Finite Group Databasebeta

← C2×C3×#63769 #13173238

Subgroups — C2×C3×#63769 #13173238

Order 1728.

456 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 120 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⋊#157C212yesmaximal
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
3C3×#63769C214yesmaximal, direct product factor
4C2×#63769C311yesmaximal, characteristic, direct product factor
5C22×C3×#157—22no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C3×C4⋊#157—22no—
8C3×C4⋊#157C2214yes—
9C2×C33×C8C411yescharacteristic, complemented
10C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
11C3×C8×C3⋊D3C412yescomplemented
12C3×C8×C3⋊D3C2212yes—
13C2×C4⋊#157C2×C312yes—
14C2×C8×C3⋊D3C2×C311yescharacteristic
15C2×C3×D3×C8—62no—
16#63769C2×C314yesdirect product factor
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3×#157—24no—
19C2×C3×#157—44no—
20C2×C33×C4C2×C411yescharacteristic
21C2×C3×C3⋊Dic3Q811yescharacteristic
22C3×C4×C3⋊D3C2×C411yescharacteristic
23C3×C4×C3⋊D3D411yescharacteristic
24C3×C4×C3⋊D3Q811yescharacteristic
25C3×C4×C3⋊D3C2311yescharacteristic
26C33×C8C2×C412yescomplemented
27C3×C3⋊(C3⋊C8)C2×C412yescomplemented
28C2×C3×#107—91nomaximal
29C22×#157—22no—
30C2×C4×C3⋊D3C22×C311yescharacteristic
31C2×C3×C4×D3—62no—
32C4⋊#157—22no—
33C4⋊#157C22×C314yes—
34C2×C32×C8—22no—
35C2×C32×C8—42no—
36C2×C32×C8C3×C411yescharacteristic, complemented
37C2×C3⋊(C3⋊C8)C3×C411yescharacteristic, complemented
38C8×C3⋊D3C3×C412yescomplemented
39C8×C3⋊D3C22×C312yes—
40C2×C3×C3⋊C8—62no—
41C3×D3×C8—64no—
42C3×D3×C8—64no—
43C22×C33C4⋊C411yescharacteristic
44C2×C3×C3⋊D3C4⋊C411yescharacteristic
45C2×C3×C3⋊D3D812yes—
46C2×C3×C3⋊D3Q1612yes—
47C2×C3×C3⋊D3C2×D411yescharacteristic
48C3×#157—44no—
49C33×C4C4⋊C411yescharacteristic
50C33×C4C22×C411yescharacteristic
51C3×C3⋊Dic3C4⋊C411yescharacteristic
52C3×C3⋊Dic3C2×Q811yescharacteristic
53C22×C3×C8—91no—
54C2×C3×C4⋊C4—92no—
55C2×D3×C8—62no—
56C3×#107—94no—
57C22×C3⋊D3C3×D411yescharacteristic
58C22×C3×D3—62no—
59C2×#157—24no—
60C2×#157—44no—
61C2×C32×C4—22no—
62C2×C32×C4—42no—
63C2×C32×C4C2×C3×C411yescharacteristic
64C2×C3⋊Dic3C3×Q811yescharacteristic
65C4×C3⋊D3C2×C3×C411yescharacteristic
66C4×C3⋊D3C3×D411yescharacteristic
67C4×C3⋊D3C3×Q811yescharacteristic
68C4×C3⋊D3C23×C311yescharacteristic
69C2×C3×Dic3—62no—
70C3×C4×D3—62no—
71C3×C4×D3—62no—
72C3×C4×D3—62no—
73C3×C4×D3—62no—
74C32×C8—24no—
75C32×C8—44no—
76C32×C8C2×C3×C412yescomplemented
77C3⋊(C3⋊C8)C2×C3×C412yescomplemented
78C3×C3⋊C8—64no—
79C2×#107—91noSylow
80C2×C33#10712yescomplemented
81C2×C33C2×C4⋊C411yescharacteristic
82C3×C3⋊D3#10712yescomplemented
83C3×C3⋊D3C2×D811yescharacteristic
84C3×C3⋊D3C2×Q1611yescharacteristic
85C22×C3×C4—91no—
86C22×C3×C4—182no—
87C2×C4×D3—62no—
88C2×C3×C8—22no—
89C2×C3×C8—42no—
90C2×C3×C8—91no—
91C2×C3×C8—92no—
92C2×C3×C8—92no—
93C2×C3×C8#15711yescharacteristic, complemented
94C3×C4⋊C4—94no—
95C3×C4⋊C4—182no—
96C2×C3⋊C8—62no—
97D3×C8—64no—
98D3×C8—64no—
99C22×C32—22no—
100C22×C32—42no—
101C22×C32C3×C4⋊C411yescharacteristic
102C2×C3⋊D3C3×C4⋊C411yescharacteristic
103C2×C3⋊D3C3×D812yes—
104C2×C3⋊D3C3×Q1612yes—
105C2×C3⋊D3C2×C3×D411yescharacteristic
106C2×C3×D3—62no—
107C2×C3×D3—64no—
108C2×C3×D3—64no—
109C2×C3×D3—62no—
110#157—44no—
111C32×C4—22no—
112C32×C4—22no—
113C32×C4—42no—
114C32×C4—42no—
115C32×C4C3×C4⋊C411yescharacteristic
116C32×C4C22×C3×C411yescharacteristic
117C3⋊Dic3C3×C4⋊C411yescharacteristic
118C3⋊Dic3C2×C3×Q811yescharacteristic
119C3×Dic3—62no—
120C3×Dic3—62no—
121C22×C8—91no—
122C2×C4⋊C4—92no—
123#107—94no—
124C33C2×#10711yescharacteristic, complemented, Sylow
125C23×C3—91no—
126C22×D3—62no—
127C2×C3×C4—22no—
128C2×C3×C4—42no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—184no—
135C2×C3×C4—364no—
136C2×C3×C4C2×#15711yescharacteristic
137C2×Dic3—62no—
138C4×D3—62no—
139C4×D3—62no—
140C4×D3—62no—
141C4×D3—62no—
142C3×C8—24no—
143C3×C8—44no—
144C3×C8—92no—
145C3×C8C2×#15712yescomplemented
146C3⋊C8—64no—
147C2×C32—24no—
148C2×C32—22no—
149C2×C32—44no—
150C2×C32—42no—
151C2×C32C3×#10712yescomplemented
152C2×C32C2×C3×C4⋊C411yescharacteristic
153C3⋊D3C3×#10712yescomplemented
154C3⋊D3C2×C3×D811yescharacteristic
155C3⋊D3C2×C3×Q1611yescharacteristic
156C3×D3—64no—
157C3×D3—62no—
158C3×D3—62no—
159C22×C4—91no—
160C22×C4—182no—
161C2×C8—91no—
162C2×C8—92no—
163C2×C8—92no—
164C2×C8C3×#15711yescharacteristic, complemented
165C4⋊C4—94no—
166C4⋊C4—182no—
167C22×C3—22no—
168C22×C3—42no—
169C22×C3—92no—
170C22×C3—92no—
171C22×C3—91no—
172C22×C3—91no—
173C22×C3C4⋊#15711yescharacteristic
174C2×D3—62no—
175C2×D3—62no—
176C2×D3—64no—
177C2×D3—64no—
178C3×C4—22no—
179C3×C4—22no—
180C3×C4—42no—
181C3×C4—42no—
182C3×C4—91no—
183C3×C4—91no—
184C3×C4—364no—
185C3×C4C4⋊#15711yescharacteristic
186C3×C4C22×#15711yescharacteristic
187Dic3—62no—
188Dic3—62no—
189C32—22no—
190C32—42no—
191C32C2×C3×#10711yescharacteristic, complemented
192C23—91no—
193C2×C4—91no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4—91no—
198C2×C4—184no—
199C2×C4—364no—
200C2×C4C2×C3×#15711yescharacteristic
201C8—92no—
202C8C2×C3×#15712yescomplemented
203C2×C3—24no—
204C2×C3—22no—
205C2×C3—44no—
206C2×C3—42no—
207C2×C3—91no—
208C2×C3—92no—
209C2×C3—91no—
210C2×C3#6376912yesdirect product factor
211C2×C3C2×C4⋊#15711yescharacteristic
212D3—64no—
213D3—62no—
214D3—62no—
215C22—92no—
216C22—92no—
217C22—91no—
218C22—91no—
219C22C3×C4⋊#15711yescharacteristic
220C4—91no—
221C4—91no—
222C4—364no—
223C4C3×C4⋊#15711yescharacteristic
224C4C22×C3×#15711yescharacteristic
225C3—22no—
226C3—42no—
227C3C2×#6376911yescharacteristic, direct product factor
228C2—91no—
229C2—92no—
230C2—91no—
231C2C3×#6376912yesdirect product factor
232C2C2×C3×C4⋊#15711yescharacteristic
subgroupG/H
centerC22×C3C4⋊#157
commutatorC32×C4C22×C3×C4
FrattiniC4C22×C3×#157
FittingC2×C33×C8C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C8#157
hypocenterC32C2×C3×#107
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63769C2direct (N × H)
4. C2×#63769C3direct (N × H)
9. C2×C33×C8C4semidirect
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
11. C3×C8×C3⋊D3C4semidirect
16. #63769C2×C3direct (N × H)
26. C33×C8C2×C4semidirect
27. C3×C3⋊(C3⋊C8)C2×C4semidirect
36. C2×C32×C8C3×C4semidirect
37. C2×C3⋊(C3⋊C8)C3×C4semidirect
38. C8×C3⋊D3C3×C4semidirect
76. C32×C8C2×C3×C4semidirect
77. C3⋊(C3⋊C8)C2×C3×C4semidirect
80. C2×C33#107semidirect
82. C3×C3⋊D3#107semidirect
93. C2×C3×C8#157semidirect
124. C33C2×#107semidirect
145. C3×C8C2×#157semidirect
151. C2×C32C3×#107semidirect
153. C3⋊D3C3×#107semidirect
164. C2×C8C3×#157semidirect
191. C32C2×C3×#107semidirect
202. C8C2×C3×#157semidirect
210. C2×C3#63769direct (N × H)
227. C3C2×#63769direct (N × H)
231. C2C3×#63769direct (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⋊#1572
C2×C3×C8×C3⋊D32
C3×#637692
C2×#637693
C2×C3×#1079