Finite Group Databasebeta

← C3×#10589529 #13146920

Subgroups — C3×#10589529 #13146920

Order 1728.

331 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 48 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
2C2×C3×#3689C211yesmaximal, characteristic, complemented
3C3×#63421C211yesmaximal, characteristic, complemented
4C3×#5238—31nomaximal
5#10589529C311yesmaximal, characteristic, direct product factor
6C3×#41⋊C9—21no—
7C2×C3×Q8⋊C9C2211yescharacteristic
8C3×#3689—21no—
9C3×(C2×D9)⋊C4—41nomaximal
10C2×C32×#41D311yescharacteristic
11C2×C3×#215—31no—
12C2×C3×#213—31no—
13C3×#930—31no—
14C2×#41⋊C9—21no—
15C2×#41⋊C9C2×C311yescharacteristic, complemented
16C2×#3689C2×C311yescharacteristic, complemented
17C3×#835—31no—
18C3×#832—31no—
19C3×#830—31no—
20#63421C2×C311yescharacteristic, complemented
21C22×C3×D9—41no—
22C2×C3×C4×C9—41no—
23C2×C3×Dic9—41no—
24C3×Q8⋊C9D411yescharacteristic, complemented
25C3×#449—91no—
26#5238—31no—
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C3×#67—61no—
33C2×C3×D12—31no—
34C3×#215—61no—
35C3×#215—61no—
36C3×#213—62no—
37C3×(C2×D3)⋊C4—61no—
38C3×C4⋊Dic3—31no—
39C2×C3×C3⋊C8—31no—
40C2×C3×C3⋊C8—31no—
41#41⋊C9—22no—
42#41⋊C9—21no—
43C2×Q8⋊C9—21no—
44C2×Q8⋊C9C22×C311yescharacteristic
45#3689—21no—
46(C2×D9)⋊C4—41no—
47C22×C3×C9—41no—
48C2×C3×D9—42no—
49C2×C3×D9—81no—
50C2×C3×D9—81no—
51C3×C4×C9—81no—
52C3×Dic9—81no—
53C2×C3×#41—21no—
54C2×C3×#41D911yescharacteristic, complemented
55C2×C3×#41C3×D311yescharacteristic
56C2×C3×SD16—91no—
57C2×C3×D8—91no—
58C3×#121—91no—
59C2×#215—31no—
60C2×#213—31no—
61#930—31no—
62C3×#103—91no—
63C3×#102—91no—
64C3×C22⋊C8—91no—
65#835—31no—
66#832—31no—
67#830—31no—
68C23×C32—31no—
69C22×C3×D3—61no—
70C32×Q8—31no—
71C32×Q8#6711yescharacteristic
72C32×D4—32no—
73C32×D4—61no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77C2×C32×C4S411yescharacteristic
78C3×#67—122no—
79C2×C3×Dic3—61no—
80C3×D12—61no—
81C3×D12—61no—
82C22×D9—41no—
83C3×C3⋊C8—61no—
84C3×C3⋊C8—61no—
85C2×C4×C9—41no—
86C2×C4×C9—81no—
87C2×Dic9—41no—
88Q8⋊C9—21no—
89Q8⋊C9C3×D411yescharacteristic, complemented
90#449—91noSylow
91C2×C3×C9—41no—
92C2×C3×C9—41no—
93C2×C3×C9—41no—
94C3×D9—82no—
95C3×#41—22no—
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—61no—
99C3×#41—61no—
100C3×#41—62no—
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—91no—
108C2×C3×D4—181no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—61no—
112C2×#67—61no—
113C2×D12—31no—
114C3×SD16—181no—
115C3×SD16—181no—
116C3×D8—182no—
117C2×C3×C8—91no—
118C2×C3×C8—91no—
119C3×C4⋊C4—91no—
120C3×C22⋊C4—181no—
121#215—61no—
122#215—61no—
123#213—62no—
124(C2×D3)⋊C4—61no—
125C4⋊Dic3—31no—
126C2×C3⋊C8—31no—
127C2×C3⋊C8—31no—
128C22×C32—61no—
129C22×C32—61no—
130C22×C32—61no—
131C22×C32C2×S411yescharacteristic
132C2×C3×D3—121no—
133C2×C3×D3—121no—
134C2×C3×D3—121no—
135C32×C4—21no—
136C32×C4—31no—
137C32×C4—31no—
138C3×Dic3—121no—
139C22×C9—41no—
140C22×C9—81no—
141C2×D9—42no—
142C2×D9—81no—
143C2×D9—81no—
144C4×C9—81no—
145C4×C9—82no—
146Dic9—81no—
147C2×#41C3×D911yescharacteristic, complemented
148C2×SD16—91no—
149C2×D8—91no—
150#121—91no—
151#103—91no—
152#102—91no—
153C22⋊C8—91no—
154C3×C9—41noSylow
155C23×C3—31no—
156C23×C3—31no—
157C23×C3—61no—
158C23×C3—181no—
159C22×D3—61no—
160C3×Q8—21no—
161C3×Q8—31no—
162C3×Q8—31no—
163C3×Q8—61no—
164C3×Q8#61111yescharacteristic
165C3×Q8C3×#6711yescharacteristic
166C3×D4—32no—
167C3×D4—32no—
168C3×D4—62no—
169C3×D4—61no—
170C3×D4—61no—
171C3×D4—62no—
172C3×D4—181no—
173C3×D4—181no—
174C3×D4—362no—
175C2×C3×C4—21no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4—181no—
186C2×C3×C4(C22⋊C9)⋊C211yescharacteristic
187C2×C3×C4C3×S411yescharacteristic
188#67—122no—
189C2×Dic3—61no—
190D12—61no—
191D12—61no—
192C3×C8—181no—
193C3×C8—181no—
194C3⋊C8—61no—
195C3⋊C8—61no—
196C2×C32—121no—
197C2×C32#98511yescharacteristic
198C2×C32#98611yescharacteristic
199C2×C32GL(2,ℤ/4)11yescharacteristic
200C3×D3—241no—
201C2×C9—41no—
202C2×C9—41no—
203C2×C9—41no—
204C2×C9—81no—
205C2×C9—81no—
206C2×C9—81no—
207D9—82no—
208#41—21no—
209#41—61no—
210C2×Q8C2×C3×D911yescharacteristic
211C2×D4—31no—
212C2×D4—91no—
213C2×D4—181no—
214C22×C4—31no—
215SD16—181no—
216SD16—181no—
217D8—182no—
218C2×C8—91no—
219C2×C8—91no—
220C4⋊C4—91no—
221C22⋊C4—181no—
222C22×C3—21no—
223C22×C3—62no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3—61no—
227C22×C3—61no—
228C22×C3—61no—
229C22×C3—61no—
230C22×C3—121no—
231C22×C3—121no—
232C22×C3—361no—
233C22×C3—361no—
234C22×C3—361no—
235C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
236C22×C3C2×C3×S411yescharacteristic
237C2×D3—121no—
238C2×D3—121no—
239C2×D3—121no—
240C3×C4—22no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—361no—
250Dic3—121no—
251C32#542711yescharacteristic
252C9—41no—
253C9—81no—
254C23—31no—
255C23—181no—
256Q8—31no—
257Q8C3×#61111yescharacteristic
258D4—32no—
259D4—61no—
260D4—181no—
261D4—181no—
262D4—362no—
263C2×C4—31no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—181no—
267C2×C4C3×(C22⋊C9)⋊C211yescharacteristic
268C8—181no—
269C8—181no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—122no—
274C2×C3—121no—
275C2×C3—121no—
276C2×C3—721no—
277C2×C3#6369111yescharacteristic
278C2×C3#6369211yescharacteristic
279C2×C3#6369411yescharacteristic
280C2×C3C3×#98511yescharacteristic
281C2×C3C3×#98611yescharacteristic
282C2×C3C3×GL(2,ℤ/4)11yescharacteristic
283D3—241no—
284C22—61no—
285C22—61no—
286C22—61no—
287C22—361no—
288C22—361no—
289C22—361no—
290C22C2×C3×(C22⋊C9)⋊C211yescharacteristic
291C4—21no—
292C4—31no—
293C4—31no—
294C4—361no—
295C3—21no—
296C3#1058952911yescharacteristic, direct product factor
297C3C3×#542711yescharacteristic
298C2—121no—
299C2—721no—
300C2C3×#6369111yescharacteristic
301C2C3×#6369211yescharacteristic
302C2C3×#6369411yescharacteristic
subgroupG/H
centerC22×C3C2×(C22⋊C9)⋊C2
commutator, cosocleC2×Q8⋊C9C22×C3
FrattiniC22×C3C2×C3×S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C3×C4(C22⋊C9)⋊C2
hypocenterQ8⋊C9C3×D4
LCSG ▷ C2×Q8⋊C9 ▷ 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. C2×C3×#3689C2semidirect
3. C3×#63421C2semidirect
5. #10589529C3direct (N × H)
15. C2×#41⋊C9C2×C3semidirect
16. C2×#3689C2×C3semidirect
20. #63421C2×C3semidirect
24. C3×Q8⋊C9D4semidirect
54. C2×C3×#41D9semidirect
89. Q8⋊C9C3×D4semidirect
147. C2×#41C3×D9semidirect
296. C3#10589529direct (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
C2×C3×#36892
C3×#634212
C3×#52383
#105895293
C3×(C2×D9)⋊C44