Finite Group Databasebeta

← C2×C3×#64107 #13179263

Subgroups — C2×C3×#64107 #13179263

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic
3C2×C3×#3752C211yesmaximal, characteristic
4C2×C3×#3752C211yesmaximal, characteristic
5C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6C2×C3×#3750C211yesmaximal, characteristic, complemented
7C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
8C3×#64107C218yesmaximal, direct product factor
9C2×#64107C311yesmaximal, characteristic, direct product factor
10C2×C3×#898—34nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C33×C4C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C33×C4⋊C4C2214yescomplemented
19C3×#3752C2214yes—
20C3×#3752C2214yes—
21C3×#3752C2214yes—
22C3×C4⋊(C3⋊Dic3)C2214yescomplemented
23C3×#3750C2214yescomplemented
24C3×C4×C3⋊Dic3C2214yescomplemented
25C2×C32×C4⋊C4—24no—
26C2×C32×C4⋊C4D314yescomplemented
27C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
28C2×#3752C2×C311yescharacteristic
29C2×#3752C2×C311yescharacteristic
30C2×#3752C2×C311yescharacteristic
31C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
32C2×#3750C2×C311yescharacteristic, complemented
33C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
34#64107C2×C318yesdirect product factor
35C2×C3×(C2×D3)⋊C4—34no—
36C2×C3×(C2×D3)⋊C4—34no—
37C2×C3×(C2×D3)⋊C4—34no—
38C2×C3×C4⋊Dic3—34no—
39C2×C3×Dic3⋊C4—34no—
40C2×C3×C4×Dic3—34no—
41C3×#898—332no—
42C23×C33C2311yescharacteristic
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3—22no—
45C22×C3×C3⋊D3—22no—
46C22×C3×C3⋊D3C2312yes—
47C2×C33×C4—22no—
48C2×C33×C4—22no—
49C2×C33×C4—22no—
50C2×C33×C4C2312yes—
51C2×C33×C4C2312yes—
52C2×C33×C4C2312yes—
53C2×C3×C3⋊Dic3—22no—
54C2×C3×C3⋊Dic3—22no—
55C2×C3×C3⋊Dic3—22no—
56C2×C3×C3⋊Dic3C2312yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×#126—91no—
60C2×#898—34no—
61C23×C3⋊D3C22×C311yescharacteristic
62C23×C3×D3—34no—
63C22×C32×C4—24no—
64C22×C32×C4—24no—
65C22×C32×C4—24no—
66C22×C32×C4C2×D314yes—
67C22×C32×C4C2×D314yes—
68C22×C32×C4C2×D314yes—
69C22×C32×C4C22×C311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C32×C4C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3⋊Dic3C22×C311yescharacteristic
75C22×C3×Dic3—34no—
76C22×C3×Dic3—34no—
77C22×C3×Dic3—34no—
78C32×C4⋊C4—216no—
79C32×C4⋊C4C2×D3116yescomplemented
80C32×C4⋊C4C22×C314yescomplemented
81#3752C22×C314yes—
82#3752C22×C314yes—
83#3752C22×C314yes—
84C4⋊(C3⋊Dic3)C22×C314yescomplemented
85#3750C22×C314yescomplemented
86C4×C3⋊Dic3C22×C314yescomplemented
87C3×(C2×D3)⋊C4—316no—
88C3×(C2×D3)⋊C4—316no—
89C3×(C2×D3)⋊C4—316no—
90C3×C4⋊Dic3—316no—
91C3×Dic3⋊C4—316no—
92C3×C4×Dic3—316no—
93C22×C33#4112yes—
94C22×C33#4112yes—
95C22×C33#4112yes—
96C22×C33C2411yescharacteristic
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—44no—
101C33×C4—22no—
102C33×C4—22no—
103C33×C4—22no—
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—22no—
106C3×C3⋊Dic3—22no—
107C3×#126—98no—
108C2×C3×C4⋊C4—24no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C4⋊C4C3×D314yescomplemented
112C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C42—91no—
117C2×(C2×D3)⋊C4—34no—
118C2×(C2×D3)⋊C4—34no—
119C2×(C2×D3)⋊C4—34no—
120C2×C4⋊Dic3—34no—
121C2×Dic3⋊C4—34no—
122C2×C4×Dic3—34no—
123#898—332no—
124C23×C32—24no—
125C23×C32C22×D314yes—
126C23×C32C23×C311yescharacteristic
127C22×C3⋊D3—22no—
128C22×C3⋊D3—22no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3C23×C312yes—
131C22×C3×D3—38no—
132C22×C3×D3—68no—
133C22×C3×D3—68no—
134C22×C3×D3—68no—
135C2×C32×C4—216no—
136C2×C32×C4—28no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—28no—
141C2×C32×C4—28no—
142C2×C32×C4—28no—
143C2×C32×C4—28no—
144C2×C32×C4—28no—
145C2×C32×C4—48no—
146C2×C32×C4—48no—
147C2×C32×C4C22×D318yes—
148C2×C32×C4C22×D318yes—
149C2×C32×C4C22×D318yes—
150C2×C32×C4C23×C312yes—
151C2×C32×C4C23×C312yes—
152C2×C32×C4C23×C312yes—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—22no—
156C2×C3⋊Dic3C23×C312yes—
157C2×C3⋊Dic3C23×C312yes—
158C2×C3⋊Dic3C23×C312yes—
159C2×C3×Dic3—38no—
160C2×C3×Dic3—38no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—68no—
163C2×C3×Dic3—68no—
164C2×C3×Dic3—68no—
165C2×#126—91noSylow
166C2×C33#12614yescomplemented
167C2×C33C2×#4111yescharacteristic
168C2×C33C2×#4111yescharacteristic
169C2×C33C2×#4111yescharacteristic
170C3×C3⋊D3—42no—
171C24×C3—91no—
172C23×D3—34no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—91no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C22×C3×C4C2×C3×D314yes—
180C22×C3×C4C2×C3×D314yes—
181C22×C3×C4C2×C3×D314yes—
182C22×C3×C4C2×C3⋊D311yescharacteristic
183C22×C3×C4C2×C3⋊D311yescharacteristic
184C22×C3×C4C2×C3⋊D311yescharacteristic
185C22×Dic3—34no—
186C22×Dic3—34no—
187C22×Dic3—34no—
188C3×C4⋊C4—216no—
189C3×C4⋊C4—94no—
190C3×C4⋊C4—94no—
191C3×C4⋊C4C2×C3×D3116yescomplemented
192C3×C4⋊C4C2×C3⋊D314yescomplemented
193C3×C22⋊C4—94no—
194C3×C22⋊C4—94no—
195C3×C22⋊C4—94no—
196C3×C42—94no—
197(C2×D3)⋊C4—316no—
198(C2×D3)⋊C4—316no—
199(C2×D3)⋊C4—316no—
200C4⋊Dic3—316no—
201Dic3⋊C4—316no—
202C4×Dic3—316no—
203C22×C32—24no—
204C22×C32—28no—
205C22×C32—28no—
206C22×C32—28no—
207C22×C32#23518yes—
208C22×C32#23718yes—
209C22×C32#23918yes—
210C22×C32C3×#4112yes—
211C22×C32C3×#4112yes—
212C22×C32C3×#4112yes—
213C22×C32C23×D314yes—
214C22×C32C24×C311yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—22no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—44no—
219C2×C3×D3—68no—
220C2×C3×D3—68no—
221C2×C3×D3—68no—
222C2×C3×D3—1216no—
223C32×C4—216no—
224C32×C4—28no—
225C32×C4—22no—
226C32×C4—22no—
227C32×C4—22no—
228C32×C4—28no—
229C32×C4—28no—
230C32×C4—48no—
231C32×C4—48no—
232C3⋊Dic3—22no—
233C3⋊Dic3—22no—
234C3⋊Dic3—22no—
235C3×Dic3—68no—
236C3×Dic3—68no—
237C3×Dic3—68no—
238#126—98no—
239C2×C4⋊C4—91no—
240C2×C4⋊C4—91no—
241C2×C4⋊C4C3×C3⋊D311yescharacteristic, complemented
242C2×C22⋊C4—91no—
243C2×C22⋊C4—91no—
244C2×C22⋊C4—91no—
245C2×C42—91no—
246C33C2×#12611yescharacteristic, complemented, Sylow
247C23×C3—24no—
248C23×C3—92no—
249C23×C3—182no—
250C23×C3—182no—
251C23×C3—182no—
252C23×C3C22×C3×D314yes—
253C23×C3C22×C3⋊D311yescharacteristic
254C22×D3—38no—
255C22×D3—68no—
256C22×D3—68no—
257C22×D3—68no—
258C2×C3×C4—216no—
259C2×C3×C4—28no—
260C2×C3×C4—28no—
261C2×C3×C4—28no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—28no—
266C2×C3×C4—28no—
267C2×C3×C4—28no—
268C2×C3×C4—48no—
269C2×C3×C4—48no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—182no—
274C2×C3×C4—182no—
275C2×C3×C4—182no—
276C2×C3×C4C22×C3×D318yes—
277C2×C3×C4C22×C3×D318yes—
278C2×C3×C4C22×C3×D318yes—
279C2×C3×C4C22×C3⋊D312yes—
280C2×C3×C4C22×C3⋊D312yes—
281C2×C3×C4C22×C3⋊D312yes—
282C2×Dic3—38no—
283C2×Dic3—38no—
284C2×Dic3—38no—
285C2×Dic3—68no—
286C2×Dic3—68no—
287C2×Dic3—68no—
288C2×C32—216no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32#898116yescomplemented
293C2×C32C3×#12614yescomplemented
294C2×C32C2×#23514yes—
295C2×C32C2×#23714yes—
296C2×C32C2×#23914yes—
297C2×C32C2×C3×#4111yescharacteristic
298C2×C32C2×C3×#4111yescharacteristic
299C2×C32C2×C3×#4111yescharacteristic
300C3⋊D3—42no—
301C3×D3—128no—
302C24—91no—
303C22×C4—91no—
304C22×C4—91no—
305C22×C4—91no—
306C22×C4C2×C3×C3⋊D311yescharacteristic
307C22×C4C2×C3×C3⋊D311yescharacteristic
308C22×C4C2×C3×C3⋊D311yescharacteristic
309C4⋊C4—94no—
310C4⋊C4—94no—
311C4⋊C4C2×C3×C3⋊D314yescomplemented
312C22⋊C4—94no—
313C22⋊C4—94no—
314C22⋊C4—94no—
315C42—94no—
316C22×C3—24no—
317C22×C3—28no—
318C22×C3—28no—
319C22×C3—28no—
320C22×C3—182no—
321C22×C3—182no—
322C22×C3—182no—
323C22×C3—364no—
324C22×C3C3×#23518yes—
325C22×C3C3×#23718yes—
326C22×C3C3×#23918yes—
327C22×C3#382812yes—
328C22×C3#383012yes—
329C22×C3#383212yes—
330C22×C3C23×C3×D314yes—
331C22×C3C23×C3⋊D311yescharacteristic
332C2×D3—68no—
333C2×D3—68no—
334C2×D3—68no—
335C2×D3—1216no—
336C3×C4—216no—
337C3×C4—28no—
338C3×C4—28no—
339C3×C4—28no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—48no—
344C3×C4—48no—
345C3×C4—182no—
346C3×C4—182no—
347C3×C4—182no—
348Dic3—68no—
349Dic3—68no—
350Dic3—68no—
351C32—24no—
352C32C2×#89814yescomplemented
353C32C2×C3×#12611yescharacteristic, complemented
354C23—92no—
355C23—182no—
356C23—182no—
357C23—182no—
358C23C22×C3×C3⋊D311yescharacteristic
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—92no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4C22×C3×C3⋊D312yes—
369C2×C4C22×C3×C3⋊D312yes—
370C2×C4C22×C3×C3⋊D312yes—
371C2×C3—216no—
372C2×C3—24no—
373C2×C3—24no—
374C2×C3—24no—
375C2×C3—362no—
376C2×C3C3×#898116yescomplemented
377C2×C3#6410714yesdirect product factor
378C2×C3C2×C3×#23514yes—
379C2×C3C2×C3×#23714yes—
380C2×C3C2×C3×#23914yes—
381C2×C3C2×#382811yescharacteristic
382C2×C3C2×#383011yescharacteristic
383C2×C3C2×#383211yescharacteristic
384D3—128no—
385C22—182no—
386C22—182no—
387C22—182no—
388C22—364no—
389C22C3×#382812yes—
390C22C3×#383012yes—
391C22C3×#383212yes—
392C22C23×C3×C3⋊D311yescharacteristic
393C4—22no—
394C4—22no—
395C4—22no—
396C4—182no—
397C4—182no—
398C4—182no—
399C3—24no—
400C3C2×C3×#89814yescomplemented
401C3C2×#6410711yescharacteristic, direct product factor
402C2—362no—
403C2C3×#6410714yesdirect product factor
404C2C2×C3×#382811yescharacteristic
405C2C2×C3×#383011yescharacteristic
406C2C2×C3×#383211yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C2×C3×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C33×C4⋊C4C2semidirect
5. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
6. C2×C3×#3750C2semidirect
7. C2×C3×C4×C3⋊Dic3C2semidirect
8. C3×#64107C2direct (N × H)
9. C2×#64107C3direct (N × H)
18. C33×C4⋊C4C22semidirect
22. C3×C4⋊(C3⋊Dic3)C22semidirect
23. C3×#3750C22semidirect
24. C3×C4×C3⋊Dic3C22semidirect
26. C2×C32×C4⋊C4D3semidirect
27. C2×C32×C4⋊C4C2×C3semidirect
31. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
32. C2×#3750C2×C3semidirect
33. C2×C4×C3⋊Dic3C2×C3semidirect
34. #64107C2×C3direct (N × H)
79. C32×C4⋊C4C2×D3semidirect
80. C32×C4⋊C4C22×C3semidirect
84. C4⋊(C3⋊Dic3)C22×C3semidirect
85. #3750C22×C3semidirect
86. C4×C3⋊Dic3C22×C3semidirect
111. C2×C3×C4⋊C4C3×D3semidirect
112. C2×C3×C4⋊C4C3⋊D3semidirect
166. C2×C33#126semidirect
191. C3×C4⋊C4C2×C3×D3semidirect
192. C3×C4⋊C4C2×C3⋊D3semidirect
241. C2×C4⋊C4C3×C3⋊D3semidirect
246. C33C2×#126semidirect
292. C2×C32#898semidirect
293. C2×C32C3×#126semidirect
311. C4⋊C4C2×C3×C3⋊D3semidirect
352. C32C2×#898semidirect
353. C32C2×C3×#126semidirect
376. C2×C3C3×#898semidirect
377. C2×C3#64107direct (N × H)
400. C3C2×C3×#898semidirect
401. C3C2×#64107direct (N × H)
403. C2C3×#64107direct (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×#37522
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37502
C2×C3×C4×C3⋊Dic32
C3×#641072
C2×#641073
C2×C3×#8983