Finite Group Databasebeta

← C3×#10593301 #13166670

Subgroups — C3×#10593301 #13166670

Order 1728.

398 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 58 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×#64224C212yesmaximal, complemented
2C3×#63803C211yesmaximal, characteristic, complemented
3C3×#63797C211yesmaximal, characteristic, complemented
4C3×#63771C211yesmaximal, characteristic, complemented
5C3×#63727C212yesmaximal, complemented
6#10593301C311yesmaximal, characteristic, direct product factor
7C3×#3802C2212yes—
8C3×#3798C2211yescharacteristic
9C3×#3788C2212yes—
10C3×#3775—22no—
11C3×#3774—22no—
12C3×C8×C3⋊D3C2211yescharacteristic
13C32×D24—21no—
14C32×D3×C8—21no—
15C3×#3713—21no—
16C3×#3712C2211yescharacteristic
17C3×#3711—21no—
18#64224C2×C312yescomplemented
19C3×D3×D8—61no—
20C3×#901—61no—
21#63803C2×C311yescharacteristic, complemented
22#63797C2×C311yescharacteristic, complemented
23#63771C2×C311yescharacteristic, complemented
24#63727C2×C312yescomplemented
25C2×C3×D32—22no—
26C3×C4×C3⋊D3C2311yescharacteristic
27C32×D12—22no—
28C32×C4×D3—21no—
29C3×#627D411yescharacteristic
30C3×#626—21no—
31C3×#625—22no—
32C3×#625D411yescharacteristic
33C3×#622—22no—
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)D411yescharacteristic
36C32×C3⋊C8—21no—
37C3×#472—91nomaximal
38C3×D3×D4—62no—
39C3×#235—61no—
40#3802C22×C312yes—
41#3798C22×C311yescharacteristic
42#3788C22×C312yes—
43#3775—22no—
44#3774—22no—
45C32×D8—61no—
46C2×C32×C8—61no—
47C8×C3⋊D3C22×C311yescharacteristic
48C3×#213—62no—
49C3×#208—61no—
50C3×D24—21no—
51C3×D24—21no—
52C3×D24—41no—
53C3×ΓL(1,𝔽25)—61no—
54C3×ΓL(1,𝔽25)—61no—
55C3×D3×C8—21no—
56C3×D3×C8—21no—
57C3×D3×C8—41no—
58C3×D3×C8—61no—
59C3×D3×C8—61no—
60#3713—21no—
61#3712C22×C311yescharacteristic
62#3711—21no—
63C2×C3×C3⋊D3#4111yescharacteristic
64C2×C32×D3—21no—
65C2×C32×D3—42no—
66C3×D32—82no—
67C33×C4C2×D411yescharacteristic
68C3×C3⋊Dic3C2×D411yescharacteristic
69C32×Dic3—21no—
70C3×AGL(1,ℤ/8)—92no—
71C2×C3×D8—91no—
72C3×#131—91no—
73D3×D8—61no—
74#901—61no—
75C3×#108—91no—
76C3×#100—92no—
77C22×C3×D3—122no—
78C2×D32—22no—
79C32×D4—62no—
80C2×C32×C4—61no—
81C4×C3⋊D3C23×C311yescharacteristic
82C3×#67—61no—
83C3×#67—61no—
84C3×#67—122no—
85C3×D12—22no—
86C3×D12—22no—
87C3×D12—42no—
88C3×D12—61no—
89C3×C4×D3—21no—
90C3×C4×D3—21no—
91C3×C4×D3—41no—
92C3×C4×D3—61no—
93C3×C4×D3—61no—
94C3×Dic6—61no—
95#627C3×D411yescharacteristic
96#626—21no—
97#625—22no—
98#625C3×D411yescharacteristic
99#622—22no—
100C32×C8—21no—
101C32×C8—21no—
102C32×C8—41no—
103C32×C8—41no—
104C32×C8—61no—
105C32×C8C3×D411yescharacteristic
106C3⋊(C3⋊C8)C3×D411yescharacteristic
107C3×C3⋊C8—21no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8—61no—
111C3×C3⋊C8—61no—
112#472—91noSylow
113C2×C33#12111yescharacteristic
114C3×C3⋊D3—21no—
115C32×D3—41no—
116C32×D3—82no—
117C3×#41—91no—
118C2×C3×D4—92no—
119D3×D4—62no—
120#235—61no—
121C3×SD16—182no—
122C3×D8—61no—
123C3×D8—61no—
124C3×D8—121no—
125C3×D8—181no—
126C3×D8—182no—
127C3×M16—91no—
128C3×M16—92no—
129C3×M16—181no—
130C2×C3×C8—61no—
131C2×C3×C8—61no—
132C2×C3×C8—91no—
133C2×C3×C8—121no—
134#213—62no—
135#208—61no—
136D24—21no—
137ΓL(1,𝔽25)—61no—
138ΓL(1,𝔽25)—61no—
139D3×C8—21no—
140D3×C8—61no—
141D3×C8—61no—
142C22×C32—61no—
143C22×C32—122no—
144C2×C3⋊D3C3×#4111yescharacteristic
145C2×C3×D3—21no—
146C2×C3×D3—21no—
147C2×C3×D3—41no—
148C2×C3×D3—42no—
149C2×C3×D3—42no—
150C2×C3×D3—61no—
151C2×C3×D3—61no—
152C2×C3×D3—82no—
153C2×C3×D3—244no—
154D32—82no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4—61no—
160C32×C4C2×C3×D411yescharacteristic
161C3⋊Dic3C2×C3×D411yescharacteristic
162C3×Dic3—21no—
163C3×Dic3—21no—
164C3×Dic3—41no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167AGL(1,ℤ/8)—92no—
168C2×D8—91no—
169#131—91no—
170#108—91no—
171#100—92no—
172C33#47211yescharacteristic, complemented, Sylow
173C23×C3—182no—
174C22×D3—122no—
175C3×Q8—91no—
176C3×D4—62no—
177C3×D4—62no—
178C3×D4—91no—
179C3×D4—122no—
180C3×D4—181no—
181C3×D4—182no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—91no—
185C2×C3×C4—121no—
186#67—61no—
187#67—61no—
188#67—122no—
189D12—22no—
190D12—61no—
191C4×D3—21no—
192C4×D3—61no—
193C4×D3—61no—
194Dic6—61no—
195C3×C8—21no—
196C3×C8—21no—
197C3×C8—41no—
198C3×C8—41no—
199C3×C8—61no—
200C3×C8—61no—
201C3×C8—91no—
202C3×C8—121no—
203C3×C8—182no—
204C3×C8GO+(4,𝔽2)11yescharacteristic
205C3⋊C8—21no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—121no—
213C2×C32—242no—
214C2×C32C3×#12111yescharacteristic
215C3⋊D3—21no—
216C3×D3—41no—
217C3×D3—42no—
218C3×D3—41no—
219C3×D3—62no—
220C3×D3—82no—
221C3×D3—82no—
222C3×D3—121no—
223C3×D3—162no—
224#41—91no—
225C2×D4—92no—
226SD16—182no—
227D8—61no—
228D8—181no—
229D8—182no—
230M16—91no—
231M16—92no—
232M16—181no—
233C2×C8—61no—
234C2×C8—91no—
235C22×C3—61no—
236C22×C3—61no—
237C22×C3—91no—
238C22×C3—121no—
239C22×C3—122no—
240C22×C3—122no—
241C22×C3—242no—
242C22×C3—722no—
243C2×D3—21no—
244C2×D3—42no—
245C2×D3—61no—
246C2×D3—61no—
247C2×D3—244no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—91no—
255C3×C4—121no—
256C3×C4C2×GO+(4,𝔽2)11yescharacteristic
257Dic3—21no—
258Dic3—61no—
259Dic3—61no—
260C32—21no—
261C32—21no—
262C32—41no—
263C32—41no—
264C32C3×#47211yescharacteristic, complemented
265C23—182no—
266Q8—91no—
267D4—62no—
268D4—91no—
269D4—181no—
270D4—182no—
271C2×C4—61no—
272C2×C4—91no—
273C8—61no—
274C8—91no—
275C8—182no—
276C8C3×GO+(4,𝔽2)11yescharacteristic
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—121no—
282C2×C3—122no—
283C2×C3—121no—
284C2×C3—181no—
285C2×C3—242no—
286C2×C3—242no—
287C2×C3—482no—
288C2×C3#6423011yescharacteristic
289D3—41no—
290D3—62no—
291D3—82no—
292D3—121no—
293C22—61no—
294C22—91no—
295C22—122no—
296C22—722no—
297C4—61no—
298C4—91no—
299C4C2×C3×GO+(4,𝔽2)11yescharacteristic
300C3—21no—
301C3—21no—
302C3—41no—
303C3—41no—
304C3#1059330111yescharacteristic, direct product factor
305C2—121no—
306C2—181no—
307C2—242no—
308C2C3×#6423011yescharacteristic
subgroupG/H
centerC2×C3#64230
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#121
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#472
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#64224C2semidirect
2. C3×#63803C2semidirect
3. C3×#63797C2semidirect
4. C3×#63771C2semidirect
5. C3×#63727C2semidirect
6. #10593301C3direct (N × H)
18. #64224C2×C3semidirect
21. #63803C2×C3semidirect
22. #63797C2×C3semidirect
23. #63771C2×C3semidirect
24. #63727C2×C3semidirect
172. C33#472semidirect
264. C32C3×#472semidirect
304. C3#10593301direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642242
C3×#638032
C3×#637972
C3×#637712
C3×#637272
#105933013
C3×#4729