Finite Group Databasebeta

← C2×C3×#63733 #13166749

Subgroups — C2×C3×#63733 #13166749

Order 1728.

392 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3787C211yesmaximal, characteristic
2C2×C3×#3712C212yesmaximal
3C3×#63733C214yesmaximal, direct product factor
4C2×#63733C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6C2×C3×#622C412yes—
7C3×#3787C412yes—
8C3×#3787C2212yes—
9C2×C32×C3⋊C8—22no—
10C3×#3712—22no—
11C3×#3712C2214yes—
12C2×#3787C2×C311yescharacteristic
13C2×C3×ΓL(1,𝔽25)—62no—
14C2×#3712C2×C312yes—
15#63733C2×C314yesdirect product factor
16C22×C3×C3⋊D3Q811yescharacteristic
17C2×C33×C4D411yescharacteristic
18C2×C3×C3⋊Dic3C2×C411yescharacteristic
19C3×C4×C3⋊D3C2×C411yescharacteristic
20C3×C4×C3⋊D3D411yescharacteristic
21C3×C4×C3⋊D3Q811yescharacteristic
22C3×C4×C3⋊D3C2311yescharacteristic
23C3×#622C2×C414yes—
24C32×C3⋊C8—24no—
25C2×C3×#108—91nomaximal
26C2×C4×C3⋊D3C22×C311yescharacteristic
27C2×C3×C4×D3—62no—
28C2×#622C3×C412yes—
29#3787C3×C412yes—
30#3787C22×C312yes—
31C2×C32×C8—62no—
32C2×C3×C3⋊C8—22no—
33C2×C3×C3⋊C8—22no—
34C2×C3×C3⋊C8—42no—
35C3×ΓL(1,𝔽25)—64no—
36C3×ΓL(1,𝔽25)—64no—
37#3712—22no—
38#3712C22×C314yes—
39C22×C33C4⋊C411yescharacteristic
40C2×C3×C3⋊D3—22no—
41C2×C3×C3⋊D3C4⋊C411yescharacteristic
42C2×C3×C3⋊D3C2×Q811yescharacteristic
43C33×C4C4⋊C411yescharacteristic
44C33×C4C2×D411yescharacteristic
45C3×C3⋊Dic3C4⋊C411yescharacteristic
46C3×C3⋊Dic3C22×C411yescharacteristic
47C2×C3×M16—92no—
48C22×C3×C8—91no—
49C2×ΓL(1,𝔽25)—62no—
50C3×#108—94no—
51C22×C3⋊D3C3×Q811yescharacteristic
52C22×C3×D3—62no—
53C2×C32×C4—22no—
54C2×C32×C4—42no—
55C2×C32×C4C3×D411yescharacteristic
56C2×C3⋊Dic3C2×C3×C411yescharacteristic
57C4×C3⋊D3C2×C3×C411yescharacteristic
58C4×C3⋊D3C3×D411yescharacteristic
59C4×C3⋊D3C3×Q811yescharacteristic
60C4×C3⋊D3C23×C311yescharacteristic
61C2×C3×Dic3—62no—
62C3×C4×D3—62no—
63C3×C4×D3—62no—
64C3×C4×D3—62no—
65C3×C4×D3—62no—
66#622C2×C3×C414yes—
67C32×C8—64no—
68C3×C3⋊C8—24no—
69C3×C3⋊C8—24no—
70C3×C3⋊C8—44no—
71C2×#108—91noSylow
72C2×C33#10812yescomplemented
73C2×C33C2×C4⋊C411yescharacteristic
74C3×C3⋊D3—21no—
75C3×C3⋊D3—21no—
76C22×C3×C4—91no—
77C2×C4×D3—62no—
78C3×M16—94no—
79C3×M16—182no—
80C2×C3×C8—62no—
81C2×C3×C8—62no—
82C2×C3×C8—92no—
83C2×C3×C8—92no—
84C2×C3×C8—92no—
85C2×C3×C8—122no—
86C2×C3⋊C8—22no—
87ΓL(1,𝔽25)—64no—
88ΓL(1,𝔽25)—64no—
89C22×C32—22no—
90C22×C32—42no—
91C22×C32C3×C4⋊C411yescharacteristic
92C2×C3⋊D3—22no—
93C2×C3⋊D3C3×C4⋊C411yescharacteristic
94C2×C3⋊D3C2×C3×Q811yescharacteristic
95C2×C3×D3—62no—
96C2×C3×D3—62no—
97C2×C3×D3—124no—
98C32×C4—22no—
99C32×C4—22no—
100C32×C4—42no—
101C32×C4—42no—
102C32×C4C3×C4⋊C411yescharacteristic
103C32×C4C2×C3×D411yescharacteristic
104C3⋊Dic3C3×C4⋊C411yescharacteristic
105C3⋊Dic3C22×C3×C411yescharacteristic
106C3×Dic3—62no—
107C3×Dic3—62no—
108C2×M16—92no—
109C22×C8—91no—
110#108—94no—
111C33C2×#10811yescharacteristic, complemented, Sylow
112C23×C3—91no—
113C22×D3—62no—
114C2×C3×C4—22no—
115C2×C3×C4—42no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—91no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4GO+(4,𝔽2)11yescharacteristic
122C2×Dic3—62no—
123C4×D3—62no—
124C4×D3—62no—
125C4×D3—62no—
126C4×D3—62no—
127C3×C8—64no—
128C3×C8—64no—
129C3×C8—94no—
130C3×C8—124no—
131C3⋊C8—24no—
132C2×C32—22no—
133C2×C32—24no—
134C2×C32—44no—
135C2×C32—42no—
136C2×C32C3×#10812yescomplemented
137C2×C32C2×C3×C4⋊C411yescharacteristic
138C3⋊D3—21no—
139C3⋊D3—21no—
140C3×D3—122no—
141C3×D3—122no—
142C22×C4—91no—
143M16—94no—
144M16—182no—
145C2×C8—62no—
146C2×C8—92no—
147C2×C8—92no—
148C2×C8—92no—
149C22×C3—22no—
150C22×C3—42no—
151C22×C3—91no—
152C22×C3—91no—
153C22×C3—182no—
154C22×C3#377311yescharacteristic
155C2×D3—62no—
156C2×D3—62no—
157C2×D3—124no—
158C3×C4—22no—
159C3×C4—22no—
160C3×C4—42no—
161C3×C4—42no—
162C3×C4—91no—
163C3×C4—91no—
164C3×C4#377311yescharacteristic
165C3×C4C2×GO+(4,𝔽2)11yescharacteristic
166Dic3—62no—
167Dic3—62no—
168C32—22no—
169C32—42no—
170C32C2×C3×#10811yescharacteristic, complemented
171C23—91no—
172C2×C4—91no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4C3×GO+(4,𝔽2)11yescharacteristic
178C8—64no—
179C8—94no—
180C2×C3—22no—
181C2×C3—24no—
182C2×C3—42no—
183C2×C3—44no—
184C2×C3—181no—
185C2×C3—181no—
186C2×C3#6373312yesdirect product factor
187C2×C3C2×#377311yescharacteristic
188D3—122no—
189D3—122no—
190C22—91no—
191C22—91no—
192C22—182no—
193C22C3×#377311yescharacteristic
194C4—91no—
195C4—91no—
196C4C3×#377311yescharacteristic
197C4C2×C3×GO+(4,𝔽2)11yescharacteristic
198C3—22no—
199C3—42no—
200C3C2×#6373311yescharacteristic, direct product factor
201C2—181no—
202C2—181no—
203C2C3×#6373312yesdirect product factor
204C2C2×C3×#377311yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC4×C3⋊D3C23×C3
hypocenterC32C2×C3×#108
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63733C2direct (N × H)
4. C2×#63733C3direct (N × H)
15. #63733C2×C3direct (N × H)
72. C2×C33#108semidirect
111. C33C2×#108semidirect
136. C2×C32C3×#108semidirect
170. C32C2×C3×#108semidirect
186. C2×C3#63733direct (N × H)
200. C3C2×#63733direct (N × H)
203. C2C3×#63733direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C2×C3×#37122
C3×#637332
C2×#637333
C2×C3×#1089