Finite Group Databasebeta

← C3×#10595981 #13179261

Subgroups — C3×#10595981 #13179261

Order 1728.

2132 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C212yesmaximal, complemented
2C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#64139C214yesmaximal, complemented
4C3×#64106C214yesmaximal, complemented
5C3×#64104C214yesmaximal, complemented
6#10595981C311yesmaximal, characteristic, direct product factor
7C3×#5510—34nomaximal
8C22×C33×C4C2212yes—
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638C2214yes—
11C3×#3828—28no—
12C2×C3×C4⋊(C3⋊D3)C2212yes—
13C2×C3×C4⋊(C3⋊D3)C2214yes—
14C2×C3×C4×C3⋊D3C2214yes—
15C2×C3×#634C2212yes—
16C33×C4⋊C4C2214yes—
17C3×#3752C2218yes—
18C3×C4⋊(C3⋊Dic3)C2214yes—
19C2×#3828C2×C312yescomplemented
20C2×C3×#235—38no—
21C2×C32×C4⋊C4—24no—
22C2×C32×C4⋊C4D314yescomplemented
23C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
24#64139C2×C314yescomplemented
25#64106C2×C314yescomplemented
26#64104C2×C314yescomplemented
27C3×#930—316no—
28C3×#897—316no—
29C3×#895—316no—
30C23×C33C2311yescharacteristic
31C22×C3×C3⋊D3C2314yes—
32C2×C33×C4—42no—
33C2×C33×C4D414yescomplemented
34C2×C33×C4C2314yes—
35C2×C33×C4C2312yes—
36C3×#638—28no—
37C2×C3×C3⋊Dic3C2314yes—
38C3×C4⋊(C3⋊D3)—24no—
39C3×C4⋊(C3⋊D3)—44no—
40C3×C4×C3⋊D3—28no—
41C3×#634—24no—
42C3×#537—91no—
43#5510—34no—
44C22×C32×C4—24no—
45C22×C32×C4—28no—
46C22×C32×C4C2×D318yes—
47C22×C32×C4C2×D314yes—
48C22×C32×C4C22×C312yes—
49C22×C32×C4C22×C311yescharacteristic
50C2×#638C22×C314yes—
51#3828—28no—
52C2×C4⋊(C3⋊D3)C22×C312yes—
53C2×C4⋊(C3⋊D3)C22×C314yes—
54C2×C4×C3⋊D3C22×C314yes—
55C2×#634C22×C312yes—
56C2×C3×#67—316no—
57C3×#235—632no—
58C2×C3×D12—38no—
59C2×C3×D12—316no—
60C2×C3×C4×D3—316no—
61C2×C3×Dic6—38no—
62C32×C4⋊C4—216no—
63C32×C4⋊C4C2×D3116yes—
64C32×C4⋊C4C22×C314yes—
65#3752C22×C318yes—
66C4⋊(C3⋊Dic3)C22×C314yes—
67C3×(C2×D3)⋊C4—332no—
68C3×C4⋊Dic3—316no—
69C22×C33—21no—
70C22×C33—21no—
71C22×C33C2×D412yes—
72C22×C33C2411yescharacteristic
73C2×C3×C3⋊D3—24no—
74C2×C3×C3⋊D3—24no—
75C2×C3×C3⋊D3—24no—
76C33×C4—24no—
77C33×C4C2×D414yes—
78C3×C3⋊Dic3—24no—
79C2×C3×#41—92no—
80C2×#235—38no—
81C3×C22⋊Q8—94no—
82C3×#121—94no—
83C3×#121—94no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4C3×D314yescomplemented
86C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
87#930—316no—
88#897—316no—
89#895—316no—
90C23×C32—24no—
91C23×C32C22×D314yes—
92C23×C32C23×C311yescharacteristic
93C22×C3⋊D3C23×C314yes—
94C22×C3×D3—316no—
95C2×C32×C4—216no—
96C2×C32×C4—28no—
97C2×C32×C4—216no—
98C2×C32×C4—48no—
99C2×C32×C4—42no—
100C2×C32×C4—416no—
101C2×C32×C4D12116yescomplemented
102C2×C32×C4C3×D414yescomplemented
103C2×C32×C4C22×D3116yes—
104C2×C32×C4C22×D318yes—
105C2×C32×C4C23×C314yes—
106C2×C32×C4C23×C312yes—
107#638—28no—
108C2×C3⋊Dic3C23×C314yes—
109C4⋊(C3⋊D3)—24no—
110C4⋊(C3⋊D3)—44no—
111C4×C3⋊D3—28no—
112#634—24no—
113C3×#67—632no—
114C2×C3×Dic3—316no—
115C3×D12—616no—
116C3×D12—1216no—
117C3×C4×D3—632no—
118C3×Dic6—616no—
119#537—91noSylow
120C2×C33—22no—
121C2×C33C22×D411yescharacteristic
122C2×C33H(5,𝔽2)11yescharacteristic
123C2×C332−1+411yescharacteristic
124C3×C3⋊D3—44no—
125C3×#41—188no—
126C2×C3×Q8—92no—
127C2×C3×D4—94no—
128C2×C3×D4—92no—
129C2×C3×D4—94no—
130C22×C3×C4—24no—
131C22×C3×C4—28no—
132C22×C3×C4—94no—
133C22×C3×C4C2×C3×D318yes—
134C22×C3×C4C2×C3×D314yes—
135C22×C3×C4C2×C3⋊D312yes—
136C22×C3×C4C2×C3⋊D311yescharacteristic
137C2×#67—316no—
138#235—632no—
139C2×D12—38no—
140C2×D12—316no—
141C2×C4×D3—316no—
142C2×Dic6—38no—
143C3×C4⋊C4—216no—
144C3×C4⋊C4—94no—
145C3×C4⋊C4C2×C3×D3116yes—
146C3×C4⋊C4C2×C3⋊D314yes—
147C3×C22⋊C4—98no—
148(C2×D3)⋊C4—332no—
149C4⋊Dic3—316no—
150C22×C32—28no—
151C22×C32—28no—
152C22×C32—24no—
153C22×C32—24no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—24no—
157C22×C32—28no—
158C22×C32C2×D1218yes—
159C22×C32C2×C3×D412yes—
160C22×C32C23×D314yes—
161C22×C32C24×C311yescharacteristic
162C2×C3⋊D3—24no—
163C2×C3⋊D3—24no—
164C2×C3⋊D3—24no—
165C2×C3×D3—616no—
166C2×C3×D3—616no—
167C2×C3×D3—616no—
168C32×C4—24no—
169C32×C4—216no—
170C32×C4—216no—
171C32×C4—416no—
172C32×C4C2×D12116yes—
173C32×C4C2×C3×D414yes—
174C3⋊Dic3—24no—
175C3×Dic3—616no—
176C2×#41—92no—
177C22⋊Q8—94no—
178#121—94no—
179#121—94no—
180C2×C4⋊C4C3×C3⋊D311yescharacteristic, complemented
181C33#53711yescharacteristic, complemented, Sylow
182C23×C3—24no—
183C23×C3—94no—
184C23×C3C22×C3×D314yes—
185C23×C3C22×C3⋊D311yescharacteristic
186C22×D3—316no—
187C3×Q8—184no—
188C3×D4—188no—
189C3×D4—184no—
190C3×D4—364no—
191C2×C3×C4—216no—
192C2×C3×C4—28no—
193C2×C3×C4—216no—
194C2×C3×C4—48no—
195C2×C3×C4—42no—
196C2×C3×C4—416no—
197C2×C3×C4—94no—
198C2×C3×C4—188no—
199C2×C3×C4C3×D12116yescomplemented
200C2×C3×C4C4⋊(C3⋊D3)14yescomplemented
201C2×C3×C4C22×C3×D3116yes—
202C2×C3×C4C22×C3×D318yes—
203C2×C3×C4C22×C3⋊D314yes—
204C2×C3×C4C22×C3⋊D312yes—
205#67—632no—
206C2×Dic3—316no—
207D12—616no—
208D12—1216no—
209C4×D3—632no—
210Dic6—616no—
211C2×C32—216no—
212C2×C32—28no—
213C2×C32—22no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32C22×D1214yes—
218C2×C32#100414yes—
219C2×C32#100714yes—
220C2×C32C22×C3×D411yescharacteristic
221C2×C32C3×H(5,𝔽2)11yescharacteristic
222C2×C32C3×2−1+411yescharacteristic
223C3⋊D3—44no—
224C3×D3—1216no—
225#41—188no—
226C2×Q8—92no—
227C2×D4—94no—
228C2×D4—92no—
229C2×D4—94no—
230C22×C4—94no—
231C22×C4C2×C3×C3⋊D312yes—
232C22×C4C2×C3×C3⋊D311yescharacteristic
233C4⋊C4—94no—
234C4⋊C4C2×C3×C3⋊D314yes—
235C22⋊C4—98no—
236C22×C3—28no—
237C22×C3—28no—
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—24no—
243C22×C3—28no—
244C22×C3—184no—
245C22×C3—184no—
246C22×C3—184no—
247C22×C3C2×C3×D1218yes—
248C22×C3C2×C4⋊(C3⋊D3)12yes—
249C22×C3C23×C3×D314yes—
250C22×C3C23×C3⋊D311yescharacteristic
251C2×D3—616no—
252C2×D3—616no—
253C2×D3—616no—
254C3×C4—216no—
255C3×C4—24no—
256C3×C4—216no—
257C3×C4—416no—
258C3×C4—184no—
259C3×C4C2×C3×D12116yes—
260C3×C4C2×C4⋊(C3⋊D3)14yes—
261Dic3—616no—
262C32—24no—
263C32#551014yescomplemented
264C32C3×#53711yescharacteristic, complemented
265C23—94no—
266C23C22×C3×C3⋊D311yescharacteristic
267Q8—184no—
268D4—188no—
269D4—184no—
270D4—364no—
271C2×C4—42no—
272C2×C4—94no—
273C2×C4—188no—
274C2×C4C3×C4⋊(C3⋊D3)14yescomplemented
275C2×C4C22×C3×C3⋊D314yes—
276C2×C4C22×C3×C3⋊D312yes—
277C2×C3—216no—
278C2×C3—28no—
279C2×C3—22no—
280C2×C3—24no—
281C2×C3—24no—
282C2×C3—24no—
283C2×C3—364no—
284C2×C3C22×C3×D1214yes—
285C2×C3C3×#100414yes—
286C2×C3C3×#100714yes—
287C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
288C2×C3#6436111yescharacteristic
289C2×C3#6436411yescharacteristic
290D3—1216no—
291C22—21no—
292C22—21no—
293C22—184no—
294C22—184no—
295C22—184no—
296C22C2×C3×C4⋊(C3⋊D3)12yes—
297C22C23×C3×C3⋊D311yescharacteristic
298C4—24no—
299C4—184no—
300C4C2×C3×C4⋊(C3⋊D3)14yes—
301C3—24no—
302C3C3×#551014yescomplemented
303C3#1059598111yescharacteristic, direct product factor
304C2—22no—
305C2—364no—
306C2C22×C3×C4⋊(C3⋊D3)11yescharacteristic
307C2C3×#6436111yescharacteristic
308C2C3×#6436411yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC22×C33C24
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C2×C33×C4⋊C4C2semidirect
3. C3×#64139C2semidirect
4. C3×#64106C2semidirect
5. C3×#64104C2semidirect
6. #10595981C3direct (N × H)
19. C2×#3828C2×C3semidirect
22. C2×C32×C4⋊C4D3semidirect
23. C2×C32×C4⋊C4C2×C3semidirect
24. #64139C2×C3semidirect
25. #64106C2×C3semidirect
26. #64104C2×C3semidirect
33. C2×C33×C4D4semidirect
85. C2×C3×C4⋊C4C3×D3semidirect
86. C2×C3×C4⋊C4C3⋊D3semidirect
101. C2×C32×C4D12semidirect
102. C2×C32×C4C3×D4semidirect
180. C2×C4⋊C4C3×C3⋊D3semidirect
181. C33#537semidirect
199. C2×C3×C4C3×D12semidirect
200. C2×C3×C4C4⋊(C3⋊D3)semidirect
263. C32#5510semidirect
264. C32C3×#537semidirect
274. C2×C4C3×C4⋊(C3⋊D3)semidirect
302. C3C3×#5510semidirect
303. C3#10595981direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C33×C4⋊C42
C3×#641392
C3×#641062
C3×#641042
#105959813
C3×#55103