Finite Group Databasebeta

← C3×#10593771 #13173242

Subgroups — C3×#10593771 #13173242

Order 1728.

408 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 100 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×#64283C212yesmaximal
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
3C3×#63771C212yesmaximal, complemented
4C3×#63770C212yesmaximal, complemented
5#10593771C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×#3792—22no—
8C3×#3788C2214yes—
9C3×#3787—22no—
10C2×C33×C8C411yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
12C3×C8×C3⋊D3C411yescharacteristic
13C3×C8×C3⋊D3C411yescharacteristic
14C3×C8×C3⋊D3C2211yescharacteristic
15C3×C8×C3⋊D3C2211yescharacteristic
16#64283C2×C312yes—
17C2×C8×C3⋊D3C2×C311yescharacteristic
18C2×C3×D3×C8—62no—
19#63771C2×C312yescomplemented
20#63770C2×C312yescomplemented
21C22×C3×C3⋊D3Q811yescharacteristic
22C2×C33×C4C2×C411yescharacteristic
23C2×C3×C3⋊Dic3D411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3Q811yescharacteristic
27C3×C4×C3⋊D3C2311yescharacteristic
28C3×#622—24no—
29C33×C8C2×C411yescharacteristic
30C33×C8C2×C411yescharacteristic
31C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33C2×C3×#108—91nomaximal
34C2×C4×C3⋊D3C22×C311yescharacteristic
35C2×C3×C4×D3—62no—
36#3792—22no—
37#3788C22×C314yes—
38#3787—22no—
39C2×C32×C8—22no—
40C2×C32×C8—42no—
41C2×C32×C8C3×C411yescharacteristic
42C2×C3⋊(C3⋊C8)C3×C411yescharacteristic
43C8×C3⋊D3C3×C411yescharacteristic
44C8×C3⋊D3C3×C411yescharacteristic
45C8×C3⋊D3C22×C311yescharacteristic
46C8×C3⋊D3C22×C311yescharacteristic
47C2×C3×C3⋊C8—62no—
48C3×D3×C8—62no—
49C3×D3×C8—62no—
50C3×D3×C8—62no—
51C3×D3×C8—62no—
52C22×C33C4⋊C411yescharacteristic
53C2×C3×C3⋊D3—22no—
54C2×C3×C3⋊D3C4⋊C411yescharacteristic
55C2×C3×C3⋊D3C2×Q811yescharacteristic
56C33×C4C4⋊C411yescharacteristic
57C33×C4C22×C411yescharacteristic
58C3×C3⋊Dic3C4⋊C411yescharacteristic
59C3×C3⋊Dic3C2×D411yescharacteristic
60C2×C3×M16—92no—
61C22×C3×C8—91no—
62C2×D3×C8—62no—
63C3×#108—92no—
64C3×#108—92no—
65C22×C3⋊D3C3×Q811yescharacteristic
66C22×C3×D3—62no—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4C2×C3×C411yescharacteristic
70C2×C3⋊Dic3C3×D411yescharacteristic
71C4×C3⋊D3C2×C3×C411yescharacteristic
72C4×C3⋊D3C3×D411yescharacteristic
73C4×C3⋊D3C3×Q811yescharacteristic
74C4×C3⋊D3C23×C311yescharacteristic
75C2×C3×Dic3—62no—
76C3×C4×D3—62no—
77C3×C4×D3—62no—
78C3×C4×D3—62no—
79C3×C4×D3—62no—
80#622—24no—
81C32×C8—22no—
82C32×C8—22no—
83C32×C8—42no—
84C32×C8—42no—
85C32×C8C2×C3×C411yescharacteristic
86C32×C8C2×C3×C411yescharacteristic
87C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
88C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
89C3×C3⋊C8—62no—
90C3×C3⋊C8—62no—
91C2×#108—91noSylow
92C2×C33—21no—
93C2×C33C2×C4⋊C411yescharacteristic
94C3×C3⋊D3—21no—
95C3×C3⋊D3#10812yescomplemented
96C22×C3×C4—91no—
97C2×C4×D3—62no—
98C3×M16—94no—
99C3×M16—182no—
100C2×C3×C8—22no—
101C2×C3×C8—42no—
102C2×C3×C8—91no—
103C2×C3×C8—91no—
104C2×C3×C8—91no—
105C2×C3×C8—91no—
106C2×C3×C8—91no—
107C2×C3×C8—182no—
108C2×C3×C8#15711yescharacteristic
109C2×C3⋊C8—62no—
110D3×C8—62no—
111D3×C8—62no—
112D3×C8—62no—
113D3×C8—62no—
114C22×C32—22no—
115C22×C32—42no—
116C22×C32C3×C4⋊C411yescharacteristic
117C2×C3⋊D3—22no—
118C2×C3⋊D3C3×C4⋊C411yescharacteristic
119C2×C3⋊D3C2×C3×Q811yescharacteristic
120C2×C3×D3—62no—
121C2×C3×D3—62no—
122C2×C3×D3—68no—
123C32×C4—22no—
124C32×C4—22no—
125C32×C4—42no—
126C32×C4—42no—
127C32×C4C3×C4⋊C411yescharacteristic
128C32×C4C22×C3×C411yescharacteristic
129C3⋊Dic3C3×C4⋊C411yescharacteristic
130C3⋊Dic3C2×C3×D411yescharacteristic
131C3×Dic3—62no—
132C3×Dic3—62no—
133C2×M16—92no—
134C22×C8—91no—
135#108—92no—
136#108—92no—
137C33C2×#10811yescharacteristic, complemented, Sylow
138C23×C3—91no—
139C22×D3—62no—
140C2×C3×C4—22no—
141C2×C3×C4—42no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4C2×#15711yescharacteristic
148C2×Dic3—62no—
149C4×D3—62no—
150C4×D3—62no—
151C4×D3—62no—
152C4×D3—62no—
153C3×C8—22no—
154C3×C8—22no—
155C3×C8—42no—
156C3×C8—42no—
157C3×C8—91no—
158C3×C8—91no—
159C3×C8—184no—
160C3×C8C2×#15711yescharacteristic
161C3×C8C2×#15711yescharacteristic
162C3⋊C8—62no—
163C3⋊C8—62no—
164C2×C32—21no—
165C2×C32—24no—
166C2×C32—22no—
167C2×C32—44no—
168C2×C32—42no—
169C2×C32C2×C3×C4⋊C411yescharacteristic
170C3⋊D3—21no—
171C3⋊D3C3×#10812yescomplemented
172C3×D3—64no—
173C3×D3—64no—
174C22×C4—91no—
175M16—94no—
176M16—182no—
177C2×C8—91no—
178C2×C8—91no—
179C2×C8—91no—
180C2×C8—91no—
181C2×C8—91no—
182C2×C8—182no—
183C2×C8C3×#15711yescharacteristic
184C22×C3—22no—
185C22×C3—42no—
186C22×C3—91no—
187C22×C3—91no—
188C22×C3—182no—
189C22×C3C4⋊#15711yescharacteristic
190C2×D3—62no—
191C2×D3—62no—
192C2×D3—68no—
193C3×C4—22no—
194C3×C4—22no—
195C3×C4—42no—
196C3×C4—42no—
197C3×C4—91no—
198C3×C4—91no—
199C3×C4C4⋊#15711yescharacteristic
200C3×C4C22×#15711yescharacteristic
201Dic3—62no—
202Dic3—62no—
203C32—22no—
204C32—42no—
205C32C2×C3×#10811yescharacteristic, complemented
206C23—91no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4C2×C3×#15711yescharacteristic
213C8—91no—
214C8—91no—
215C8—184no—
216C8C2×C3×#15711yescharacteristic
217C8C2×C3×#15711yescharacteristic
218C2×C3—24no—
219C2×C3—21no—
220C2×C3—22no—
221C2×C3—44no—
222C2×C3—42no—
223C2×C3—92no—
224C2×C3—181no—
225C2×C3C2×C4⋊#15711yescharacteristic
226D3—64no—
227D3—64no—
228C22—91no—
229C22—91no—
230C22—182no—
231C22C3×C4⋊#15711yescharacteristic
232C4—91no—
233C4—91no—
234C4C3×C4⋊#15711yescharacteristic
235C4C22×C3×#15711yescharacteristic
236C3—22no—
237C3—42no—
238C3#1059377111yescharacteristic, direct product factor
239C2—21no—
240C2—92no—
241C2—181no—
242C2C2×C3×C4⋊#15711yescharacteristic
subgroupG/H
centerC3×C4C4⋊#157
commutatorC32×C4C22×C3×C4
FrattiniC4C22×C3×#157
FittingC2×C33×C8C4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C8#157
hypocenterC32C2×C3×#108
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63771C2semidirect
4. C3×#63770C2semidirect
5. #10593771C3direct (N × H)
19. #63771C2×C3semidirect
20. #63770C2×C3semidirect
95. C3×C3⋊D3#108semidirect
137. C33C2×#108semidirect
171. C3⋊D3C3×#108semidirect
205. C32C2×C3×#108semidirect
238. C3#10593771direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642832
C2×C3×C8×C3⋊D32
C3×#637712
C3×#637702
#105937713
C2×C3×#1089