Finite Group Databasebeta

← C3×#10590181 #13157106

Subgroups — C3×#10590181 #13157106

Order 1728.

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

normal subgroups only

This group has 432 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×C4×#157C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic
3C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
4#10590181C311yesmaximal, characteristic, direct product factor
5C22×C3×#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#622C412yescomplemented
8C3×C4×#157—21no—
9C3×C4×#157—21no—
10C3×#3787—21no—
11C3×#3787—21no—
12C2×C33×C8C411yescharacteristic, complemented
13C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
14C3×C8×C3⋊D3—21no—
15C3×C8×C3⋊D3—21no—
16C2×C4×#157C2×C311yescharacteristic
17C2×#3787C2×C311yescharacteristic
18C2×C8×C3⋊D3C2×C311yescharacteristic
19C2×C3×D3×C8—62no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C3×#157—22no—
22C2×C3×#157—22no—
23C2×C3×#157C814yescomplemented
24C2×C33×C4C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C3×C4×C3⋊D3D411yescharacteristic
29C3×C4×C3⋊D3Q811yescharacteristic
30C3×#622—22no—
31C33×C8—21no—
32C3×C3⋊(C3⋊C8)—21no—
33C3×#336—91nomaximal
34C22×#157C3×C412yes—
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C2×C3×C4×D3—62no—
37C2×#622C3×C412yescomplemented
38C4×#157—21no—
39C4×#157—21no—
40#3787—21no—
41#3787—21no—
42C2×C32×C8—22no—
43C2×C32×C8—42no—
44C2×C32×C8C3×C411yescharacteristic, complemented
45C2×C3⋊(C3⋊C8)C3×C411yescharacteristic, complemented
46C8×C3⋊D3—21no—
47C8×C3⋊D3—21no—
48C2×C3×C3⋊C8—62no—
49C3×D3×C8—64no—
50C3×D3×C8—64no—
51C22×C33C4211yescharacteristic
52C2×C3×C3⋊D3C22⋊C411yescharacteristic
53C2×C3×C3⋊D3C4⋊C411yescharacteristic
54C2×C3×C3⋊D3C2×C812yes—
55C2×C3×C3⋊D3M1612yes—
56C3×#157—24no—
57C33×C4C22⋊C411yescharacteristic
58C33×C4C4⋊C411yescharacteristic
59C3×C3⋊Dic3C22⋊C411yescharacteristic
60C3×C3⋊Dic3C4⋊C411yescharacteristic
61C22×C3×C8—91no—
62C22×C3×C8—91no—
63C2×C3×C42—91no—
64C2×D3×C8—62no—
65C22×C3⋊D3C2×C3×C411yescharacteristic
66C22×C3×D3—62no—
67C2×#157—22no—
68C2×#157—22no—
69C2×#157C3×C814yescomplemented
70C2×C32×C4—22no—
71C2×C32×C4—42no—
72C2×C32×C4C2×C3×C411yescharacteristic
73C2×C3⋊Dic3C2×C3×C411yescharacteristic
74C4×C3⋊D3C3×D411yescharacteristic
75C4×C3⋊D3C3×D411yescharacteristic
76C4×C3⋊D3C3×D411yescharacteristic
77C4×C3⋊D3C3×Q811yescharacteristic
78C2×C3×Dic3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83#622—22no—
84C32×C8—21no—
85C32×C8—24no—
86C32×C8—44no—
87C3⋊(C3⋊C8)—21no—
88C3×C3⋊C8—64no—
89#336—91noSylow
90C2×C33#9511yescharacteristic
91C2×C33C4×C811yescharacteristic
92C2×C33#9711yescharacteristic
93C3×C3⋊D3C22⋊C812yes—
94C3×C3⋊D3C4⋊C812yes—
95C22×C3×C4—91no—
96C22×C3×C4—92no—
97C2×C4×D3—62no—
98C2×C3×C8—22no—
99C2×C3×C8—42no—
100C2×C3×C8—92no—
101C2×C3×C8—91no—
102C2×C3×C8—181no—
103C2×C3×C8—181no—
104C2×C3×C8—181no—
105C2×C3×C8—181no—
106C2×C3×C8#15711yescharacteristic, complemented
107C3×C42—181no—
108C3×C42—181no—
109C2×C3⋊C8—62no—
110D3×C8—64no—
111D3×C8—64no—
112C22×C32—22no—
113C22×C32—42no—
114C22×C32C3×C4211yescharacteristic
115C2×C3⋊D3C3×C22⋊C411yescharacteristic
116C2×C3⋊D3C3×C4⋊C411yescharacteristic
117C2×C3⋊D3C2×C3×C812yes—
118C2×C3⋊D3C3×M1612yes—
119C2×C3×D3—62no—
120C2×C3×D3—62no—
121C2×C3×D3—64no—
122C2×C3×D3—64no—
123#157—24no—
124C32×C4—22no—
125C32×C4—22no—
126C32×C4—42no—
127C32×C4—42no—
128C32×C4C3×C22⋊C411yescharacteristic
129C32×C4C3×C4⋊C411yescharacteristic
130C3⋊Dic3C3×C22⋊C411yescharacteristic
131C3⋊Dic3C3×C4⋊C411yescharacteristic
132C3×Dic3—62no—
133C3×Dic3—62no—
134C22×C8—91no—
135C22×C8—91no—
136C2×C42—91no—
137C33#33611yescharacteristic, complemented, Sylow
138C23×C3—91no—
139C22×D3—62no—
140C2×C3×C4—22no—
141C2×C3×C4—42no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—94no—
148C2×C3×C4—182no—
149C2×C3×C4—182no—
150C2×C3×C4C2×#15711yescharacteristic
151C2×Dic3—62no—
152C4×D3—62no—
153C4×D3—62no—
154C4×D3—62no—
155C4×D3—62no—
156C3×C8—24no—
157C3×C8—21no—
158C3×C8—44no—
159C3×C8—182no—
160C3×C8—181no—
161C3⋊C8—64no—
162C2×C32—22no—
163C2×C32—22no—
164C2×C32—22no—
165C2×C32—42no—
166C2×C32—42no—
167C2×C32—42no—
168C2×C32C3×#9511yescharacteristic
169C2×C32C3×C4×C811yescharacteristic
170C2×C32C3×#9711yescharacteristic
171C3⋊D3C3×C22⋊C812yes—
172C3⋊D3C3×C4⋊C812yes—
173C3×D3—64no—
174C3×D3—64no—
175C22×C4—91no—
176C22×C4—92no—
177C2×C8—92no—
178C2×C8—91no—
179C2×C8—181no—
180C2×C8—181no—
181C2×C8—181no—
182C2×C8—181no—
183C2×C8C3×#15711yescharacteristic, complemented
184C42—181no—
185C42—181no—
186C22×C3—22no—
187C22×C3—42no—
188C22×C3—92no—
189C22×C3—92no—
190C22×C3—91no—
191C22×C3—91no—
192C22×C3C4×#15711yescharacteristic
193C2×D3—62no—
194C2×D3—62no—
195C2×D3—64no—
196C2×D3—64no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4—91no—
202C3×C4—91no—
203C3×C4—184no—
204C3×C4C4⋊#15711yescharacteristic
205C3×C4C22⋊#15711yescharacteristic
206Dic3—62no—
207Dic3—62no—
208C32—22no—
209C32—42no—
210C32C3×#33611yescharacteristic, complemented
211C23—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—94no—
218C2×C4—182no—
219C2×C4—182no—
220C2×C4C2×C3×#15711yescharacteristic
221C8—21no—
222C8—182no—
223C8—181no—
224C2×C3—22no—
225C2×C3—22no—
226C2×C3—22no—
227C2×C3—42no—
228C2×C3—42no—
229C2×C3—42no—
230C2×C3—92no—
231C2×C3—92no—
232C2×C3C8×#15711yescharacteristic
233C2×C3#6376711yescharacteristic
234C2×C3#6378111yescharacteristic
235D3—64no—
236D3—64no—
237C22—92no—
238C22—92no—
239C22—91no—
240C22—91no—
241C22C3×C4×#15711yescharacteristic
242C4—91no—
243C4—91no—
244C4—184no—
245C4C3×C4⋊#15711yescharacteristic
246C4C3×C22⋊#15711yescharacteristic
247C3—22no—
248C3—42no—
249C3#1059018111yescharacteristic, direct product factor
250C2—92no—
251C2—92no—
252C2C3×C8×#15711yescharacteristic
253C2C3×#6376711yescharacteristic
254C2C3×#6378111yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×C32C3×C4×C8
FrattiniC2×C4C2×C3×#157
FittingC2×C33×C8C4
socleC22×C33C42
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC2×C3×C8#157
hypocenterC32C3×#336
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. #10590181C3direct (N × H)
7. C2×C3×#622C4semidirect
12. C2×C33×C8C4semidirect
13. C2×C3×C3⋊(C3⋊C8)C4semidirect
23. C2×C3×#157C8semidirect
37. C2×#622C3×C4semidirect
44. C2×C32×C8C3×C4semidirect
45. C2×C3⋊(C3⋊C8)C3×C4semidirect
69. C2×#157C3×C8semidirect
106. C2×C3×C8#157semidirect
137. C33#336semidirect
183. C2×C8C3×#157semidirect
210. C32C3×#336semidirect
249. C3#10590181direct (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×#37872
C2×C3×C8×C3⋊D32
#105901813
C3×#3369