Finite Group Databasebeta

← C2×C3×#63742 #13166870

Subgroups — C2×C3×#63742 #13166870

Order 1728.

552 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22×C3×#622C211yesmaximal, characteristic
2C2×C3×#3724C212yesmaximal
3C3×#63742C214yesmaximal, direct product factor
4C2×#63742C311yesmaximal, characteristic, direct product factor
5C22×C3×C3⋊Dic3C2211yescharacteristic
6C22×C32×Dic3—22no—
7C2×C3×#622C414yescomplemented
8C2×C3×#622C2212yes—
9C3×#3724—22no—
10C3×#3724C2214yes—
11C22×#622C2×C311yescharacteristic
12C2×C3×Dic3⋊C4—62no—
13C2×#3724C2×C312yes—
14#63742C2×C314yesdirect product factor
15C23×C33D411yescharacteristic
16C2×C3×C3⋊Dic3C2×C412yes—
17C2×C3×C3⋊Dic3D411yescharacteristic
18C2×C3×C3⋊Dic3Q812yes—
19C2×C3×C3⋊Dic3C2311yescharacteristic
20C2×C32×Dic3—24no—
21C2×C32×Dic3—44no—
22C3×#622C2×C414yescomplemented
23C2×C3×#106—91nomaximal
24C22×C32×C4—62no—
25C22×C3⋊Dic3C22×C311yescharacteristic
26C22×C3×Dic3—22no—
27C22×C3×Dic3—22no—
28C22×C3×Dic3—42no—
29C22×C3×Dic3—62no—
30C2×#622C3×C414yescomplemented
31C2×#622C22×C312yes—
32C3×Dic3⋊C4—64no—
33C3×Dic3⋊C4—64no—
34#3724—22no—
35#3724C22×C314yes—
36C22×C33C4⋊C412yes—
37C22×C33SD1614yes—
38C22×C33C2×D411yescharacteristic
39C3×C3⋊Dic3C4⋊C412yes—
40C3×C3⋊Dic3C22×C411yescharacteristic
41C3×C3⋊Dic3C2×Q811yescharacteristic
42C32×Dic3—44no—
43C22×C3×C8—91no—
44C2×C3×C4⋊C4—92no—
45C2×Dic3⋊C4—62no—
46C3×#106—94no—
47C23×C32—22no—
48C23×C32—42no—
49C23×C32C3×D411yescharacteristic
50C2×C32×C4—64no—
51C2×C32×C4—124no—
52C2×C3⋊Dic3C2×C3×C412yes—
53C2×C3⋊Dic3C3×D411yescharacteristic
54C2×C3⋊Dic3C3×Q812yes—
55C2×C3⋊Dic3C23×C311yescharacteristic
56C2×C3×Dic3—24no—
57C2×C3×Dic3—24no—
58C2×C3×Dic3—44no—
59C2×C3×Dic3—44no—
60C2×C3×Dic3—48no—
61C2×C3×Dic3—44no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3—62no—
64C2×C3×Dic3—64no—
65C2×C3×Dic3—64no—
66#622C2×C3×C414yescomplemented
67C2×#106—91noSylow
68C2×C33#10614yescomplemented
69C2×C33C2×C4⋊C411yescharacteristic
70C2×C33C2×SD1612yes—
71C22×C3×C4—62no—
72C22×C3×C4—62no—
73C22×C3×C4—91no—
74C22×C3×C4—122no—
75C22×Dic3—22no—
76C22×Dic3—62no—
77C2×C3×C8—92no—
78C2×C3×C8—94no—
79C3×C4⋊C4—94no—
80C3×C4⋊C4—182no—
81Dic3⋊C4—64no—
82Dic3⋊C4—64no—
83C22×C32—22no—
84C22×C32—24no—
85C22×C32—24no—
86C22×C32—24no—
87C22×C32—44no—
88C22×C32—44no—
89C22×C32—44no—
90C22×C32—42no—
91C22×C32C3×C4⋊C412yes—
92C22×C32C3×SD1614yes—
93C22×C32C2×C3×D411yescharacteristic
94C32×C4—124no—
95C3⋊Dic3C3×C4⋊C412yes—
96C3⋊Dic3C22×C3×C411yescharacteristic
97C3⋊Dic3C2×C3×Q811yescharacteristic
98C3×Dic3—44no—
99C3×Dic3—48no—
100C3×Dic3—44no—
101C3×Dic3—64no—
102C3×Dic3—62no—
103C3×Dic3—62no—
104C22×C8—91no—
105C2×C4⋊C4—92no—
106#106—94no—
107C33C2×#10611yescharacteristic, complemented, Sylow
108C23×C3—22no—
109C23×C3—42no—
110C23×C3GO+(4,𝔽2)11yescharacteristic
111C2×C3×C4—64no—
112C2×C3×C4—64no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—92no—
116C2×C3×C4—92no—
117C2×C3×C4—124no—
118C2×C3×C4—124no—
119C2×C3×C4—128no—
120C2×C3×C4—124no—
121C2×Dic3—24no—
122C2×Dic3—44no—
123C2×Dic3—64no—
124C2×Dic3—62no—
125C2×Dic3—64no—
126C2×Dic3—62no—
127C3×C8—94no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—22no—
131C2×C32—28no—
132C2×C32—48no—
133C2×C32—42no—
134C2×C32—42no—
135C2×C32—42no—
136C2×C32C3×#10614yescomplemented
137C2×C32C2×C3×C4⋊C411yescharacteristic
138C2×C32C2×C3×SD1612yes—
139C22×C4—62no—
140C22×C4—91no—
141C2×C8—92no—
142C2×C8—94no—
143C4⋊C4—94no—
144C4⋊C4—182no—
145C22×C3—22no—
146C22×C3—24no—
147C22×C3—24no—
148C22×C3—24no—
149C22×C3—42no—
150C22×C3—44no—
151C22×C3—44no—
152C22×C3—44no—
153C22×C3#377312yes—
154C22×C3#377514yes—
155C22×C3C2×GO+(4,𝔽2)11yescharacteristic
156C3×C4—91no—
157C3×C4—91no—
158C3×C4—92no—
159C3×C4—124no—
160C3×C4—128no—
161C3×C4—124no—
162Dic3—44no—
163Dic3—64no—
164Dic3—62no—
165Dic3—62no—
166C32—22no—
167C32—42no—
168C32C2×C3×#10611yescharacteristic, complemented
169C23C3×GO+(4,𝔽2)11yescharacteristic
170C2×C4—64no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4—92no—
174C2×C4—92no—
175C2×C4—124no—
176C8—94no—
177C2×C3—22no—
178C2×C3—22no—
179C2×C3—22no—
180C2×C3—28no—
181C2×C3—42no—
182C2×C3—42no—
183C2×C3—42no—
184C2×C3—48no—
185C2×C3#6374214yesdirect product factor
186C2×C3C2×#377311yescharacteristic
187C2×C3C2×#377512yes—
188C22C3×#377312yes—
189C22C3×#377514yes—
190C22C2×C3×GO+(4,𝔽2)11yescharacteristic
191C4—91no—
192C4—91no—
193C4—92no—
194C4—124no—
195C3—22no—
196C3—42no—
197C3C2×#6374211yescharacteristic, direct product factor
198C2C3×#6374214yesdirect product factor
199C2C2×C3×#377311yescharacteristic
200C2C2×C3×#377512yes—
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, socle, prime radicalC23×C33D4
cosocleC2×C3⋊Dic3C23×C3
hypocenterC32C2×C3×#106
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
3. C3×#63742C2direct (N × H)
4. C2×#63742C3direct (N × H)
7. C2×C3×#622C4semidirect
14. #63742C2×C3direct (N × H)
22. C3×#622C2×C4semidirect
30. C2×#622C3×C4semidirect
66. #622C2×C3×C4semidirect
68. C2×C33#106semidirect
107. C33C2×#106semidirect
136. C2×C32C3×#106semidirect
168. C32C2×C3×#106semidirect
185. C2×C3#63742direct (N × H)
197. C3C2×#63742direct (N × H)
198. C2C3×#63742direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6222
C2×C3×#37242
C3×#637422
C2×#637423
C2×C3×#1069