Finite Group Databasebeta

← C3×#10593762 #13173233

Subgroups — C3×#10593762 #13173233

Order 1728.

376 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 100 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×C8×C3⋊D3C211yesmaximal, characteristic
2C3×#63773C212yesmaximal, complemented
3C3×#63765C212yesmaximal, complemented
4C3×#63764C212yesmaximal, complemented
5#10593762C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C411yescharacteristic
7C2×C33×C8C411yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9C3×C8×C3⋊D3C411yescharacteristic
10C3×C8×C3⋊D3C411yescharacteristic
11C3×C8×C3⋊D3C2211yescharacteristic
12C3×C8×C3⋊D3C2211yescharacteristic
13C3×#3708C2214yescomplemented
14C2×C8×C3⋊D3C2×C311yescharacteristic
15C2×C3×D3×C8—62no—
16#63773C2×C312yescomplemented
17#63765C2×C312yescomplemented
18#63764C2×C312yescomplemented
19C22×C3×C3⋊D3C811yescharacteristic
20C2×C33×C4C2×C411yescharacteristic
21C2×C3×C3⋊Dic3C811yescharacteristic
22C3×C4×C3⋊D3C811yescharacteristic
23C3×C4×C3⋊D3C811yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C33×C8C2×C411yescharacteristic
27C33×C8C2×C411yescharacteristic
28C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
29C3×C3⋊(C3⋊C8)C2311yescharacteristic
30C2×C3×M32—91nomaximal
31C2×C4×C3⋊D3C3×C411yescharacteristic
32C2×C3×C4×D3—62no—
33C2×C32×C8—22no—
34C2×C32×C8—42no—
35C2×C32×C8C3×C411yescharacteristic
36C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
37C8×C3⋊D3C3×C411yescharacteristic
38C8×C3⋊D3C3×C411yescharacteristic
39C8×C3⋊D3C22×C311yescharacteristic
40C8×C3⋊D3C22×C311yescharacteristic
41C2×C3×C3⋊C8—62no—
42C3×D3×C8—62no—
43C3×D3×C8—62no—
44C3×D3×C8—62no—
45C3×D3×C8—62no—
46#3708C22×C314yescomplemented
47C22×C33C2×C811yescharacteristic
48C2×C3×C3⋊D3—22no—
49C2×C3×C3⋊D3C2×C811yescharacteristic
50C2×C3×C3⋊D3C2×C811yescharacteristic
51C33×C4C2×C811yescharacteristic
52C33×C4C22×C411yescharacteristic
53C3×C3⋊Dic3C2×C811yescharacteristic
54C3×C3⋊Dic3C2×C811yescharacteristic
55C22×C3×C8—91no—
56C2×D3×C8—62no—
57C3×M32—92no—
58C3×M32—92no—
59C2×C3×C16—92no—
60C22×C3⋊D3C3×C811yescharacteristic
61C22×C3×D3—62no—
62C2×C32×C4—22no—
63C2×C32×C4—42no—
64C2×C32×C4C2×C3×C411yescharacteristic
65C2×C3⋊Dic3C3×C811yescharacteristic
66C4×C3⋊D3C3×C811yescharacteristic
67C4×C3⋊D3C3×C811yescharacteristic
68C4×C3⋊D3C2×C3×C411yescharacteristic
69C4×C3⋊D3C2×C3×C411yescharacteristic
70C2×C3×Dic3—62no—
71C3×C4×D3—62no—
72C3×C4×D3—62no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C32×C8—22no—
76C32×C8—22no—
77C32×C8—42no—
78C32×C8—42no—
79C32×C8C2×C3×C411yescharacteristic
80C32×C8C2×C3×C411yescharacteristic
81C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
82C3⋊(C3⋊C8)C23×C311yescharacteristic
83C3×C3⋊C8—62no—
84C3×C3⋊C8—62no—
85C2×M32—91noSylow
86C2×C33—21no—
87C2×C33C22×C811yescharacteristic
88C3×C3⋊D3—21no—
89C3×C3⋊D3M3212yescomplemented
90C22×C3×C4—91no—
91C2×C4×D3—62no—
92C2×C3×C8—22no—
93C2×C3×C8—42no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C2×C3×C8—91no—
97C2×C3×C8—91no—
98C2×C3×C8—91no—
99C2×C3×C8#15711yescharacteristic
100C2×C3⋊C8—62no—
101D3×C8—62no—
102D3×C8—62no—
103D3×C8—62no—
104D3×C8—62no—
105C3×C16—94no—
106C22×C32—22no—
107C22×C32—42no—
108C22×C32C2×C3×C811yescharacteristic
109C2×C3⋊D3—22no—
110C2×C3⋊D3C2×C3×C811yescharacteristic
111C2×C3⋊D3C2×C3×C811yescharacteristic
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—68no—
115C32×C4—22no—
116C32×C4—22no—
117C32×C4—42no—
118C32×C4—42no—
119C32×C4C2×C3×C811yescharacteristic
120C32×C4C22×C3×C411yescharacteristic
121C3⋊Dic3C2×C3×C811yescharacteristic
122C3⋊Dic3C2×C3×C811yescharacteristic
123C3×Dic3—62no—
124C3×Dic3—62no—
125C22×C8—91no—
126M32—92no—
127M32—92no—
128C2×C16—92no—
129C33C2×M3211yescharacteristic, complemented, Sylow
130C23×C3—91no—
131C22×D3—62no—
132C2×C3×C4—22no—
133C2×C3×C4—42no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4C2×#15711yescharacteristic
140C2×Dic3—62no—
141C4×D3—62no—
142C4×D3—62no—
143C4×D3—62no—
144C4×D3—62no—
145C3×C8—22no—
146C3×C8—22no—
147C3×C8—42no—
148C3×C8—42no—
149C3×C8—91no—
150C3×C8—91no—
151C3×C8C2×#15711yescharacteristic
152C3×C8C2×#15711yescharacteristic
153C3⋊C8—62no—
154C3⋊C8—62no—
155C2×C32—21no—
156C2×C32—24no—
157C2×C32—22no—
158C2×C32—44no—
159C2×C32—42no—
160C2×C32C22×C3×C811yescharacteristic
161C3⋊D3—21no—
162C3⋊D3C3×M3212yescomplemented
163C3×D3—64no—
164C3×D3—64no—
165C22×C4—91no—
166C2×C8—91no—
167C2×C8—91no—
168C2×C8—91no—
169C2×C8—91no—
170C2×C8—91no—
171C2×C8C3×#15711yescharacteristic
172C16—94no—
173C22×C3—22no—
174C22×C3—42no—
175C22×C3—91no—
176C22×C3—91no—
177C22×C3—182no—
178C22×C3#378711yescharacteristic
179C2×D3—62no—
180C2×D3—62no—
181C2×D3—68no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—42no—
185C3×C4—42no—
186C3×C4—91no—
187C3×C4—91no—
188C3×C4#378711yescharacteristic
189C3×C4C22×#15711yescharacteristic
190Dic3—62no—
191Dic3—62no—
192C32—22no—
193C32—42no—
194C32C2×C3×M3211yescharacteristic, complemented
195C23—91no—
196C2×C4—91no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4C2×C3×#15711yescharacteristic
202C8—91no—
203C8—91no—
204C8C2×C3×#15711yescharacteristic
205C8C2×C3×#15711yescharacteristic
206C2×C3—24no—
207C2×C3—21no—
208C2×C3—22no—
209C2×C3—44no—
210C2×C3—42no—
211C2×C3—92no—
212C2×C3—181no—
213C2×C3C2×#378711yescharacteristic
214D3—64no—
215D3—64no—
216C22—91no—
217C22—91no—
218C22—182no—
219C22C3×#378711yescharacteristic
220C4—91no—
221C4—91no—
222C4C3×#378711yescharacteristic
223C4C22×C3×#15711yescharacteristic
224C3—22no—
225C3—42no—
226C3#1059376211yescharacteristic, direct product factor
227C2—21no—
228C2—92no—
229C2—181no—
230C2C2×C3×#378711yescharacteristic
subgroupG/H
centerC3×C8C2×#157
commutatorC2×C32C22×C3×C8
FrattiniC4C22×C3×#157
FittingC2×C33×C8C4
socleC2×C33C22×C8
prime radicalC22×C3×C3⋊D3C8
cosocleC3⋊(C3⋊C8)C23×C3
hypercenterC2×C3×C8#157
hypocenterC32C2×C3×M32
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C8 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63773C2semidirect
3. C3×#63765C2semidirect
4. C3×#63764C2semidirect
5. #10593762C3direct (N × H)
13. C3×#3708C22semidirect
16. #63773C2×C3semidirect
17. #63765C2×C3semidirect
18. #63764C2×C3semidirect
46. #3708C22×C3semidirect
89. C3×C3⋊D3M32semidirect
129. C33C2×M32semidirect
162. C3⋊D3C3×M32semidirect
194. C32C2×C3×M32semidirect
226. C3#10593762direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C3×#637732
C3×#637652
C3×#637642
#105937623
C2×C3×M329