Finite Group Databasebeta

← C3×#10591359 #13158310

Subgroups — C3×#10591359 #13158310

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 338 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
3C2×C3×#3724C214yesmaximal
4C3×#4975—32nomaximal
5#10591359C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C2212yes—
8C22×C32×Dic3C2214yes—
9C3×C4×C3⋊Dic3C414yescomplemented
10C32×C4⋊Dic3—24no—
11C3×#3724—28no—
12C2×C32×C4⋊C4—32no—
13C2×C4×C3⋊Dic3C2×C311yescharacteristic
14C2×C3×C4⋊Dic3—22no—
15C2×C3×C4⋊Dic3D312yes—
16C2×C3×C4⋊Dic3C2×C312yes—
17C2×C3×Dic3⋊C4—38no—
18C2×C3×C4×Dic3—32no—
19C2×C3×C4×Dic3—61no—
20C2×#3724C2×C314yes—
21C23×C33C2311yescharacteristic
22C2×C33×C4C2×C412yes—
23C2×C33×C4D412yes—
24C2×C33×C4Q812yes—
25C2×C3×C3⋊Dic3C2×C414yes—
26C2×C3×C3⋊Dic3D414yes—
27C2×C3×C3⋊Dic3Q814yes—
28C2×C32×Dic3—24no—
29C2×C32×Dic3—28no—
30C3×#384—91no—
31#4975—32no—
32C22×C32×C4—21no—
33C22×C32×C4—22no—
34C22×C32×C4—34no—
35C22×C32×C4—41no—
36C22×C32×C4C2×D312yes—
37C22×C32×C4C22×C311yescharacteristic
38C22×C3⋊Dic3C22×C312yes—
39C22×C3×Dic3—24no—
40C22×C3×Dic3—34no—
41C22×C3×Dic3—62no—
42C22×C3×Dic3C2×D314yes—
43C22×C3×Dic3C22×C314yes—
44C32×C4⋊C4—64no—
45C4×C3⋊Dic3C3×C414yescomplemented
46C3×C4⋊Dic3—28no—
47C3×C4⋊Dic3—24no—
48C3×C4⋊Dic3—24no—
49C3×Dic3⋊C4—616no—
50C3×C4×Dic3—38no—
51C3×C4×Dic3—64no—
52#3724—28no—
53C22×C33C22×C411yescharacteristic
54C22×C33C2×D412yes—
55C22×C33C2×D411yescharacteristic
56C22×C33C2×Q811yescharacteristic
57C22×C33C2×Q812yes—
58C33×C4C4⋊C414yescomplemented
59C3×C3⋊Dic3C4⋊C418yescomplemented
60C32×Dic3—44no—
61C2×C3×C4⋊C4—32no—
62C2×C3×C4⋊C4—32no—
63C2×C3×C4⋊C4—62no—
64C2×C3×C4⋊C4—94no—
65C2×C3×C42—91no—
66C2×C4⋊Dic3C3×D312yes—
67C2×Dic3⋊C4—38no—
68C2×C4×Dic3—32no—
69C2×C4×Dic3—61no—
70C23×C32—21no—
71C23×C32—22no—
72C23×C32—41no—
73C23×C32C22×D312yes—
74C23×C32C23×C311yescharacteristic
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—24no—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4—42no—
82C2×C32×C4—42no—
83C2×C32×C4—42no—
84C2×C32×C4—64no—
85C2×C32×C4—68no—
86C2×C32×C4Dic614yes—
87C2×C32×C4C4×D314yes—
88C2×C32×C4#6714yes—
89C2×C32×C4C2×C3×C412yes—
90C2×C32×C4C3×D412yes—
91C2×C32×C4C3×Q812yes—
92C2×C3⋊Dic3C2×C3×C414yes—
93C2×C3⋊Dic3C3×D414yes—
94C2×C3⋊Dic3C3×Q814yes—
95C2×C3×Dic3—28no—
96C2×C3×Dic3—28no—
97C2×C3×Dic3—24no—
98C2×C3×Dic3—28no—
99C2×C3×Dic3—24no—
100C2×C3×Dic3—38no—
101C2×C3×Dic3—38no—
102C2×C3×Dic3—38no—
103C2×C3×Dic3—48no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—64no—
107#384—91noSylow
108C2×C33C2×C4⋊C412yes—
109C2×C33C2×C4⋊C411yescharacteristic
110C2×C33#12711yescharacteristic
111C2×C33C4⋊Q812yes—
112C2×C33C4⋊Q811yescharacteristic
113C22×C3×C4—21no—
114C22×C3×C4—22no—
115C22×C3×C4—34no—
116C22×C3×C4—34no—
117C22×C3×C4—41no—
118C22×C3×C4—64no—
119C22×C3×C4—92no—
120C22×C3×C4D3211yescharacteristic
121C22×C3×C4C2×C3×D312yes—
122C22×Dic3—34no—
123C22×Dic3—62no—
124C22×Dic3C2×C3×D314yes—
125C3×C4⋊C4—68no—
126C3×C4⋊C4—64no—
127C3×C4⋊C4—64no—
128C3×C4⋊C4—188no—
129C3×C42—94no—
130C4⋊Dic3—24no—
131Dic3⋊C4—616no—
132C4×Dic3—38no—
133C4×Dic3—64no—
134C22×C32—22no—
135C22×C32—22no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—24no—
142C22×C32—24no—
143C22×C32—22no—
144C22×C32—41no—
145C22×C32—41no—
146C22×C32—42no—
147C22×C32—41no—
148C22×C32—42no—
149C22×C32C2×Dic612yes—
150C22×C32C2×C4×D312yes—
151C22×C32D3×D414yes—
152C22×C32D3×Q814yes—
153C22×C32C2×#6712yes—
154C22×C32C22×C3×C411yescharacteristic
155C22×C32C2×C3×D412yes—
156C22×C32C2×C3×D411yescharacteristic
157C22×C32C2×C3×Q811yescharacteristic
158C22×C32C2×C3×Q812yes—
159C32×C4—24no—
160C32×C4—28no—
161C32×C4—44no—
162C32×C4—124no—
163C32×C4Dic3⋊C418yescomplemented
164C32×C4C3×C4⋊C414yescomplemented
165C3⋊Dic3C3×C4⋊C418yescomplemented
166C3×Dic3—316no—
167C3×Dic3—48no—
168C3×Dic3—44no—
169C3×Dic3—44no—
170C3×Dic3—68no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—94no—
173C2×C42—91no—
174C33#38411yescharacteristic, complemented, Sylow
175C23×C3—22no—
176C23×C3—21no—
177C23×C3—41no—
178C23×C3C2×D3211yescharacteristic
179C23×C3C22×C3×D312yes—
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—24no—
183C2×C3×C4—22no—
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—68no—
190C2×C3×C4—68no—
191C2×C3×C4—64no—
192C2×C3×C4—68no—
193C2×C3×C4—64no—
194C2×C3×C4—94no—
195C2×C3×C4—94no—
196C2×C3×C4—94no—
197C2×C3×C4—128no—
198C2×C3×C4#62412yes—
199C2×C3×C4#62512yes—
200C2×C3×C4#62712yes—
201C2×C3×C4C3×Dic614yes—
202C2×C3×C4C3×C4×D314yes—
203C2×C3×C4C3×#6714yes—
204C2×Dic3—28no—
205C2×Dic3—24no—
206C2×Dic3—38no—
207C2×Dic3—38no—
208C2×Dic3—38no—
209C2×Dic3—64no—
210C2×Dic3—64no—
211C2×Dic3—64no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32—22no—
215C2×C32—22no—
216C2×C32—22no—
217C2×C32—21no—
218C2×C32—22no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32—42no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—42no—
227C2×C32#88814yes—
228C2×C32D3×C4⋊C414yes—
229C2×C32C2×Dic3⋊C412yes—
230C2×C32#94012yes—
231C2×C32#94412yes—
232C2×C32C2×C3×C4⋊C412yes—
233C2×C32C2×C3×C4⋊C411yescharacteristic
234C2×C32C3×#12711yescharacteristic
235C2×C32C3×C4⋊Q812yes—
236C2×C32C3×C4⋊Q811yescharacteristic
237C22×C4—34no—
238C22×C4—92no—
239C22×C4C3×D3211yescharacteristic
240C4⋊C4—64no—
241C4⋊C4—188no—
242C42—94no—
243C22×C3—22no—
244C22×C3—24no—
245C22×C3—22no—
246C22×C3—24no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3#379712yes—
259C22×C3#380212yes—
260C22×C3C2×#62411yescharacteristic
261C22×C3C2×#62511yescharacteristic
262C22×C3C2×#62711yescharacteristic
263C22×C3C2×C3×Dic612yes—
264C22×C3C2×C3×C4×D312yes—
265C22×C3C3×D3×D414yes—
266C22×C3C3×D3×Q814yes—
267C22×C3C2×C3×#6712yes—
268C3×C4—28no—
269C3×C4—24no—
270C3×C4—44no—
271C3×C4—98no—
272C3×C4—128no—
273C3×C4—124no—
274C3×C4—124no—
275C3×C4#372414yescomplemented
276C3×C4C3×Dic3⋊C418yescomplemented
277Dic3—316no—
278Dic3—44no—
279Dic3—68no—
280C32—22no—
281C32—21no—
282C32—41no—
283C32#497512yescomplemented
284C32C3×#38411yescharacteristic, complemented
285C23C2×C3×D3211yescharacteristic
286C2×C4—68no—
287C2×C4—64no—
288C2×C4—94no—
289C2×C4—94no—
290C2×C4—94no—
291C2×C4C3×#62412yes—
292C2×C4C3×#62512yes—
293C2×C4C3×#62712yes—
294C2×C3—22no—
295C2×C3—21no—
296C2×C3—22no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—24no—
300C2×C3—22no—
301C2×C3—21no—
302C2×C3—24no—
303C2×C3—22no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—42no—
307C2×C3—41no—
308C2×C3—42no—
309C2×C3#6387311yescharacteristic
310C2×C3#6390012yes—
311C2×C3#6391411yescharacteristic
312C2×C3#6391912yes—
313C2×C3C2×#372411yescharacteristic
314C2×C3C3×#88814yes—
315C2×C3C3×D3×C4⋊C414yes—
316C2×C3C2×C3×Dic3⋊C412yes—
317C2×C3C3×#94012yes—
318C2×C3C3×#94412yes—
319C22C3×#379712yes—
320C22C3×#380212yes—
321C22C2×C3×#62411yescharacteristic
322C22C2×C3×#62511yescharacteristic
323C22C2×C3×#62711yescharacteristic
324C4—98no—
325C4—124no—
326C4C3×#372414yescomplemented
327C3—22no—
328C3—21no—
329C3—41no—
330C3#1059135911yescharacteristic, direct product factor
331C3C3×#497512yescomplemented
332C2C3×#6387311yescharacteristic
333C2C3×#6390012yes—
334C2C3×#6391411yescharacteristic
335C2C3×#6391912yes—
336C2C2×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×#384
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. #10591359C3direct (N × H)
9. C3×C4×C3⋊Dic3C4semidirect
45. C4×C3⋊Dic3C3×C4semidirect
58. C33×C4C4⋊C4semidirect
59. C3×C3⋊Dic3C4⋊C4semidirect
163. C32×C4Dic3⋊C4semidirect
164. C32×C4C3×C4⋊C4semidirect
165. C3⋊Dic3C3×C4⋊C4semidirect
174. C33#384semidirect
275. C3×C4#3724semidirect
276. C3×C4C3×Dic3⋊C4semidirect
283. C32#4975semidirect
284. C32C3×#384semidirect
326. C4C3×#3724semidirect
330. C3#10591359direct (N × H)
331. C3C3×#4975semidirect

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
C2×C3×#37242
C3×#49753
#105913593