Finite Group Databasebeta

← C3×C4×#3776 #13166811

Subgroups — C3×C4×#3776 #13166811

Order 1728.

420 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3776C211yesmaximal, characteristic
2C3×C4×#627C212yesmaximal
3C3×C4×#622C211yesmaximal, characteristic
4C3×#63743C211yesmaximal, characteristic
5C3×#63740C212yesmaximal
6C4×#3776C311yesmaximal, characteristic, direct product factor
7C2×C3×#627C2212yes—
8C2×C3×#622C2211yescharacteristic
9C2×C3×#622C2211yescharacteristic
10C3×#3776C414yesdirect product factor
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4×Dic3—22no—
13C3×#3724C2212yes—
14C3×Dic3⋊Dic3—22no—
15C2×#3776C2×C311yescharacteristic
16C3×C4×Dic6—62no—
17C4×#627C2×C312yes—
18C4×#622C2×C311yescharacteristic
19#63743C2×C311yescharacteristic
20#63740C2×C312yes—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C2×C32×Dic3—22no—
25C2×C32×Dic3—22no—
26C3×#627—22no—
27C3×#627C2×C414yes—
28C3×#622—21no—
29C3×#622C2×C412yes—
30C3×C4×Q16—91nomaximal
31C2×C3×Dic6—62no—
32C2×#627C22×C312yes—
33C2×#622C22×C311yescharacteristic
34C2×#622C22×C311yescharacteristic
35#3776C3×C414yesdirect product factor
36C32×C42—62no—
37C4×C3⋊Dic3C22×C311yescharacteristic
38C3×C4⋊Dic3—62no—
39C3×Dic3⋊C4—62no—
40C3×Dic3⋊C4—62no—
41C3×C4×Dic3—22no—
42C3×C4×Dic3—22no—
43C3×C4×Dic3—42no—
44C3×C4×Dic3—62no—
45#3724C22×C312yes—
46Dic3⋊Dic3—22no—
47C22×C33C2×D411yescharacteristic
48C33×C4Q1612yescomplemented
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3C22×C411yescharacteristic
51C3×C3⋊Dic3#4111yescharacteristic
52C32×Dic3—24no—
53C32×Dic3—42no—
54C2×C3×Q16—91no—
55C3×C4×Q8—92no—
56C4×Dic6—62no—
57C3×#107—91no—
58C3×#103—92no—
59C3×C4×C8—91no—
60C2×C32×C4—22no—
61C2×C32×C4—42no—
62C2×C32×C4—62no—
63C2×C32×C4—62no—
64C2×C32×C4C3×D411yescharacteristic
65C2×C3⋊Dic3C3×D411yescharacteristic
66C2×C3⋊Dic3C23×C311yescharacteristic
67C2×C3×Dic3—22no—
68C2×C3×Dic3—22no—
69C2×C3×Dic3—22no—
70C2×C3×Dic3—22no—
71C2×C3×Dic3—42no—
72C2×C3×Dic3—42no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C3×Dic6—64no—
76C3×Dic6—64no—
77#627—22no—
78#627C2×C3×C414yes—
79#622—21no—
80#622C2×C3×C412yes—
81C4×Q16—91noSylow
82C2×C33C4×D411yescharacteristic
83C2×C33C2×Q1611yescharacteristic
84C2×C33#13511yescharacteristic
85C2×C3×Q8—92no—
86C2×Dic6—62no—
87C3×Q16—94no—
88C2×C3×C8—91no—
89C2×C3×C8—91no—
90C3×C4⋊C4—92no—
91C3×C4⋊C4—182no—
92C3×C42—62no—
93C3×C42—62no—
94C3×C42—91no—
95C3×C42—122no—
96C4⋊Dic3—62no—
97Dic3⋊C4—62no—
98Dic3⋊C4—62no—
99C4×Dic3—22no—
100C4×Dic3—62no—
101C22×C32—22no—
102C22×C32—42no—
103C22×C32C2×C3×D411yescharacteristic
104C32×C4—24no—
105C32×C4—44no—
106C32×C4—64no—
107C32×C4—122no—
108C32×C4C3×Q1612yescomplemented
109C3⋊Dic3—21no—
110C3⋊Dic3C22×C3×C411yescharacteristic
111C3⋊Dic3C3×#4111yescharacteristic
112C3×Dic3—24no—
113C3×Dic3—24no—
114C3×Dic3—44no—
115C3×Dic3—42no—
116C3×Dic3—44no—
117C3×Dic3—42no—
118C3×Dic3—62no—
119C3×Dic3—62no—
120C3×Dic3—122no—
121C2×Q16—91no—
122C4×Q8—92no—
123#107—91no—
124#103—92no—
125C4×C8—91no—
126C33C4×Q1611yescharacteristic, complemented, Sylow
127C3×Q8—94no—
128C3×Q8—182no—
129C2×C3×C4—22no—
130C2×C3×C4—42no—
131C2×C3×C4—62no—
132C2×C3×C4—62no—
133C2×C3×C4—62no—
134C2×C3×C4—62no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—122no—
138C2×C3×C4—122no—
139C2×C3×C4GO+(4,𝔽2)11yescharacteristic
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—62no—
143C2×Dic3—62no—
144Dic6—64no—
145Dic6—64no—
146C3×C8—92no—
147C3×C8—181no—
148C2×C32—22no—
149C2×C32—22no—
150C2×C32—22no—
151C2×C32—42no—
152C2×C32—42no—
153C2×C32—42no—
154C2×C32C3×C4×D411yescharacteristic
155C2×C32C2×C3×Q1611yescharacteristic
156C2×C32C3×#13511yescharacteristic
157C2×Q8—92no—
158Q16—94no—
159C2×C8—91no—
160C2×C8—91no—
161C4⋊C4—92no—
162C4⋊C4—182no—
163C42—62no—
164C42—91no—
165C22×C3—22no—
166C22×C3—42no—
167C22×C3C2×GO+(4,𝔽2)11yescharacteristic
168C3×C4—24no—
169C3×C4—44no—
170C3×C4—64no—
171C3×C4—64no—
172C3×C4—91no—
173C3×C4—91no—
174C3×C4—124no—
175C3×C4—122no—
176C3×C4—124no—
177C3×C4—122no—
178C3×C4—181no—
179C3×C4#377612yesdirect product factor
180Dic3—24no—
181Dic3—42no—
182Dic3—62no—
183Dic3—62no—
184Dic3—122no—
185C32—22no—
186C32—42no—
187C32C3×C4×Q1611yescharacteristic, complemented
188Q8—94no—
189Q8—182no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4C3×GO+(4,𝔽2)11yescharacteristic
195C8—92no—
196C8—181no—
197C2×C3—22no—
198C2×C3—22no—
199C2×C3—22no—
200C2×C3—42no—
201C2×C3—42no—
202C2×C3—42no—
203C2×C3#6422311yescharacteristic
204C2×C3C4×GO+(4,𝔽2)11yescharacteristic
205C2×C3C2×#377611yescharacteristic
206C22C2×C3×GO+(4,𝔽2)11yescharacteristic
207C4—64no—
208C4—91no—
209C4—91no—
210C4—122no—
211C4—181no—
212C4C3×#377612yesdirect product factor
213C3—22no—
214C3—42no—
215C3C4×#377611yescharacteristic, direct product factor
216C2C3×#6422311yescharacteristic
217C2C3×C4×GO+(4,𝔽2)11yescharacteristic
218C2C2×C3×#377611yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
cosocleC2×C3⋊Dic3C23×C3
hypocenterC32C3×C4×Q16
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. C4×#3776C3direct (N × H)
10. C3×#3776C4direct (N × H)
35. #3776C3×C4direct (N × H)
48. C33×C4Q16semidirect
108. C32×C4C3×Q16semidirect
126. C33C4×Q16semidirect
179. C3×C4#3776direct (N × H)
187. C32C3×C4×Q16semidirect
212. C4C3×#3776direct (N × H)
215. C3C4×#3776direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C3×C4×#6272
C3×C4×#6222
C3×#637432
C3×#637402
C4×#37763
C3×C4×Q169