Finite Group Databasebeta

← C3×#10593423 #13166789

Subgroups — C3×#10593423 #13166789

Order 1728.

524 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3787C211yesmaximal, characteristic, complemented
2C2×C3×#3774C211yesmaximal, characteristic, complemented
3C3×#63912C212yesmaximal, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593423C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#626—22no—
9C2×C3×#625C2212yes—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3787—22no—
13C3×#3774—22no—
14C32×(C2×D3)⋊C4—22no—
15C3×#3724C2212yes—
16C2×#3787C2×C311yescharacteristic, complemented
17C2×#3774C2×C311yescharacteristic, complemented
18C3×#884—62no—
19#63912C2×C312yescomplemented
20#63743C2×C311yescharacteristic, complemented
21#63739C2×C312yescomplemented
22C22×C3×C3⋊D3D411yescharacteristic
23C22×C32×D3—22no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4×C3⋊D3—21no—
27C3×C4×C3⋊D3—21no—
28C2×C32×Dic3—22no—
29C3×#626—44no—
30C3×#625—22no—
31C3×#625—22no—
32C3×#622—21no—
33C3×#622D412yescomplemented
34C3×#466—91nomaximal
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C2×C3×#67—62no—
37C2×C3×#67—62no—
38C2×C3×D12—62no—
39C2×C3×C4×D3—62no—
40C2×#626—22no—
41C2×#625C22×C312yes—
42C2×#622C22×C311yescharacteristic
43C2×#622C22×C311yescharacteristic
44#3787—22no—
45#3774—22no—
46C32×C22⋊C4—62no—
47C3×(C2×D3)⋊C4—22no—
48C3×(C2×D3)⋊C4—22no—
49C3×(C2×D3)⋊C4—42no—
50C3×Dic3⋊C4—62no—
51#3724C22×C312yes—
52C22×C33C2×D411yescharacteristic
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—21no—
55C2×C3×C3⋊D3D812yescomplemented
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C2×C32×D3—42no—
59C33×C4—21no—
60C3×C3⋊Dic3C2×D411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C32×Dic3—42no—
63C2×C3×D8—91no—
64C22×C3×C8—91no—
65C3×#121—92no—
66#884—62no—
67C3×#107—91no—
68C3×#102—92no—
69C23×C32—62no—
70C22×C3⋊D3C3×D411yescharacteristic
71C22×C3×D3—22no—
72C22×C3×D3—22no—
73C22×C3×D3—42no—
74C22×C3×D3—62no—
75C2×C32×C4—22no—
76C2×C32×C4—42no—
77C2×C32×C4—62no—
78C2×C32×C4C3×D411yescharacteristic
79C2×C3⋊Dic3C23×C311yescharacteristic
80C4×C3⋊D3—21no—
81C4×C3⋊D3—21no—
82C3×#67—122no—
83C3×#67—122no—
84C3×#67—124no—
85C2×C3×Dic3—22no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—42no—
88C2×C3×Dic3—62no—
89C3×D12—124no—
90C3×C4×D3—122no—
91C3×C4×D3—122no—
92#626—44no—
93#625—22no—
94#625—22no—
95#622—21no—
96#622C3×D412yescomplemented
97#466—91noSylow
98C2×C33#12111yescharacteristic
99C2×C33C2×D811yescharacteristic
100C2×C33#13511yescharacteristic
101C3×C3⋊D3—22no—
102C32×D3—82no—
103C2×C3×D4—92no—
104C2×C3×D4—182no—
105C22×C3×C4—91no—
106C2×#67—62no—
107C2×#67—62no—
108C2×D12—62no—
109C2×C4×D3—62no—
110C3×D8—182no—
111C2×C3×C8—91no—
112C2×C3×C8—91no—
113C2×C3×C8—182no—
114C3×C4⋊C4—92no—
115C3×C22⋊C4—62no—
116C3×C22⋊C4—62no—
117C3×C22⋊C4—122no—
118(C2×D3)⋊C4—22no—
119Dic3⋊C4—62no—
120C22×C32—22no—
121C22×C32—42no—
122C22×C32—122no—
123C22×C32—122no—
124C22×C32—122no—
125C22×C32C2×C3×D411yescharacteristic
126C2×C3⋊D3—21no—
127C2×C3⋊D3—21no—
128C2×C3⋊D3C3×D812yescomplemented
129C2×C3×D3—42no—
130C2×C3×D3—42no—
131C2×C3×D3—42no—
132C2×C3×D3—44no—
133C2×C3×D3—42no—
134C2×C3×D3—42no—
135C2×C3×D3—42no—
136C2×C3×D3—64no—
137C2×C3×D3—82no—
138C2×C3×D3—82no—
139C2×C3×D3—122no—
140C2×C3×D3—122no—
141C32×C4—21no—
142C32×C4—42no—
143C32×C4—82no—
144C32×C4—122no—
145C3⋊Dic3C2×C3×D411yescharacteristic
146C3⋊Dic3C3×#4111yescharacteristic
147C3×Dic3—42no—
148C3×Dic3—42no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—82no—
152C2×D8—91no—
153C22×C8—91no—
154#121—92no—
155#107—91no—
156#102—92no—
157C33#46611yescharacteristic, complemented, Sylow
158C23×C3—62no—
159C23×C3—62no—
160C23×C3—91no—
161C23×C3—122no—
162C22×D3—22no—
163C22×D3—62no—
164C3×D4—182no—
165C3×D4—182no—
166C3×D4—364no—
167C2×C3×C4—22no—
168C2×C3×C4—42no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—91no—
172C2×C3×C4—122no—
173C2×C3×C4—181no—
174C2×C3×C4—181no—
175C2×C3×C4GO+(4,𝔽2)11yescharacteristic
176#67—122no—
177#67—122no—
178#67—124no—
179C2×Dic3—22no—
180C2×Dic3—62no—
181D12—124no—
182C4×D3—122no—
183C4×D3—122no—
184C3×C8—92no—
185C3×C8—181no—
186C2×C32—22no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—42no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32—242no—
193C2×C32C3×#12111yescharacteristic
194C2×C32C2×C3×D811yescharacteristic
195C2×C32C3×#13511yescharacteristic
196C3⋊D3—22no—
197C3×D3—82no—
198C3×D3—84no—
199C3×D3—82no—
200C3×D3—124no—
201C2×D4—92no—
202C2×D4—182no—
203C22×C4—91no—
204D8—182no—
205C2×C8—91no—
206C2×C8—91no—
207C2×C8—182no—
208C4⋊C4—92no—
209C22⋊C4—62no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—92no—
213C22×C3—122no—
214C22×C3—122no—
215C22×C3—122no—
216C22×C3—124no—
217C22×C3—122no—
218C22×C3—122no—
219C22×C3—122no—
220C22×C3—181no—
221C22×C3—181no—
222C22×C3—242no—
223C22×C3—242no—
224C22×C3C2×GO+(4,𝔽2)11yescharacteristic
225C2×D3—42no—
226C2×D3—42no—
227C2×D3—42no—
228C2×D3—64no—
229C2×D3—122no—
230C2×D3—122no—
231C3×C4—21no—
232C3×C4—42no—
233C3×C4—82no—
234C3×C4—91no—
235C3×C4—91no—
236C3×C4—122no—
237C3×C4—122no—
238C3×C4—242no—
239Dic3—42no—
240Dic3—62no—
241Dic3—62no—
242C32—22no—
243C32—42no—
244C32C3×#46611yescharacteristic, complemented
245C23—62no—
246C23—91no—
247D4—182no—
248D4—182no—
249D4—364no—
250C2×C4—62no—
251C2×C4—91no—
252C2×C4—181no—
253C2×C4—181no—
254C2×C4C3×GO+(4,𝔽2)11yescharacteristic
255C8—92no—
256C8—181no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—42no—
263C2×C3—182no—
264C2×C3—242no—
265C2×C3—244no—
266C2×C3—242no—
267C2×C3#6422311yescharacteristic
268C2×C3#6422511yescharacteristic
269C2×C3#6424211yescharacteristic
270D3—82no—
271D3—124no—
272C22—92no—
273C22—122no—
274C22—122no—
275C22—122no—
276C22—181no—
277C22—181no—
278C22C2×C3×GO+(4,𝔽2)11yescharacteristic
279C4—21no—
280C4—91no—
281C4—91no—
282C4—122no—
283C3—22no—
284C3—42no—
285C3#1059342311yescharacteristic, direct product factor
286C2—182no—
287C2—242no—
288C2C3×#6422311yescharacteristic
289C2C3×#6422511yescharacteristic
290C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#466
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3787C2semidirect
2. C2×C3×#3774C2semidirect
3. C3×#63912C2semidirect
4. C3×#63743C2semidirect
5. C3×#63739C2semidirect
6. #10593423C3direct (N × H)
16. C2×#3787C2×C3semidirect
17. C2×#3774C2×C3semidirect
19. #63912C2×C3semidirect
20. #63743C2×C3semidirect
21. #63739C2×C3semidirect
33. C3×#622D4semidirect
55. C2×C3×C3⋊D3D8semidirect
96. #622C3×D4semidirect
128. C2×C3⋊D3C3×D8semidirect
157. C33#466semidirect
244. C32C3×#466semidirect
285. C3#10593423direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C2×C3×#37742
C3×#639122
C3×#637432
C3×#637392
#105934233
C3×#4669