Finite Group Databasebeta

← C3×C4×#3713 #13157357

Subgroups — C3×C4×#3713 #13157357

Order 1728.

798 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D12C212yesmaximal, complemented
2C2×C3×#3713C211yesmaximal, characteristic
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C3×#63576C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6C3×C4×#213—32nomaximal
7C4×#3713C311yesmaximal, characteristic, direct product factor
8C2×C32×D12C2212yes—
9C2×C32×C4×D3—22no—
10C33×C42C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×(C2×D3)⋊C4—22no—
14C32×C4⋊Dic3C2212yes—
15C3×#3713C414yesdirect product factor
16C32×C4×D4—32no—
17C2×C3×#213—32no—
18C3×C4×D12—22no—
19C3×C4×D12D312yescomplemented
20C3×C4×D12C2×C312yescomplemented
21C2×#3713C2×C311yescharacteristic
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
23C3×#832—32no—
24C3×#809—32no—
25C3×#807—32no—
26C3×C4×C3⋊C8—32no—
27C3×C4×C3⋊C8—61no—
28#63576C2×C311yescharacteristic, complemented
29#63562C2×C312yescomplemented
30C22×C32×D3—22no—
31C2×C33×C4D411yescharacteristic
32C2×C33×C4D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C32×Dic3—22no—
35C32×D12—22no—
36C32×D12C2×C414yescomplemented
37C32×C4×D3—44no—
38C3×C3⋊(C3⋊C8)—21no—
39C3×C3⋊(C3⋊C8)C2×C412yescomplemented
40C3×C4×D8—91no—
41C4×#213—32no—
42C2×C32×D4—32no—
43C22×C32×C4—62no—
44C2×C3×D12—22no—
45C2×C3×D12C2×D312yes—
46C2×C3×D12C22×C312yes—
47C2×C3×C4×D3—24no—
48C2×C3×C4×D3—22no—
49C2×C3×C4×D3—22no—
50C32×C4⋊C4—32no—
51C32×C22⋊C4—62no—
52C32×C42—21no—
53C32×C42—22no—
54C32×C42—41no—
55C32×C42C2×D312yes—
56C32×C42C22×C311yescharacteristic
57C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
58C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
59C3×#213—38no—
60C3×(C2×D3)⋊C4—24no—
61C3×(C2×D3)⋊C4—22no—
62C3×(C2×D3)⋊C4—22no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C22×C312yes—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—61no—
69C2×C3×C3⋊C8—61no—
70#3713C3×C414yesdirect product factor
71C22×C33C2×D411yescharacteristic
72C2×C32×D3—24no—
73C2×C32×D3—42no—
74C2×C32×D3—42no—
75C33×C4—21no—
76C33×C4D812yescomplemented
77C33×C4C22×C411yescharacteristic
78C33×C4#4111yescharacteristic
79C32×Dic3—42no—
80C2×C3×D8—91no—
81C3×C4×D4—32no—
82C3×C4×D4—32no—
83C3×C4×D4—62no—
84C2×#213—32no—
85C4×D12C3×D312yescomplemented
86C3×#107—91no—
87C3×#102—92no—
88C3×C4×C8—91no—
89#832—32no—
90#809—32no—
91#807—32no—
92C4×C3⋊C8—32no—
93C4×C3⋊C8—61no—
94C23×C32—62no—
95C22×C3×D3—24no—
96C22×C3×D3—22no—
97C22×C3×D3—22no—
98C32×D4—34no—
99C32×D4—62no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4—62no—
110C2×C32×C4—124no—
111C2×C32×C4#6712yes—
112C2×C32×C4#6712yes—
113C2×C32×C4C3×D411yescharacteristic
114C2×C32×C4C3×D411yescharacteristic
115C2×C32×C4C22×D312yes—
116C2×C32×C4C23×C311yescharacteristic
117C2×C3×Dic3—24no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C3×D12—24no—
121C3×D12—22no—
122C3×D12—22no—
123C3×D12—24no—
124C3×D12C4×D314yescomplemented
125C3×D12C2×C3×C414yescomplemented
126C3×C4×D3—48no—
127C3×C4×D3—44no—
128C3×C4×D3—44no—
129C3⋊(C3⋊C8)—21no—
130C3⋊(C3⋊C8)C2×C3×C412yescomplemented
131C3×C3⋊C8—34no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—62no—
135C4×D8—91noSylow
136C2×C33C4×D411yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33#13511yescharacteristic
139C32×D3—44no—
140C2×C3×D4—32no—
141C2×C3×D4—32no—
142C2×C3×D4—62no—
143C22×C3×C4—64no—
144C22×C3×C4—62no—
145C22×C3×C4—62no—
146C2×D12C2×C3×D312yes—
147C2×C4×D3—22no—
148C3×D8—94no—
149C2×C3×C8—91no—
150C2×C3×C8—91no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—62no—
154C3×C22⋊C4—64no—
155C3×C22⋊C4—62no—
156C3×C22⋊C4—62no—
157C3×C42—21no—
158C3×C42—22no—
159C3×C42—41no—
160C3×C42D3211yescharacteristic
161C3×C42C2×C3×D312yes—
162#213—38no—
163(C2×D3)⋊C4—22no—
164C4⋊Dic3C2×C3×D312yes—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—61no—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—41no—
172C22×C32—64no—
173C22×C32—122no—
174C22×C32—122no—
175C22×C32C2×#6712yes—
176C22×C32C2×C3×D411yescharacteristic
177C2×C3×D3—28no—
178C2×C3×D3—24no—
179C2×C3×D3—24no—
180C2×C3×D3—44no—
181C2×C3×D3—44no—
182C2×C3×D3—42no—
183C2×C3×D3—42no—
184C2×C3×D3—42no—
185C2×C3×D3—42no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—24no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—42no—
196C32×C4—41no—
197C32×C4—41no—
198C32×C4—42no—
199C32×C4—42no—
200C32×C4—122no—
201C32×C4#21314yescomplemented
202C32×C4C3×D812yescomplemented
203C32×C4C2×C4×D312yes—
204C32×C4#23512yes—
205C32×C4C22×C3×C411yescharacteristic
206C32×C4C3×#4111yescharacteristic
207C3×Dic3—44no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C2×D8—91no—
211C4×D4—32no—
212#107—91no—
213#102—92no—
214C4×C8—91no—
215C33C4×D811yescharacteristic, complemented, Sylow
216C23×C3—64no—
217C23×C3—62no—
218C23×C3—62no—
219C22×D3—22no—
220C3×D4—34no—
221C3×D4—34no—
222C3×D4—64no—
223C3×D4—64no—
224C3×D4—62no—
225C3×D4—62no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—64no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—128no—
239C2×C3×C4—124no—
240C2×C3×C4—124no—
241C2×C3×C4#62511yescharacteristic
242C2×C3×C4#62511yescharacteristic
243C2×C3×C4C3×#6712yes—
244C2×C3×C4C3×#6712yes—
245C2×C3×C4C2×D3211yescharacteristic
246C2×C3×C4C22×C3×D312yes—
247C2×Dic3—22no—
248D12—22no—
249D12C3×C4×D314yescomplemented
250C4×D3—44no—
251C3×C8—92no—
252C3×C8—181no—
253C3⋊C8—34no—
254C3⋊C8—62no—
255C3⋊C8—62no—
256C3⋊C8—62no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—124no—
267C2×C32C4×#6712yes—
268C2×C32C2×#21312yes—
269C2×C32#95012yes—
270C2×C32C3×C4×D411yescharacteristic
271C2×C32C2×C3×D811yescharacteristic
272C2×C32C3×#13511yescharacteristic
273C3×D3—48no—
274C3×D3—44no—
275C3×D3—44no—
276C2×D4—32no—
277C22×C4—62no—
278D8—94no—
279C2×C8—91no—
280C2×C8—91no—
281C4⋊C4—32no—
282C22⋊C4—62no—
283C42C3×D3211yescharacteristic
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—41no—
287C22×C3—68no—
288C22×C3—64no—
289C22×C3—64no—
290C22×C3—124no—
291C22×C3—124no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3C2×#62511yescharacteristic
297C22×C3C2×C3×#6712yes—
298C2×D3—24no—
299C2×D3—42no—
300C2×D3—42no—
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—21no—
307C3×C4—22no—
308C3×C4—24no—
309C3×C4—22no—
310C3×C4—42no—
311C3×C4—42no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—42no—
315C3×C4—124no—
316C3×C4—122no—
317C3×C4—122no—
318C3×C4#371312yesdirect product factor
319C3×C4C3×#21314yescomplemented
320C3×C4#379811yescharacteristic
321C3×C4C4×D3211yescharacteristic
322C3×C4C2×C3×C4×D312yes—
323C3×C4C3×#23512yes—
324Dic3—42no—
325C32—22no—
326C32—21no—
327C32—41no—
328C32C4×#21312yescomplemented
329C32C3×C4×D811yescharacteristic, complemented
330C23—62no—
331D4—34no—
332D4—62no—
333C2×C4—62no—
334C2×C4—124no—
335C2×C4C3×#62511yescharacteristic
336C2×C4C3×#62511yescharacteristic
337C2×C4C2×C3×D3211yescharacteristic
338C8—92no—
339C8—181no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—128no—
350C2×C3—124no—
351C2×C3—124no—
352C2×C3C2×#371311yescharacteristic
353C2×C3#6382211yescharacteristic
354C2×C3C4×#62511yescharacteristic
355C2×C3C3×C4×#6712yes—
356C2×C3C2×C3×#21312yes—
357C2×C3C3×#95012yes—
358D3—44no—
359C22—64no—
360C22—122no—
361C22—122no—
362C22C2×C3×#62511yescharacteristic
363C4—21no—
364C4—122no—
365C4C3×#371312yesdirect product factor
366C4C3×#379811yescharacteristic
367C4C3×C4×D3211yescharacteristic
368C3—22no—
369C3—21no—
370C3—41no—
371C3C4×#371311yescharacteristic, direct product factor
372C3C3×C4×#21312yescomplemented
373C2—124no—
374C2C2×C3×#371311yescharacteristic
375C2C3×#6382211yescharacteristic
376C2C3×C4×#62511yescharacteristic
subgroupG/H
centerC2×C3×C4#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×C4×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C32×C4×D12C2semidirect
3. C3×C4×C3⋊(C3⋊C8)C2semidirect
4. C3×#63576C2semidirect
5. C3×#63562C2semidirect
7. C4×#3713C3direct (N × H)
15. C3×#3713C4direct (N × H)
19. C3×C4×D12D3semidirect
20. C3×C4×D12C2×C3semidirect
22. C4×C3⋊(C3⋊C8)C2×C3semidirect
28. #63576C2×C3semidirect
29. #63562C2×C3semidirect
36. C32×D12C2×C4semidirect
39. C3×C3⋊(C3⋊C8)C2×C4semidirect
70. #3713C3×C4direct (N × H)
76. C33×C4D8semidirect
85. C4×D12C3×D3semidirect
124. C3×D12C4×D3semidirect
125. C3×D12C2×C3×C4semidirect
130. C3⋊(C3⋊C8)C2×C3×C4semidirect
201. C32×C4#213semidirect
202. C32×C4C3×D8semidirect
215. C33C4×D8semidirect
249. D12C3×C4×D3semidirect
318. C3×C4#3713direct (N × H)
319. C3×C4C3×#213semidirect
328. C32C4×#213semidirect
329. C32C3×C4×D8semidirect
365. C4C3×#3713direct (N × H)
371. C3C4×#3713direct (N × H)
372. C3C3×C4×#213semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D122
C2×C3×#37132
C3×C4×C3⋊(C3⋊C8)2
C3×#635762
C3×#635622
C3×C4×#2133
C4×#37133