Finite Group Databasebeta

← C3×#10592363 #13162706

Subgroups — C3×#10592363 #13162706

Order 1728.

1612 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3752C212yesmaximal
3C33×#95C211yesmaximal, characteristic, complemented
4C3×#63658C211yesmaximal, characteristic, complemented
5C3×#63658C212yesmaximal, complemented
6#10592363C311yesmaximal, characteristic, direct product factor
7C3×#4664—34nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C33×C4C2212yes—
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C4×C3⋊D3—24no—
14C2×C3×C4×C3⋊D3C414yescomplemented
15C3×#3752—24no—
16C22×C4×C3⋊D3C2×C311yescharacteristic
17C22×C3×C4×D3—34no—
18C2×#3752C2×C312yes—
19C2×C3×(C2×D3)⋊C4—38no—
20C32×#95—24no—
21C32×#95D314yescomplemented
22C32×#95C2×C311yescharacteristic, complemented
23#63658C2×C311yescharacteristic, complemented
24#63658C2×C312yescomplemented
25C3×C3⋊#95—38no—
26C3×C3⋊#95—34no—
27C23×C33C2311yescharacteristic
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3—21no—
30C22×C3×C3⋊D3—21no—
31C22×C3×C3⋊D3C2×C412yes—
32C22×C3×C3⋊D3D414yes—
33C2×C33×C4—22no—
34C2×C33×C4—22no—
35C2×C33×C4—22no—
36C2×C33×C4—22no—
37C2×C33×C4C2×C412yes—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3C2×C412yes—
43C3×C4×C3⋊D3—28no—
44C3×#381—91no—
45#4664—34no—
46C23×C3⋊D3C22×C311yescharacteristic
47C23×C3×D3—34no—
48C22×C32×C4—28no—
49C22×C32×C4—24no—
50C22×C32×C4C2×D318yes—
51C22×C32×C4C2×D314yes—
52C22×C32×C4C22×C312yes—
53C22×C32×C4C22×C311yescharacteristic
54C22×C3⋊Dic3C22×C312yes—
55C22×C3⋊Dic3C22×C311yescharacteristic
56C2×C4×C3⋊D3—24no—
57C2×C4×C3⋊D3C3×C414yescomplemented
58C22×C3×Dic3—34no—
59C22×C3×Dic3—38no—
60C2×C3×C4×D3—316no—
61C2×C3×C4×D3—616no—
62#3752—24no—
63C3×(C2×D3)⋊C4—616no—
64C22×C33C22×C411yescharacteristic
65C22×C33C2×D412yes—
66C22×C33#4112yes—
67C22×C33#4111yescharacteristic
68C22×C33#4111yescharacteristic
69C2×C3×C3⋊D3—22no—
70C2×C3×C3⋊D3—22no—
71C2×C3×C3⋊D3—24no—
72C2×C3×C3⋊D3—24no—
73C2×C3×C3⋊D3C22⋊C414yes—
74C33×C4—22no—
75C33×C4—42no—
76C3×C3⋊Dic3—22no—
77C3×C3⋊Dic3—42no—
78C23×C3×C4—91no—
79C22×C4×D3—34no—
80C2×C3×C22⋊C4—92no—
81C2×(C2×D3)⋊C4—38no—
82C3×#95—24no—
83C3×#95—92no—
84C3×#95—91no—
85C3×#95C3×D314yescomplemented
86C3×#95C3⋊D311yescharacteristic, complemented
87C3⋊#95—38no—
88C3⋊#95—34no—
89C23×C32—24no—
90C23×C32C22×D314yes—
91C23×C32C23×C311yescharacteristic
92C22×C3⋊D3—22no—
93C22×C3⋊D3—21no—
94C22×C3⋊D3—21no—
95C22×C3⋊D3C2×C3×C412yes—
96C22×C3⋊D3C3×D414yes—
97C22×C3×D3—38no—
98C22×C3×D3—316no—
99C22×C3×D3—68no—
100C22×C3×D3—64no—
101C22×C3×D3—64no—
102C2×C32×C4—22no—
103C2×C32×C4—216no—
104C2×C32×C4—28no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—28no—
109C2×C32×C4—28no—
110C2×C32×C4—28no—
111C2×C32×C4—28no—
112C2×C32×C4—48no—
113C2×C32×C4—48no—
114C2×C32×C4—48no—
115C2×C32×C4C4×D318yes—
116C2×C32×C4C2×C3×C412yes—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3C2×C3×C412yes—
122C4×C3⋊D3—28no—
123C2×C3×Dic3—38no—
124C2×C3×Dic3—68no—
125C2×C3×Dic3—68no—
126C2×C3×Dic3—68no—
127C2×C3×Dic3—68no—
128C3×C4×D3—632no—
129#381—91noSylow
130C2×C33C2×C22⋊C411yescharacteristic
131C2×C33#11711yescharacteristic
132C2×C33#11711yescharacteristic
133C2×C33#12312yes—
134C2×C33#12312yes—
135C3×C3⋊D3—24no—
136C24×C3—91no—
137C23×D3—34no—
138C22×C3×C4—28no—
139C22×C3×C4—24no—
140C22×C3×C4—91no—
141C22×C3×C4—94no—
142C22×C3×C4—92no—
143C22×C3×C4—184no—
144C22×C3×C4C2×C3×D318yes—
145C22×C3×C4C2×C3×D314yes—
146C22×C3×C4C2×C3⋊D312yes—
147C22×C3×C4C2×C3⋊D311yescharacteristic
148C22×Dic3—34no—
149C22×Dic3—38no—
150C2×C4×D3—316no—
151C2×C4×D3—616no—
152C3×C22⋊C4—184no—
153(C2×D3)⋊C4—616no—
154C22×C32—24no—
155C22×C32—28no—
156C22×C32—28no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32C2×C4×D314yes—
160C22×C32#23518yes—
161C22×C32D3×D418yes—
162C22×C32#23714yes—
163C22×C32#23914yes—
164C22×C32C22×C3×C411yescharacteristic
165C22×C32C2×C3×D412yes—
166C22×C32C3×#4112yes—
167C22×C32C3×#4111yescharacteristic
168C22×C32C3×#4111yescharacteristic
169C2×C3⋊D3—22no—
170C2×C3⋊D3—22no—
171C2×C3⋊D3—24no—
172C2×C3⋊D3—24no—
173C2×C3⋊D3C3×C22⋊C414yes—
174C2×C3×D3—316no—
175C2×C3×D3—68no—
176C2×C3×D3—616no—
177C2×C3×D3—68no—
178C2×C3×D3—616no—
179C32×C4—22no—
180C32×C4—28no—
181C32×C4—48no—
182C32×C4—416no—
183C32×C4—48no—
184C32×C4—42no—
185C3⋊Dic3—22no—
186C3⋊Dic3—42no—
187C3×Dic3—68no—
188C3×Dic3—128no—
189C23×C4—91no—
190C2×C22⋊C4—92no—
191#95—92no—
192#95—91no—
193#95C3×C3⋊D311yescharacteristic, complemented
194C33#38111yescharacteristic, complemented, Sylow
195C23×C3—24no—
196C23×C3—92no—
197C23×C3—94no—
198C23×C3—182no—
199C23×C3—181no—
200C23×C3—181no—
201C23×C3C22×C3×D314yes—
202C23×C3C22×C3⋊D311yescharacteristic
203C22×D3—38no—
204C22×D3—316no—
205C22×D3—68no—
206C22×D3—64no—
207C22×D3—64no—
208C2×C3×C4—28no—
209C2×C3×C4—22no—
210C2×C3×C4—216no—
211C2×C3×C4—28no—
212C2×C3×C4—28no—
213C2×C3×C4—28no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—28no—
218C2×C3×C4—48no—
219C2×C3×C4—48no—
220C2×C3×C4—48no—
221C2×C3×C4—92no—
222C2×C3×C4—188no—
223C2×C3×C4—182no—
224C2×C3×C4—182no—
225C2×C3×C4—182no—
226C2×C3×C4—182no—
227C2×C3×C4C3×C4×D318yes—
228C2×C3×C4C4×C3⋊D312yes—
229C2×Dic3—38no—
230C2×Dic3—68no—
231C2×Dic3—68no—
232C2×Dic3—68no—
233C2×Dic3—68no—
234C4×D3—632no—
235C2×C32—24no—
236C2×C32—28no—
237C2×C32—24no—
238C2×C32—28no—
239C2×C32—24no—
240C2×C32#87214yes—
241C2×C32D3×C22⋊C414yes—
242C2×C32#88318yes—
243C2×C32#89214yes—
244C2×C32#89418yes—
245C2×C32C2×C3×C22⋊C411yescharacteristic
246C2×C32C3×#11711yescharacteristic
247C2×C32C3×#11711yescharacteristic
248C2×C32C3×#12312yes—
249C2×C32C3×#12312yes—
250C3⋊D3—24no—
251C3×D3—616no—
252C24—91no—
253C22×C4—91no—
254C22×C4—94no—
255C22×C4—92no—
256C22×C4—184no—
257C22×C4C2×C3×C3⋊D312yes—
258C22×C4C2×C3×C3⋊D311yescharacteristic
259C22⋊C4—184no—
260C22×C3—24no—
261C22×C3—28no—
262C22×C3—28no—
263C22×C3—24no—
264C22×C3—24no—
265C22×C3—94no—
266C22×C3—182no—
267C22×C3—184no—
268C22×C3—182no—
269C22×C3—184no—
270C22×C3C2×C3×C4×D314yes—
271C22×C3C3×#23518yes—
272C22×C3C3×D3×D418yes—
273C22×C3C3×#23714yes—
274C22×C3C3×#23914yes—
275C22×C3C2×C4×C3⋊D311yescharacteristic
276C22×C3#382812yes—
277C22×C3D4×C3⋊D312yes—
278C22×C3#383011yescharacteristic
279C22×C3#383211yescharacteristic
280C2×D3—316no—
281C2×D3—616no—
282C2×D3—68no—
283C2×D3—616no—
284C2×D3—68no—
285C3×C4—28no—
286C3×C4—22no—
287C3×C4—48no—
288C3×C4—416no—
289C3×C4—48no—
290C3×C4—42no—
291C3×C4—182no—
292C3×C4—362no—
293Dic3—68no—
294Dic3—128no—
295C32—24no—
296C32#466414yescomplemented
297C32C3×#38111yescharacteristic, complemented
298C23—92no—
299C23—94no—
300C23—182no—
301C23—181no—
302C23—181no—
303C23C22×C3×C3⋊D311yescharacteristic
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—92no—
309C2×C4—188no—
310C2×C4—182no—
311C2×C4—182no—
312C2×C4—182no—
313C2×C4—182no—
314C2×C4C3×C4×C3⋊D312yes—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—28no—
318C2×C3—24no—
319C2×C3—28no—
320C2×C3—184no—
321C2×C3C3×#87214yes—
322C2×C3C3×D3×C22⋊C414yes—
323C2×C3C3×#88318yes—
324C2×C3C3×#89214yes—
325C2×C3C3×#89418yes—
326C2×C3#6408111yescharacteristic
327C2×C3C22⋊C4×C3⋊D311yescharacteristic
328C2×C3#6409212yes—
329C2×C3#6410111yescharacteristic
330C2×C3#6410312yes—
331D3—616no—
332C22—94no—
333C22—182no—
334C22—184no—
335C22—182no—
336C22—184no—
337C22C2×C3×C4×C3⋊D311yescharacteristic
338C22C3×#382812yes—
339C22C3×D4×C3⋊D312yes—
340C22C3×#383011yescharacteristic
341C22C3×#383211yescharacteristic
342C4—22no—
343C4—42no—
344C4—182no—
345C4—362no—
346C3—24no—
347C3C3×#466414yescomplemented
348C3#1059236311yescharacteristic, direct product factor
349C2—184no—
350C2C3×#6408111yescharacteristic
351C2C3×C22⋊C4×C3⋊D311yescharacteristic
352C2C3×#6409212yes—
353C2C3×#6410111yescharacteristic
354C2C3×#6410312yes—
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC33×#95C2
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC3×#95C3⋊D3
hypocenterC32C3×#381
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
3. C33×#95C2semidirect
4. C3×#63658C2semidirect
5. C3×#63658C2semidirect
6. #10592363C3direct (N × H)
14. C2×C3×C4×C3⋊D3C4semidirect
21. C32×#95D3semidirect
22. C32×#95C2×C3semidirect
23. #63658C2×C3semidirect
24. #63658C2×C3semidirect
57. C2×C4×C3⋊D3C3×C4semidirect
85. C3×#95C3×D3semidirect
86. C3×#95C3⋊D3semidirect
193. #95C3×C3⋊D3semidirect
194. C33#381semidirect
296. C32#4664semidirect
297. C32C3×#381semidirect
347. C3C3×#4664semidirect
348. C3#10592363direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C3×#37522
C33×#952
C3×#636582
#105923633
C3×#46643