Finite Group Databasebeta

← C3×#10594828 #13174381

Subgroups — C3×#10594828 #13174381

Order 1728.

1426 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C32×#926C212yesmaximal
3C3×#63957C214yesmaximal
4C3×#63844C218yesmaximal, complemented
5C3×#5516—32nomaximal
6#10594828C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2212yes—
9C2×C3×#634C418yescomplemented
10C2×C3×#634C2214yes—
11C32×#217C2214yes—
12C32×C4⋊Dic3C2214yes—
13C32×C4×Dic3C2214yes—
14C3×#3724C2218yes—
15C3×Dic32C2218yes—
16C22×#634C2×C311yescharacteristic
17C22×C3×Dic6—32no—
18C22×C3×Dic6—61no—
19C32×#117—32no—
20C3×#926—22no—
21C3×#926D312yes—
22C3×#926C2×C312yes—
23C3×#887—38no—
24C3×#877—38no—
25C3×C4×Dic6—38no—
26#63957C2×C314yes—
27#63844C2×C318yescomplemented
28C23×C33C2311yescharacteristic
29C2×C33×C4C2×C414yes—
30C2×C33×C4C2312yes—
31C2×C3×C3⋊Dic3C2×C418yes—
32C2×C3×C3⋊Dic3C2314yes—
33C3×#634C2×C4116yes—
34C2×C32×Dic3C2318yes—
35C3×#519—91no—
36#5516—32no—
37C22×C32×C4—21no—
38C22×C32×C4—22no—
39C22×C32×C4—41no—
40C22×C32×C4C2×D312yes—
41C22×C32×C4C22×C311yescharacteristic
42C22×C3⋊Dic3C22×C312yes—
43C2×#634C3×C418yescomplemented
44C2×#634C22×C314yes—
45C22×C3×Dic3—34no—
46C22×C3×Dic3—62no—
47C2×C3×Dic6—316no—
48C2×C3×Dic6—34no—
49C2×C3×Dic6—34no—
50C2×C3×Dic6—68no—
51C2×C3×Dic6—64no—
52C32×C4⋊C4—34no—
53C32×C22⋊C4—34no—
54C32×C42—34no—
55C3×#217—24no—
56C3×#217C2×D314yes—
57C3×#217C22×C314yes—
58C3×C4⋊Dic3—24no—
59C3×C4⋊Dic3C2×D314yes—
60C3×C4⋊Dic3C22×C314yes—
61C3×Dic3⋊C4—316no—
62C3×C4×Dic3—24no—
63C3×C4×Dic3—316no—
64C3×C4×Dic3C2×D314yes—
65C3×C4×Dic3C22×C314yes—
66#3724C22×C318yes—
67Dic32C22×C318yes—
68C22×C33—22no—
69C22×C33C22×C412yes—
70C22×C33C2411yescharacteristic
71C33×C4C22×C414yes—
72C3×C3⋊Dic3C22×C418yes—
73C32×Dic3—28no—
74C22×C3×Q8—91no—
75C22×Dic6—32no—
76C22×Dic6—61no—
77C3×C4×Q8—98no—
78C3×#117—32no—
79C3×#117—32no—
80C3×#117—62no—
81C3×#117—94no—
82#926C3×D312yes—
83#887—38no—
84#877—38no—
85C4×Dic6—38no—
86C23×C32—21no—
87C23×C32—22no—
88C23×C32—41no—
89C23×C32C22×D312yes—
90C23×C32C23×C311yescharacteristic
91C2×C32×C4—24no—
92C2×C32×C4—28no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—38no—
97C2×C32×C4—44no—
98C2×C32×C4—42no—
99C2×C32×C4C4×D318yes—
100C2×C32×C4C2×C3×C414yes—
101C2×C32×C4C22×D312yes—
102C2×C32×C4C22×D312yes—
103C2×C32×C4C23×C312yes—
104C2×C3⋊Dic3C2×C3×C418yes—
105C2×C3⋊Dic3C23×C314yes—
106#634C2×C3×C4116yes—
107C2×C3×Dic3—28no—
108C2×C3×Dic3—316no—
109C2×C3×Dic3—38no—
110C2×C3×Dic3—68no—
111C2×C3×Dic3—64no—
112C2×C3×Dic3C22×D318yes—
113C2×C3×Dic3C23×C318yes—
114C3×Dic6—316no—
115C3×Dic6—316no—
116C3×Dic6—616no—
117#519—91noSylow
118C2×C33—22no—
119C2×C33C23×C411yescharacteristic
120C2×C332−1+412yes—
121C2×C3×Q8—98no—
122C2×C3×Q8—94no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—92no—
127C22×C3×C4D3211yescharacteristic
128C22×C3×C4C2×C3×D312yes—
129C22×Dic3—34no—
130C22×Dic3—62no—
131C2×Dic6—316no—
132C2×Dic6—34no—
133C2×Dic6—34no—
134C2×Dic6—68no—
135C2×Dic6—64no—
136C3×C4⋊C4—34no—
137C3×C4⋊C4—34no—
138C3×C4⋊C4—64no—
139C3×C4⋊C4—98no—
140C3×C22⋊C4—34no—
141C3×C22⋊C4—34no—
142C3×C22⋊C4—64no—
143C3×C42—34no—
144C3×C42—34no—
145C3×C42—64no—
146C3×C42—98no—
147#217C2×C3×D314yes—
148C4⋊Dic3C2×C3×D314yes—
149Dic3⋊C4—316no—
150C4×Dic3—316no—
151C4×Dic3C2×C3×D314yes—
152C22×C32—22no—
153C22×C32—24no—
154C22×C32—24no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—22no—
158C22×C32—24no—
159C22×C32—42no—
160C22×C32—41no—
161C22×C32—44no—
162C22×C32—44no—
163C22×C32C2×C4×D314yes—
164C22×C32C22×C3×C412yes—
165C22×C32C23×D312yes—
166C22×C32C24×C311yescharacteristic
167C32×C4—24no—
168C32×C4—24no—
169C32×C4—24no—
170C32×C4—44no—
171C32×C4—68no—
172C32×C4C2×C4×D314yes—
173C32×C4C2×C4×D314yes—
174C32×C4C22×C3×C414yes—
175C3⋊Dic3C22×C3×C418yes—
176C3×Dic3—28no—
177C3×Dic3—28no—
178C3×Dic3—316no—
179C3×Dic3—48no—
180C3×Dic3—68no—
181C22×Q8—91no—
182C4×Q8—98no—
183#117—32no—
184#117—94no—
185C33#51911yescharacteristic, complemented, Sylow
186C23×C3—21no—
187C23×C3—22no—
188C23×C3—41no—
189C23×C3C2×D3211yescharacteristic
190C23×C3C22×C3×D312yes—
191C3×Q8—916no—
192C2×C3×C4—28no—
193C2×C3×C4—24no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—38no—
198C2×C3×C4—38no—
199C2×C3×C4—44no—
200C2×C3×C4—42no—
201C2×C3×C4—68no—
202C2×C3×C4—94no—
203C2×C3×C4—98no—
204C2×C3×C4#62414yes—
205C2×C3×C4C3×C4×D318yes—
206C2×C3×C4C2×D3212yes—
207C2×C3×C4C22×C3×D312yes—
208C2×C3×C4C22×C3×D312yes—
209C2×Dic3—38no—
210C2×Dic3—316no—
211C2×Dic3—68no—
212C2×Dic3—64no—
213C2×Dic3C22×C3×D318yes—
214Dic6—316no—
215Dic6—316no—
216Dic6—616no—
217C2×C32—22no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—22no—
223C2×C32—24no—
224C2×C32—41no—
225C2×C32—42no—
226C2×C32—44no—
227C2×C32—44no—
228C2×C32C22×C4×D312yes—
229C2×C32#101014yes—
230C2×C32C23×C3×C411yescharacteristic
231C2×C32C3×2−1+412yes—
232C2×Q8—98no—
233C2×Q8—94no—
234C22×C4—92no—
235C22×C4C3×D3211yescharacteristic
236C4⋊C4—34no—
237C4⋊C4—98no—
238C22⋊C4—34no—
239C42—34no—
240C42—98no—
241C22×C3—24no—
242C22×C3—22no—
243C22×C3—24no—
244C22×C3—21no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—24no—
248C22×C3—44no—
249C22×C3—44no—
250C22×C3—41no—
251C22×C3—42no—
252C22×C3C2×#62412yes—
253C22×C3C2×C3×C4×D314yes—
254C22×C3C22×D3211yescharacteristic
255C22×C3C23×C3×D312yes—
256C3×C4—24no—
257C3×C4—24no—
258C3×C4—24no—
259C3×C4—44no—
260C3×C4—68no—
261C3×C4—68no—
262C3×C4—98no—
263C3×C4—128no—
264C3×C4C2×#62414yes—
265C3×C4C2×C3×C4×D314yes—
266C3×C4C2×C3×C4×D314yes—
267Dic3—28no—
268Dic3—316no—
269Dic3—68no—
270C32—21no—
271C32—22no—
272C32—41no—
273C32#551612yescomplemented
274C32C3×#51911yescharacteristic, complemented
275C23C2×C3×D3211yescharacteristic
276Q8—916no—
277C2×C4—38no—
278C2×C4—94no—
279C2×C4—98no—
280C2×C4C3×#62414yes—
281C2×C4C2×C3×D3212yes—
282C2×C3—24no—
283C2×C3—22no—
284C2×C3—24no—
285C2×C3—21no—
286C2×C3—22no—
287C2×C3—24no—
288C2×C3—22no—
289C2×C3—44no—
290C2×C3—44no—
291C2×C3—41no—
292C2×C3—42no—
293C2×C3#6429812yes—
294C2×C3C22×#62411yescharacteristic
295C2×C3C22×C3×C4×D312yes—
296C2×C3C3×#101014yes—
297C22—22no—
298C22C2×C3×#62412yes—
299C22C22×C3×D3211yescharacteristic
300C4—68no—
301C4—98no—
302C4C2×C3×#62414yes—
303C3—21no—
304C3—22no—
305C3—41no—
306C3#1059482811yescharacteristic, direct product factor
307C3C3×#551612yescomplemented
308C2—22no—
309C2C3×#6429812yes—
310C2C22×C3×#62411yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC23×C33C23
cosocleC22×C32C24×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#519
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63844C2semidirect
6. #10594828C3direct (N × H)
9. C2×C3×#634C4semidirect
27. #63844C2×C3semidirect
43. C2×#634C3×C4semidirect
185. C33#519semidirect
273. C32#5516semidirect
274. C32C3×#519semidirect
306. C3#10594828direct (N × H)
307. C3C3×#5516semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C32×#9262
C3×#639572
C3×#638442
C3×#55163
#105948283