Finite Group Databasebeta

← C3×#10591230 #13158079

Subgroups — C3×#10591230 #13158079

Order 1728.

962 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×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C212yesmaximal
3C3×#63579C214yesmaximal
4C3×#4977—32nomaximal
5#10591230C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C2212yes—
8C22×C32×Dic3C2214yes—
9C3×C4×C3⋊Dic3C414yescomplemented
10C32×C4⋊Dic3—24no—
11C2×C32×C4⋊C4—32no—
12C2×C4×C3⋊Dic3C2×C311yescharacteristic
13C2×C3×C4⋊Dic3—22no—
14C2×C3×C4⋊Dic3D312yes—
15C2×C3×C4⋊Dic3C2×C312yes—
16C2×C3×C4×Dic3—32no—
17C2×C3×C4×Dic3—61no—
18C3×C3⋊#95—38no—
19#63579C2×C314yes—
20C23×C33C2311yescharacteristic
21C2×C33×C4C2×C412yes—
22C2×C33×C4D412yes—
23C2×C33×C4Q812yes—
24C2×C3×C3⋊Dic3—24no—
25C2×C3×C3⋊Dic3C2×C414yes—
26C2×C32×Dic3—24no—
27C2×C32×Dic3—28no—
28C3×#382—91no—
29#4977—32no—
30C22×C32×C4—21no—
31C22×C32×C4—22no—
32C22×C32×C4—34no—
33C22×C32×C4—41no—
34C22×C32×C4C2×D312yes—
35C22×C32×C4C22×C311yescharacteristic
36C22×C3⋊Dic3C22×C312yes—
37C22×C3×Dic3—24no—
38C22×C3×Dic3—34no—
39C22×C3×Dic3—62no—
40C22×C3×Dic3C2×D314yes—
41C22×C3×Dic3C22×C314yes—
42C32×C4⋊C4—64no—
43C4×C3⋊Dic3C3×C414yescomplemented
44C3×C4⋊Dic3—28no—
45C3×C4⋊Dic3—24no—
46C3×C4⋊Dic3—24no—
47C3×C4×Dic3—38no—
48C3×C4×Dic3—64no—
49C22×C33C22×C411yescharacteristic
50C22×C33C2×D411yescharacteristic
51C22×C33C2×Q811yescharacteristic
52C22×C33#4114yes—
53C33×C4C4⋊C414yes—
54C3×C3⋊Dic3—24no—
55C32×Dic3—44no—
56C2×C3×C4⋊C4—32no—
57C2×C3×C4⋊C4—32no—
58C2×C3×C4⋊C4—62no—
59C2×C3×C42—91no—
60C2×C4⋊Dic3C3×D312yes—
61C2×C4×Dic3—32no—
62C2×C4×Dic3—61no—
63C3×#95—94no—
64C3⋊#95—38no—
65C23×C32—21no—
66C23×C32—22no—
67C23×C32—41no—
68C23×C32C22×D312yes—
69C23×C32C23×C311yescharacteristic
70C2×C32×C4—22no—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4—24no—
74C2×C32×C4—24no—
75C2×C32×C4—24no—
76C2×C32×C4—42no—
77C2×C32×C4—42no—
78C2×C32×C4—42no—
79C2×C32×C4—64no—
80C2×C32×C4—68no—
81C2×C32×C4Dic614yes—
82C2×C32×C4C4×D314yes—
83C2×C32×C4#6714yes—
84C2×C32×C4C2×C3×C412yes—
85C2×C32×C4C3×D412yes—
86C2×C32×C4C3×Q812yes—
87C2×C3⋊Dic3—24no—
88C2×C3⋊Dic3C2×C3×C414yes—
89C2×C3×Dic3—28no—
90C2×C3×Dic3—28no—
91C2×C3×Dic3—24no—
92C2×C3×Dic3—24no—
93C2×C3×Dic3—28no—
94C2×C3×Dic3—38no—
95C2×C3×Dic3—48no—
96C2×C3×Dic3—64no—
97C2×C3×Dic3—68no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—64no—
100#382—91noSylow
101C2×C33C2×C4⋊C411yescharacteristic
102C2×C33#11712yes—
103C2×C33#12412yes—
104C2×C33#12512yes—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—34no—
108C22×C3×C4—34no—
109C22×C3×C4—41no—
110C22×C3×C4—64no—
111C22×C3×C4—92no—
112C22×C3×C4D3211yescharacteristic
113C22×C3×C4C2×C3×D312yes—
114C22×Dic3—34no—
115C22×Dic3—62no—
116C22×Dic3C2×C3×D314yes—
117C3×C4⋊C4—68no—
118C3×C4⋊C4—64no—
119C3×C4⋊C4—64no—
120C3×C42—94no—
121C4⋊Dic3—24no—
122C4×Dic3—38no—
123C4×Dic3—64no—
124C22×C32—24no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—24no—
131C22×C32—24no—
132C22×C32—22no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—44no—
137C22×C32C2×Dic612yes—
138C22×C32C2×C4×D312yes—
139C22×C32#23714yes—
140C22×C32#23914yes—
141C22×C32C2×#6712yes—
142C22×C32C22×C3×C411yescharacteristic
143C22×C32C2×C3×D411yescharacteristic
144C22×C32C2×C3×Q811yescharacteristic
145C22×C32C3×#4114yes—
146C32×C4—24no—
147C32×C4—28no—
148C32×C4—44no—
149C32×C4—124no—
150C32×C4Dic3⋊C418yes—
151C32×C4C3×C4⋊C414yes—
152C3⋊Dic3—24no—
153C3×Dic3—48no—
154C3×Dic3—44no—
155C3×Dic3—44no—
156C3×Dic3—68no—
157C3×Dic3—68no—
158C2×C4⋊C4—32no—
159C2×C42—91no—
160#95—94no—
161C33#38211yescharacteristic, complemented, Sylow
162C23×C3—22no—
163C23×C3—21no—
164C23×C3—41no—
165C23×C3C2×D3211yescharacteristic
166C23×C3C22×C3×D312yes—
167C2×C3×C4—24no—
168C2×C3×C4—22no—
169C2×C3×C4—24no—
170C2×C3×C4—22no—
171C2×C3×C4—24no—
172C2×C3×C4—22no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—68no—
177C2×C3×C4—68no—
178C2×C3×C4—64no—
179C2×C3×C4—68no—
180C2×C3×C4—64no—
181C2×C3×C4—94no—
182C2×C3×C4—128no—
183C2×C3×C4—184no—
184C2×C3×C4#62412yes—
185C2×C3×C4#62512yes—
186C2×C3×C4#62712yes—
187C2×C3×C4C3×Dic614yes—
188C2×C3×C4C3×C4×D314yes—
189C2×C3×C4C3×#6714yes—
190C2×Dic3—28no—
191C2×Dic3—24no—
192C2×Dic3—38no—
193C2×Dic3—64no—
194C2×Dic3—68no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197C2×C32—24no—
198C2×C32—24no—
199C2×C32—22no—
200C2×C32—22no—
201C2×C32—22no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—21no—
206C2×C32—41no—
207C2×C32—42no—
208C2×C32—42no—
209C2×C32—42no—
210C2×C32#89014yes—
211C2×C32#89214yes—
212C2×C32C2×Dic3⋊C412yes—
213C2×C32#93612yes—
214C2×C32#94712yes—
215C2×C32C2×C3×C4⋊C411yescharacteristic
216C2×C32C3×#11712yes—
217C2×C32C3×#12412yes—
218C2×C32C3×#12512yes—
219C22×C4—34no—
220C22×C4—92no—
221C22×C4C3×D3211yescharacteristic
222C4⋊C4—64no—
223C42—94no—
224C22×C3—24no—
225C22×C3—24no—
226C22×C3—24no—
227C22×C3—21no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—21no—
233C22×C3—44no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3#379614yes—
238C22×C3C2×#62411yescharacteristic
239C22×C3C2×#62511yescharacteristic
240C22×C3C2×#62711yescharacteristic
241C22×C3C2×C3×Dic612yes—
242C22×C3C2×C3×C4×D312yes—
243C22×C3C3×#23714yes—
244C22×C3C3×#23914yes—
245C22×C3C2×C3×#6712yes—
246C3×C4—28no—
247C3×C4—24no—
248C3×C4—44no—
249C3×C4—128no—
250C3×C4—124no—
251C3×C4—124no—
252C3×C4—184no—
253C3×C4#372414yes—
254C3×C4C3×Dic3⋊C418yes—
255Dic3—44no—
256Dic3—68no—
257Dic3—68no—
258C32—22no—
259C32—21no—
260C32—41no—
261C32#497712yescomplemented
262C32C3×#38211yescharacteristic, complemented
263C23C2×C3×D3211yescharacteristic
264C2×C4—68no—
265C2×C4—64no—
266C2×C4—94no—
267C2×C4—184no—
268C2×C4C3×#62412yes—
269C2×C4C3×#62512yes—
270C2×C4C3×#62712yes—
271C2×C3—22no—
272C2×C3—21no—
273C2×C3—22no—
274C2×C3—24no—
275C2×C3—22no—
276C2×C3—22no—
277C2×C3—22no—
278C2×C3—24no—
279C2×C3—22no—
280C2×C3—41no—
281C2×C3—42no—
282C2×C3—42no—
283C2×C3—42no—
284C2×C3#6384912yes—
285C2×C3#6386312yes—
286C2×C3#6387212yes—
287C2×C3C2×#372411yescharacteristic
288C2×C3C3×#89014yes—
289C2×C3C3×#89214yes—
290C2×C3C2×C3×Dic3⋊C412yes—
291C2×C3C3×#93612yes—
292C2×C3C3×#94712yes—
293C22C3×#379614yes—
294C22C2×C3×#62411yescharacteristic
295C22C2×C3×#62511yescharacteristic
296C22C2×C3×#62711yescharacteristic
297C4—124no—
298C4—184no—
299C4C3×#372414yes—
300C3—22no—
301C3—21no—
302C3—41no—
303C3#1059123011yescharacteristic, direct product factor
304C3C3×#497712yescomplemented
305C2C3×#6384912yes—
306C2C3×#6386312yes—
307C2C3×#6387212yes—
308C2C2×C3×#372411yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#382
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. #10591230C3direct (N × H)
9. C3×C4×C3⋊Dic3C4semidirect
43. C4×C3⋊Dic3C3×C4semidirect
161. C33#382semidirect
261. C32#4977semidirect
262. C32C3×#382semidirect
303. C3#10591230direct (N × H)
304. C3C3×#4977semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C2×C32×C4⋊Dic32
C3×#635792
C3×#49773
#105912303