Finite Group Databasebeta

← C2×C3⋊(#131⋊C9) #13147284

Subgroups — C2×C3⋊(#131⋊C9) #13147284

Order 1728.

315 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 77 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⋊C9C211yesmaximal, characteristic
2C3⋊(#131⋊C9)C212yesmaximal, direct product factor
3C2×C3×#948C311yesmaximal, characteristic
4C2×#131⋊C9—31nomaximal
5C3×#41⋊C9C411yescharacteristic
6C3×#41⋊C9C2211yescharacteristic
7C2×C3×Q8⋊C9C411yescharacteristic
8C2×C9×C3⋊C8—41nomaximal
9C2×C32×#41C2×C311yescharacteristic
10C3×#948—32no—
11C3×#948C2×C312yes—
12C2×C3×#208—31no—
13C22×C3×C3⋊C8—31no—
14C2×#41⋊C9—21no—
15C2×#41⋊C9D311yescharacteristic
16#131⋊C9—32no—
17C2×C3×C4×C9—41no—
18C3×Q8⋊C9C2×C411yescharacteristic
19C9×C3⋊C8—42no—
20C2×C3×#131—31no—
21C2×#948C911yescharacteristic, complemented
22C32×#41—31no—
23C32×#41—31no—
24C32×#41C3×C411yescharacteristic
25C32×#41C22×C311yescharacteristic
26C2×C32×Q8C3×C411yescharacteristic
27C2×C32×D4—31no—
28C22×C32×C4—31no—
29C3×#208—32no—
30C3×#208—32no—
31C2×C3×C3⋊C8—31no—
32C2×C3×C3⋊C8—32no—
33C2×C3×C3⋊C8—32no—
34C2×C3×C3⋊C8A411yescharacteristic
35#41⋊C9—21no—
36#41⋊C9—21no—
37#41⋊C9Dic311yescharacteristic
38#41⋊C9C2×D311yescharacteristic
39C2×Q8⋊C9—21no—
40C2×Q8⋊C9Dic311yescharacteristic
41C2×C8×C9—121no—
42C22×C3×C9—41no—
43C3×C4×C9—41no—
44C3×C4×C9—41no—
45C2×C3×#41—21no—
46C2×C3×#41C2×C911yescharacteristic
47C2×C3×#41C3×D311yescharacteristic
48C3×#131—32no—
49C3×#131—92no—
50C2×C3×M16—91no—
51C22×C3×C8—91no—
52#948—32no—
53#948C2×C912yescomplemented
54C2×#208—31no—
55C22×C3⋊C8—31no—
56C23×C32—31no—
57C32×Q8—31no—
58C32×Q8C2×C3×C411yescharacteristic
59C32×D4—31no—
60C32×D4—31no—
61C32×D4—32no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4C2×A411yescharacteristic
68C3×C3⋊C8—32no—
69C3×C3⋊C8C2×A412yes—
70C2×C4×C9—41no—
71C2×C4×C9—81no—
72Q8⋊C9—21no—
73Q8⋊C9C2×Dic311yescharacteristic
74C8×C9—122no—
75C2×#131—31noSylow
76C2×C3×C9—42no—
77C2×C3×C9—41no—
78C3×#41—21no—
79C3×#41—21no—
80C3×#41—31no—
81C3×#41—31no—
82C3×#41—31no—
83C3×#41—31no—
84C3×#41—61no—
85C3×#41—61no—
86C3×#41C4×C911yescharacteristic
87C3×#41C22×C911yescharacteristic
88C3×#41C3×Dic311yescharacteristic
89C3×#41C2×C3×D311yescharacteristic
90C2×C3×Q8—21no—
91C2×C3×Q8C4×C911yescharacteristic
92C2×C3×Q8C3×Dic311yescharacteristic
93C2×C3×D4—31no—
94C2×C3×D4—31no—
95C2×C3×D4—61no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—61no—
99C3×M16—92no—
100C3×M16—92no—
101C2×C3×C8—31no—
102C2×C3×C8—92no—
103C2×C3×C8—92no—
104C2×C3×C8—91no—
105#208—32no—
106#208—32no—
107C2×C3⋊C8—31no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—32no—
110C2×C3⋊C8C22⋊C911yescharacteristic
111C22×C32—31no—
112C22×C32—31no—
113C22×C32—62no—
114C22×C32C4×A411yescharacteristic
115C32×C4—31no—
116C32×C4—31no—
117C32×C4C4×A411yescharacteristic
118C32×C4C22×A411yescharacteristic
119C22×C9—41no—
120C22×C9—81no—
121C4×C9—41no—
122C4×C9—41no—
123C4×C9—81no—
124C4×C9—81no—
125C2×#41D3×C911yescharacteristic
126#131—32no—
127#131—92no—
128C2×M16—91no—
129C22×C8—91no—
130C3×C9—41noSylow
131C23×C3—31no—
132C23×C3—31no—
133C23×C3—61no—
134C3×Q8—21no—
135C3×Q8—31no—
136C3×Q8—31no—
137C3×Q8—61no—
138C3×Q8C2×C4×C911yescharacteristic
139C3×Q8C2×C3×Dic311yescharacteristic
140C3×D4—31no—
141C3×D4—31no—
142C3×D4—31no—
143C3×D4—31no—
144C3×D4—32no—
145C3×D4—32no—
146C3×D4—61no—
147C3×D4—61no—
148C3×D4—62no—
149C2×C3×C4—21no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4C2×C22⋊C911yescharacteristic
166C2×C3×C4D3×A411yescharacteristic
167C3×C8—32no—
168C3×C8—92no—
169C3⋊C8—32no—
170C3⋊C8C2×C22⋊C912yes—
171C2×C32—61no—
172C2×C32—61no—
173C2×C32#131⋊C312yes—
174C2×C32C2×C4×A411yescharacteristic
175C2×C9—42no—
176C2×C9—41no—
177C2×C9—82no—
178C2×C9—81no—
179#41—31no—
180#41—31no—
181#41C9×Dic311yescharacteristic
182#41C2×D3×C911yescharacteristic
183C2×Q8C9×Dic311yescharacteristic
184C2×D4—31no—
185C22×C4—31no—
186M16—92no—
187M16—92no—
188C2×C8—31no—
189C2×C8—92no—
190C2×C8—92no—
191C2×C8—91no—
192C22×C3—21no—
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—31no—
197C22×C3—61no—
198C22×C3—61no—
199C22×C3—62no—
200C22×C3—62no—
201C22×C3—122no—
202C22×C3C4×C22⋊C911yescharacteristic
203C22×C3Dic3×A411yescharacteristic
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—61no—
211C3×C4—61no—
212C3×C4C4×C22⋊C911yescharacteristic
213C3×C4C22×C22⋊C911yescharacteristic
214C3×C4Dic3×A411yescharacteristic
215C3×C4C2×D3×A411yescharacteristic
216C32C2×#131⋊C311yescharacteristic
217C9—41no—
218C9—81no—
219C23—31no—
220Q8—31no—
221Q8C2×C9×Dic311yescharacteristic
222D4—31no—
223D4—31no—
224D4—32no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—31no—
230C2×C4D3×C22⋊C911yescharacteristic
231C8—32no—
232C8—92no—
233C2×C3—22no—
234C2×C3—21no—
235C2×C3—61no—
236C2×C3—61no—
237C2×C3—61no—
238C2×C3—61no—
239C2×C3—121no—
240C2×C3—121no—
241C2×C3#131⋊C912yescomplemented
242C2×C3C2×C4×C22⋊C911yescharacteristic
243C2×C3C3⋊(#131⋊C3)12yes—
244C2×C3C2×Dic3×A411yescharacteristic
245C22—31no—
246C22—31no—
247C22—62no—
248C22Dic3×C22⋊C911yescharacteristic
249C4—31no—
250C4—31no—
251C4Dic3×C22⋊C911yescharacteristic
252C4C2×D3×C22⋊C911yescharacteristic
253C3—21no—
254C3C2×#131⋊C911yescharacteristic, complemented
255C3C2×C3⋊(#131⋊C3)11yescharacteristic
256C2—61no—
257C2—61no—
258C2C3⋊(#131⋊C9)12yesdirect product factor
259C2C2×Dic3×C22⋊C911yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
FrattiniC3×C4C2×D3×A4
Fitting, prime radicalC2×C32×#41C2×C3
socleC22×C32C4×A4
cosocleC32×#41C22×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(#131⋊C9)C2direct (N × H)
21. C2×#948C9semidirect
53. #948C2×C9semidirect
241. C2×C3#131⋊C9semidirect
254. C3C2×#131⋊C9semidirect
258. C2C3⋊(#131⋊C9)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C3⋊(#131⋊C9)2
C2×C3×#9483
C2×#131⋊C93
C2×C9×C3⋊C84