Finite Group Databasebeta

← C2×C3×#64127 #13179515

Subgroups — C2×C3×#64127 #13179515

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C33×Q16C211yesmaximal, characteristic, complemented
4C2×C3×#3756C211yesmaximal, characteristic, complemented
5C2×C3×#3755C211yesmaximal, characteristic, complemented
6C3×#64127C218yesmaximal, direct product factor
7C2×C3×#3744C211yesmaximal, characteristic, complemented
8C2×C3×#3743C211yesmaximal, characteristic, complemented
9C2×#64127C311yesmaximal, characteristic, direct product factor
10C2×C3×#918—34nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C33×Q8C2211yescharacteristic
13C3×#3832—22no—
14C3×#3832C2214yes—
15C3×Q8×C3⋊D3—22no—
16C3×Q8×C3⋊D3C2214yescomplemented
17C2×C3×C4⋊(C3⋊D3)—21no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×C4×C3⋊D3—21no—
20C2×C3×C4×C3⋊D3—21no—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C3×#634—21no—
23C2×C3×#634C2211yescharacteristic
24C33×Q16C2214yescomplemented
25C2×C33×C8C2211yescharacteristic
26C3×#3756C2214yescomplemented
27C3×#3755C2214yescomplemented
28C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
29C3×#3744C2214yescomplemented
30C3×#3743C2214yescomplemented
31C2×#3832C2×C311yescharacteristic
32C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#239—34no—
34C2×C3×D3×Q8—34no—
35C2×C32×Q16—24no—
36C2×C32×Q16D314yescomplemented
37C2×C32×Q16C2×C311yescharacteristic, complemented
38C2×#3756C2×C311yescharacteristic, complemented
39C2×#3755C2×C311yescharacteristic, complemented
40#64127C2×C318yesdirect product factor
41C2×#3744C2×C311yescharacteristic, complemented
42C2×#3743C2×C311yescharacteristic, complemented
43C2×C3×#216—34no—
44C2×C3×#215—34no—
45C3×#918—332no—
46C2×C3×#204—34no—
47C2×C3×ΓL(1,𝔽25)—34no—
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3D411yescharacteristic
50C33×Q8—21no—
51C33×Q8—21no—
52C33×Q8C2312yes—
53C33×Q8C2312yes—
54C2×C33×C4—21no—
55C2×C33×C4—21no—
56C2×C33×C4C2311yescharacteristic
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3D411yescharacteristic
59C3×C4⋊(C3⋊D3)—24no—
60C3×C4⋊(C3⋊D3)—21no—
61C3×C4⋊(C3⋊D3)C2312yes—
62C3×C4×C3⋊D3—24no—
63C3×C4×C3⋊D3—24no—
64C3×C4×C3⋊D3D412yes—
65C3×C4×C3⋊D3C2312yes—
66C3×#634—21no—
67C3×#634—24no—
68C3×#634C2312yes—
69C33×C8C2312yes—
70C3×C3⋊(C3⋊C8)C2312yes—
71C2×C3×#137—91no—
72C2×#918—34no—
73C2×C32×Q8—24no—
74C2×C32×Q8—24no—
75C2×C32×Q8C2×D314yes—
76C2×C32×Q8C2×D314yes—
77C2×C32×Q8C22×C311yescharacteristic
78C2×C32×Q8C22×C311yescharacteristic
79#3832—22no—
80#3832C22×C314yes—
81Q8×C3⋊D3—22no—
82Q8×C3⋊D3C22×C314yescomplemented
83C2×C4⋊(C3⋊D3)—21no—
84C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
85C2×C4×C3⋊D3—21no—
86C2×C4×C3⋊D3—21no—
87C2×C4×C3⋊D3C22×C311yescharacteristic
88C2×#634—21no—
89C2×#634C22×C311yescharacteristic
90C3×#239—316no—
91C3×#239—68no—
92C3×D3×Q8—316no—
93C3×D3×Q8—68no—
94C2×C3×D12—34no—
95C2×C3×D12—64no—
96C2×C3×C4×D3—34no—
97C2×C3×C4×D3—64no—
98C2×C3×C4×D3—64no—
99C2×C3×Dic6—34no—
100C2×C3×Dic6—64no—
101C32×Q16—216no—
102C32×Q16C2×D3116yescomplemented
103C32×Q16C22×C314yescomplemented
104C2×C32×C8—24no—
105C2×C32×C8C2×D314yes—
106C2×C32×C8C22×C311yescharacteristic
107#3756C22×C314yescomplemented
108#3755C22×C314yescomplemented
109C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
110#3744C22×C314yescomplemented
111#3743C22×C314yescomplemented
112C3×#216—316no—
113C3×#215—316no—
114C2×C3×C3⋊C8—34no—
115C3×#204—316no—
116C3×ΓL(1,𝔽25)—316no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—42no—
121C2×C3×C3⋊D3C2×D412yes—
122C33×C4—22no—
123C33×C4—22no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2411yescharacteristic
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3C2×D412yes—
128C2×C3×#41—91no—
129C22×C3×Q8—91no—
130C2×#239—34no—
131C2×D3×Q8—34no—
132C3×#137—98no—
133C2×C3×Q16—24no—
134C2×C3×Q16—91no—
135C2×C3×Q16C3×D314yescomplemented
136C2×C3×Q16C3⋊D311yescharacteristic, complemented
137C2×C3×SD16—91no—
138C2×C3×SD16—91no—
139C2×C3×M16—91no—
140C2×#216—34no—
141C2×#215—34no—
142#918—332no—
143C2×#204—34no—
144C2×ΓL(1,𝔽25)—34no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3C3×D411yescharacteristic
147C22×C3×D3—34no—
148C22×C3×D3—64no—
149C32×Q8—21no—
150C32×Q8—21no—
151C32×Q8—28no—
152C32×Q8—28no—
153C32×Q8—24no—
154C32×Q8—24no—
155C32×Q8—44no—
156C32×Q8—44no—
157C32×Q8C22×D318yes—
158C32×Q8C22×D318yes—
159C32×Q8C23×C312yes—
160C32×Q8C23×C312yes—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—24no—
164C2×C32×C4—24no—
165C2×C32×C4—24no—
166C2×C32×C4—44no—
167C2×C32×C4—44no—
168C2×C32×C4C22×D314yes—
169C2×C32×C4C23×C311yescharacteristic
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3C3×D411yescharacteristic
172C4⋊(C3⋊D3)—24no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)C23×C312yes—
175C4×C3⋊D3—24no—
176C4×C3⋊D3—24no—
177C4×C3⋊D3C3×D412yes—
178C4×C3⋊D3C23×C312yes—
179#634—21no—
180#634—24no—
181#634C23×C312yes—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—64no—
184C3×D12—38no—
185C3×D12—616no—
186C3×D12—64no—
187C3×C4×D3—38no—
188C3×C4×D3—38no—
189C3×C4×D3—616no—
190C3×C4×D3—616no—
191C3×Dic6—38no—
192C3×Dic6—616no—
193C3×Dic6—64no—
194C32×C8—28no—
195C32×C8C22×D318yes—
196C32×C8C23×C312yes—
197C3⋊(C3⋊C8)C23×C312yes—
198C3×C3⋊C8—38no—
199C2×#137—91noSylow
200C2×C33#13712yescomplemented
201C2×C33C22×D411yescharacteristic
202C3×C3⋊D3—22no—
203C3×C3⋊D3—42no—
204C3×#41—94no—
205C3×#41—182no—
206C2×C3×Q8—24no—
207C2×C3×Q8—24no—
208C2×C3×Q8—94no—
209C2×C3×Q8—91no—
210C2×C3×Q8—181no—
211C2×C3×Q8—182no—
212C2×C3×Q8C2×C3×D314yes—
213C2×C3×Q8C2×C3×D314yes—
214C2×C3×Q8C2×C3⋊D311yescharacteristic
215C2×C3×Q8C2×C3⋊D311yescharacteristic
216C2×C3×D4—91no—
217C2×C3×D4—181no—
218C22×C3×C4—91no—
219C22×C3×C4—181no—
220C22×C3×C4—181no—
221#239—316no—
222#239—68no—
223D3×Q8—316no—
224D3×Q8—68no—
225C2×D12—34no—
226C2×D12—64no—
227C2×C4×D3—34no—
228C2×C4×D3—64no—
229C2×C4×D3—64no—
230C2×Dic6—34no—
231C2×Dic6—64no—
232C3×Q16—216no—
233C3×Q16—94no—
234C3×Q16C2×C3×D3116yescomplemented
235C3×Q16C2×C3⋊D314yescomplemented
236C3×SD16—94no—
237C3×SD16—94no—
238C3×M16—94no—
239C2×C3×C8—24no—
240C2×C3×C8—91no—
241C2×C3×C8C2×C3×D314yes—
242C2×C3×C8C2×C3⋊D311yescharacteristic
243#216—316no—
244#215—316no—
245C2×C3⋊C8—34no—
246#204—316no—
247ΓL(1,𝔽25)—316no—
248C22×C32—24no—
249C22×C32D3×D414yes—
250C22×C32C2×C3×D411yescharacteristic
251C2×C3⋊D3—22no—
252C2×C3⋊D3—22no—
253C2×C3⋊D3—42no—
254C2×C3⋊D3C2×C3×D412yes—
255C2×C3×D3—38no—
256C2×C3×D3—68no—
257C2×C3×D3—68no—
258C2×C3×D3—128no—
259C32×C4—22no—
260C32×C4—22no—
261C32×C4—24no—
262C32×C4—24no—
263C32×C4—28no—
264C32×C4—28no—
265C32×C4—48no—
266C32×C4—48no—
267C32×C4D3×D414yes—
268C32×C4C2×C3×D411yescharacteristic
269C32×C4C23×D314yes—
270C32×C4C24×C311yescharacteristic
271C3⋊Dic3—22no—
272C3⋊Dic3C2×C3×D412yes—
273C3×Dic3—38no—
274C3×Dic3—68no—
275C2×#41—91no—
276C22×Q8—91no—
277#137—98no—
278C2×Q16—91no—
279C2×Q16C3×C3⋊D311yescharacteristic, complemented
280C2×SD16—91no—
281C2×SD16—91no—
282C2×M16—91no—
283C33C2×#13711yescharacteristic, complemented, Sylow
284C23×C3—91no—
285C23×C3—181no—
286C22×D3—34no—
287C22×D3—64no—
288C3×Q8—24no—
289C3×Q8—24no—
290C3×Q8—21no—
291C3×Q8—21no—
292C3×Q8—28no—
293C3×Q8—28no—
294C3×Q8—44no—
295C3×Q8—44no—
296C3×Q8—92no—
297C3×Q8—184no—
298C3×Q8—181no—
299C3×Q8C22×C3×D318yes—
300C3×Q8C22×C3×D318yes—
301C3×Q8C22×C3⋊D312yes—
302C3×Q8C22×C3⋊D312yes—
303C3×D4—92no—
304C3×D4—184no—
305C3×D4—181no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—91no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—184no—
317C2×C3×C4—184no—
318C2×C3×C4—181no—
319C2×C3×C4C22×C3×D314yes—
320C2×C3×C4C22×C3⋊D311yescharacteristic
321C2×Dic3—34no—
322C2×Dic3—64no—
323D12—38no—
324D12—616no—
325D12—64no—
326C4×D3—38no—
327C4×D3—38no—
328C4×D3—616no—
329C4×D3—616no—
330Dic6—38no—
331Dic6—616no—
332Dic6—64no—
333C3×C8—28no—
334C3×C8—92no—
335C3×C8C22×C3×D318yes—
336C3×C8C22×C3⋊D312yes—
337C3⋊C8—38no—
338C2×C32—28no—
339C2×C32—24no—
340C2×C32#91818yescomplemented
341C2×C32C3×#13712yescomplemented
342C2×C32C2×D3×D414yes—
343C2×C32C22×C3×D411yescharacteristic
344C3⋊D3—22no—
345C3⋊D3—42no—
346C3×D3—68no—
347C3×D3—128no—
348#41—94no—
349#41—182no—
350C2×Q8—94no—
351C2×Q8—91no—
352C2×Q8—181no—
353C2×Q8—182no—
354C2×Q8C2×C3×C3⋊D311yescharacteristic
355C2×Q8C2×C3×C3⋊D311yescharacteristic
356C2×D4—91no—
357C2×D4—181no—
358C22×C4—91no—
359C22×C4—181no—
360C22×C4—181no—
361Q16—94no—
362Q16C2×C3×C3⋊D314yescomplemented
363SD16—94no—
364SD16—94no—
365M16—94no—
366C2×C8—91no—
367C2×C8C2×C3×C3⋊D311yescharacteristic
368C22×C3—24no—
369C22×C3—92no—
370C22×C3—182no—
371C22×C3—182no—
372C22×C3—362no—
373C22×C3C3×D3×D414yes—
374C22×C3D4×C3⋊D311yescharacteristic
375C2×D3—38no—
376C2×D3—68no—
377C2×D3—68no—
378C2×D3—128no—
379C3×C4—28no—
380C3×C4—28no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—24no—
384C3×C4—24no—
385C3×C4—48no—
386C3×C4—48no—
387C3×C4—92no—
388C3×C4—182no—
389C3×C4C3×D3×D414yes—
390C3×C4D4×C3⋊D311yescharacteristic
391C3×C4C23×C3×D314yes—
392C3×C4C23×C3⋊D311yescharacteristic
393Dic3—38no—
394Dic3—68no—
395C32—24no—
396C32C2×#91814yescomplemented
397C32C2×C3×#13711yescharacteristic, complemented
398C23—91no—
399C23—181no—
400Q8—21no—
401Q8—21no—
402Q8—92no—
403Q8—184no—
404Q8—181no—
405Q8C22×C3×C3⋊D312yes—
406Q8C22×C3×C3⋊D312yes—
407D4—92no—
408D4—184no—
409D4—181no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—91no—
413C2×C4—92no—
414C2×C4—92no—
415C2×C4—184no—
416C2×C4—184no—
417C2×C4—181no—
418C2×C4C22×C3×C3⋊D311yescharacteristic
419C8—92no—
420C8C22×C3×C3⋊D312yes—
421C2×C3—28no—
422C2×C3—24no—
423C2×C3—182no—
424C2×C3—362no—
425C2×C3C3×#91818yescomplemented
426C2×C3#6412712yesdirect product factor
427C2×C3C2×C3×D3×D414yes—
428C2×C3C2×D4×C3⋊D311yescharacteristic
429D3—68no—
430D3—128no—
431C22—92no—
432C22—182no—
433C22—182no—
434C22—362no—
435C22C3×D4×C3⋊D311yescharacteristic
436C4—22no—
437C4—22no—
438C4—92no—
439C4—182no—
440C4C3×D4×C3⋊D311yescharacteristic
441C4C23×C3×C3⋊D311yescharacteristic
442C3—24no—
443C3C2×C3×#91814yescomplemented
444C3C2×#6412711yescharacteristic, direct product factor
445C2—182no—
446C2—362no—
447C2C3×#6412712yesdirect product factor
448C2C2×C3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC2×C33×Q16C2
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
hypercenterC2×C3×Q16C3⋊D3
hypocenterC32C2×C3×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C2×C33×Q16C2semidirect
4. C2×C3×#3756C2semidirect
5. C2×C3×#3755C2semidirect
6. C3×#64127C2direct (N × H)
7. C2×C3×#3744C2semidirect
8. C2×C3×#3743C2semidirect
9. C2×#64127C3direct (N × H)
16. C3×Q8×C3⋊D3C22semidirect
24. C33×Q16C22semidirect
26. C3×#3756C22semidirect
27. C3×#3755C22semidirect
29. C3×#3744C22semidirect
30. C3×#3743C22semidirect
32. C2×Q8×C3⋊D3C2×C3semidirect
36. C2×C32×Q16D3semidirect
37. C2×C32×Q16C2×C3semidirect
38. C2×#3756C2×C3semidirect
39. C2×#3755C2×C3semidirect
40. #64127C2×C3direct (N × H)
41. C2×#3744C2×C3semidirect
42. C2×#3743C2×C3semidirect
82. Q8×C3⋊D3C22×C3semidirect
102. C32×Q16C2×D3semidirect
103. C32×Q16C22×C3semidirect
107. #3756C22×C3semidirect
108. #3755C22×C3semidirect
110. #3744C22×C3semidirect
111. #3743C22×C3semidirect
135. C2×C3×Q16C3×D3semidirect
136. C2×C3×Q16C3⋊D3semidirect
200. C2×C33#137semidirect
234. C3×Q16C2×C3×D3semidirect
235. C3×Q16C2×C3⋊D3semidirect
279. C2×Q16C3×C3⋊D3semidirect
283. C33C2×#137semidirect
340. C2×C32#918semidirect
341. C2×C32C3×#137semidirect
362. Q16C2×C3×C3⋊D3semidirect
396. C32C2×#918semidirect
397. C32C2×C3×#137semidirect
425. C2×C3C3×#918semidirect
426. C2×C3#64127direct (N × H)
443. C3C2×C3×#918semidirect
444. C3C2×#64127direct (N × H)
447. C2C3×#64127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×Q8×C3⋊D32
C2×C33×Q162
C2×C3×#37562
C2×C3×#37552
C3×#641272
C2×C3×#37442
C2×C3×#37432
C2×#641273
C2×C3×#9183