Finite Group Databasebeta

← C3×#10593778 #13173249

Subgroups — C3×#10593778 #13173249

Order 1728.

460 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 84 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×D4×#157C212yesmaximal, complemented
2C3×D8×C3⋊D3C211yesmaximal, characteristic
3C3×#63782C212yesmaximal, complemented
4C3×#63768C211yesmaximal, characteristic, complemented
5C3×#63767C211yesmaximal, characteristic, complemented
6#10593778C311yesmaximal, characteristic, direct product factor
7C22×C3×#157—22no—
8C3×D4×C3⋊D3C2212yes—
9C3×C22⋊#157—22no—
10C3×C4⋊#157C2212yes—
11C3×C4×#157C2211yescharacteristic
12C3×#3787C2211yescharacteristic
13C33×D8C411yescharacteristic, complemented
14C3×#3753C412yescomplemented
15C3×#3745C411yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17D4×#157C2×C312yescomplemented
18D8×C3⋊D3C2×C311yescharacteristic
19C3×D3×D8—62no—
20#63782C2×C312yescomplemented
21#63768C2×C311yescharacteristic, complemented
22#63767C2×C311yescharacteristic, complemented
23C22×C3×C3⋊D3—22no—
24C2×C3×#157—22no—
25C2×C3×#157—44no—
26C2×C3×#157D412yes—
27C33×D4C2×C412yescomplemented
28C3×#638—22no—
29C3×C4⋊(C3⋊D3)C2×C412yescomplemented
30C3×C4×C3⋊D3C2311yescharacteristic
31C3×#622—21no—
32C33×C8C2×C411yescharacteristic, complemented
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic, complemented
34C3×#442—91nomaximal
35C22×#157—22no—
36D4×C3⋊D3C22×C312yes—
37C3×D3×D4—64no—
38C22⋊#157—22no—
39C4⋊#157C22×C312yes—
40C4×#157C22×C311yescharacteristic
41#3787C22×C311yescharacteristic
42C32×D8—22no—
43C32×D8—42no—
44C32×D8C3×C411yescharacteristic, complemented
45#3753C3×C412yescomplemented
46#3745C3×C411yescharacteristic, complemented
47C8×C3⋊D3C22×C311yescharacteristic
48C3×#213—64no—
49C3×D24—62no—
50C3×D3×C8—62no—
51C22×C33—22no—
52C2×C3×C3⋊D3—22no—
53C2×C3×C3⋊D3—42no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3C2×D411yescharacteristic
56C3×#157—22no—
57C3×#157—42no—
58C33×C4C22×C411yescharacteristic
59C3×C3⋊Dic3#4111yescharacteristic
60C2×C3×D8—91no—
61C3×C4×D4—92no—
62D3×D8—62no—
63C3×#106—91no—
64C3×#102—92no—
65C3×#97—91no—
66C22×C3⋊D3—22no—
67C22×C3×D3—124no—
68C2×#157—22no—
69C2×#157—44no—
70C2×#157C3×D412yes—
71C32×D4—24no—
72C32×D4—44no—
73C32×D4C2×C3×C412yescomplemented
74#638—22no—
75C4⋊(C3⋊D3)C2×C3×C412yescomplemented
76C4×C3⋊D3C23×C311yescharacteristic
77C3×#67—124no—
78C3×D12—64no—
79C3×C4×D3—62no—
80#622—21no—
81C32×C8—22no—
82C32×C8—42no—
83C32×C8C2×C3×C411yescharacteristic, complemented
84C3⋊(C3⋊C8)C2×C3×C411yescharacteristic, complemented
85C3×C3⋊C8—62no—
86#442—91noSylow
87C2×C33—42no—
88C2×C33C4×D411yescharacteristic
89C3×C3⋊D3—42no—
90C3×C3⋊D3AGL(1,ℤ/8)12yes—
91C2×C3×D4—92no—
92C22×C3×C4—182no—
93D3×D4—64no—
94C3×D8—22no—
95C3×D8—42no—
96C3×D8—92no—
97C3×D8—91no—
98C3×D8#15711yescharacteristic, complemented
99C2×C3×C8—91no—
100C2×C3×C8—91no—
101C3×C4⋊C4—92no—
102C3×C22⋊C4—182no—
103C3×C42—91no—
104#213—64no—
105D24—62no—
106D3×C8—62no—
107C22×C32—22no—
108C22×C32—44no—
109C22×C32—84no—
110C2×C3⋊D3—22no—
111C2×C3⋊D3—42no—
112C2×C3⋊D3—42no—
113C2×C3⋊D3C2×C3×D411yescharacteristic
114C2×C3×D3—62no—
115C2×C3×D3—124no—
116C2×C3×D3—244no—
117C2×C3×D3—244no—
118#157—22no—
119#157—42no—
120C32×C4—22no—
121C32×C4—42no—
122C32×C4C22×C3×C411yescharacteristic
123C3⋊Dic3C3×#4111yescharacteristic
124C3×Dic3—62no—
125C2×D8—91no—
126C4×D4—92no—
127#106—91no—
128#102—92no—
129#97—91no—
130C33#44211yescharacteristic, complemented, Sylow
131C23×C3—182no—
132C22×D3—124no—
133C3×D4—24no—
134C3×D4—44no—
135C3×D4—92no—
136C3×D4—182no—
137C3×D4C2×#15712yescomplemented
138C2×C3×C4—91no—
139C2×C3×C4—92no—
140C2×C3×C4—182no—
141C2×C3×C4—364no—
142#67—124no—
143D12—64no—
144C4×D3—62no—
145C3×C8—22no—
146C3×C8—42no—
147C3×C8—91no—
148C3×C8—181no—
149C3×C8C2×#15711yescharacteristic, complemented
150C3⋊C8—62no—
151C2×C32—22no—
152C2×C32—42no—
153C2×C32—42no—
154C2×C32—84no—
155C2×C32—164no—
156C2×C32C3×C4×D411yescharacteristic
157C3⋊D3—42no—
158C3⋊D3C3×AGL(1,ℤ/8)12yes—
159C3×D3—64no—
160C3×D3—244no—
161C2×D4—92no—
162C22×C4—182no—
163D8—92no—
164D8—91no—
165D8C3×#15711yescharacteristic, complemented
166C2×C8—91no—
167C2×C8—91no—
168C4⋊C4—92no—
169C22⋊C4—182no—
170C42—91no—
171C22×C3—22no—
172C22×C3—44no—
173C22×C3—84no—
174C22×C3—91no—
175C22×C3—182no—
176C22×C3—362no—
177C22×C3—362no—
178C2×D3—62no—
179C2×D3—124no—
180C2×D3—244no—
181C2×D3—244no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—91no—
185C3×C4—182no—
186C3×C4—362no—
187C3×C4C22×#15711yescharacteristic
188Dic3—62no—
189C32—22no—
190C32—42no—
191C32C3×#44211yescharacteristic, complemented
192C23—182no—
193D4—92no—
194D4—182no—
195D4C2×C3×#15712yescomplemented
196C2×C4—91no—
197C2×C4—92no—
198C2×C4—182no—
199C2×C4—364no—
200C8—91no—
201C8—181no—
202C8C2×C3×#15711yescharacteristic, complemented
203C2×C3—22no—
204C2×C3—42no—
205C2×C3—42no—
206C2×C3—84no—
207C2×C3—92no—
208C2×C3—164no—
209C2×C3—362no—
210C2×C3D4×#15711yescharacteristic
211D3—64no—
212D3—244no—
213C22—22no—
214C22—91no—
215C22—182no—
216C22—362no—
217C22—362no—
218C4—91no—
219C4—182no—
220C4—362no—
221C4C22×C3×#15711yescharacteristic
222C3—22no—
223C3—42no—
224C3#1059377811yescharacteristic, direct product factor
225C2—42no—
226C2—92no—
227C2—362no—
228C2C3×D4×#15711yescharacteristic
subgroupG/H
centerC2×C3D4×#157
commutatorC32×C4C22×C3×C4
FrattiniC4C22×C3×#157
FittingC33×D8C4
socleC2×C33C4×D4
prime radicalC3×D8×C3⋊D3C2
cosocleC4×C3⋊D3C23×C3
hypercenterC3×D8#157
hypocenterC32C3×#442
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8

Complements

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

NH ≅ G/Ntype
1. C3×D4×#157C2semidirect
3. C3×#63782C2semidirect
4. C3×#63768C2semidirect
5. C3×#63767C2semidirect
6. #10593778C3direct (N × H)
13. C33×D8C4semidirect
14. C3×#3753C4semidirect
15. C3×#3745C4semidirect
17. D4×#157C2×C3semidirect
20. #63782C2×C3semidirect
21. #63768C2×C3semidirect
22. #63767C2×C3semidirect
27. C33×D4C2×C4semidirect
29. C3×C4⋊(C3⋊D3)C2×C4semidirect
32. C33×C8C2×C4semidirect
33. C3×C3⋊(C3⋊C8)C2×C4semidirect
44. C32×D8C3×C4semidirect
45. #3753C3×C4semidirect
46. #3745C3×C4semidirect
73. C32×D4C2×C3×C4semidirect
75. C4⋊(C3⋊D3)C2×C3×C4semidirect
83. C32×C8C2×C3×C4semidirect
84. C3⋊(C3⋊C8)C2×C3×C4semidirect
98. C3×D8#157semidirect
130. C33#442semidirect
137. C3×D4C2×#157semidirect
149. C3×C8C2×#157semidirect
165. D8C3×#157semidirect
191. C32C3×#442semidirect
195. D4C2×C3×#157semidirect
202. C8C2×C3×#157semidirect
224. C3#10593778direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×#1572
C3×D8×C3⋊D32
C3×#637822
C3×#637682
C3×#637672
#105937783
C3×#4429