⋊D3—82no—
162C2×C3×D3—68no—
163C2×C3×D3—248no—
164C32×C4—28no—
165C32×C4D3×D418yes—
166C32×C4C2×C3×D412yes—
167C3⋊Dic3—22no—
168C3⋊Dic3—41no—
169C3⋊Dic3—41no—
170C3×Dic3—68no—
171C3×Dic3—124no—
172C3×Dic3—124no—
173H(5,𝔽2)—91no—
174#137—92no—
175#124—91no—
176#104—92no—
177#100C3×C3⋊D311yescharacteristic, complemented
178C33#45511yescharacteristic, complemented, Sylow
179C23×C3—28no—
180C23×C3—182no—
181C23×C3C3×D1218yes—
182C23×C3C4⋊(C3⋊D3)12yes—
183C22×D3—68no—
184C3×Q8—92no—
185C3×Q8—182no—
186C3×D4—28no—
187C3×D4—22no—
188C3×D4—48no—
189C3×D4—92no—
190C3×D4—184no—
191C3×D4—182no—
192C2×C3×C4—24no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4—181no—
196C2×C3×C4—181no—
197C2×C3×C4C22×C3×D314yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199#67—616no—
200#67—68no—
201C2×Dic3—68no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204D12—38no—
205C4×D3—68no—
206Dic6—38no—
207Dic6—68no—
208C3×C8—28no—
209C3×C8—22no—
210C3×C8—48no—
211C2×C32—21no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32—44no—
215C2×C32—42no—
216C2×C32—48no—
217C2×C32—88no—
218C2×C32#88214yes—
219C2×C32C3×#12011yescharacteristic
220C3⋊D3—42no—
221C3×D3—128no—
222#41—92no—
223#41—182no—
224C2×Q8—91no—
225C2×D4—182no—
226C2×D4—182no—
227C2×D4C2×C3×C3⋊D311yescharacteristic
228Q16—182no—
229SD16—182no—
230M16C2×C3×C3⋊D312yescomplemented
231C22⋊C4—182no—
232C42—91no—
233C22×C3—28no—
234C22×C3—22no—
235C22×C3—24no—
236C22×C3—48no—
237C22×C3—48no—
238C22×C3—42no—
239C22×C3—88no—
240C22×C3—182no—
241C22×C3—722no—
242C22×C3C2×C3×D1214yes—
243C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
244C2×D3—68no—
245C2×D3—248no—
246C3×C4—28no—
247C3×C4—182no—
248C3×C4—361no—
249C3×C4—361no—
250C3×C4C3×D3×D418yes—
251C3×C4D4×C3⋊D312yes—
252Dic3—68no—
253Dic3—124no—
254Dic3—124no—
255C32—24no—
256C32#474614yescomplemented
257C32C3×#45511yescharacteristic, complemented
258C23—182no—
259C23C3×C4⋊(C3⋊D3)12yes—
260Q8—92no—
261Q8—182no—
262D4—22no—
263D4—92no—
264D4—184no—
265D4—182no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4C22×C3×C3⋊D311yescharacteristic
271C8—22no—
272C2×C3—24no—
273C2×C3—21no—
274C2×C3—24no—
275C2×C3—44no—
276C2×C3—48no—
277C2×C3—42no—
278C2×C3—88no—
279C2×C3—362no—
280C2×C3C3×#88214yes—
281C2×C3#6409111yescharacteristic
282D3—128no—
283C22—22no—
284C22—42no—
285C22—182no—
286C22—722no—
287C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
288C4—182no—
289C4—361no—
290C4—361no—
291C4C3×D4×C3⋊D312yes—
292C3—24no—
293C3C3×#474614yescomplemented
294C3#1059244511yescharacteristic, direct product factor
295C2—21no—
296C2—42no—
297C2—362no—
298C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#100C2
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC3×#100C3⋊D3
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#100

Complements

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

NH ≅ G/Ntype
2. C3×#64145C2semidirect
3. C3×#64118C2semidirect
4. C33×#100C2semidirect
5. C3×#63652C2semidirect
6. #10592445C3direct (N × H)
13. C2×C3×#634C22semidirect
14. C33×M16C22semidirect
16. C3×C4×C3⋊Dic3C22semidirect
21. #64145C2×C3semidirect
22. #64118C2×C3semidirect
26. C32×#100D3semidirect
27. C32×#100C2×C3semidirect
28. #63652C2×C3semidirect
53. C2×#634C22×C3semidirect
60. C32×M16C2×D3semidirect
61. C32×M16C22×C3semidirect
63. C4×C3⋊Dic3C22×C3semidirect
87. C3×#100C3×D3semidirect
88. C3×#100C3⋊D3semidirect
143. C3×M16C2×C3×D3semidirect
144. C3×M16C2×C3⋊D3semidirect
177. #100C3×C3⋊D3semidirect
178. C33#455semidirect
230. M16C2×C3×C3⋊D3semidirect
256. C32#4746semidirect
257. C32C3×#455semidirect
293. C3C3×#4746semidirect
294. C3#10592445direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643612
C3×#641452
C3×#641182
C33×#1002
C3×#636522
#105924453
C3×#47463