Finite Group Databasebeta

← C3×#10589671 #13152670

Subgroups — C3×#10589671 #13152670

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×Q8×C9C211yesmaximal, characteristic
2C3×Q8×Dic9C211yesmaximal, characteristic
3C3×Q8×Dic9C211yesmaximal, characteristic
4C3×#63452C213yesmaximal
5C3×#63450C213yesmaximal
6C3×#63432C213yesmaximal
7C3×C4×Dic18C213yesmaximal
8C3×#5569—31nomaximal
9#10589671C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×Dic18C2213yes—
12C3×C9×C4⋊C4C2213yes—
13C3×C42×C9C2213yes—
14C3×C4⋊Dic9C2213yes—
15C3×C4⋊Dic9C2213yes—
16C3×C4⋊Dic9C2213yes—
17C3×Dic9⋊C4C2211yescharacteristic
18C3×Dic9⋊C4C2213yes—
19C3×Dic9⋊C4C2216yes—
20C3×C4×Dic9C2213yes—
21C3×C4×Dic9C2213yes—
22C32×C4×Q8D311yescharacteristic
23C3×Q8×Dic3—31no—
24C3×Q8×Dic3—31no—
25C3×#890—33no—
26C3×#888—33no—
27C3×#870—33no—
28C3×C4×Dic6—33no—
29C4×Q8×C9—21no—
30C4×Q8×C9C2×C311yescharacteristic
31Q8×Dic9C2×C311yescharacteristic
32Q8×Dic9C2×C311yescharacteristic
33#63452C2×C313yes—
34#63450C2×C313yes—
35#63432C2×C313yes—
36C4×Dic18C2×C313yes—
37C3×Q8×C9Q814yescomplemented
38C2×C3×C4×C9C2313yes—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3×C4×C9C2313yes—
41C2×C3×Dic9C2313yes—
42C2×C3×Dic9C2311yescharacteristic
43C2×C3×Dic9C2313yes—
44C2×C3×Dic9C2311yescharacteristic
45C3×Dic18—26no—
46C3×#557—91no—
47#5569—31no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C3×Dic6—33no—
50C32×C4⋊C4C2×D313yes—
51C32×C42C2×D313yes—
52C3×C4⋊Dic3—33no—
53C3×C4⋊Dic3—33no—
54C3×C4⋊Dic3—33no—
55C3×Dic3⋊C4—36no—
56C3×Dic3⋊C4—31no—
57C3×Dic3⋊C4—33no—
58C3×C4×Dic3—33no—
59C3×C4×Dic3—33no—
60C2×Q8×C9—21no—
61C2×Q8×C9C22×C311yescharacteristic
62C2×Dic18C22×C313yes—
63C9×C4⋊C4—23no—
64C9×C4⋊C4C22×C313yes—
65C42×C9—23no—
66C42×C9C22×C313yes—
67C4⋊Dic9C22×C313yes—
68C4⋊Dic9C22×C313yes—
69C4⋊Dic9C22×C313yes—
70Dic9⋊C4C22×C311yescharacteristic
71Dic9⋊C4C22×C313yes—
72Dic9⋊C4C22×C316yes—
73C4×Dic9C22×C313yes—
74C4×Dic9C22×C313yes—
75C22×C3×C9C2411yescharacteristic
76C3×C4×C9—23no—
77C3×C4×C9—21no—
78C3×C4×C9C2×Q816yes—
79C3×Dic9—23no—
80C3×Dic9—21no—
81C3×Dic9—23no—
82C3×Dic9#4112yes—
83C3×C4⋊Q8—93no—
84C3×#125—93no—
85C3×#125—93no—
86C3×C4×Q8—21no—
87C3×C4×Q8—91no—
88C3×C4×Q8—91no—
89C3×C4×Q8—93no—
90C3×C4×Q8D911yescharacteristic
91C3×C4×Q8C3×D311yescharacteristic
92Q8×Dic3—31no—
93Q8×Dic3—31no—
94#890—33no—
95#888—33no—
96#870—33no—
97C4×Dic6—33no—
98C32×Q8Dic614yes—
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D313yes—
101C2×C32×C4C22×D313yes—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—33no—
105C2×C3×Dic3—33no—
106C3×Dic6—66no—
107Q8×C9—24no—
108Q8×C9C3×Q814yescomplemented
109C2×C4×C9—21no—
110C2×C4×C9—23no—
111C2×C4×C9—23no—
112C2×C4×C9C23×C311yescharacteristic
113C2×C4×C9C23×C313yes—
114C2×C4×C9C23×C313yes—
115C2×Dic9C23×C311yescharacteristic
116C2×Dic9C23×C313yes—
117C2×Dic9C23×C313yes—
118C2×Dic9C23×C311yescharacteristic
119Dic18—26no—
120#557—91noSylow
121C2×C3×C9C22×Q811yescharacteristic
122C2×C3×C9C2×#4111yescharacteristic
123C2×C3×C92−1+411yescharacteristic
124C2×C3×Q8—21no—
125C2×C3×Q8—93no—
126C2×C3×Q8C2×D911yescharacteristic
127C2×C3×Q8C2×C3×D311yescharacteristic
128C2×Dic6—33no—
129C3×C4⋊C4—23no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—93no—
132C3×C4⋊C4—96no—
133C3×C4⋊C4—93no—
134C3×C4⋊C4—93no—
135C3×C4⋊C4—93no—
136C3×C4⋊C4C2×D913yes—
137C3×C4⋊C4C2×C3×D313yes—
138C3×C42—23no—
139C3×C42—93no—
140C3×C42—93no—
141C3×C42C2×D913yes—
142C3×C42C2×C3×D313yes—
143C4⋊Dic3—33no—
144C4⋊Dic3—33no—
145C4⋊Dic3—33no—
146Dic3⋊C4—31no—
147Dic3⋊C4—33no—
148Dic3⋊C4—36no—
149C4×Dic3—33no—
150C4×Dic3—33no—
151C22×C32C23×D311yescharacteristic
152C32×C4—21no—
153C32×C4—23no—
154C32×C4C2×Dic616yes—
155C3×Dic3—32no—
156C3×Dic3—61no—
157C3×Dic3—63no—
158C3×Dic3—63no—
159C22×C9—21no—
160C22×C9C24×C311yescharacteristic
161C4×C9—22no—
162C4×C9—23no—
163C4×C9—21no—
164C4×C9—26no—
165C4×C9—43no—
166C4×C9C2×C3×Q816yes—
167Dic9—23no—
168Dic9—21no—
169Dic9—23no—
170Dic9C3×#4112yes—
171C4⋊Q8—93no—
172#125—93no—
173#125—93no—
174C4×Q8—91no—
175C4×Q8—91no—
176C4×Q8—93no—
177C4×Q8C3×D911yescharacteristic
178C3×C9#55711yescharacteristic, complemented, Sylow
179C3×Q8—24no—
180C3×Q8—186no—
181C3×Q8Dic1814yescomplemented
182C3×Q8C3×Dic614yes—
183C2×C3×C4—21no—
184C2×C3×C4—23no—
185C2×C3×C4—23no—
186C2×C3×C4—91no—
187C2×C3×C4—91no—
188C2×C3×C4—93no—
189C2×C3×C4—93no—
190C2×C3×C4C22×D911yescharacteristic
191C2×C3×C4C22×D913yes—
192C2×C3×C4C22×D913yes—
193C2×C3×C4C22×C3×D311yescharacteristic
194C2×C3×C4C22×C3×D313yes—
195C2×C3×C4C22×C3×D313yes—
196C2×Dic3—31no—
197C2×Dic3—31no—
198C2×Dic3—33no—
199C2×Dic3—33no—
200Dic6—66no—
201C2×C32C22×Dic611yescharacteristic
202C2×C32#100711yescharacteristic
203C2×C32D3×#4111yescharacteristic
204C2×C9—21no—
205C2×C9—21no—
206C2×C9—21no—
207C2×C9C22×C3×Q811yescharacteristic
208C2×C9C2×C3×#4111yescharacteristic
209C2×C9C3×2−1+411yescharacteristic
210C2×Q8—93no—
211C2×Q8C2×C3×D911yescharacteristic
212C4⋊C4—91no—
213C4⋊C4—93no—
214C4⋊C4—93no—
215C4⋊C4—93no—
216C4⋊C4—93no—
217C4⋊C4—96no—
218C4⋊C4C2×C3×D913yes—
219C42—93no—
220C42—93no—
221C42C2×C3×D913yes—
222C22×C3—21no—
223C22×C3C23×D911yescharacteristic
224C22×C3C23×C3×D311yescharacteristic
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—23no—
228C3×C4—21no—
229C3×C4—23no—
230C3×C4—26no—
231C3×C4—43no—
232C3×C4—92no—
233C3×C4—181no—
234C3×C4—183no—
235C3×C4—183no—
236C3×C4C2×Dic1816yes—
237C3×C4C2×C3×Dic616yes—
238Dic3—32no—
239Dic3—61no—
240Dic3—63no—
241Dic3—63no—
242C32#556911yescharacteristic
243C9—21no—
244C9C3×#55711yescharacteristic, complemented
245Q8—186no—
246Q8C3×Dic1814yescomplemented
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—93no—
250C2×C4—93no—
251C2×C4C22×C3×D911yescharacteristic
252C2×C4C22×C3×D913yes—
253C2×C4C22×C3×D913yes—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3C22×Dic1811yescharacteristic
258C2×C3#6371311yescharacteristic
259C2×C3#41×D911yescharacteristic
260C2×C3C22×C3×Dic611yescharacteristic
261C2×C3C3×#100711yescharacteristic
262C2×C3C3×D3×#4111yescharacteristic
263C22C23×C3×D911yescharacteristic
264C4—21no—
265C4—23no—
266C4—92no—
267C4—181no—
268C4—183no—
269C4—183no—
270C4C2×C3×Dic1816yes—
271C3—21no—
272C3#1058967111yescharacteristic, direct product factor
273C3C3×#556911yescharacteristic
274C2C22×C3×Dic1811yescharacteristic
275C2C3×#6371311yescharacteristic
276C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socle, prime radicalC22×C32C23×D3
hypercenterC3×C4×Q8D9
hypocenterC9C3×#557
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
9. #10589671C3direct (N × H)
37. C3×Q8×C9Q8semidirect
108. Q8×C9C3×Q8semidirect
178. C3×C9#557semidirect
181. C3×Q8Dic18semidirect
244. C9C3×#557semidirect
246. Q8C3×Dic18semidirect
272. C3#10589671direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×Q8×C92
C3×Q8×Dic92
C3×#634522
C3×#634502
C3×#634322
C3×C4×Dic182
C3×#55693
#105896713