Finite Group Databasebeta

← C3×#10592396 #13162735

Subgroups — C3×#10592396 #13162735

Order 1728.

752 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 190 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×#64084C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C33×C4×C8C211yesmaximal, characteristic, complemented
4C3×#63648C212yesmaximal
5C3×#63643C212yesmaximal, complemented
6#10592396C311yesmaximal, characteristic, direct product factor
7C3×#4697—34nomaximal
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×C33×C8C2212yes—
11C33×C42C2211yescharacteristic
12C3×#3752—22no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×#3750—22no—
15#64084C2×C311yescharacteristic
16#64079C2×C311yescharacteristic, complemented
17C3×#875—34no—
18C3×#870—34no—
19C32×C4×C8—24no—
20C32×C4×C8D314yescomplemented
21C32×C4×C8C2×C311yescharacteristic, complemented
22#63648C2×C312yes—
23#63643C2×C312yescomplemented
24C3×#821—38no—
25C3×#816—38no—
26C22×C3×C3⋊D3—21no—
27C2×C33×C4D412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3—21no—
31C3×C4⋊(C3⋊D3)—22no—
32C3×#634—22no—
33C33×C8—22no—
34C3×#488—91no—
35#4697—34no—
36C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
37C2×#634C22×C311yescharacteristic
38C2×C3×D12—34no—
39C2×C3×Dic6—34no—
40C2×C32×C8—28no—
41C2×C32×C8C2×D318yes—
42C2×C32×C8C22×C312yes—
43C32×C42—24no—
44C32×C42C2×D314yes—
45C32×C42C22×C311yescharacteristic
46#3752—22no—
47C4⋊(C3⋊Dic3)C22×C312yes—
48#3750—22no—
49C3×(C2×D3)⋊C4—68no—
50C3×C4⋊Dic3—38no—
51C3×Dic3⋊C4—68no—
52C22×C33C2×D411yescharacteristic
53C2×C3×C3⋊D3—41no—
54C2×C3×C3⋊D3—42no—
55C33×C4—22no—
56C33×C4#4112yes—
57C3×C3⋊Dic3—41no—
58C3×C3⋊Dic3—42no—
59C3×#125—91no—
60C3×#124—91no—
61#875—34no—
62#870—34no—
63C3×#103—92no—
64C3×#102—92no—
65C3×C4×C8—24no—
66C3×C4×C8C3×D314yescomplemented
67C3×C4×C8C3⋊D311yescharacteristic, complemented
68#821—38no—
69#816—38no—
70C22×C3⋊D3—21no—
71C22×C3×D3—64no—
72C2×C32×C4—28no—
73C2×C32×C4—24no—
74C2×C32×C4D1218yes—
75C2×C32×C4C3×D412yes—
76C2×C32×C4C22×D314yes—
77C2×C32×C4C23×C311yescharacteristic
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3—21no—
80C4⋊(C3⋊D3)—22no—
81#634—22no—
82C2×C3×Dic3—68no—
83C2×C3×Dic3—64no—
84C3×D12—68no—
85C3×Dic6—68no—
86C32×C8—216no—
87C32×C8—28no—
88C32×C8—22no—
89#488—91noSylow
90C2×C33#12411yescharacteristic
91C2×C33#13512yes—
92C3×C3⋊D3—81no—
93C2×C3×Q8—91no—
94C2×C3×D4—91no—
95C2×D12—34no—
96C2×Dic6—34no—
97C2×C3×C8—28no—
98C2×C3×C8C2×C3×D318yes—
99C2×C3×C8C2×C3⋊D312yes—
100C3×C4⋊C4—92no—
101C3×C4⋊C4—182no—
102C3×C22⋊C4—182no—
103C3×C42—24no—
104C3×C42C2×C3×D314yes—
105C3×C42C2×C3⋊D311yescharacteristic
106(C2×D3)⋊C4—68no—
107C4⋊Dic3—38no—
108Dic3⋊C4—68no—
109C22×C32—24no—
110C22×C32C2×D1214yes—
111C22×C32C2×C3×D411yescharacteristic
112C2×C3⋊D3—41no—
113C2×C3⋊D3—42no—
114C2×C3×D3—124no—
115C2×C3×D3—128no—
116C32×C4—216no—
117C32×C4—28no—
118C32×C4—22no—
119C32×C4—28no—
120C32×C4#23518yes—
121C32×C4C3×#4112yes—
122C3⋊Dic3—41no—
123C3⋊Dic3—42no—
124C3×Dic3—124no—
125C3×Dic3—128no—
126#125—91no—
127#124—91no—
128#103—92no—
129#102—92no—
130C4×C8C3×C3⋊D311yescharacteristic, complemented
131C33#48811yescharacteristic, complemented, Sylow
132C23×C3—181no—
133C22×D3—64no—
134C3×Q8—182no—
135C3×D4—182no—
136C2×C3×C4—28no—
137C2×C3×C4—24no—
138C2×C3×C4—181no—
139C2×C3×C4—182no—
140C2×C3×C4C3×D1218yes—
141C2×C3×C4C4⋊(C3⋊D3)12yes—
142C2×C3×C4C22×C3×D314yes—
143C2×C3×C4C22×C3⋊D311yescharacteristic
144C2×Dic3—68no—
145C2×Dic3—64no—
146D12—68no—
147Dic6—68no—
148C3×C8—216no—
149C3×C8—28no—
150C3×C8—22no—
151C2×C32—28no—
152C2×C32—24no—
153C2×C32#87514yes—
154C2×C32#90418yes—
155C2×C32C3×#12411yescharacteristic
156C2×C32C3×#13512yes—
157C3⋊D3—81no—
158C3×D3—244no—
159C2×Q8—91no—
160C2×D4—91no—
161C2×C8C2×C3×C3⋊D312yes—
162C4⋊C4—92no—
163C4⋊C4—182no—
164C22⋊C4—182no—
165C42C2×C3×C3⋊D311yescharacteristic
166C22×C3—24no—
167C22×C3—361no—
168C22×C3—362no—
169C22×C3C2×C3×D1214yes—
170C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
171C2×D3—128no—
172C2×D3—124no—
173C3×C4—216no—
174C3×C4—28no—
175C3×C4—22no—
176C3×C4—28no—
177C3×C4—361no—
178C3×C4—362no—
179C3×C4C3×#23518yes—
180C3×C4#382812yes—
181Dic3—128no—
182Dic3—124no—
183C32—24no—
184C32#469714yescomplemented
185C32C3×#48811yescharacteristic, complemented
186C23—181no—
187Q8—182no—
188D4—182no—
189C2×C4—181no—
190C2×C4—182no—
191C2×C4C3×C4⋊(C3⋊D3)12yes—
192C2×C4C22×C3×C3⋊D311yescharacteristic
193C8—22no—
194C2×C3—24no—
195C2×C3—28no—
196C2×C3—721no—
197C2×C3C3×#87514yes—
198C2×C3C3×#90418yes—
199C2×C3#6408411yescharacteristic
200C2×C3#6411312yes—
201D3—244no—
202C22—361no—
203C22—362no—
204C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
205C4—22no—
206C4—361no—
207C4—362no—
208C4C3×#382812yes—
209C3—24no—
210C3C3×#469714yescomplemented
211C3#1059239611yescharacteristic, direct product factor
212C2—721no—
213C2C3×#6408411yescharacteristic
214C2C3×#6411312yes—
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×C8C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4×C8C3⋊D3
hypocenterC32C3×#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8

Complements

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

NH ≅ G/Ntype
2. C3×#64079C2semidirect
3. C33×C4×C8C2semidirect
5. C3×#63643C2semidirect
6. #10592396C3direct (N × H)
16. #64079C2×C3semidirect
20. C32×C4×C8D3semidirect
21. C32×C4×C8C2×C3semidirect
23. #63643C2×C3semidirect
66. C3×C4×C8C3×D3semidirect
67. C3×C4×C8C3⋊D3semidirect
130. C4×C8C3×C3⋊D3semidirect
131. C33#488semidirect
184. C32#4697semidirect
185. C32C3×#488semidirect
210. C3C3×#4697semidirect
211. C3#10592396direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640842
C3×#640792
C33×C4×C82
C3×#636482
C3×#636432
#105923963
C3×#46973