Finite Group Databasebeta

← C2×C3×#63767 #13173235

Subgroups — C2×C3×#63767 #13173235

Order 1728.

488 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 152 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×#157C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic
3C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
4C3×#63767C214yesmaximal, direct product factor
5C2×#63767C311yesmaximal, characteristic, direct product factor
6C22×C3×#157C412yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#622C412yescomplemented
9C3×C4×#157C412yes—
10C3×C4×#157C2212yes—
11C3×#3787C412yescomplemented
12C3×#3787C2212yes—
13C2×C33×C8C411yescharacteristic, complemented
14C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
15C3×C8×C3⋊D3C412yescomplemented
16C3×C8×C3⋊D3C2212yes—
17C2×C4×#157C2×C311yescharacteristic
18C2×#3787C2×C311yescharacteristic
19C2×C8×C3⋊D3C2×C311yescharacteristic
20C2×C3×D3×C8—62no—
21#63767C2×C314yesdirect product factor
22C22×C3×C3⋊D3C2×C411yescharacteristic
23C2×C3×#157—24no—
24C2×C3×#157C2×C414yes—
25C2×C33×C4C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C3×C4×C3⋊D3C2×C411yescharacteristic
28C3×C4×C3⋊D3C2×C411yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2311yescharacteristic
31C3×#622C2×C414yescomplemented
32C33×C8C2×C412yescomplemented
33C3×C3⋊(C3⋊C8)C2×C412yescomplemented
34C2×C3×#97—91nomaximal
35C22×#157C3×C412yes—
36C2×C4×C3⋊D3C22×C311yescharacteristic
37C2×C3×C4×D3—62no—
38C2×#622C3×C412yescomplemented
39C4×#157C3×C412yes—
40C4×#157C22×C312yes—
41#3787C3×C412yescomplemented
42#3787C22×C312yes—
43C2×C32×C8—22no—
44C2×C32×C8—42no—
45C2×C32×C8C3×C411yescharacteristic, complemented
46C2×C3⋊(C3⋊C8)C3×C411yescharacteristic, complemented
47C8×C3⋊D3C3×C412yescomplemented
48C8×C3⋊D3C22×C312yes—
49C2×C3×C3⋊C8—62no—
50C3×D3×C8—64no—
51C3×D3×C8—64no—
52C22×C33C4211yescharacteristic
53C2×C3×C3⋊D3C4211yescharacteristic
54C2×C3×C3⋊D3M1614yes—
55C2×C3×C3⋊D3C22×C411yescharacteristic
56C3×#157—24no—
57C33×C4C4211yescharacteristic
58C33×C4C22×C411yescharacteristic
59C3×C3⋊Dic3C4211yescharacteristic
60C3×C3⋊Dic3C22×C411yescharacteristic
61C22×C3×C8—91no—
62C22×C3×C8—91no—
63C2×C3×C42—91no—
64C2×D3×C8—62no—
65C3×#97—94no—
66C22×C3⋊D3C2×C3×C411yescharacteristic
67C22×C3×D3—62no—
68C2×#157—24no—
69C2×#157C2×C3×C414yes—
70C2×C32×C4—22no—
71C2×C32×C4—42no—
72C2×C32×C4C2×C3×C411yescharacteristic
73C2×C3⋊Dic3C2×C3×C411yescharacteristic
74C4×C3⋊D3C2×C3×C411yescharacteristic
75C4×C3⋊D3C2×C3×C411yescharacteristic
76C4×C3⋊D3C2×C3×C411yescharacteristic
77C4×C3⋊D3C23×C311yescharacteristic
78C2×C3×Dic3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83#622C2×C3×C414yescomplemented
84C32×C8—24no—
85C32×C8—44no—
86C32×C8C2×C3×C412yescomplemented
87C3⋊(C3⋊C8)C2×C3×C412yescomplemented
88C3×C3⋊C8—64no—
89C2×#97—91noSylow
90C2×C33#9712yescomplemented
91C2×C33C2×C4211yescharacteristic
92C3×C3⋊D3#9712yescomplemented
93C3×C3⋊D3C2×M1612yes—
94C22×C3×C4—91no—
95C22×C3×C4—92no—
96C2×C4×D3—62no—
97C2×C3×C8—22no—
98C2×C3×C8—42no—
99C2×C3×C8—92no—
100C2×C3×C8—92no—
101C2×C3×C8—92no—
102C2×C3×C8—91no—
103C2×C3×C8—92no—
104C2×C3×C8—92no—
105C2×C3×C8#15711yescharacteristic, complemented
106C3×C42—92no—
107C3×C42—92no—
108C2×C3⋊C8—62no—
109D3×C8—64no—
110D3×C8—64no—
111C22×C32—22no—
112C22×C32—42no—
113C22×C32C3×C4211yescharacteristic
114C2×C3⋊D3C3×C4211yescharacteristic
115C2×C3⋊D3C3×M1614yes—
116C2×C3⋊D3C22×C3×C411yescharacteristic
117C2×C3×D3—62no—
118C2×C3×D3—68no—
119C2×C3×D3—62no—
120#157—24no—
121C32×C4—22no—
122C32×C4—22no—
123C32×C4—42no—
124C32×C4—42no—
125C32×C4C3×C4211yescharacteristic
126C32×C4C22×C3×C411yescharacteristic
127C3⋊Dic3C3×C4211yescharacteristic
128C3⋊Dic3C22×C3×C411yescharacteristic
129C3×Dic3—62no—
130C3×Dic3—62no—
131C22×C8—91no—
132C22×C8—91no—
133C2×C42—91no—
134#97—94no—
135C33C2×#9711yescharacteristic, complemented, Sylow
136C23×C3—91no—
137C22×D3—62no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—94no—
146C2×C3×C4—184no—
147C2×C3×C4C2×#15711yescharacteristic
148C2×Dic3—62no—
149C4×D3—62no—
150C4×D3—62no—
151C4×D3—62no—
152C4×D3—62no—
153C3×C8—24no—
154C3×C8—44no—
155C3×C8—94no—
156C3×C8—92no—
157C3×C8C2×#15712yescomplemented
158C3⋊C8—64no—
159C2×C32—24no—
160C2×C32—22no—
161C2×C32—44no—
162C2×C32—42no—
163C2×C32C3×#9712yescomplemented
164C2×C32C2×C3×C4211yescharacteristic
165C3⋊D3C3×#9712yescomplemented
166C3⋊D3C2×C3×M1612yes—
167C3×D3—64no—
168C3×D3—64no—
169C22×C4—91no—
170C22×C4—92no—
171C2×C8—92no—
172C2×C8—92no—
173C2×C8—92no—
174C2×C8—91no—
175C2×C8—92no—
176C2×C8—92no—
177C2×C8C3×#15711yescharacteristic, complemented
178C42—92no—
179C42—92no—
180C22×C3—22no—
181C22×C3—42no—
182C22×C3—94no—
183C22×C3—91no—
184C22×C3—91no—
185C22×C3C4×#15711yescharacteristic
186C2×D3—62no—
187C2×D3—62no—
188C2×D3—68no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—42no—
192C3×C4—42no—
193C3×C4—91no—
194C3×C4—91no—
195C3×C4—184no—
196C3×C4C4×#15711yescharacteristic
197C3×C4C22×#15711yescharacteristic
198Dic3—62no—
199Dic3—62no—
200C32—22no—
201C32—42no—
202C32C2×C3×#9711yescharacteristic, complemented
203C23—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—94no—
210C2×C4—184no—
211C2×C4C2×C3×#15711yescharacteristic
212C8—94no—
213C8—92no—
214C8C2×C3×#15712yescomplemented
215C2×C3—24no—
216C2×C3—22no—
217C2×C3—44no—
218C2×C3—42no—
219C2×C3—92no—
220C2×C3—92no—
221C2×C3#6376712yesdirect product factor
222C2×C3C2×C4×#15711yescharacteristic
223D3—64no—
224D3—64no—
225C22—94no—
226C22—91no—
227C22—91no—
228C22C3×C4×#15711yescharacteristic
229C4—91no—
230C4—91no—
231C4—184no—
232C4C3×C4×#15711yescharacteristic
233C4C22×C3×#15711yescharacteristic
234C3—22no—
235C3—42no—
236C3C2×#6376711yescharacteristic, direct product factor
237C2—92no—
238C2—92no—
239C2C3×#6376712yesdirect product factor
240C2C2×C3×C4×#15711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×C32C2×C3×C42
FrattiniC4C22×C3×#157
FittingC2×C33×C8C4
socleC22×C33C42
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C8#157
hypocenterC32C2×C3×#97
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63767C2direct (N × H)
5. C2×#63767C3direct (N × H)
8. C2×C3×#622C4semidirect
11. C3×#3787C4semidirect
13. C2×C33×C8C4semidirect
14. C2×C3×C3⋊(C3⋊C8)C4semidirect
15. C3×C8×C3⋊D3C4semidirect
21. #63767C2×C3direct (N × H)
31. C3×#622C2×C4semidirect
32. C33×C8C2×C4semidirect
33. C3×C3⋊(C3⋊C8)C2×C4semidirect
38. C2×#622C3×C4semidirect
41. #3787C3×C4semidirect
45. C2×C32×C8C3×C4semidirect
46. C2×C3⋊(C3⋊C8)C3×C4semidirect
47. C8×C3⋊D3C3×C4semidirect
83. #622C2×C3×C4semidirect
86. C32×C8C2×C3×C4semidirect
87. C3⋊(C3⋊C8)C2×C3×C4semidirect
90. C2×C33#97semidirect
92. C3×C3⋊D3#97semidirect
105. C2×C3×C8#157semidirect
135. C33C2×#97semidirect
157. C3×C8C2×#157semidirect
163. C2×C32C3×#97semidirect
165. C3⋊D3C3×#97semidirect
177. C2×C8C3×#157semidirect
202. C32C2×C3×#97semidirect
214. C8C2×C3×#157semidirect
221. C2×C3#63767direct (N × H)
236. C3C2×#63767direct (N × H)
239. C2C3×#63767direct (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×#37872
C2×C3×C8×C3⋊D32
C3×#637672
C2×#637673
C2×C3×#979