Finite Group Databasebeta

← C3×#10592420 #13162760

Subgroups — C3×#10592420 #13162760

Order 1728.

824 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C33×C22⋊C8C211yesmaximal, characteristic
4C3×#63644C212yesmaximal, complemented
5C3×#63643C212yesmaximal, complemented
6#10592420C311yesmaximal, characteristic, direct product factor
7C3×#4721—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C3×#634C2211yescharacteristic
11C2×C33×C8C2212yes—
12C3×#3757—21no—
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C3×#3750—21no—
17C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
18#64133C2×C311yescharacteristic
19C2×C3×C4⋊Dic3—34no—
20C3×#924—34no—
21C32×C22⋊C8—24no—
22C32×C22⋊C8D314yes—
23C32×C22⋊C8C2×C311yescharacteristic
24#63644C2×C312yescomplemented
25#63643C2×C312yescomplemented
26C3×#817—38no—
27C3×#816—38no—
28C23×C33D411yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4D411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—42no—
36C3×#634—22no—
37C33×C8—22no—
38C3×#483—91no—
39#4721—34no—
40C22×C32×C4—24no—
41C22×C32×C4C2×D314yes—
42C22×C32×C4C22×C311yescharacteristic
43C22×C3⋊Dic3—21no—
44C2×#634C22×C311yescharacteristic
45C22×C3×Dic3—64no—
46C2×C3×Dic6—34no—
47C2×C32×C8—28no—
48C2×C32×C8C2×D318yes—
49C2×C32×C8C22×C312yes—
50#3757—21no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)C22×C311yescharacteristic
53C4⋊(C3⋊Dic3)C22×C312yes—
54#3750—21no—
55C3×#217—64no—
56C3×C4⋊Dic3—34no—
57C3×C4⋊Dic3—38no—
58C3×C4⋊Dic3—64no—
59C3×Dic3⋊C4—64no—
60C22×C33—21no—
61C22×C33—21no—
62C22×C33SD1612yes—
63C22×C33C2×D411yescharacteristic
64C33×C4—21no—
65C33×C4#4112yes—
66C3×C3⋊Dic3—41no—
67C3×C3⋊Dic3—42no—
68C3×C3⋊Dic3—41no—
69C3×C22⋊Q8—91no—
70C2×C3×C4⋊C4—91no—
71C2×C4⋊Dic3—34no—
72#924—34no—
73C3×#106—92no—
74C3×#103—92no—
75C3×C22⋊C8—24no—
76C3×C22⋊C8C3×D314yes—
77C3×C22⋊C8C3⋊D311yescharacteristic
78#817—38no—
79#816—38no—
80C23×C32—24no—
81C23×C32D1214yes—
82C23×C32C3×D411yescharacteristic
83C2×C32×C4—22no—
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4—28no—
87C2×C32×C4—48no—
88C2×C32×C4D1214yes—
89C2×C32×C4C3×D411yescharacteristic
90C2×C32×C4C22×D314yes—
91C2×C32×C4C23×C311yescharacteristic
92C2×C3⋊Dic3—21no—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—42no—
96#634—22no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—68no—
100C2×C3×Dic3—128no—
101C3×Dic6—68no—
102C32×C8—22no—
103C32×C8—28no—
104C32×C8—48no—
105#483—91noSylow
106C2×C33—22no—
107C2×C33#12311yescharacteristic
108C2×C33C2×SD1611yescharacteristic
109C2×C33#13711yescharacteristic
110C2×C3×Q8—91no—
111C22×C3×C4—24no—
112C22×C3×C4—181no—
113C22×C3×C4C2×C3×D314yes—
114C22×C3×C4C2×C3⋊D311yescharacteristic
115C22×Dic3—64no—
116C2×Dic6—34no—
117C2×C3×C8—28no—
118C2×C3×C8C2×C3×D318yes—
119C2×C3×C8C2×C3⋊D312yes—
120C3×C4⋊C4—92no—
121C3×C4⋊C4—91no—
122C3×C4⋊C4—181no—
123C3×C4⋊C4—181no—
124C3×C22⋊C4—181no—
125#217—64no—
126C4⋊Dic3—38no—
127C4⋊Dic3—34no—
128C4⋊Dic3—64no—
129Dic3⋊C4—64no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—28no—
133C22×C32—24no—
134C22×C32—24no—
135C22×C32—24no—
136C22×C32—44no—
137C22×C32—44no—
138C22×C32#20418yes—
139C22×C32C3×SD1612yes—
140C22×C32C2×D1214yes—
141C22×C32C2×C3×D411yescharacteristic
142C32×C4—21no—
143C32×C4—28no—
144C32×C4—24no—
145C32×C4—44no—
146C32×C4#23718yes—
147C32×C4C3×#4112yes—
148C3⋊Dic3—41no—
149C3⋊Dic3—42no—
150C3⋊Dic3—41no—
151C3×Dic3—124no—
152C3×Dic3—128no—
153C3×Dic3—124no—
154C22⋊Q8—91no—
155C2×C4⋊C4—91no—
156#106—92no—
157#103—92no—
158C22⋊C8C3×C3⋊D311yescharacteristic
159C33#48311yescharacteristic, complemented, Sylow
160C23×C3—24no—
161C23×C3C3×D1214yes—
162C23×C3C4⋊(C3⋊D3)11yescharacteristic
163C3×Q8—182no—
164C2×C3×C4—28no—
165C2×C3×C4—22no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4—48no—
169C2×C3×C4—181no—
170C2×C3×C4—181no—
171C2×C3×C4—182no—
172C2×C3×C4—362no—
173C2×C3×C4C3×D1214yes—
174C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
175C2×C3×C4C22×C3×D314yes—
176C2×C3×C4C22×C3⋊D311yescharacteristic
177C2×Dic3—64no—
178C2×Dic3—64no—
179C2×Dic3—68no—
180C2×Dic3—128no—
181Dic6—68no—
182C3×C8—28no—
183C3×C8—22no—
184C3×C8—48no—
185C2×C32—22no—
186C2×C32—24no—
187C2×C32—24no—
188C2×C32—24no—
189C2×C32—28no—
190C2×C32—48no—
191C2×C32#88614yes—
192C2×C32C2×#20414yes—
193C2×C32#90914yes—
194C2×C32C3×#12311yescharacteristic
195C2×C32C2×C3×SD1611yescharacteristic
196C2×C32C3×#13711yescharacteristic
197C2×Q8—91no—
198C22×C4—181no—
199C22×C4C2×C3×C3⋊D311yescharacteristic
200C2×C8C2×C3×C3⋊D312yes—
201C4⋊C4—92no—
202C4⋊C4—91no—
203C4⋊C4—181no—
204C4⋊C4—181no—
205C22⋊C4—181no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—28no—
211C22×C3—24no—
212C22×C3—44no—
213C22×C3—44no—
214C22×C3C3×#20418yes—
215C22×C3#374412yes—
216C22×C3C2×C3×D1214yes—
217C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
218C3×C4—24no—
219C3×C4—21no—
220C3×C4—28no—
221C3×C4—44no—
222C3×C4—361no—
223C3×C4—362no—
224C3×C4—361no—
225C3×C4C3×#23718yes—
226C3×C4#383012yes—
227Dic3—124no—
228Dic3—128no—
229Dic3—124no—
230C32—24no—
231C32#472114yescomplemented
232C32C3×#48311yescharacteristic, complemented
233C23C3×C4⋊(C3⋊D3)11yescharacteristic
234Q8—182no—
235C2×C4—22no—
236C2×C4—181no—
237C2×C4—181no—
238C2×C4—182no—
239C2×C4—362no—
240C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
241C2×C4C22×C3×C3⋊D311yescharacteristic
242C8—22no—
243C2×C3—28no—
244C2×C3—22no—
245C2×C3—24no—
246C2×C3—24no—
247C2×C3—24no—
248C2×C3—48no—
249C2×C3C3×#88614yes—
250C2×C3C2×C3×#20414yes—
251C2×C3C3×#90914yes—
252C2×C3#6409511yescharacteristic
253C2×C3C2×#374411yescharacteristic
254C2×C3#6411811yescharacteristic
255C22—21no—
256C22—21no—
257C22C3×#374412yes—
258C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
259C4—21no—
260C4—361no—
261C4—362no—
262C4—361no—
263C4C3×#383012yes—
264C3—24no—
265C3C3×#472114yescomplemented
266C3#1059242011yescharacteristic, direct product factor
267C2—22no—
268C2C3×#6409511yescharacteristic
269C2C2×C3×#374411yescharacteristic
270C2C3×#6411811yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#483
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C3×#63644C2semidirect
5. C3×#63643C2semidirect
6. #10592420C3direct (N × H)
24. #63644C2×C3semidirect
25. #63643C2×C3semidirect
159. C33#483semidirect
231. C32#4721semidirect
232. C32C3×#483semidirect
265. C3C3×#4721semidirect
266. C3#10592420direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C3×#641332
C33×C22⋊C82
C3×#636442
C3×#636432
#105924203
C3×#47213