Finite Group Databasebeta

← C2×C3×#63743 #13166871

Subgroups — C2×C3×#63743 #13166871

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×#63743C214yesmaximal, direct product factor
4C2×#63743C311yesmaximal, 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#63743C2×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×#107—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×C33D812yes—
38C22×C33Q1612yes—
39C22×C33C2×D411yescharacteristic
40C3×C3⋊Dic3C4⋊C412yes—
41C3×C3⋊Dic3C22×C411yescharacteristic
42C3×C3⋊Dic3C2×Q811yescharacteristic
43C32×Dic3—44no—
44C22×C3×C8—91no—
45C2×C3×C4⋊C4—92no—
46C2×Dic3⋊C4—62no—
47C3×#107—94no—
48C23×C32—22no—
49C23×C32—42no—
50C23×C32C3×D411yescharacteristic
51C2×C32×C4—64no—
52C2×C32×C4—124no—
53C2×C3⋊Dic3C2×C3×C412yes—
54C2×C3⋊Dic3C3×D411yescharacteristic
55C2×C3⋊Dic3C3×Q812yes—
56C2×C3⋊Dic3C23×C311yescharacteristic
57C2×C3×Dic3—24no—
58C2×C3×Dic3—24no—
59C2×C3×Dic3—44no—
60C2×C3×Dic3—44no—
61C2×C3×Dic3—48no—
62C2×C3×Dic3—44no—
63C2×C3×Dic3—62no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—64no—
66C2×C3×Dic3—64no—
67#622C2×C3×C414yescomplemented
68C2×#107—91noSylow
69C2×C33#10714yescomplemented
70C2×C33C2×C4⋊C411yescharacteristic
71C2×C33C2×D811yescharacteristic
72C2×C33C2×Q1611yescharacteristic
73C22×C3×C4—62no—
74C22×C3×C4—62no—
75C22×C3×C4—91no—
76C22×C3×C4—122no—
77C22×Dic3—22no—
78C22×Dic3—62no—
79C2×C3×C8—92no—
80C2×C3×C8—94no—
81C3×C4⋊C4—94no—
82C3×C4⋊C4—182no—
83Dic3⋊C4—64no—
84Dic3⋊C4—64no—
85C22×C32—22no—
86C22×C32—24no—
87C22×C32—24no—
88C22×C32—24no—
89C22×C32—44no—
90C22×C32—44no—
91C22×C32—44no—
92C22×C32—42no—
93C22×C32C3×C4⋊C412yes—
94C22×C32C3×D812yes—
95C22×C32C3×Q1612yes—
96C22×C32C2×C3×D411yescharacteristic
97C32×C4—124no—
98C3⋊Dic3C3×C4⋊C412yes—
99C3⋊Dic3C22×C3×C411yescharacteristic
100C3⋊Dic3C2×C3×Q811yescharacteristic
101C3×Dic3—44no—
102C3×Dic3—48no—
103C3×Dic3—44no—
104C3×Dic3—64no—
105C3×Dic3—62no—
106C3×Dic3—62no—
107C22×C8—91no—
108C2×C4⋊C4—92no—
109#107—94no—
110C33C2×#10711yescharacteristic, complemented, Sylow
111C23×C3—22no—
112C23×C3—42no—
113C23×C3GO+(4,𝔽2)11yescharacteristic
114C2×C3×C4—64no—
115C2×C3×C4—64no—
116C2×C3×C4—91no—
117C2×C3×C4—91no—
118C2×C3×C4—92no—
119C2×C3×C4—92no—
120C2×C3×C4—124no—
121C2×C3×C4—124no—
122C2×C3×C4—128no—
123C2×C3×C4—124no—
124C2×Dic3—24no—
125C2×Dic3—44no—
126C2×Dic3—64no—
127C2×Dic3—62no—
128C2×Dic3—64no—
129C2×Dic3—62no—
130C3×C8—94no—
131C2×C32—22no—
132C2×C32—22no—
133C2×C32—22no—
134C2×C32—28no—
135C2×C32—48no—
136C2×C32—42no—
137C2×C32—42no—
138C2×C32—42no—
139C2×C32C3×#10714yescomplemented
140C2×C32C2×C3×C4⋊C411yescharacteristic
141C2×C32C2×C3×D811yescharacteristic
142C2×C32C2×C3×Q1611yescharacteristic
143C22×C4—62no—
144C22×C4—91no—
145C2×C8—92no—
146C2×C8—94no—
147C4⋊C4—94no—
148C4⋊C4—182no—
149C22×C3—22no—
150C22×C3—24no—
151C22×C3—24no—
152C22×C3—24no—
153C22×C3—42no—
154C22×C3—44no—
155C22×C3—44no—
156C22×C3—44no—
157C22×C3#377312yes—
158C22×C3#377412yes—
159C22×C3#377612yes—
160C22×C3C2×GO+(4,𝔽2)11yescharacteristic
161C3×C4—91no—
162C3×C4—91no—
163C3×C4—92no—
164C3×C4—124no—
165C3×C4—128no—
166C3×C4—124no—
167Dic3—44no—
168Dic3—64no—
169Dic3—62no—
170Dic3—62no—
171C32—22no—
172C32—42no—
173C32C2×C3×#10711yescharacteristic, complemented
174C23C3×GO+(4,𝔽2)11yescharacteristic
175C2×C4—64no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—92no—
179C2×C4—92no—
180C2×C4—124no—
181C8—94no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—22no—
185C2×C3—28no—
186C2×C3—42no—
187C2×C3—42no—
188C2×C3—42no—
189C2×C3—48no—
190C2×C3#6374314yesdirect product factor
191C2×C3C2×#377311yescharacteristic
192C2×C3C2×#377411yescharacteristic
193C2×C3C2×#377611yescharacteristic
194C22C3×#377312yes—
195C22C3×#377412yes—
196C22C3×#377612yes—
197C22C2×C3×GO+(4,𝔽2)11yescharacteristic
198C4—91no—
199C4—91no—
200C4—92no—
201C4—124no—
202C3—22no—
203C3—42no—
204C3C2×#6374311yescharacteristic, direct product factor
205C2C3×#6374314yesdirect product factor
206C2C2×C3×#377311yescharacteristic
207C2C2×C3×#377411yescharacteristic
208C2C2×C3×#377611yescharacteristic
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×#107
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×#63743C2direct (N × H)
4. C2×#63743C3direct (N × H)
7. C2×C3×#622C4semidirect
14. #63743C2×C3direct (N × H)
22. C3×#622C2×C4semidirect
30. C2×#622C3×C4semidirect
67. #622C2×C3×C4semidirect
69. C2×C33#107semidirect
110. C33C2×#107semidirect
139. C2×C32C3×#107semidirect
173. C32C2×C3×#107semidirect
190. C2×C3#63743direct (N × H)
204. C3C2×#63743direct (N × H)
205. C2C3×#63743direct (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×#637432
C2×#637433
C2×C3×#1079