Finite Group Databasebeta

← C3×#10592452 #13162791

Subgroups — C3×#10592452 #13162791

Order 1728.

1028 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1028 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×#64364C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3C3×#64117C212yesmaximal, complemented
4C33×#101C211yesmaximal, characteristic, complemented
5C3×#63652C212yesmaximal, complemented
6#10592452C311yesmaximal, characteristic, direct product factor
7C3×#4753—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×#3832—22no—
10C3×Q8×C3⋊D3—22no—
11C3×#3828—22no—
12C3×#3828C2212yes—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C33×M16C2212yes—
15C3×#3752—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C3×#3745—22no—
18C3×#3744—22no—
19#64364C2×C311yescharacteristic, complemented
20C3×#1007—34no—
21#64156C2×C311yescharacteristic, complemented
22#64117C2×C312yescomplemented
23C3×#947—34no—
24C3×#908—38no—
25C32×#101—24no—
26C32×#101D314yescomplemented
27C32×#101C2×C311yescharacteristic, complemented
28#63652C2×C312yescomplemented
29C3×#825—38no—
30C22×C3×C3⋊D3—21no—
31C33×Q8—22no—
32C2×C33×C4D412yes—
33C2×C33×C4C2311yescharacteristic
34C3×#638—22no—
35C2×C3×C3⋊Dic3—21no—
36C3×C4⋊(C3⋊D3)—22no—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4⋊(C3⋊D3)D412yes—
39C3×C4×C3⋊D3—24no—
40C3×C4×C3⋊D3—22no—
41C3×#634—22no—
42C3×#634D412yes—
43C33×C8—22no—
44C3×#454—91no—
45#4753—34no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C22×C311yescharacteristic
49#3832—22no—
50Q8×C3⋊D3—22no—
51#3828—22no—
52#3828C22×C312yes—
53C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
54C3×#239—68no—
55C3×D3×Q8—68no—
56C3×#235—38no—
57C3×#235—68no—
58C2×C3×D12—34no—
59C32×M16—28no—
60C32×M16C2×D318yes—
61C32×M16C22×C312yes—
62#3752—22no—
63C4×C3⋊Dic3C22×C311yescharacteristic
64#3745—22no—
65#3744—22no—
66C3×(C2×D3)⋊C4—68no—
67C3×C4×Dic3—34no—
68C3×D24—68no—
69C3×#204—68no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—22no—
72C2×C3×C3⋊D3—42no—
73C2×C3×C3⋊D3—41no—
74C33×C4—22no—
75C33×C4C2×D412yes—
76C3×C3⋊Dic3—22no—
77C3×C3⋊Dic3—41no—
78C3×2−1+4—91no—
79#1007—34no—
80C3×AGL(1,ℤ/8)—92no—
81C3×#124—91no—
82#947—34no—
83#908—38no—
84C3×#104—92no—
85C3×#101—24no—
86C3×#101C3×D314yescomplemented
87C3×#101C3⋊D311yescharacteristic, complemented
88#825—38no—
89C22×C3⋊D3—21no—
90C22×C3×D3—64no—
91C32×Q8—22no—
92C32×Q8—28no—
93C32×Q8—48no—
94C2×C32×C4—28no—
95C2×C32×C4—24no—
96C2×C32×C4D1218yes—
97C2×C32×C4C3×D412yes—
98C2×C32×C4C22×D314yes—
99C2×C32×C4C23×C311yescharacteristic
100#638—22no—
101C2×C3⋊Dic3—21no—
102C4⋊(C3⋊D3)—22no—
103C4⋊(C3⋊D3)—22no—
104C4⋊(C3⋊D3)C3×D412yes—
105C4×C3⋊D3—24no—
106C4×C3⋊D3—22no—
107#634—22no—
108#634C3×D412yes—
109C3×#67—68no—
110C2×C3×Dic3—64no—
111C3×D12—38no—
112C3×D12—68no—
113C3×D12—68no—
114C3×C4×D3—616no—
115C3×C4×D3—68no—
116C3×Dic6—38no—
117C3×Dic6—68no—
118C32×C8—22no—
119C32×C8—28no—
120C32×C8—48no—
121#454—91noSylow
122C2×C33—21no—
123C2×C33#12011yescharacteristic
124C3×C3⋊D3—42no—
125C3×C3⋊D3—81no—
126C3×#41—92no—
127C3×#41—182no—
128C3×#41—182no—
129C2×C3×Q8—24no—
130C2×C3×Q8—182no—
131C2×C3×Q8C2×C3×D314yes—
132C2×C3×Q8C2×C3⋊D311yescharacteristic
133C2×C3×D4—91no—
134#239—68no—
135D3×Q8—68no—
136#235—38no—
137#235—68no—
138C2×D12—34no—
139C3×SD16—182no—
140C3×D8—182no—
141C3×M16—28no—
142C3×M16C2×C3×D318yes—
143C3×M16C2×C3⋊D312yes—
144C3×C22⋊C4—182no—
145C3×C42—91no—
146(C2×D3)⋊C4—68no—
147C4×Dic3—34no—
148D24—68no—
149#204—68no—
150C22×C32—24no—
151C22×C32C2×D1214yes—
152C22×C32C2×C3×D411yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3—42no—
155C2×C3⋊D3—41no—
156C2×C3×D3—68no—
157C2×C3×D3—128no—
158C2×C3×D3—124no—
159C32×C4—22no—
160C32×C4—28no—
161C32×C4—28no—
162C32×C4—48no—
163C32×C4D3×D418yes—
164C32×C4C2×C3×D412yes—
165C3⋊Dic3—22no—
166C3⋊Dic3—41no—
167C3×Dic3—68no—
168C3×Dic3—124no—
1692−1+4—91no—
170AGL(1,ℤ/8)—92no—
171#124—91no—
172#104—92no—
173#101C3×C3⋊D311yescharacteristic, complemented
174C33#45411yescharacteristic, complemented, Sylow
175C23×C3—181no—
176C22×D3—64no—
177C3×Q8—28no—
178C3×Q8—22no—
179C3×Q8—48no—
180C3×Q8—92no—
181C3×Q8—182no—
182C3×D4—92no—
183C3×D4—182no—
184C3×D4—182no—
185C3×D4—182no—
186C2×C3×C4—28no—
187C2×C3×C4—24no—
188C2×C3×C4—184no—
189C2×C3×C4—182no—
190C2×C3×C4—181no—
191C2×C3×C4C3×D1218yes—
192C2×C3×C4C4⋊(C3⋊D3)12yes—
193C2×C3×C4C22×C3×D314yes—
194C2×C3×C4C22×C3⋊D311yescharacteristic
195#67—68no—
196C2×Dic3—64no—
197D12—38no—
198D12—68no—
199D12—68no—
200C4×D3—616no—
201C4×D3—68no—
202Dic6—38no—
203Dic6—68no—
204C3×C8—28no—
205C3×C8—22no—
206C3×C8—48no—
207C2×C32—21no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32—44no—
211C2×C32#88214yes—
212C2×C32C3×#12011yescharacteristic
213C3⋊D3—42no—
214C3⋊D3—81no—
215C3×D3—128no—
216C3×D3—244no—
217#41—92no—
218#41—182no—
219#41—182no—
220C2×Q8—182no—
221C2×Q8C2×C3×C3⋊D311yescharacteristic
222C2×D4—91no—
223SD16—182no—
224D8—182no—
225M16C2×C3×C3⋊D312yes—
226C22⋊C4—182no—
227C42—91no—
228C22×C3—24no—
229C22×C3—182no—
230C22×C3—362no—
231C22×C3—361no—
232C22×C3C2×C3×D1214yes—
233C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
234C2×D3—68no—
235C2×D3—128no—
236C2×D3—124no—
237C3×C4—28no—
238C3×C4—22no—
239C3×C4—28no—
240C3×C4—48no—
241C3×C4—182no—
242C3×C4—361no—
243C3×C4C3×D3×D418yes—
244C3×C4D4×C3⋊D312yes—
245Dic3—68no—
246Dic3—124no—
247C32—24no—
248C32#475314yescomplemented
249C32C3×#45411yescharacteristic, complemented
250C23—181no—
251Q8—22no—
252Q8—92no—
253Q8—182no—
254D4—92no—
255D4—182no—
256D4—182no—
257D4—182no—
258C2×C4—184no—
259C2×C4—182no—
260C2×C4—181no—
261C2×C4C3×C4⋊(C3⋊D3)12yes—
262C2×C4C22×C3×C3⋊D311yescharacteristic
263C8—22no—
264C2×C3—24no—
265C2×C3—21no—
266C2×C3—24no—
267C2×C3—44no—
268C2×C3—362no—
269C2×C3—721no—
270C2×C3C3×#88214yes—
271C2×C3#6409111yescharacteristic
272D3—128no—
273D3—244no—
274C22—182no—
275C22—362no—
276C22—361no—
277C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
278C4—22no—
279C4—182no—
280C4—361no—
281C4C3×D4×C3⋊D312yes—
282C3—24no—
283C3C3×#475314yescomplemented
284C3#1059245211yescharacteristic, direct product factor
285C2—21no—
286C2—362no—
287C2—721no—
288C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#101C2
socleC2×C33#120
hypercenterC3×#101C3⋊D3
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
1. C3×#64364C2semidirect
2. C3×#64156C2semidirect
3. C3×#64117C2semidirect
4. C33×#101C2semidirect
5. C3×#63652C2semidirect
6. #10592452C3direct (N × H)
19. #64364C2×C3semidirect
21. #64156C2×C3semidirect
22. #64117C2×C3semidirect
26. C32×#101D3semidirect
27. C32×#101C2×C3semidirect
28. #63652C2×C3semidirect
86. C3×#101C3×D3semidirect
87. C3×#101C3⋊D3semidirect
173. #101C3×C3⋊D3semidirect
174. C33#454semidirect
248. C32#4753semidirect
249. C32C3×#454semidirect
283. C3C3×#4753semidirect
284. C3#10592452direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643642
C3×#641562
C3×#641172
C33×#1012
C3×#636522
#105924523
C3×#47533