Finite Group Databasebeta

← C2×C3×#63692 #13164423

Subgroups — C2×C3×#63692 #13164423

Order 1728.

639 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 84 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, complemented
2C3×#63692C214yesmaximal, direct product factor
3C2×C3×#3689C212yesmaximal, complemented
4C2×C3×#949—31nomaximal
5C2×#63692C311yesmaximal, characteristic, direct product factor
6C2×C3×D36—41nomaximal
7C3×#41⋊C9C2212yescomplemented
8C2×C3×Q8⋊C9C2211yescharacteristic, complemented
9C3×#3689C2214yescomplemented
10C2×C32×#41D311yescharacteristic
11C22×C3×D12—31no—
12C3×#949—34no—
13C3×#949—34no—
14C2×C3×#215—32no—
15C2×C3×#213—32no—
16C2×C3×#208—31no—
17C2×#41⋊C9—21no—
18C2×#41⋊C9C2×C311yescharacteristic, complemented
19#63692C2×C314yesdirect product factor
20C2×#3689C2×C312yescomplemented
21C22×C3×D9—42no—
22C2×C3×C4×C9—41no—
23C3×D36—44no—
24C3×Q8⋊C9C2311yescharacteristic, complemented
25C2×C3×AGL(1,ℤ/8)—91no—
26C2×#949—31no—
27C23×C3×D3—61no—
28C32×#41—32no—
29C32×#41C2×D312yes—
30C2×C32×Q8C2×D311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×C3×D12—32no—
34C2×C3×D12—34no—
35C2×C3×D12—62no—
36C2×C3×D12—61no—
37C3×#215—34no—
38C3×#215—34no—
39C3×#213—38no—
40C3×#208—34no—
41C2×C3×C3⋊C8—32no—
42C2×D36—41no—
43#41⋊C9—22no—
44#41⋊C9C22×C312yescomplemented
45C2×Q8⋊C9—21no—
46C2×Q8⋊C9C22×C311yescharacteristic, complemented
47#3689C22×C314yescomplemented
48C22×C3×C9—41no—
49C2×C3×D9—44no—
50C2×C3×D9—84no—
51C3×C4×C9—42no—
52C2×C3×#41—21no—
53C2×C3×#41D911yescharacteristic, complemented
54C2×C3×#41C3×D311yescharacteristic
55C22×C3×D4—91no—
56C22×D12—31no—
57C3×AGL(1,ℤ/8)—94no—
58C3×AGL(1,ℤ/8)—94no—
59C2×C3×SD16—92no—
60C2×C3×D8—92no—
61C2×C3×M16—91no—
62#949—34no—
63#949—34no—
64C2×#215—32no—
65C2×#213—32no—
66C2×#208—31no—
67C23×C32—31no—
68C22×C3×D3—64no—
69C22×C3×D3—62no—
70C22×C3×D3—124no—
71C32×Q8—31no—
72C32×Q8C22×D311yescharacteristic
73C32×D4—32no—
74C32×D4—61no—
75C2×C32×C4—31no—
76C2×C32×C4—32no—
77C2×C32×C4—32no—
78C2×C32×C4S411yescharacteristic
79C3×D12—34no—
80C3×D12—62no—
81C3×D12—64no—
82C22×D9—42no—
83C3×C3⋊C8—34no—
84C2×C4×C9—41no—
85C2×C4×C9—81no—
86D36—44no—
87Q8⋊C9—21no—
88Q8⋊C9C23×C311yescharacteristic, complemented
89C2×AGL(1,ℤ/8)—91noSylow
90C2×C3×C9—42no—
91C2×C3×C9—41no—
92C3×D9—84no—
93C24×C3—181no—
94C23×D3—61no—
95C3×#41—22no—
96C3×#41—32no—
97C3×#41—32no—
98C3×#41—62no—
99C3×#41C2×D912yescomplemented
100C3×#41C2×C3×D312yes—
101C2×C3×Q8—21no—
102C2×C3×Q8C2×D911yescharacteristic
103C2×C3×Q8C2×C3×D311yescharacteristic
104C2×C3×D4—31no—
105C2×C3×D4—31no—
106C2×C3×D4—61no—
107C2×C3×D4—92no—
108C2×C3×D4—94no—
109C2×C3×D4—182no—
110C2×C3×D4—181no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—61no—
114C2×D12—32no—
115C2×D12—34no—
116C2×D12—62no—
117C2×D12—61no—
118C3×SD16—94no—
119C3×SD16—94no—
120C3×D8—98no—
121C3×M16—94no—
122C2×C3×C8—92no—
123#215—34no—
124#215—34no—
125#213—38no—
126#208—34no—
127C2×C3⋊C8—32no—
128C22×C32—32no—
129C22×C32—62no—
130C22×C32C2×S411yescharacteristic
131C2×C3×D3—64no—
132C2×C3×D3—128no—
133C2×C3×D3—124no—
134C32×C4—31no—
135C32×C4—31no—
136C32×C4C2×S412yes—
137C22×C9—41no—
138C22×C9—81no—
139C2×D9—44no—
140C2×D9—84no—
141C4×C9—42no—
142C4×C9—82no—
143C2×#41C3×D911yescharacteristic, complemented
144C22×D4—91no—
145AGL(1,ℤ/8)—94no—
146AGL(1,ℤ/8)—94no—
147C2×SD16—92no—
148C2×D8—92no—
149C2×M16—91no—
150C3×C9—41noSylow
151C23×C3—31no—
152C23×C3—31no—
153C23×C3—61no—
154C23×C3—184no—
155C23×C3—182no—
156C23×C3—364no—
157C22×D3—64no—
158C22×D3—62no—
159C22×D3—124no—
160C3×Q8—21no—
161C3×Q8—31no—
162C3×Q8—31no—
163C3×Q8—61no—
164C3×Q8C22×D911yescharacteristic
165C3×Q8C22×C3×D311yescharacteristic
166C3×D4—32no—
167C3×D4—32no—
168C3×D4—62no—
169C3×D4—61no—
170C3×D4—61no—
171C3×D4—62no—
172C3×D4—94no—
173C3×D4—182no—
174C3×D4—184no—
175C2×C3×C4—21no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—61no—
185C2×C3×C4(C22⋊C9)⋊C211yescharacteristic
186C2×C3×C4C3×S411yescharacteristic
187D12—34no—
188D12—62no—
189D12—64no—
190C3×C8—94no—
191C3⋊C8—34no—
192C2×C32—62no—
193C2×C32#98612yes—
194C2×C32C22×S411yescharacteristic
195C3×D3—124no—
196C2×C9—42no—
197C2×C9—41no—
198C2×C9—82no—
199C2×C9—81no—
200D9—84no—
201C24—181no—
202#41—32no—
203#41C2×C3×D912yescomplemented
204C2×Q8C2×C3×D911yescharacteristic
205C2×D4—31no—
206C2×D4—92no—
207C2×D4—94no—
208C2×D4—182no—
209C2×D4—181no—
210C22×C4—31no—
211SD16—94no—
212SD16—94no—
213D8—98no—
214M16—94no—
215C2×C8—92no—
216C22×C3—21no—
217C22×C3—32no—
218C22×C3—32no—
219C22×C3—62no—
220C22×C3—62no—
221C22×C3—62no—
222C22×C3—122no—
223C22×C3—184no—
224C22×C3—368no—
225C22×C3—364no—
226C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
227C22×C3C2×C3×S411yescharacteristic
228C2×D3—64no—
229C2×D3—128no—
230C2×D3—124no—
231C3×C4—22no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4C2×(C22⋊C9)⋊C212yes—
239C3×C4C2×C3×S412yes—
240C32C2×#98611yescharacteristic
241C9—41no—
242C9—81no—
243C23—31no—
244C23—184no—
245C23—182no—
246C23—364no—
247Q8—31no—
248Q8C22×C3×D911yescharacteristic
249D4—32no—
250D4—61no—
251D4—94no—
252D4—182no—
253D4—184no—
254C2×C4—31no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4C3×(C22⋊C9)⋊C211yescharacteristic
258C8—94no—
259C2×C3—21no—
260C2×C3—22no—
261C2×C3—62no—
262C2×C3—62no—
263C2×C3—122no—
264C2×C3—364no—
265C2×C3#6369212yesdirect product factor
266C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
267C2×C3C3×#98612yes—
268C2×C3C22×C3×S411yescharacteristic
269D3—124no—
270C22—32no—
271C22—62no—
272C22—184no—
273C22—368no—
274C22—364no—
275C22C2×C3×(C22⋊C9)⋊C211yescharacteristic
276C4—31no—
277C4—31no—
278C4C2×C3×(C22⋊C9)⋊C212yes—
279C3—21no—
280C3C2×#6369211yescharacteristic, direct product factor
281C3C2×C3×#98611yescharacteristic
282C2—62no—
283C2—364no—
284C2C3×#6369212yesdirect product factor
285C2C22×C3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
centerC22×C3C2×(C22⋊C9)⋊C2
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C3×C4(C22⋊C9)⋊C2
LCSG ▷ Q8⋊C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. C3×#63692C2direct (N × H)
3. C2×C3×#3689C2semidirect
5. C2×#63692C3direct (N × H)
7. C3×#41⋊C9C22semidirect
8. C2×C3×Q8⋊C9C22semidirect
9. C3×#3689C22semidirect
18. C2×#41⋊C9C2×C3semidirect
19. #63692C2×C3direct (N × H)
20. C2×#3689C2×C3semidirect
24. C3×Q8⋊C9C23semidirect
44. #41⋊C9C22×C3semidirect
46. C2×Q8⋊C9C22×C3semidirect
47. #3689C22×C3semidirect
53. C2×C3×#41D9semidirect
88. Q8⋊C9C23×C3semidirect
99. C3×#41C2×D9semidirect
143. C2×#41C3×D9semidirect
203. #41C2×C3×D9semidirect
265. C2×C3#63692direct (N × H)
280. C3C2×#63692direct (N × H)
284. C2C3×#63692direct (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×#636922
C2×C3×#36892
C2×C3×#9493
C2×#636923
C2×C3×D364