Finite Group Databasebeta

← C2×C3×#63768 #13173237

Subgroups — C2×C3×#63768 #13173237

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×#63768C214yesmaximal, direct product factor
4C2×#63768C311yesmaximal, 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#63768C2×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×#106—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⋊D3SD1614yes—
46C2×C3×C3⋊D3C2×D411yescharacteristic
47C3×#157—44no—
48C33×C4C4⋊C411yescharacteristic
49C33×C4C22×C411yescharacteristic
50C3×C3⋊Dic3C4⋊C411yescharacteristic
51C3×C3⋊Dic3C2×Q811yescharacteristic
52C22×C3×C8—91no—
53C2×C3×C4⋊C4—92no—
54C2×D3×C8—62no—
55C3×#106—94no—
56C22×C3⋊D3C3×D411yescharacteristic
57C22×C3×D3—62no—
58C2×#157—24no—
59C2×#157—44no—
60C2×C32×C4—22no—
61C2×C32×C4—42no—
62C2×C32×C4C2×C3×C411yescharacteristic
63C2×C3⋊Dic3C3×Q811yescharacteristic
64C4×C3⋊D3C2×C3×C411yescharacteristic
65C4×C3⋊D3C3×D411yescharacteristic
66C4×C3⋊D3C3×Q811yescharacteristic
67C4×C3⋊D3C23×C311yescharacteristic
68C2×C3×Dic3—62no—
69C3×C4×D3—62no—
70C3×C4×D3—62no—
71C3×C4×D3—62no—
72C3×C4×D3—62no—
73C32×C8—24no—
74C32×C8—44no—
75C32×C8C2×C3×C412yescomplemented
76C3⋊(C3⋊C8)C2×C3×C412yescomplemented
77C3×C3⋊C8—64no—
78C2×#106—91noSylow
79C2×C33#10612yescomplemented
80C2×C33C2×C4⋊C411yescharacteristic
81C3×C3⋊D3#10612yescomplemented
82C3×C3⋊D3C2×SD1612yes—
83C22×C3×C4—91no—
84C22×C3×C4—182no—
85C2×C4×D3—62no—
86C2×C3×C8—22no—
87C2×C3×C8—42no—
88C2×C3×C8—91no—
89C2×C3×C8—92no—
90C2×C3×C8—92no—
91C2×C3×C8#15711yescharacteristic, complemented
92C3×C4⋊C4—94no—
93C3×C4⋊C4—182no—
94C2×C3⋊C8—62no—
95D3×C8—64no—
96D3×C8—64no—
97C22×C32—22no—
98C22×C32—42no—
99C22×C32C3×C4⋊C411yescharacteristic
100C2×C3⋊D3C3×C4⋊C411yescharacteristic
101C2×C3⋊D3C3×SD1614yes—
102C2×C3⋊D3C2×C3×D411yescharacteristic
103C2×C3×D3—62no—
104C2×C3×D3—68no—
105C2×C3×D3—62no—
106#157—44no—
107C32×C4—22no—
108C32×C4—22no—
109C32×C4—42no—
110C32×C4—42no—
111C32×C4C3×C4⋊C411yescharacteristic
112C32×C4C22×C3×C411yescharacteristic
113C3⋊Dic3C3×C4⋊C411yescharacteristic
114C3⋊Dic3C2×C3×Q811yescharacteristic
115C3×Dic3—62no—
116C3×Dic3—62no—
117C22×C8—91no—
118C2×C4⋊C4—92no—
119#106—94no—
120C33C2×#10611yescharacteristic, complemented, Sylow
121C23×C3—91no—
122C22×D3—62no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4—91no—
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—184no—
131C2×C3×C4—364no—
132C2×C3×C4C2×#15711yescharacteristic
133C2×Dic3—62no—
134C4×D3—62no—
135C4×D3—62no—
136C4×D3—62no—
137C4×D3—62no—
138C3×C8—24no—
139C3×C8—44no—
140C3×C8—92no—
141C3×C8C2×#15712yescomplemented
142C3⋊C8—64no—
143C2×C32—24no—
144C2×C32—22no—
145C2×C32—44no—
146C2×C32—42no—
147C2×C32C3×#10612yescomplemented
148C2×C32C2×C3×C4⋊C411yescharacteristic
149C3⋊D3C3×#10612yescomplemented
150C3⋊D3C2×C3×SD1612yes—
151C3×D3—64no—
152C3×D3—64no—
153C22×C4—91no—
154C22×C4—182no—
155C2×C8—91no—
156C2×C8—92no—
157C2×C8—92no—
158C2×C8C3×#15711yescharacteristic, complemented
159C4⋊C4—94no—
160C4⋊C4—182no—
161C22×C3—22no—
162C22×C3—42no—
163C22×C3—94no—
164C22×C3—91no—
165C22×C3—91no—
166C22×C3C4⋊#15711yescharacteristic
167C2×D3—62no—
168C2×D3—62no—
169C2×D3—68no—
170C3×C4—22no—
171C3×C4—22no—
172C3×C4—42no—
173C3×C4—42no—
174C3×C4—91no—
175C3×C4—91no—
176C3×C4—364no—
177C3×C4C4⋊#15711yescharacteristic
178C3×C4C22×#15711yescharacteristic
179Dic3—62no—
180Dic3—62no—
181C32—22no—
182C32—42no—
183C32C2×C3×#10611yescharacteristic, complemented
184C23—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—91no—
190C2×C4—184no—
191C2×C4—364no—
192C2×C4C2×C3×#15711yescharacteristic
193C8—92no—
194C8C2×C3×#15712yescomplemented
195C2×C3—24no—
196C2×C3—22no—
197C2×C3—44no—
198C2×C3—42no—
199C2×C3—92no—
200C2×C3—92no—
201C2×C3#6376812yesdirect product factor
202C2×C3C2×C4⋊#15711yescharacteristic
203D3—64no—
204D3—64no—
205C22—94no—
206C22—91no—
207C22—91no—
208C22C3×C4⋊#15711yescharacteristic
209C4—91no—
210C4—91no—
211C4—364no—
212C4C3×C4⋊#15711yescharacteristic
213C4C22×C3×#15711yescharacteristic
214C3—22no—
215C3—42no—
216C3C2×#6376811yescharacteristic, direct product factor
217C2—92no—
218C2—92no—
219C2C3×#6376812yesdirect product factor
220C2C2×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×#106
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×#63768C2direct (N × H)
4. C2×#63768C3direct (N × H)
9. C2×C33×C8C4semidirect
10. C2×C3×C3⋊(C3⋊C8)C4semidirect
11. C3×C8×C3⋊D3C4semidirect
16. #63768C2×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
75. C32×C8C2×C3×C4semidirect
76. C3⋊(C3⋊C8)C2×C3×C4semidirect
79. C2×C33#106semidirect
81. C3×C3⋊D3#106semidirect
91. C2×C3×C8#157semidirect
120. C33C2×#106semidirect
141. C3×C8C2×#157semidirect
147. C2×C32C3×#106semidirect
149. C3⋊D3C3×#106semidirect
158. C2×C8C3×#157semidirect
183. C32C2×C3×#106semidirect
194. C8C2×C3×#157semidirect
201. C2×C3#63768direct (N × H)
216. C3C2×#63768direct (N × H)
219. C2C3×#63768direct (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×#637682
C2×#637683
C2×C3×#1069