Finite Group Databasebeta

← C3×#10592559 #13162898

Subgroups — C3×#10592559 #13162898

Order 1728.

828 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 222 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×C8×C3⋊D3C211yesmaximal, characteristic
2C3×#64101C212yesmaximal
3C33×#106C211yesmaximal, characteristic, complemented
4C3×#63644C211yesmaximal, characteristic, complemented
5C3×#63634C212yesmaximal, complemented
6#10592559C311yesmaximal, characteristic, direct product factor
7C3×#4860—34nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C33×C8C2211yescharacteristic
10C33×C4⋊C4C2212yes—
11C3×#3752—22no—
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C3×C4×C3⋊Dic3—22no—
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×C8×C3⋊D3C414yescomplemented
16C2×C8×C3⋊D3C2×C311yescharacteristic
17#64101C2×C312yes—
18C2×C3×D3×C8—34no—
19C3×#892—38no—
20C32×#106—24no—
21C32×#106D314yescomplemented
22C32×#106C2×C311yescharacteristic, complemented
23#63644C2×C311yescharacteristic, complemented
24#63634C2×C312yescomplemented
25C3×#817—34no—
26C3×#807—38no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C33×C4—22no—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3D411yescharacteristic
32C3×C4×C3⋊D3C2×C412yes—
33C3×C4×C3⋊D3Q812yes—
34C33×C8C2×C412yes—
35C3×C3⋊(C3⋊C8)C2×C412yes—
36C3×#427—91no—
37#4860—34no—
38C2×C4×C3⋊D3C22×C311yescharacteristic
39C2×C3×C4×D3—34no—
40C2×C32×C8—24no—
41C2×C32×C8C2×D314yes—
42C2×C32×C8C22×C311yescharacteristic
43C32×C4⋊C4—28no—
44C32×C4⋊C4C2×D318yes—
45C32×C4⋊C4C22×C312yes—
46#3752—22no—
47C4⋊(C3⋊Dic3)C22×C312yes—
48C4×C3⋊Dic3—22no—
49C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
50C8×C3⋊D3C3×C414yescomplemented
51C3×(C2×D3)⋊C4—68no—
52C3×C4⋊Dic3—38no—
53C3×C4×Dic3—68no—
54C2×C3×C3⋊C8—34no—
55C3×D3×C8—316no—
56C22×C33C2×D411yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3C4⋊C412yes—
59C33×C4—42no—
60C33×C4C22×C411yescharacteristic
61C33×C4C2×Q811yescharacteristic
62C3×C3⋊Dic3—42no—
63C3×C3⋊Dic3C4⋊C412yes—
64C22×C3×C8—91no—
65C3×#117—92no—
66C2×D3×C8—34no—
67#892—38no—
68C3×#107—92no—
69C3×#106—24no—
70C3×#106—91no—
71C3×#106C3×D314yescomplemented
72C3×#106C3⋊D311yescharacteristic, complemented
73#817—34no—
74#807—38no—
75C22×C3⋊D3C3×D411yescharacteristic
76C22×C3×D3—34no—
77C2×C32×C4—22no—
78C2×C32×C4—24no—
79C2×C32×C4—28no—
80C2×C32×C4—48no—
81C2×C32×C4C22×D314yes—
82C2×C32×C4C23×C311yescharacteristic
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3C3×D411yescharacteristic
85C4×C3⋊D3C2×C3×C412yes—
86C4×C3⋊D3C3×Q812yes—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—68no—
89C3×C4×D3—38no—
90C3×C4×D3—38no—
91C32×C8—28no—
92C32×C8C4×D318yes—
93C32×C8C2×C3×C412yes—
94C3⋊(C3⋊C8)C2×C3×C412yes—
95C3×C3⋊C8—38no—
96#427—91noSylow
97C2×C33C2×C4⋊C411yescharacteristic
98C2×C33#13512yes—
99C3×C3⋊D3—22no—
100C22×C3×C4—91no—
101C2×C4×D3—34no—
102C2×C3×C8—24no—
103C2×C3×C8—91no—
104C2×C3×C8—94no—
105C2×C3×C8C2×C3×D314yes—
106C2×C3×C8C2×C3⋊D311yescharacteristic
107C3×C4⋊C4—28no—
108C3×C4⋊C4—92no—
109C3×C4⋊C4C2×C3×D318yes—
110C3×C4⋊C4C2×C3⋊D312yes—
111C3×C22⋊C4—182no—
112C3×C42—182no—
113(C2×D3)⋊C4—68no—
114C4⋊Dic3—38no—
115C4×Dic3—68no—
116C2×C3⋊C8—34no—
117D3×C8—316no—
118C22×C32—24no—
119C22×C32D3×D414yes—
120C22×C32C2×C3×D411yescharacteristic
121C2×C3⋊D3—22no—
122C2×C3⋊D3C3×C4⋊C412yes—
123C2×C3×D3—38no—
124C2×C3×D3—68no—
125C32×C4—24no—
126C32×C4—24no—
127C32×C4—42no—
128C32×C4—48no—
129C32×C4—88no—
130C32×C4C2×C4×D314yes—
131C32×C4D3×Q814yes—
132C32×C4C22×C3×C411yescharacteristic
133C32×C4C2×C3×Q811yescharacteristic
134C3⋊Dic3—42no—
135C3⋊Dic3C3×C4⋊C412yes—
136C3×Dic3—38no—
137C3×Dic3—128no—
138C22×C8—91no—
139#117—92no—
140#107—92no—
141#106—91no—
142#106C3×C3⋊D311yescharacteristic, complemented
143C33#42711yescharacteristic, complemented, Sylow
144C23×C3—91no—
145C22×D3—34no—
146C2×C3×C4—28no—
147C2×C3×C4—22no—
148C2×C3×C4—24no—
149C2×C3×C4—48no—
150C2×C3×C4—91no—
151C2×C3×C4—92no—
152C2×C3×C4—92no—
153C2×C3×C4—182no—
154C2×C3×C4C22×C3×D314yes—
155C2×C3×C4C22×C3⋊D311yescharacteristic
156C2×Dic3—34no—
157C2×Dic3—68no—
158C4×D3—38no—
159C4×D3—38no—
160C3×C8—28no—
161C3×C8—92no—
162C3×C8C3×C4×D318yes—
163C3×C8C4×C3⋊D312yes—
164C3⋊C8—38no—
165C2×C32—24no—
166C2×C32—28no—
167C2×C32D3×C4⋊C414yes—
168C2×C32#91618yes—
169C2×C32C2×C3×C4⋊C411yescharacteristic
170C2×C32C3×#13512yes—
171C3⋊D3—22no—
172C3×D3—68no—
173C22×C4—91no—
174C2×C8—91no—
175C2×C8—94no—
176C2×C8C2×C3×C3⋊D311yescharacteristic
177C4⋊C4—92no—
178C4⋊C4C2×C3×C3⋊D312yes—
179C22⋊C4—182no—
180C42—182no—
181C22×C3—24no—
182C22×C3—92no—
183C22×C3—182no—
184C22×C3C3×D3×D414yes—
185C22×C3D4×C3⋊D311yescharacteristic
186C2×D3—38no—
187C2×D3—68no—
188C3×C4—24no—
189C3×C4—24no—
190C3×C4—48no—
191C3×C4—42no—
192C3×C4—88no—
193C3×C4—92no—
194C3×C4—362no—
195C3×C4C2×C3×C4×D314yes—
196C3×C4C3×D3×Q814yes—
197C3×C4C2×C4×C3⋊D311yescharacteristic
198C3×C4Q8×C3⋊D311yescharacteristic
199Dic3—38no—
200Dic3—128no—
201C32—24no—
202C32#486014yescomplemented
203C32C3×#42711yescharacteristic, complemented
204C23—91no—
205C2×C4—22no—
206C2×C4—91no—
207C2×C4—92no—
208C2×C4—92no—
209C2×C4—182no—
210C2×C4C22×C3×C3⋊D311yescharacteristic
211C8—92no—
212C8C3×C4×C3⋊D312yes—
213C2×C3—24no—
214C2×C3—28no—
215C2×C3—182no—
216C2×C3C3×D3×C4⋊C414yes—
217C2×C3C3×#91618yes—
218C2×C3C4⋊C4×C3⋊D311yescharacteristic
219C2×C3#6412512yes—
220D3—68no—
221C22—92no—
222C22—182no—
223C22C3×D4×C3⋊D311yescharacteristic
224C4—42no—
225C4—92no—
226C4—362no—
227C4C2×C3×C4×C3⋊D311yescharacteristic
228C4C3×Q8×C3⋊D311yescharacteristic
229C3—24no—
230C3C3×#486014yescomplemented
231C3#1059255911yescharacteristic, direct product factor
232C2—182no—
233C2C3×C4⋊C4×C3⋊D311yescharacteristic
234C2C3×#6412512yes—
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#427
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
3. C33×#106C2semidirect
4. C3×#63644C2semidirect
5. C3×#63634C2semidirect
6. #10592559C3direct (N × H)
15. C3×C8×C3⋊D3C4semidirect
21. C32×#106D3semidirect
22. C32×#106C2×C3semidirect
23. #63644C2×C3semidirect
24. #63634C2×C3semidirect
50. C8×C3⋊D3C3×C4semidirect
71. C3×#106C3×D3semidirect
72. C3×#106C3⋊D3semidirect
142. #106C3×C3⋊D3semidirect
143. C33#427semidirect
202. C32#4860semidirect
203. C32C3×#427semidirect
230. C3C3×#4860semidirect
231. C3#10592559direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C3×#641012
C33×#1062
C3×#636442
C3×#636342
#105925593
C3×#48603