Finite Group Databasebeta

← C3×C4⋊C4×C3⋊Dic3 #13163006

Subgroups — C3×C4⋊C4×C3⋊Dic3 #13163006

Order 1728.

1532 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C3×C4×C3⋊Dic3C212yesmaximal
4C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
5C3×#63658C212yesmaximal
6C4⋊C4×C3⋊Dic3C311yesmaximal, characteristic, direct product factor
7C3×Dic3×C4⋊C4—34nomaximal
8C22×C33×C4C2212yes—
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C33×C4⋊C4C414yescomplemented
14C3×C4⋊(C3⋊Dic3)C414yescomplemented
15C3×C4×C3⋊Dic3—24no—
16C3×C4×C3⋊Dic3C414yescomplemented
17C2×C32×C4⋊C4—24no—
18C2×C32×C4⋊C4D314yes—
19C2×C32×C4⋊C4C2×C311yescharacteristic
20C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
21C2×C4×C3⋊Dic3C2×C312yes—
22C2×C4×C3⋊Dic3C2×C311yescharacteristic
23C2×C3×C4⋊Dic3—34no—
24C2×C3×C4×Dic3—34no—
25C2×C3×C4×Dic3—38no—
26#63658C2×C312yes—
27C3×C3⋊#95—38no—
28C23×C33C2311yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4—22no—
31C2×C33×C4C2×C414yes—
32C2×C33×C4C2×C412yes—
33C2×C33×C4C2×C412yes—
34C2×C33×C4C2×C412yes—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3C2×C412yes—
40C2×C3×C3⋊Dic3C2×C414yes—
41C2×C3×C3⋊Dic3C2×C412yes—
42C2×C3×C3⋊Dic3D412yes—
43C2×C3×C3⋊Dic3Q812yes—
44C3×C4×C4⋊C4—91no—
45Dic3×C4⋊C4—34no—
46C22×C32×C4—24no—
47C22×C32×C4—28no—
48C22×C32×C4C2×D318yes—
49C22×C32×C4C2×D314yes—
50C22×C32×C4C22×C312yes—
51C22×C32×C4C22×C311yescharacteristic
52C22×C3⋊Dic3C22×C312yes—
53C22×C3⋊Dic3C22×C311yescharacteristic
54C22×C3⋊Dic3C22×C311yescharacteristic
55C22×C3×Dic3—34no—
56C22×C3×Dic3—38no—
57C22×C3×Dic3—34no—
58C32×C4⋊C4—216no—
59C32×C4⋊C4Dic3116yescomplemented
60C32×C4⋊C4C3×C414yescomplemented
61C4⋊(C3⋊Dic3)C3×C414yescomplemented
62C4×C3⋊Dic3—24no—
63C4×C3⋊Dic3C3×C414yescomplemented
64C3×C4⋊Dic3—316no—
65C3×C4×Dic3—316no—
66C3×C4×Dic3—616no—
67C22×C33C22×C411yescharacteristic
68C22×C33C22×C411yescharacteristic
69C22×C33C22×C411yescharacteristic
70C22×C33C2×D411yescharacteristic
71C22×C33C2×Q811yescharacteristic
72C22×C33#4111yescharacteristic
73C22×C33#4111yescharacteristic
74C33×C4—24no—
75C33×C4C4214yescomplemented
76C3×C3⋊Dic3—22no—
77C3×C3⋊Dic3—24no—
78C3×C3⋊Dic3C4⋊C414yesdirect product factor
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—91no—
81C2×C3×C4⋊C4C3×D314yes—
82C2×C3×C4⋊C4C3⋊D311yescharacteristic
83C2×C3×C42—91no—
84C2×C3×C42—92no—
85C2×C4⋊Dic3—34no—
86C2×C4×Dic3—34no—
87C2×C4×Dic3—38no—
88C3×#95—92no—
89C3⋊#95—38no—
90C23×C32—24no—
91C23×C32C22×D314yes—
92C23×C32C23×C311yescharacteristic
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—28no—
96C2×C32×C4—28no—
97C2×C32×C4—28no—
98C2×C32×C4—216no—
99C2×C32×C4—28no—
100C2×C32×C4—28no—
101C2×C32×C4—48no—
102C2×C32×C4—48no—
103C2×C32×C4C4×D318yes—
104C2×C32×C4C4×D318yes—
105C2×C32×C4C2×Dic3116yes—
106C2×C32×C4C2×Dic318yes—
107C2×C32×C4C2×C3×C412yes—
108C2×C32×C4C2×C3×C412yes—
109C2×C32×C4C2×C3×C414yes—
110C2×C32×C4C2×C3×C412yes—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3C2×C3×C412yes—
116C2×C3⋊Dic3C2×C3×C414yes—
117C2×C3⋊Dic3C2×C3×C412yes—
118C2×C3⋊Dic3C3×D412yes—
119C2×C3⋊Dic3C3×Q812yes—
120C2×C3×Dic3—38no—
121C2×C3×Dic3—38no—
122C2×C3×Dic3—38no—
123C2×C3×Dic3—38no—
124C2×C3×Dic3—316no—
125C2×C3×Dic3—68no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—68no—
129C4×C4⋊C4—91noSylow
130C2×C33C2×C4211yescharacteristic
131C2×C33C2×C4⋊C411yescharacteristic
132C2×C33#11711yescharacteristic
133C2×C33C4×D411yescharacteristic
134C2×C33C4×D411yescharacteristic
135C2×C33C4×Q811yescharacteristic
136C2×C33C4×Q811yescharacteristic
137C22×C3×C4—24no—
138C22×C3×C4—28no—
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4—92no—
142C22×C3×C4C2×C3×D318yes—
143C22×C3×C4C2×C3×D314yes—
144C22×C3×C4C2×C3⋊D312yes—
145C22×C3×C4C2×C3⋊D311yescharacteristic
146C22×Dic3—34no—
147C22×Dic3—34no—
148C22×Dic3—38no—
149C3×C4⋊C4—216no—
150C3×C4⋊C4—94no—
151C3×C4⋊C4C3×Dic3116yescomplemented
152C3×C4⋊C4C3⋊Dic314yesdirect product factor
153C3×C42—94no—
154C3×C42—184no—
155C4⋊Dic3—316no—
156C4×Dic3—316no—
157C4×Dic3—616no—
158C22×C32—24no—
159C22×C32—24no—
160C22×C32—24no—
161C22×C32—24no—
162C22×C32—24no—
163C22×C32—24no—
164C22×C32—24no—
165C22×C32C2×C4×D314yes—
166C22×C32C2×C4×D314yes—
167C22×C32D3×D414yes—
168C22×C32#23714yes—
169C22×C32D3×Q814yes—
170C22×C32#23914yes—
171C22×C32C22×Dic314yes—
172C22×C32C22×C3×C411yescharacteristic
173C22×C32C22×C3×C411yescharacteristic
174C22×C32C22×C3×C411yescharacteristic
175C22×C32C2×C3×D411yescharacteristic
176C22×C32C2×C3×Q811yescharacteristic
177C22×C32C3×#4111yescharacteristic
178C22×C32C3×#4111yescharacteristic
179C32×C4—24no—
180C32×C4—216no—
181C32×C4—216no—
182C32×C4—416no—
183C32×C4C4×Dic3116yescomplemented
184C32×C4C3×C4214yescomplemented
185C3⋊Dic3—22no—
186C3⋊Dic3—24no—
187C3⋊Dic3C3×C4⋊C414yesdirect product factor
188C3×Dic3—316no—
189C3×Dic3—68no—
190C3×Dic3—616no—
191C2×C4⋊C4—91no—
192C2×C4⋊C4C3×C3⋊D311yescharacteristic
193C2×C42—91no—
194C2×C42—92no—
195#95—92no—
196C33C4×C4⋊C411yescharacteristic, complemented, Sylow
197C23×C3—24no—
198C23×C3C22×C3×D314yes—
199C23×C3C22×C3⋊D311yescharacteristic
200C2×C3×C4—28no—
201C2×C3×C4—28no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—28no—
205C2×C3×C4—28no—
206C2×C3×C4—28no—
207C2×C3×C4—216no—
208C2×C3×C4—48no—
209C2×C3×C4—48no—
210C2×C3×C4—92no—
211C2×C3×C4—92no—
212C2×C3×C4—92no—
213C2×C3×C4—92no—
214C2×C3×C4—94no—
215C2×C3×C4—182no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4—182no—
219C2×C3×C4C3×C4×D318yes—
220C2×C3×C4C3×C4×D318yes—
221C2×C3×C4C2×C3×Dic3116yes—
222C2×C3×C4C2×C3×Dic318yes—
223C2×C3×C4C4×C3⋊D312yes—
224C2×C3×C4C4×C3⋊D312yes—
225C2×C3×C4C2×C3⋊Dic314yes—
226C2×C3×C4C2×C3⋊Dic312yes—
227C2×Dic3—38no—
228C2×Dic3—38no—
229C2×Dic3—38no—
230C2×Dic3—38no—
231C2×Dic3—316no—
232C2×Dic3—68no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—68no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32#88714yes—
244C2×C32D3×C4⋊C414yes—
245C2×C32#89214yes—
246C2×C32#89314yes—
247C2×C32C2×C4×Dic314yes—
248C2×C32D4×Dic314yes—
249C2×C32Q8×Dic314yes—
250C2×C32C2×C3×C4211yescharacteristic
251C2×C32C2×C3×C4⋊C411yescharacteristic
252C2×C32C3×#11711yescharacteristic
253C2×C32C3×C4×D411yescharacteristic
254C2×C32C3×C4×D411yescharacteristic
255C2×C32C3×C4×Q811yescharacteristic
256C2×C32C3×C4×Q811yescharacteristic
257C22×C4—91no—
258C22×C4—91no—
259C22×C4—92no—
260C22×C4C2×C3×C3⋊D312yes—
261C22×C4C2×C3×C3⋊D311yescharacteristic
262C4⋊C4—94no—
263C4⋊C4C3×C3⋊Dic314yesdirect product factor
264C42—94no—
265C42—184no—
266C22×C3—24no—
267C22×C3—24no—
268C22×C3—24no—
269C22×C3—24no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3C2×C3×C4×D314yes—
274C22×C3C2×C3×C4×D314yes—
275C22×C3C3×D3×D414yes—
276C22×C3C3×#23714yes—
277C22×C3C3×D3×Q814yes—
278C22×C3C3×#23914yes—
279C22×C3C22×C3×Dic314yes—
280C22×C3C2×C4×C3⋊D311yescharacteristic
281C22×C3C2×C4×C3⋊D311yescharacteristic
282C22×C3D4×C3⋊D311yescharacteristic
283C22×C3#383011yescharacteristic
284C22×C3Q8×C3⋊D311yescharacteristic
285C22×C3#383211yescharacteristic
286C22×C3C22×C3⋊Dic311yescharacteristic
287C3×C4—216no—
288C3×C4—24no—
289C3×C4—216no—
290C3×C4—416no—
291C3×C4—94no—
292C3×C4—182no—
293C3×C4—184no—
294C3×C4C3×C4×Dic3116yescomplemented
295C3×C4C4×C3⋊Dic314yescomplemented
296Dic3—316no—
297Dic3—68no—
298Dic3—616no—
299C32—24no—
300C32Dic3×C4⋊C414yescomplemented
301C32C3×C4×C4⋊C411yescharacteristic, complemented
302C23C22×C3×C3⋊D311yescharacteristic
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4—92no—
308C2×C4—92no—
309C2×C4—94no—
310C2×C4—182no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4—182no—
314C2×C4C3×C4×C3⋊D312yes—
315C2×C4C3×C4×C3⋊D312yes—
316C2×C4C2×C3×C3⋊Dic314yes—
317C2×C4C2×C3×C3⋊Dic312yes—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—24no—
324C2×C3—24no—
325C2×C3C3×#88714yes—
326C2×C3C3×D3×C4⋊C414yes—
327C2×C3C3×#89214yes—
328C2×C3C3×#89314yes—
329C2×C3C2×C3×C4×Dic314yes—
330C2×C3C3×D4×Dic314yes—
331C2×C3C3×Q8×Dic314yes—
332C2×C3#6409611yescharacteristic
333C2×C3C4⋊C4×C3⋊D311yescharacteristic
334C2×C3#6410111yescharacteristic
335C2×C3#6410211yescharacteristic
336C2×C3C2×C4×C3⋊Dic311yescharacteristic
337C2×C3D4×C3⋊Dic311yescharacteristic
338C2×C3Q8×C3⋊Dic311yescharacteristic
339C22C2×C3×C4×C3⋊D311yescharacteristic
340C22C2×C3×C4×C3⋊D311yescharacteristic
341C22C3×D4×C3⋊D311yescharacteristic
342C22C3×#383011yescharacteristic
343C22C3×Q8×C3⋊D311yescharacteristic
344C22C3×#383211yescharacteristic
345C22C22×C3×C3⋊Dic311yescharacteristic
346C4—24no—
347C4—94no—
348C4—182no—
349C4—184no—
350C4C3×C4×C3⋊Dic314yescomplemented
351C3—24no—
352C3C3×Dic3×C4⋊C414yescomplemented
353C3C4⋊C4×C3⋊Dic311yescharacteristic, direct product factor
354C2C3×#6409611yescharacteristic
355C2C3×C4⋊C4×C3⋊D311yescharacteristic
356C2C3×#6410111yescharacteristic
357C2C3×#6410211yescharacteristic
358C2C2×C3×C4×C3⋊Dic311yescharacteristic
359C2C3×D4×C3⋊Dic311yescharacteristic
360C2C3×Q8×C3⋊Dic311yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC2×C32C2×C3×C42
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C4⋊C4×C3⋊Dic3C3direct (N × H)
13. C33×C4⋊C4C4semidirect
14. C3×C4⋊(C3⋊Dic3)C4semidirect
16. C3×C4×C3⋊Dic3C4semidirect
59. C32×C4⋊C4Dic3semidirect
60. C32×C4⋊C4C3×C4semidirect
61. C4⋊(C3⋊Dic3)C3×C4semidirect
63. C4×C3⋊Dic3C3×C4semidirect
75. C33×C4C42semidirect
78. C3×C3⋊Dic3C4⋊C4direct (N × H)
151. C3×C4⋊C4C3×Dic3semidirect
152. C3×C4⋊C4C3⋊Dic3direct (N × H)
183. C32×C4C4×Dic3semidirect
184. C32×C4C3×C42semidirect
187. C3⋊Dic3C3×C4⋊C4direct (N × H)
196. C33C4×C4⋊C4semidirect
263. C4⋊C4C3×C3⋊Dic3direct (N × H)
294. C3×C4C3×C4×Dic3semidirect
295. C3×C4C4×C3⋊Dic3semidirect
300. C32Dic3×C4⋊C4semidirect
301. C32C3×C4×C4⋊C4semidirect
350. C4C3×C4×C3⋊Dic3semidirect
352. C3C3×Dic3×C4⋊C4semidirect
353. C3C4⋊C4×C3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×C4×C3⋊Dic32
C3×#636582
C4⋊C4×C3⋊Dic33
C3×Dic3×C4⋊C43