Finite Group Databasebeta

← C3×#10593446 #13166817

Subgroups — C3×#10593446 #13166817

Order 1728.

428 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3776C211yesmaximal, characteristic, complemented
3C3×#63865C212yesmaximal
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593446C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#627C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3787—22no—
12C3×#3776—22no—
13C32×Dic3⋊C4—22no—
14C3×#3724C2212yes—
15C3×#3722—22no—
16C2×#3787C2×C311yescharacteristic
17C2×#3776C2×C311yescharacteristic, complemented
18C3×#896—62no—
19#63865C2×C312yes—
20#63743C2×C311yescharacteristic, complemented
21#63740C2×C312yescomplemented
22C22×C3×C3⋊D3D411yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3×C4×C3⋊D3—21no—
26C3×C4×C3⋊D3—21no—
27C2×C32×Dic3—22no—
28C2×C32×Dic3—22no—
29C3×#627—22no—
30C3×#627—22no—
31C3×#622—21no—
32C3×#622D412yes—
33C3×#467—91nomaximal
34C2×C4×C3⋊D3C22×C311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×C3×Dic6—62no—
37C2×#627C22×C312yes—
38C2×#622C22×C311yescharacteristic
39C2×#622C22×C311yescharacteristic
40#3787—22no—
41#3776—22no—
42C32×C4⋊C4—62no—
43C3×(C2×D3)⋊C4—62no—
44C3×(C2×D3)⋊C4—62no—
45C3×Dic3⋊C4—22no—
46C3×Dic3⋊C4—22no—
47C3×Dic3⋊C4—42no—
48C3×Dic3⋊C4—62no—
49#3724C22×C312yes—
50#3722—22no—
51C22×C33C2×D411yescharacteristic
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3Q1612yescomplemented
55C33×C4—21no—
56C3×C3⋊Dic3C2×D411yescharacteristic
57C3×C3⋊Dic3#4111yescharacteristic
58C32×Dic3—42no—
59C32×Dic3—42no—
60C2×C3×Q16—91no—
61C22×C3×C8—91no—
62C3×C22⋊Q8—92no—
63#896—62no—
64C3×#107—91no—
65C3×#103—92no—
66C22×C3⋊D3C3×D411yescharacteristic
67C22×C3×D3—62no—
68C2×C32×C4—22no—
69C2×C32×C4—42no—
70C2×C32×C4—62no—
71C2×C32×C4—62no—
72C2×C32×C4C3×D411yescharacteristic
73C2×C3⋊Dic3C23×C311yescharacteristic
74C4×C3⋊D3—21no—
75C4×C3⋊D3—21no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—22no—
79C2×C3×Dic3—22no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—42no—
82C2×C3×Dic3—62no—
83C3×C4×D3—122no—
84C3×C4×D3—122no—
85C3×Dic6—122no—
86C3×Dic6—122no—
87#627—22no—
88#627—22no—
89#622—21no—
90#622C3×D412yes—
91#467—91noSylow
92C2×C33#12111yescharacteristic
93C2×C33C2×Q1611yescharacteristic
94C2×C33#13511yescharacteristic
95C3×C3⋊D3—22no—
96C2×C3×Q8—92no—
97C22×C3×C4—91no—
98C2×C4×D3—62no—
99C2×Dic6—62no—
100C3×Q16—182no—
101C2×C3×C8—91no—
102C2×C3×C8—91no—
103C2×C3×C8—182no—
104C3×C4⋊C4—62no—
105C3×C4⋊C4—62no—
106C3×C4⋊C4—92no—
107C3×C4⋊C4—122no—
108C3×C22⋊C4—182no—
109(C2×D3)⋊C4—62no—
110(C2×D3)⋊C4—62no—
111Dic3⋊C4—22no—
112Dic3⋊C4—62no—
113C22×C32—22no—
114C22×C32—42no—
115C22×C32C2×C3×D411yescharacteristic
116C2×C3⋊D3—21no—
117C2×C3⋊D3—21no—
118C2×C3⋊D3C3×Q1612yescomplemented
119C2×C3×D3—64no—
120C2×C3×D3—122no—
121C2×C3×D3—122no—
122C32×C4—21no—
123C32×C4—42no—
124C32×C4—82no—
125C32×C4—122no—
126C32×C4—122no—
127C3⋊Dic3C2×C3×D411yescharacteristic
128C3⋊Dic3C3×#4111yescharacteristic
129C3×Dic3—42no—
130C3×Dic3—42no—
131C3×Dic3—44no—
132C3×Dic3—42no—
133C3×Dic3—42no—
134C3×Dic3—62no—
135C3×Dic3—62no—
136C3×Dic3—82no—
137C2×Q16—91no—
138C22×C8—91no—
139C22⋊Q8—92no—
140#107—91no—
141#103—92no—
142C33#46711yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C22×D3—62no—
145C3×Q8—182no—
146C3×Q8—182no—
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—62no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—91no—
154C2×C3×C4—122no—
155C2×C3×C4—122no—
156C2×C3×C4—181no—
157C2×C3×C4—181no—
158C2×C3×C4GO+(4,𝔽2)11yescharacteristic
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—62no—
162C4×D3—122no—
163C4×D3—122no—
164Dic6—122no—
165Dic6—122no—
166C3×C8—92no—
167C3×C8—181no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32C3×#12111yescharacteristic
175C2×C32C2×C3×Q1611yescharacteristic
176C2×C32C3×#13511yescharacteristic
177C3⋊D3—22no—
178C3×D3—124no—
179C2×Q8—92no—
180C22×C4—91no—
181Q16—182no—
182C2×C8—91no—
183C2×C8—91no—
184C2×C8—182no—
185C4⋊C4—62no—
186C4⋊C4—92no—
187C22⋊C4—182no—
188C22×C3—22no—
189C22×C3—42no—
190C22×C3—92no—
191C22×C3—181no—
192C22×C3—181no—
193C22×C3C2×GO+(4,𝔽2)11yescharacteristic
194C2×D3—64no—
195C2×D3—122no—
196C2×D3—122no—
197C3×C4—21no—
198C3×C4—42no—
199C3×C4—82no—
200C3×C4—91no—
201C3×C4—91no—
202C3×C4—122no—
203C3×C4—122no—
204C3×C4—124no—
205C3×C4—122no—
206C3×C4—122no—
207C3×C4—242no—
208Dic3—42no—
209Dic3—42no—
210Dic3—62no—
211Dic3—62no—
212C32—22no—
213C32—42no—
214C32C3×#46711yescharacteristic, complemented
215C23—91no—
216Q8—182no—
217Q8—182no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—91no—
221C2×C4—181no—
222C2×C4—181no—
223C2×C4C3×GO+(4,𝔽2)11yescharacteristic
224C8—92no—
225C8—181no—
226C2×C3—22no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—42no—
230C2×C3—42no—
231C2×C3—42no—
232C2×C3—182no—
233C2×C3#6422311yescharacteristic
234C2×C3#6422811yescharacteristic
235C2×C3#6424211yescharacteristic
236D3—124no—
237C22—92no—
238C22—181no—
239C22—181no—
240C22C2×C3×GO+(4,𝔽2)11yescharacteristic
241C4—21no—
242C4—91no—
243C4—91no—
244C4—122no—
245C4—122no—
246C3—22no—
247C3—42no—
248C3#1059344611yescharacteristic, direct product factor
249C2—182no—
250C2C3×#6422311yescharacteristic
251C2C3×#6422811yescharacteristic
252C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#467
LCSG ▷ C2×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
2. C2×C3×#3776C2semidirect
4. C3×#63743C2semidirect
5. C3×#63740C2semidirect
6. #10593446C3direct (N × H)
17. C2×#3776C2×C3semidirect
20. #63743C2×C3semidirect
21. #63740C2×C3semidirect
54. C2×C3×C3⋊D3Q16semidirect
118. C2×C3⋊D3C3×Q16semidirect
142. C33#467semidirect
214. C32C3×#467semidirect
248. C3#10593446direct (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×#37762
C3×#638652
C3×#637432
C3×#637402
#105934463
C3×#4679