Finite Group Databasebeta

← C4×C3⋊(#41⋊C9) #13147201

Subgroups — C4×C3⋊(#41⋊C9) #13147201

Order 1728.

344 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 79 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⋊(#41⋊C9)C211yesmaximal, characteristic
2C3×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
3Dic3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C3×C4×#239C311yesmaximal, characteristic
5C4×#41⋊C9—31nomaximal
6C2×C3×Q8⋊C9C2211yescharacteristic
7C3⋊(#41⋊C9)C412yesdirect product factor
8C4×C9×Dic3—41nomaximal
9C2×C3×#239C2×C311yescharacteristic
10C32×C4×Q8C2×C311yescharacteristic
11C3×Q8×Dic3C2×C311yescharacteristic
12C3×#893—31no—
13C3×#892—31no—
14C3×C4×D12—31no—
15C3×C42×D3—31no—
16C2×#41⋊C9—31no—
17C4×Q8⋊C9—21no—
18C4×Q8⋊C9—31no—
19C4×Q8⋊C9D311yescharacteristic, complemented
20C2×C3×C4×C9—41no—
21C2×C9×Dic3—41no—
22C2×C9×Dic3—41no—
23C3×Q8⋊C9C2×C411yescharacteristic, complemented
24C3×C4×#41—31no—
25C4×#239C911yescharacteristic, complemented
26C2×C32×Q8C22×C311yescharacteristic
27C3×#239—32no—
28C3×#239C3×C412yes—
29C2×C3×D12—31no—
30C2×C3×C4×D3—31no—
31C2×C3×C4×D3—31no—
32C2×C3×C4×D3—31no—
33C32×C4⋊C4—31no—
34C32×C42—31no—
35C3×(C2×D3)⋊C4—32no—
36C3×C4⋊Dic3—31no—
37C3×C4×Dic3—31no—
38C3×C4×Dic3—31no—
39C3×C4×Dic3A411yescharacteristic
40#41⋊C9—32no—
41C2×Q8⋊C9—21no—
42C2×Q8⋊C9C2×D311yescharacteristic
43C42×C9—121no—
44C22×C3×C9—41no—
45C3×C4×C9—42no—
46C9×Dic3—42no—
47C9×Dic3—42no—
48C2×C3×#41—31no—
49C2×#239C2×C911yescharacteristic
50C3×C4×Q8—21no—
51C3×C4×Q8—31no—
52C3×C4×Q8C2×C911yescharacteristic
53C3×C4×Q8C3×D311yescharacteristic
54C3×C4×D4—91no—
55C3×C4×D4—91no—
56C3×#117—91no—
57C2×C3×C42—91no—
58Q8×Dic3C2×C911yescharacteristic
59#893—31no—
60#892—31no—
61C4×D12—31no—
62C42×D3—31no—
63C22×C3×D3—31no—
64C32×Q8—31no—
65C32×Q8C2×C3×C411yescharacteristic
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4C2×A411yescharacteristic
69C2×C3×Dic3—31no—
70C2×C3×Dic3C2×A411yescharacteristic
71C2×C3×Dic3C2×A411yescharacteristic
72C3×D12—32no—
73C3×D12—32no—
74C3×C4×D3—32no—
75C3×C4×D3—32no—
76C3×C4×D3—62no—
77C3×C4×D3—62no—
78C2×C4×C9—41no—
79C2×C4×C9—81no—
80C2×C4×C9—121no—
81C2×C4×C9—121no—
82Q8⋊C9—21no—
83Q8⋊C9C4×D311yescharacteristic, complemented
84C4×#41—31noSylow
85C2×C3×C9—41no—
86C2×C3×C9—41no—
87C2×C3×C9—41no—
88C3×#41—32no—
89C3×#41—92no—
90C2×C3×Q8—21no—
91C2×C3×Q8C22×C911yescharacteristic
92C2×C3×Q8C2×C3×D311yescharacteristic
93C2×C3×D4—91no—
94C22×C3×C4—91no—
95C22×C3×C4—91no—
96C22×C3×C4—91no—
97#239—32no—
98#239C4×C912yescomplemented
99C2×D12—31no—
100C2×C4×D3—31no—
101C2×C4×D3—31no—
102C2×C4×D3—31no—
103C3×C4⋊C4—31no—
104C3×C4⋊C4—31no—
105C3×C4⋊C4—61no—
106C3×C4⋊C4—91no—
107C3×C22⋊C4—92no—
108C3×C42—31no—
109C3×C42—31no—
110C3×C42—31no—
111C3×C42—61no—
112C3×C42—91no—
113C3×C42—91no—
114(C2×D3)⋊C4—32no—
115C4⋊Dic3—31no—
116C4×Dic3—31no—
117C4×Dic3—31no—
118C4×Dic3C22⋊C911yescharacteristic
119C22×C32C22×A411yescharacteristic
120C2×C3×D3—32no—
121C2×C3×D3—61no—
122C2×C3×D3—61no—
123C32×C4—31no—
124C32×C4—31no—
125C32×C4—61no—
126C32×C4#41⋊C312yes—
127C3×Dic3—61no—
128C3×Dic3C4×A412yes—
129C3×Dic3#41⋊C312yes—
130C22×C9—41no—
131C22×C9—81no—
132C4×C9—42no—
133C4×C9—82no—
134C4×C9—122no—
135C4×C9—122no—
136C2×#41—31no—
137C4×Q8—31no—
138C4×Q8D3×C911yescharacteristic
139C4×D4—91no—
140C4×D4—91no—
141#117—91no—
142C2×C42—91no—
143C3×C9—41noSylow
144C23×C3—91no—
145C22×D3—31no—
146C3×Q8—21no—
147C3×Q8—31no—
148C3×Q8—31no—
149C3×Q8—61no—
150C3×Q8C2×C4×C911yescharacteristic
151C3×Q8C3×C4×D311yescharacteristic
152C3×D4—92no—
153C3×D4—92no—
154C2×C3×C4—21no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—61no—
162C2×C3×C4—61no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4—91no—
166C2×C3×C4—182no—
167C2×C3×C4—182no—
168C2×C3×C4C2×C22⋊C911yescharacteristic
169C2×C3×C4D3×A411yescharacteristic
170C2×Dic3—31no—
171C2×Dic3C2×C22⋊C911yescharacteristic
172C2×Dic3C2×C22⋊C911yescharacteristic
173D12—32no—
174D12—32no—
175C4×D3—32no—
176C4×D3—32no—
177C4×D3—62no—
178C4×D3—62no—
179C2×C32C2×C4×A411yescharacteristic
180C2×C32C2×#41⋊C311yescharacteristic
181C2×C32C2×#41⋊C311yescharacteristic
182C3×D3—62no—
183C2×C9—41no—
184C2×C9—41no—
185C2×C9—41no—
186C2×C9—81no—
187C2×C9—81no—
188C2×C9—81no—
189#41—32no—
190#41—92no—
191C2×Q8C2×D3×C911yescharacteristic
192C2×D4—91no—
193C22×C4—91no—
194C22×C4—91no—
195C22×C4—91no—
196C4⋊C4—31no—
197C4⋊C4—91no—
198C22⋊C4—92no—
199C42—31no—
200C42—31no—
201C42—91no—
202C42—91no—
203C22×C3—21no—
204C22×C3—92no—
205C22×C3—181no—
206C22×C3—181no—
207C22×C3C22×C22⋊C911yescharacteristic
208C22×C3C2×D3×A411yescharacteristic
209C2×D3—32no—
210C2×D3—61no—
211C2×D3—61no—
212C3×C4—22no—
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—32no—
216C3×C4—32no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—61no—
223C3×C4—121no—
224C3×C4—181no—
225C3×C4#41⋊C912yescomplemented
226C3×C4C3⋊(#41⋊C3)12yes—
227Dic3—61no—
228Dic3C4×C22⋊C912yes—
229Dic3#41⋊C912yescomplemented
230C32C4×#41⋊C311yescharacteristic
231C9—41no—
232C9—81no—
233C23—91no—
234Q8—31no—
235Q8C4×D3×C911yescharacteristic
236D4—92no—
237D4—92no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—92no—
243C2×C4—92no—
244C2×C4—91no—
245C2×C4—182no—
246C2×C4—182no—
247C2×C4D3×C22⋊C911yescharacteristic
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—182no—
252C2×C3C2×C4×C22⋊C911yescharacteristic
253C2×C3C2×#41⋊C911yescharacteristic
254C2×C3C2×#41⋊C911yescharacteristic
255C2×C3C4×D3×A411yescharacteristic
256C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
257C2×C3D3×#41⋊C311yescharacteristic
258D3—62no—
259C22—92no—
260C22—181no—
261C22—181no—
262C22C2×D3×C22⋊C911yescharacteristic
263C4—31no—
264C4—31no—
265C4—32no—
266C4—32no—
267C4—61no—
268C4—181no—
269C4C3⋊(#41⋊C9)12yesdirect product factor
270C3—21no—
271C3C4×#41⋊C911yescharacteristic, complemented
272C3C4×C3⋊(#41⋊C3)11yescharacteristic
273C2—182no—
274C2C4×D3×C22⋊C911yescharacteristic
275C2C2×C3⋊(#41⋊C9)11yescharacteristic
276C2D3×#41⋊C911yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
FrattiniC22×C3C2×D3×A4
FittingC32×C4×Q8C2×C3
socleC22×C32C22×A4
prime radicalC2×C3×#239C2×C3
cosocleC2×C32×Q8C22×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×Q8⋊C9C2semidirect
3. Dic3×Q8⋊C9C2semidirect
7. C3⋊(#41⋊C9)C4direct (N × H)
19. C4×Q8⋊C9D3semidirect
23. C3×Q8⋊C9C2×C4semidirect
25. C4×#239C9semidirect
83. Q8⋊C9C4×D3semidirect
98. #239C4×C9semidirect
225. C3×C4#41⋊C9semidirect
229. Dic3#41⋊C9semidirect
269. C4C3⋊(#41⋊C9)direct (N × H)
271. C3C4×#41⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(#41⋊C9)2
C3×C4×Q8⋊C92
Dic3×Q8⋊C92
C3×C4×#2393
C4×#41⋊C93
C4×C9×Dic34