Finite Group Databasebeta

← C2×C3×#63732 #13166730

Subgroups — C2×C3×#63732 #13166730

Order 1728.

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

normal subgroups only

This group has 472 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
1C2×C3×C4×#157C211yesmaximal, characteristic
2C2×C3×#3710C212yesmaximal
3C3×#63732C214yesmaximal, direct product factor
4C2×#63732C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C411yescharacteristic
6C22×C3×#157C411yescharacteristic
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C3×C4×#157C412yes—
9C3×C4×#157C2212yes—
10C2×C32×C3⋊C8—22no—
11C3×#3710—22no—
12C3×#3710C2214yes—
13C2×C4×#157C2×C311yescharacteristic
14C2×C3×D3×C8—62no—
15C2×#3710C2×C312yes—
16#63732C2×C314yesdirect product factor
17C22×C3×C3⋊D3C2×C411yescharacteristic
18C2×C3×#157—22no—
19C2×C3×#157C814yescomplemented
20C2×C3×#157C2×C412yes—
21C2×C3×#157C2×C412yes—
22C2×C33×C4D411yescharacteristic
23C2×C3×C3⋊Dic3Q811yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3Q811yescharacteristic
27C3×C4×C3⋊D3C2311yescharacteristic
28C32×C3⋊C8—24no—
29C2×C3×C4⋊C8—91nomaximal
30C22×#157C3×C411yescharacteristic
31C22×#157C3×C411yescharacteristic
32C2×C4×C3⋊D3C22×C311yescharacteristic
33C2×C3×C4×D3—62no—
34C4×#157C3×C412yes—
35C4×#157C22×C312yes—
36C2×C32×C8—62no—
37C2×C3×C3⋊C8—22no—
38C2×C3×C3⋊C8—22no—
39C2×C3×C3⋊C8—42no—
40C3×D3×C8—64no—
41C3×D3×C8—64no—
42#3710—22no—
43#3710C22×C314yes—
44C22×C33C4⋊C411yescharacteristic
45C2×C3×C3⋊D3C4⋊C411yescharacteristic
46C2×C3×C3⋊D3C2×C812yes—
47C2×C3×C3⋊D3M1612yes—
48C2×C3×C3⋊D3C22×C411yescharacteristic
49C3×#157—22no—
50C3×#157C2×C814yescomplemented
51C33×C4C4⋊C411yescharacteristic
52C33×C4C2×D411yescharacteristic
53C3×C3⋊Dic3C4⋊C411yescharacteristic
54C3×C3⋊Dic3C2×Q811yescharacteristic
55C22×C3×C8—92no—
56C2×C3×C42—91no—
57C2×D3×C8—62no—
58C3×C4⋊C8—94no—
59C22×C3⋊D3C2×C3×C411yescharacteristic
60C22×C3×D3—62no—
61C2×#157—22no—
62C2×#157C3×C814yescomplemented
63C2×#157C2×C3×C412yes—
64C2×#157C2×C3×C412yes—
65C2×C32×C4—22no—
66C2×C32×C4—42no—
67C2×C32×C4C3×D411yescharacteristic
68C2×C3⋊Dic3C3×Q811yescharacteristic
69C4×C3⋊D3C2×C3×C411yescharacteristic
70C4×C3⋊D3C3×D411yescharacteristic
71C4×C3⋊D3C3×Q811yescharacteristic
72C4×C3⋊D3C23×C311yescharacteristic
73C2×C3×Dic3—62no—
74C3×C4×D3—62no—
75C3×C4×D3—62no—
76C3×C4×D3—62no—
77C3×C4×D3—62no—
78C32×C8—64no—
79C3×C3⋊C8—24no—
80C3×C3⋊C8—24no—
81C3×C3⋊C8—44no—
82C2×C4⋊C8—91noSylow
83C2×C33C4⋊C812yescomplemented
84C2×C33C2×C4⋊C411yescharacteristic
85C3×C3⋊D3C4⋊C812yescomplemented
86C3×C3⋊D3C22×C811yescharacteristic
87C3×C3⋊D3C2×M1611yescharacteristic
88C22×C3×C4—91no—
89C22×C3×C4—91no—
90C22×C3×C4—91no—
91C2×C4×D3—62no—
92C2×C3×C8—62no—
93C2×C3×C8—62no—
94C2×C3×C8—94no—
95C2×C3×C8—122no—
96C2×C3×C8—182no—
97C3×C42—92no—
98C3×C42—92no—
99C2×C3⋊C8—22no—
100D3×C8—64no—
101D3×C8—64no—
102C22×C32—22no—
103C22×C32—42no—
104C22×C32C3×C4⋊C411yescharacteristic
105C2×C3⋊D3C3×C4⋊C411yescharacteristic
106C2×C3⋊D3C2×C3×C812yes—
107C2×C3⋊D3C3×M1612yes—
108C2×C3⋊D3C22×C3×C411yescharacteristic
109C2×C3×D3—62no—
110C2×C3×D3—64no—
111C2×C3×D3—64no—
112C2×C3×D3—62no—
113#157—22no—
114#157C2×C3×C814yescomplemented
115C32×C4—22no—
116C32×C4—22no—
117C32×C4—42no—
118C32×C4—42no—
119C32×C4C3×C4⋊C411yescharacteristic
120C32×C4C2×C3×D411yescharacteristic
121C3⋊Dic3C3×C4⋊C411yescharacteristic
122C3⋊Dic3C2×C3×Q811yescharacteristic
123C3×Dic3—62no—
124C3×Dic3—62no—
125C22×C8—92no—
126C2×C42—91no—
127C4⋊C8—94no—
128C33C2×C4⋊C811yescharacteristic, complemented, Sylow
129C23×C3—91no—
130C22×D3—62no—
131C2×C3×C4—22no—
132C2×C3×C4—42no—
133C2×C3×C4—94no—
134C2×C3×C4—92no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—92no—
141C2×C3×C4—182no—
142C2×C3×C4GO+(4,𝔽2)11yescharacteristic
143C2×Dic3—62no—
144C4×D3—62no—
145C4×D3—62no—
146C4×D3—62no—
147C4×D3—62no—
148C3×C8—64no—
149C3×C8—64no—
150C3×C8—124no—
151C3⋊C8—24no—
152C2×C32—22no—
153C2×C32—24no—
154C2×C32—44no—
155C2×C32—42no—
156C2×C32C3×C4⋊C812yescomplemented
157C2×C32C2×C3×C4⋊C411yescharacteristic
158C3⋊D3C3×C4⋊C812yescomplemented
159C3⋊D3C22×C3×C811yescharacteristic
160C3⋊D3C2×C3×M1611yescharacteristic
161C3×D3—64no—
162C3×D3—62no—
163C3×D3—62no—
164C22×C4—91no—
165C22×C4—91no—
166C22×C4—91no—
167C2×C8—62no—
168C2×C8—94no—
169C2×C8—182no—
170C42—92no—
171C42—92no—
172C22×C3—22no—
173C22×C3—42no—
174C22×C3—91no—
175C22×C3—91no—
176C22×C3—92no—
177C22×C3—92no—
178C22×C3#377311yescharacteristic
179C2×D3—62no—
180C2×D3—62no—
181C2×D3—64no—
182C2×D3—64no—
183C3×C4—22no—
184C3×C4—22no—
185C3×C4—42no—
186C3×C4—42no—
187C3×C4—94no—
188C3×C4—91no—
189C3×C4—91no—
190C3×C4—182no—
191C3×C4#377311yescharacteristic
192C3×C4C2×GO+(4,𝔽2)11yescharacteristic
193Dic3—62no—
194Dic3—62no—
195C32—22no—
196C32—42no—
197C32C2×C3×C4⋊C811yescharacteristic, complemented
198C23—91no—
199C2×C4—92no—
200C2×C4—94no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—92no—
207C2×C4—182no—
208C2×C4C3×GO+(4,𝔽2)11yescharacteristic
209C8—64no—
210C2×C3—22no—
211C2×C3—24no—
212C2×C3—42no—
213C2×C3—44no—
214C2×C3—91no—
215C2×C3—91no—
216C2×C3—92no—
217C2×C3#6373212yesdirect product factor
218C2×C3C2×#377311yescharacteristic
219D3—64no—
220D3—62no—
221D3—62no—
222C22—91no—
223C22—91no—
224C22—92no—
225C22—92no—
226C22C3×#377311yescharacteristic
227C4—94no—
228C4—91no—
229C4—91no—
230C4—182no—
231C4C3×#377311yescharacteristic
232C4C2×C3×GO+(4,𝔽2)11yescharacteristic
233C3—22no—
234C3—42no—
235C3C2×#6373211yescharacteristic, direct product factor
236C2—91no—
237C2—91no—
238C2—92no—
239C2C3×#6373212yesdirect product factor
240C2C2×C3×#377311yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊D3C22×C3×C8
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63732C2direct (N × H)
4. C2×#63732C3direct (N × H)
16. #63732C2×C3direct (N × H)
19. C2×C3×#157C8semidirect
50. C3×#157C2×C8semidirect
62. C2×#157C3×C8semidirect
83. C2×C33C4⋊C8semidirect
85. C3×C3⋊D3C4⋊C8semidirect
114. #157C2×C3×C8semidirect
128. C33C2×C4⋊C8semidirect
156. C2×C32C3×C4⋊C8semidirect
158. C3⋊D3C3×C4⋊C8semidirect
197. C32C2×C3×C4⋊C8semidirect
217. C2×C3#63732direct (N × H)
235. C3C2×#63732direct (N × H)
239. C2C3×#63732direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
C2×C3×#37102
C3×#637322
C2×#637323
C2×C3×C4⋊C89