Finite Group Databasebeta

← C2×C3×#64228 #13181025

Subgroups — C2×C3×#64228 #13181025

Order 1728.

904 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 194 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×#3797C212yesmaximal
2C2×C3×#3787C211yesmaximal, characteristic
3C2×C3×#3776C214yesmaximal, complemented
4C3×#64228C218yesmaximal, direct product factor
5C2×#64228C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C32×Dic6—22no—
8C2×C3×#627C2214yes—
9C2×C3×#624—22no—
10C3×#3797—24no—
11C3×#3797C2218yes—
12C2×C3×#622C2212yes—
13C3×#3787C2214yes—
14C3×#3776C22116yescomplemented
15C2×C3×D3×Q8—62no—
16C2×#3797C2×C312yes—
17C2×#3787C2×C311yescharacteristic
18C2×#3776C2×C314yescomplemented
19#64228C2×C318yesdirect product factor
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3C2311yescharacteristic
23C3×C4×C3⋊D3D412yes—
24C3×C4×C3⋊D3C2312yes—
25C2×C32×Dic3—24no—
26C32×Dic6—28no—
27C3×#627—24no—
28C3×#627C2318yes—
29C3×#624—28no—
30C3×#622C2314yes—
31C22×C3×Q16—91nomaximal
32C2×C32×Q8—62no—
33C2×C4×C3⋊D3C22×C311yescharacteristic
34C3×D3×Q8—68no—
35C3×D3×Q8—68no—
36C2×C3×C4×D3—64no—
37C2×C3×C4×D3—62no—
38C2×C3×Dic6—22no—
39C2×C3×Dic6—22no—
40C2×C3×Dic6—42no—
41C2×C3×Dic6—64no—
42C2×#627C22×C314yes—
43C2×#624—22no—
44#3797—24no—
45#3797C22×C318yes—
46C2×#622C22×C312yes—
47#3787C22×C314yes—
48#3776C22×C3116yescomplemented
49C22×C33C2×D411yescharacteristic
50C2×C3×C3⋊D3Q1614yescomplemented
51C2×C3×C3⋊D3C2×D412yes—
52C33×C4C2×D412yes—
53C3×C3⋊Dic3C2×D411yescharacteristic
54C3×C3⋊Dic3C2411yescharacteristic
55C32×Dic3—28no—
56C22×C3×Q8—92no—
57C2×D3×Q8—62no—
58C2×C3×Q16—94no—
59C2×C3×Q16—98no—
60C22×C3×C8—91no—
61C22×C3⋊D3C3×D411yescharacteristic
62C22×C3×D3—62no—
63C32×Q8—68no—
64C2×C32×C4—22no—
65C2×C32×C4—42no—
66C2×C32×C4—64no—
67C2×C32×C4C3×D411yescharacteristic
68C2×C3⋊Dic3C23×C311yescharacteristic
69C4×C3⋊D3C3×D412yes—
70C4×C3⋊D3C23×C312yes—
71C2×C3×Dic3—24no—
72C2×C3×Dic3—24no—
73C2×C3×Dic3—44no—
74C2×C3×Dic3—62no—
75C3×C4×D3—64no—
76C3×C4×D3—64no—
77C3×C4×D3—616no—
78C3×Dic6—28no—
79C3×Dic6—28no—
80C3×Dic6—48no—
81C3×Dic6—68no—
82C3×Dic6—68no—
83#627—24no—
84#627C23×C318yes—
85#624—28no—
86#622C23×C314yes—
87C22×Q16—91noSylow
88C2×C33C2×Q1612yescomplemented
89C2×C33C22×D411yescharacteristic
90C3×C3⋊D3C2×Q1614yescomplemented
91C2×C3×Q8—62no—
92C2×C3×Q8—62no—
93C2×C3×Q8—94no—
94C2×C3×Q8—98no—
95C2×C3×Q8—122no—
96C2×C3×Q8—184no—
97C22×C3×C4—91no—
98C22×C3×C4—182no—
99D3×Q8—68no—
100D3×Q8—68no—
101C2×C4×D3—62no—
102C2×C4×D3—64no—
103C2×Dic6—22no—
104C2×Dic6—64no—
105C3×Q16—916no—
106C2×C3×C8—94no—
107C2×C3×C8—92no—
108C22×C32—22no—
109C22×C32—42no—
110C22×C32C2×C3×D411yescharacteristic
111C2×C3⋊D3C3×Q1614yescomplemented
112C2×C3⋊D3C2×C3×D412yes—
113C2×C3×D3—68no—
114C2×C3×D3—64no—
115C32×C4—24no—
116C32×C4—44no—
117C32×C4—68no—
118C32×C4C2×C3×D412yes—
119C3⋊Dic3C2×C3×D411yescharacteristic
120C3⋊Dic3C24×C311yescharacteristic
121C3×Dic3—28no—
122C3×Dic3—28no—
123C3×Dic3—48no—
124C3×Dic3—62no—
125C3×Dic3—62no—
126C22×Q8—92no—
127C2×Q16—98no—
128C2×Q16—94no—
129C22×C8—91no—
130C33C22×Q1611yescharacteristic, complemented, Sylow
131C23×C3—91no—
132C22×D3—62no—
133C3×Q8—68no—
134C3×Q8—68no—
135C3×Q8—98no—
136C3×Q8—128no—
137C3×Q8—184no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4—64no—
141C2×C3×C4—64no—
142C2×C3×C4—91no—
143C2×C3×C4—92no—
144C2×C3×C4—92no—
145C2×C3×C4—124no—
146C2×C3×C4—188no—
147C2×C3×C4GO+(4,𝔽2)11yescharacteristic
148C2×Dic3—24no—
149C2×Dic3—62no—
150C4×D3—64no—
151C4×D3—64no—
152C4×D3—616no—
153Dic6—28no—
154Dic6—68no—
155Dic6—68no—
156C3×C8—94no—
157C2×C32—22no—
158C2×C32—24no—
159C2×C32—44no—
160C2×C32—42no—
161C2×C32C2×C3×Q1612yescomplemented
162C2×C32C22×C3×D411yescharacteristic
163C3⋊D3C2×C3×Q1614yescomplemented
164C3×D3—68no—
165C2×Q8—62no—
166C2×Q8—94no—
167C2×Q8—98no—
168C2×Q8—184no—
169C22×C4—91no—
170C22×C4—182no—
171Q16—916no—
172C2×C8—94no—
173C2×C8—92no—
174C22×C3—22no—
175C22×C3—42no—
176C22×C3—92no—
177C22×C3—94no—
178C22×C3C2×GO+(4,𝔽2)11yescharacteristic
179C2×D3—64no—
180C2×D3—68no—
181C3×C4—24no—
182C3×C4—44no—
183C3×C4—68no—
184C3×C4—68no—
185C3×C4—91no—
186C3×C4—91no—
187C3×C4—128no—
188C3×C4C2×GO+(4,𝔽2)12yes—
189Dic3—28no—
190Dic3—62no—
191Dic3—62no—
192C32—22no—
193C32—42no—
194C32C22×C3×Q1611yescharacteristic, complemented
195C23—91no—
196Q8—68no—
197Q8—98no—
198Q8—184no—
199C2×C4—64no—
200C2×C4—91no—
201C2×C4—92no—
202C2×C4—92no—
203C2×C4—188no—
204C2×C4C3×GO+(4,𝔽2)11yescharacteristic
205C8—94no—
206C2×C3—22no—
207C2×C3—24no—
208C2×C3—42no—
209C2×C3—44no—
210C2×C3—94no—
211C2×C3#6422812yesdirect product factor
212C2×C3C22×GO+(4,𝔽2)11yescharacteristic
213D3—68no—
214C22—92no—
215C22—94no—
216C22C2×C3×GO+(4,𝔽2)11yescharacteristic
217C4—68no—
218C4—91no—
219C4—91no—
220C4C2×C3×GO+(4,𝔽2)12yes—
221C3—22no—
222C3—42no—
223C3C2×#6422811yescharacteristic, direct product factor
224C2—94no—
225C2C3×#6422812yesdirect product factor
226C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C22×C3×Q16
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3776C2semidirect
4. C3×#64228C2direct (N × H)
5. C2×#64228C3direct (N × H)
14. C3×#3776C22semidirect
18. C2×#3776C2×C3semidirect
19. #64228C2×C3direct (N × H)
48. #3776C22×C3semidirect
50. C2×C3×C3⋊D3Q16semidirect
88. C2×C33C2×Q16semidirect
90. C3×C3⋊D3C2×Q16semidirect
111. C2×C3⋊D3C3×Q16semidirect
130. C33C22×Q16semidirect
161. C2×C32C2×C3×Q16semidirect
163. C3⋊D3C2×C3×Q16semidirect
194. C32C22×C3×Q16semidirect
211. C2×C3#64228direct (N × H)
223. C3C2×#64228direct (N × H)
225. C2C3×#64228direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×C3×#37872
C2×C3×#37762
C3×#642282
C2×#642283
C22×C3×Q169