Finite Group Databasebeta

← C3×#10590153 #13157078

Subgroups — C3×#10590153 #13157078

Order 1728.

516 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C3×#64091C211yesmaximal, characteristic
2C3×#63788C211yesmaximal, characteristic, complemented
3C3×#63786C211yesmaximal, characteristic, complemented
4#10590153C311yesmaximal, characteristic, direct product factor
5C23×C3×C3⋊D3C411yescharacteristic, complemented
6C2×C3×#638C2211yescharacteristic
7C2×C3×C4⋊(C3⋊D3)—21no—
8C3×C22⋊#157—21no—
9C3×#3792—21no—
10C33×C22⋊C4C411yescharacteristic, complemented
11C3×#3752—21no—
12#64091C2×C311yescharacteristic
13C3×#882—62no—
14#63788C2×C311yescharacteristic, complemented
15#63786C2×C311yescharacteristic, complemented
16C23×C33C2×C411yescharacteristic
17C22×C3×C3⋊D3—21no—
18C22×C3×C3⋊D3—42no—
19C22×C3×C3⋊D3—41no—
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C3×#157—41no—
22C2×C33×C4—21no—
23C3×#638—41no—
24C2×C3×C3⋊Dic3D411yescharacteristic
25C3×C4⋊(C3⋊D3)—42no—
26C3×#622—41no—
27C3×AΣL(1,𝔽16)—91nomaximal
28C23×C3⋊D3C3×C411yescharacteristic, complemented
29C23×C3×D3—62no—
30C2×#638C22×C311yescharacteristic
31C2×C4⋊(C3⋊D3)—21no—
32C2×C3×#67—62no—
33C2×C3×D12—64no—
34C22⋊#157—21no—
35#3792—21no—
36C32×C22⋊C4—22no—
37C32×C22⋊C4—42no—
38C32×C22⋊C4C3×C411yescharacteristic, complemented
39#3752—21no—
40C3×(C2×D3)⋊C4—64no—
41C22×C33—21no—
42C22×C33—41no—
43C22×C33C22⋊C411yescharacteristic
44C2×C3×C3⋊D3—22no—
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3—41no—
47C2×C3×C3⋊D3—42no—
48C2×C3×C3⋊D3—41no—
49C2×C3×C3⋊D3—41no—
50C2×C3×C3⋊D3—42no—
51C3×#157—81no—
52C33×C4—41no—
53C3×C3⋊Dic3—21no—
54C3×#120—91no—
55#882—62no—
56C3×#100—91no—
57C3×#99—91no—
58C23×C32—22no—
59C23×C32—42no—
60C23×C32C2×C3×C411yescharacteristic
61C22×C3⋊D3—21no—
62C22×C3⋊D3—42no—
63C22×C3⋊D3—41no—
64C22×C3⋊D3C3×D411yescharacteristic
65C22×C3×D3—62no—
66C22×C3×D3—64no—
67C22×C3×D3—128no—
68C22×C3×D3—124no—
69C2×#157—41no—
70C2×C32×C4—21no—
71C2×C32×C4—24no—
72C2×C32×C4—44no—
73#638—41no—
74C2×C3⋊Dic3C3×D411yescharacteristic
75C4⋊(C3⋊D3)—42no—
76C3×#67—124no—
77C2×C3×Dic3—62no—
78C3×D12—128no—
79#622—41no—
80AΣL(1,𝔽16)—91noSylow
81C2×C33—21no—
82C2×C33—41no—
83C2×C33#9911yescharacteristic
84C3×C3⋊D3—42no—
85C3×C3⋊D3—41no—
86C24×C3—91no—
87C23×D3—62no—
88C2×C3×D4—91no—
89C2×C3×D4—181no—
90C2×#67—62no—
91C2×D12—64no—
92C3×M16—181no—
93C3×C22⋊C4—22no—
94C3×C22⋊C4—42no—
95C3×C22⋊C4—181no—
96C3×C22⋊C4—181no—
97C3×C22⋊C4#15711yescharacteristic, complemented
98(C2×D3)⋊C4—64no—
99C22×C32—21no—
100C22×C32—24no—
101C22×C32—22no—
102C22×C32—44no—
103C22×C32—42no—
104C22×C32—41no—
105C22×C32—44no—
106C22×C32—84no—
107C22×C32C3×C22⋊C411yescharacteristic
108C2×C3⋊D3—22no—
109C2×C3⋊D3—21no—
110C2×C3⋊D3—41no—
111C2×C3⋊D3—42no—
112C2×C3⋊D3—41no—
113C2×C3⋊D3—41no—
114C2×C3⋊D3—42no—
115C2×C3×D3—68no—
116C2×C3×D3—122no—
117C2×C3×D3—124no—
118C2×C3×D3—124no—
119C2×C3×D3—128no—
120C2×C3×D3—128no—
121C2×C3×D3—124no—
122#157—81no—
123C32×C4—41no—
124C32×C4—44no—
125C32×C4—84no—
126C3⋊Dic3—21no—
127C3×Dic3—122no—
128#120—91no—
129#100—91no—
130#99—91no—
131C33AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
132C23×C3—22no—
133C23×C3—42no—
134C23×C3—91no—
135C23×C3—181no—
136C23×C3—361no—
137C23×C3—362no—
138C23×C3C2×#15711yescharacteristic
139C22×D3—62no—
140C22×D3—64no—
141C22×D3—128no—
142C22×D3—124no—
143C3×D4—362no—
144C3×D4—361no—
145C2×C3×C4—24no—
146C2×C3×C4—21no—
147C2×C3×C4—44no—
148C2×C3×C4—91no—
149C2×C3×C4—361no—
150#67—124no—
151C2×Dic3—62no—
152D12—128no—
153C3×C8—361no—
154C2×C32—21no—
155C2×C32—24no—
156C2×C32—22no—
157C2×C32—44no—
158C2×C32—42no—
159C2×C32—41no—
160C2×C32—44no—
161C2×C32—84no—
162C2×C32C3×#9911yescharacteristic
163C3⋊D3—42no—
164C3⋊D3—41no—
165C3×D3—128no—
166C3×D3—242no—
167C24—91no—
168C2×D4—91no—
169C2×D4—181no—
170M16—181no—
171C22⋊C4—181no—
172C22⋊C4—181no—
173C22⋊C4C3×#15711yescharacteristic, complemented
174C22×C3—24no—
175C22×C3—21no—
176C22×C3—22no—
177C22×C3—44no—
178C22×C3—41no—
179C22×C3—44no—
180C22×C3—42no—
181C22×C3—84no—
182C22×C3—182no—
183C22×C3—181no—
184C22×C3—361no—
185C22×C3—361no—
186C22×C3—362no—
187C22×C3—362no—
188C22×C3—361no—
189C22×C3C22⋊#15711yescharacteristic
190C2×D3—68no—
191C2×D3—122no—
192C2×D3—124no—
193C2×D3—124no—
194C2×D3—128no—
195C2×D3—128no—
196C2×D3—124no—
197C3×C4—44no—
198C3×C4—41no—
199C3×C4—84no—
200C3×C4—181no—
201C3×C4—721no—
202Dic3—122no—
203C32—22no—
204C32—42no—
205C32C3×AΣL(1,𝔽16)11yescharacteristic, complemented
206C23—91no—
207C23—181no—
208C23—361no—
209C23—362no—
210C23C2×C3×#15711yescharacteristic
211D4—362no—
212D4—361no—
213C2×C4—21no—
214C2×C4—91no—
215C2×C4—361no—
216C8—361no—
217C2×C3—24no—
218C2×C3—21no—
219C2×C3—22no—
220C2×C3—44no—
221C2×C3—44no—
222C2×C3—41no—
223C2×C3—42no—
224C2×C3—84no—
225C2×C3—361no—
226C2×C3—362no—
227C2×C3#6377411yescharacteristic
228D3—128no—
229D3—242no—
230C22—21no—
231C22—41no—
232C22—182no—
233C22—181no—
234C22—361no—
235C22—361no—
236C22—362no—
237C22—362no—
238C22—361no—
239C22C3×C22⋊#15711yescharacteristic
240C4—41no—
241C4—181no—
242C4—721no—
243C3—22no—
244C3—42no—
245C3#1059015311yescharacteristic, direct product factor
246C2—21no—
247C2—41no—
248C2—361no—
249C2—362no—
250C2C3×#6377411yescharacteristic
subgroupG/H
centerC2×C3#63774
commutatorC23×C32C2×C3×C4
FrattiniC23C2×C3×#157
FittingC33×C22⋊C4C4
socleC2×C33#99
prime radicalC3×#64091C2
cosocleC2×#638C22×C3
hypercenterC3×C22⋊C4#157
hypocenterC32C3×AΣL(1,𝔽16)
LCSG ▷ C23×C32 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C23×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63788C2semidirect
3. C3×#63786C2semidirect
4. #10590153C3direct (N × H)
5. C23×C3×C3⋊D3C4semidirect
10. C33×C22⋊C4C4semidirect
14. #63788C2×C3semidirect
15. #63786C2×C3semidirect
28. C23×C3⋊D3C3×C4semidirect
38. C32×C22⋊C4C3×C4semidirect
97. C3×C22⋊C4#157semidirect
131. C33AΣL(1,𝔽16)semidirect
173. C22⋊C4C3×#157semidirect
205. C32C3×AΣL(1,𝔽16)semidirect
245. C3#10590153direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#640912
C3×#637882
C3×#637862
#105901533
C3×AΣL(1,𝔽16)9