Finite Group Databasebeta

← C2×C3×#41×C3⋊D3 #13181740

Subgroups — C2×C3×#41×C3⋊D3 #13181740

Order 1728.

5996 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C3×#41×C3⋊D3C2116yesmaximal, direct product factor
3C2×C3×#3832C211yesmaximal, characteristic, complemented
4C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
5C2×C3×#3830C213yesmaximal, complemented
6C2×C3×D4×C3⋊D3C213yesmaximal, complemented
7C2×C3×#3828C213yesmaximal, complemented
8C22×C3×C4×C3⋊D3C213yesmaximal, complemented
9C2×#41×C3⋊D3C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×#41—34nomaximal
11C23×C3×C3⋊D3C2213yes—
12C33×#41C2218yescomplemented
13C2×C33×Q8C2211yescharacteristic, complemented
14C2×C33×D4C2213yescomplemented
15C22×C33×C4C2213yescomplemented
16C2×C3×#638C2216yescomplemented
17C22×C3×C3⋊Dic3C2213yescomplemented
18C3×#3832C2218yescomplemented
19C3×Q8×C3⋊D3C2218yescomplemented
20C3×#3830C22124yescomplemented
21C3×D4×C3⋊D3C22124yescomplemented
22C3×#3828C22124yescomplemented
23C2×C3×C4⋊(C3⋊D3)C2213yescomplemented
24C2×C3×C4×C3⋊D3C2211yescharacteristic
25C2×C3×C4×C3⋊D3C2213yes—
26C2×C3×C4×C3⋊D3C22124yescomplemented
27C2×C3×C4×C3⋊D3C2213yescomplemented
28C2×C3×C4×C3⋊D3C2213yescomplemented
29C2×C3×#634C2213yescomplemented
30C2×C32×#41—24no—
31C2×C32×#41D314yescomplemented
32C2×C32×#41C2×C311yescharacteristic, complemented
33#41×C3⋊D3C2×C3116yesdirect product factor
34C2×#3832C2×C311yescharacteristic, complemented
35C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
36C2×#3830C2×C313yescomplemented
37C2×D4×C3⋊D3C2×C313yescomplemented
38C2×#3828C2×C313yescomplemented
39C22×C4×C3⋊D3C2×C313yescomplemented
40C3×D3×#41—364no—
41C2×C3×#239—34no—
42C2×C3×D3×Q8—34no—
43C2×C3×#237—312no—
44C2×C3×D3×D4—312no—
45C2×C3×#235—312no—
46C22×C3×C4×D3—312no—
47C23×C33C2313yes—
48C22×C3×C3⋊D3—212no—
49C22×C3×C3⋊D3C2313yes—
50C22×C3×C3⋊D3C23112yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C33×Q8C2314yescomplemented
53C33×D4C23112yescomplemented
54C2×C33×C4C2313yes—
55C2×C33×C4C2311yescharacteristic
56C2×C33×C4C23112yescomplemented
57C3×#638C23124yescomplemented
58C2×C3×C3⋊Dic3C2313yes—
59C2×C3×C3⋊Dic3C2311yescharacteristic
60C2×C3×C3⋊Dic3C23112yescomplemented
61C3×C4⋊(C3⋊D3)C23112yescomplemented
62C3×C4×C3⋊D3C2314yes—
63C3×C4×C3⋊D3C23112yes—
64C3×C4×C3⋊D3C23112yescomplemented
65C3×C4×C3⋊D3C23112yescomplemented
66C3×#634C23112yescomplemented
67C22×C3×#41—91no—
68C2×D3×#41—34no—
69C23×C3⋊D3C22×C313yes—
70C23×C3×D3—312no—
71C32×#41—232no—
72C32×#41C2×D3132yescomplemented
73C32×#41C22×C318yescomplemented
74C2×C32×Q8—24no—
75C2×C32×Q8C2×D314yescomplemented
76C2×C32×Q8C22×C311yescharacteristic, complemented
77C2×C32×D4—212no—
78C2×C32×D4C2×D3112yescomplemented
79C2×C32×D4C22×C313yescomplemented
80C22×C32×C4—212no—
81C22×C32×C4C2×D3112yescomplemented
82C22×C32×C4C22×C313yescomplemented
83C2×#638C22×C316yescomplemented
84C22×C3⋊Dic3C22×C313yescomplemented
85#3832C22×C318yescomplemented
86Q8×C3⋊D3C22×C318yescomplemented
87#3830C22×C3124yescomplemented
88D4×C3⋊D3C22×C3124yescomplemented
89#3828C22×C3124yescomplemented
90C2×C4⋊(C3⋊D3)C22×C313yescomplemented
91C2×C4×C3⋊D3C22×C311yescharacteristic
92C2×C4×C3⋊D3C22×C313yes—
93C2×C4×C3⋊D3C22×C3124yescomplemented
94C2×C4×C3⋊D3C22×C313yescomplemented
95C2×C4×C3⋊D3C22×C313yescomplemented
96C2×#634C22×C313yescomplemented
97C2×C3×#67—324no—
98C22×C3×Dic3—312no—
99C3×#239—332no—
100C3×D3×Q8—332no—
101C3×#237—396no—
102C3×D3×D4—396no—
103C3×#235—396no—
104C2×C3×D12—312no—
105C2×C3×C4×D3—34no—
106C2×C3×C4×D3—312no—
107C2×C3×C4×D3—312no—
108C2×C3×C4×D3—312no—
109C2×C3×C4×D3—396no—
110C2×C3×Dic6—312no—
111C22×C33—26no—
112C22×C33C2416yes—
113C22×C33C2411yescharacteristic
114C2×C3×C3⋊D3—26no—
115C2×C3×C3⋊D3—224no—
116C2×C3×C3⋊D3#4114yesdirect product factor
117C2×C3×C3⋊D3C2416yes—
118C2×C3×C3⋊D3C2412yes—
119C33×C4C2416yes—
120C33×C4C2412yes—
121C3×C3⋊Dic3C2416yes—
122C3×C3⋊Dic3C2412yes—
123C2×C3×#41—24no—
124C2×C3×#41—916no—
125C2×C3×#41—91no—
126C2×C3×#41—93no—
127C2×C3×#41—93no—
128C2×C3×#41C3×D314yescomplemented
129C2×C3×#41C3⋊D311yescharacteristic, direct product factor
130C22×C3×Q8—91no—
131C22×C3×D4—93no—
132C23×C3×C4—93no—
133D3×#41—364no—
134C2×#239—34no—
135C2×D3×Q8—34no—
136C2×#237—312no—
137C2×D3×D4—312no—
138C2×#235—312no—
139C22×C4×D3—312no—
140C23×C32—212no—
141C23×C32C22×D3112yes—
142C23×C32C23×C313yes—
143C22×C3⋊D3—212no—
144C22×C3⋊D3C23×C313yes—
145C22×C3⋊D3C23×C3112yes—
146C22×C3⋊D3C23×C311yescharacteristic
147C22×C3×D3—34no—
148C22×C3×D3—312no—
149C22×C3×D3—348no—
150C22×C3×D3—648no—
151C32×Q8—216no—
152C32×Q8C22×D3116yescomplemented
153C32×Q8C23×C314yescomplemented
154C32×D4—248no—
155C32×D4C22×D3148yescomplemented
156C32×D4C23×C3112yescomplemented
157C2×C32×C4—24no—
158C2×C32×C4—212no—
159C2×C32×C4—248no—
160C2×C32×C4C22×D3112yes—
161C2×C32×C4C22×D314yes—
162C2×C32×C4C22×D3148yescomplemented
163C2×C32×C4C23×C313yes—
164C2×C32×C4C23×C311yescharacteristic
165C2×C32×C4C23×C3112yescomplemented
166#638C23×C3124yescomplemented
167C2×C3⋊Dic3C23×C313yes—
168C2×C3⋊Dic3C23×C311yescharacteristic
169C2×C3⋊Dic3C23×C3112yescomplemented
170C4⋊(C3⋊D3)C23×C3112yescomplemented
171C4×C3⋊D3C23×C314yes—
172C4×C3⋊D3C23×C3112yes—
173C4×C3⋊D3C23×C3112yescomplemented
174C4×C3⋊D3C23×C3112yescomplemented
175#634C23×C3112yescomplemented
176C3×#67—396no—
177C2×C3×Dic3—312no—
178C2×C3×Dic3—34no—
179C2×C3×Dic3—348no—
180C3×D12—348no—
181C3×C4×D3—316no—
182C3×C4×D3—348no—
183C3×C4×D3—348no—
184C3×C4×D3—348no—
185C3×Dic6—348no—
186C22×#41—91noSylow
187C2×C33—26no—
188C2×C33C2×#4112yescomplemented
189C2×C33C2511yescharacteristic
190C3×C3⋊D3—26no—
191C3×C3⋊D3C2×#4114yesdirect product factor
192C24×C3—93no—
193C23×D3—312no—
194C3×#41—232no—
195C3×#41—98no—
196C3×#41—924no—
197C3×#41—924no—
198C3×#41C2×C3×D3132yescomplemented
199C3×#41C2×C3⋊D318yesdirect product factor
200C2×C3×Q8—24no—
201C2×C3×Q8—98no—
202C2×C3×Q8—93no—
203C2×C3×Q8C2×C3×D314yescomplemented
204C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
205C2×C3×D4—212no—
206C2×C3×D4—96no—
207C2×C3×D4—924no—
208C2×C3×D4—93no—
209C2×C3×D4C2×C3×D3112yescomplemented
210C2×C3×D4C2×C3⋊D313yescomplemented
211C22×C3×C4—212no—
212C22×C3×C4—93no—
213C22×C3×C4—93no—
214C22×C3×C4—93no—
215C22×C3×C4—91no—
216C22×C3×C4—93no—
217C22×C3×C4—924no—
218C22×C3×C4C2×C3×D3112yescomplemented
219C22×C3×C4C2×C3⋊D313yescomplemented
220C2×#67—324no—
221C22×Dic3—312no—
222#239—332no—
223D3×Q8—332no—
224#237—396no—
225D3×D4—396no—
226#235—396no—
227C2×D12—312no—
228C2×C4×D3—34no—
229C2×C4×D3—312no—
230C2×C4×D3—312no—
231C2×C4×D3—312no—
232C2×C4×D3—396no—
233C2×Dic6—312no—
234C22×C32—26no—
235C22×C32—24no—
236C22×C32—224no—
237C22×C32—224no—
238C22×C32—424no—
239C22×C32C23×D3124yes—
240C22×C32C23×D314yes—
241C22×C32C24×C316yes—
242C22×C32C24×C311yescharacteristic
243C2×C3⋊D3—26no—
244C2×C3⋊D3—224no—
245C2×C3⋊D3C3×#4114yesdirect product factor
246C2×C3⋊D3C24×C316yes—
247C2×C3⋊D3C24×C312yes—
248C2×C3×D3—316no—
249C2×C3×D3—38no—
250C2×C3×D3—324no—
251C2×C3×D3—696no—
252C2×C3×D3—624no—
253C32×C4—28no—
254C32×C4—224no—
255C32×C4C23×D3124yes—
256C32×C4C23×D318yes—
257C32×C4C24×C316yes—
258C32×C4C24×C312yes—
259C3⋊Dic3C24×C316yes—
260C3⋊Dic3C24×C312yes—
261C3×Dic3—324no—
262C3×Dic3—38no—
263C2×#41—916no—
264C2×#41—91no—
265C2×#41—93no—
266C2×#41—93no—
267C2×#41C3×C3⋊D311yescharacteristic, direct product factor
268C22×Q8—91no—
269C22×D4—93no—
270C23×C4—93no—
271C33C22×#4111yescharacteristic, complemented, Sylow
272C23×C3—212no—
273C23×C3—91no—
274C23×C3—912no—
275C23×C3—93no—
276C23×C3—1812no—
277C23×C3C22×C3×D3112yes—
278C23×C3C22×C3⋊D313yes—
279C22×D3—34no—
280C22×D3—312no—
281C22×D3—348no—
282C22×D3—648no—
283C3×Q8—216no—
284C3×Q8—912no—
285C3×Q8C22×C3×D3116yescomplemented
286C3×Q8C22×C3⋊D314yescomplemented
287C3×D4—248no—
288C3×D4—924no—
289C3×D4—912no—
290C3×D4C22×C3×D3148yescomplemented
291C3×D4C22×C3⋊D3112yescomplemented
292C2×C3×C4—24no—
293C2×C3×C4—212no—
294C2×C3×C4—248no—
295C2×C3×C4—91no—
296C2×C3×C4—93no—
297C2×C3×C4—912no—
298C2×C3×C4—94no—
299C2×C3×C4—912no—
300C2×C3×C4—912no—
301C2×C3×C4—912no—
302C2×C3×C4C22×C3×D3112yes—
303C2×C3×C4C22×C3×D314yes—
304C2×C3×C4C22×C3×D3148yescomplemented
305C2×C3×C4C22×C3⋊D313yes—
306C2×C3×C4C22×C3⋊D311yescharacteristic
307C2×C3×C4C22×C3⋊D3112yescomplemented
308#67—396no—
309C2×Dic3—312no—
310C2×Dic3—34no—
311C2×Dic3—348no—
312D12—348no—
313C4×D3—316no—
314C4×D3—348no—
315C4×D3—348no—
316C4×D3—348no—
317Dic6—348no—
318C2×C32—26no—
319C2×C32—28no—
320C2×C32—24no—
321C2×C32—224no—
322C2×C32—424no—
323C2×C32D3×#4118yescomplemented
324C2×C32C2×C3×#4112yescomplemented
325C2×C32C24×D314yes—
326C2×C32C25×C311yescharacteristic
327C3⋊D3—26no—
328C3⋊D3C2×C3×#4114yesdirect product factor
329C3×D3—316no—
330C3×D3—624no—
331C24—93no—
332#41—98no—
333#41—924no—
334#41—924no—
335#41C2×C3×C3⋊D318yesdirect product factor
336C2×Q8—98no—
337C2×Q8—93no—
338C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
339C2×D4—96no—
340C2×D4—924no—
341C2×D4—93no—
342C2×D4C2×C3×C3⋊D313yescomplemented
343C22×C4—93no—
344C22×C4—91no—
345C22×C4—93no—
346C22×C4—93no—
347C22×C4—93no—
348C22×C4—924no—
349C22×C4C2×C3×C3⋊D313yescomplemented
350C22×C3—224no—
351C22×C3—26no—
352C22×C3—24no—
353C22×C3—224no—
354C22×C3—424no—
355C22×C3—94no—
356C22×C3—92no—
357C22×C3—96no—
358C22×C3—1824no—
359C22×C3—186no—
360C22×C3C23×C3×D3124yes—
361C22×C3C23×C3×D314yes—
362C22×C3C23×C3⋊D316yes—
363C22×C3C23×C3⋊D311yescharacteristic
364C2×D3—316no—
365C2×D3—38no—
366C2×D3—324no—
367C2×D3—696no—
368C2×D3—624no—
369C3×C4—28no—
370C3×C4—224no—
371C3×C4—92no—
372C3×C4—96no—
373C3×C4C23×C3×D3124yes—
374C3×C4C23×C3×D318yes—
375C3×C4C23×C3⋊D312yes—
376C3×C4C23×C3⋊D316yes—
377Dic3—324no—
378Dic3—38no—
379C32—24no—
380C32C2×D3×#4114yescomplemented
381C32C22×C3×#4111yescharacteristic, complemented
382C23—91no—
383C23—912no—
384C23—93no—
385C23—1812no—
386C23C22×C3×C3⋊D313yes—
387Q8—912no—
388Q8C22×C3×C3⋊D314yescomplemented
389D4—924no—
390D4—912no—
391D4C22×C3×C3⋊D3112yescomplemented
392C2×C4—91no—
393C2×C4—93no—
394C2×C4—912no—
395C2×C4—94no—
396C2×C4—912no—
397C2×C4—912no—
398C2×C4—912no—
399C2×C4C22×C3×C3⋊D311yescharacteristic
400C2×C4C22×C3×C3⋊D313yes—
401C2×C4C22×C3×C3⋊D3112yescomplemented
402C2×C3—224no—
403C2×C3—26no—
404C2×C3—28no—
405C2×C3—24no—
406C2×C3—424no—
407C2×C3—94no—
408C2×C3—186no—
409C2×C3C3×D3×#4118yescomplemented
410C2×C3#41×C3⋊D312yesdirect product factor
411C2×C3C24×C3×D314yes—
412C2×C3C24×C3⋊D311yescharacteristic
413D3—316no—
414D3—624no—
415C22—26no—
416C22—94no—
417C22—92no—
418C22—96no—
419C22—1824no—
420C22—186no—
421C22C23×C3×C3⋊D316yes—
422C22C23×C3×C3⋊D311yescharacteristic
423C4—92no—
424C4—96no—
425C4C23×C3×C3⋊D312yes—
426C4C23×C3×C3⋊D316yes—
427C3—24no—
428C3C2×C3×D3×#4114yescomplemented
429C3C2×#41×C3⋊D311yescharacteristic, direct product factor
430C2—26no—
431C2—94no—
432C2—186no—
433C2C3×#41×C3⋊D312yesdirect product factor
434C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutator, cosocleC2×C32C25×C3
FrattiniC2C24×C3×C3⋊D3
FittingC2×C33×#41C2
socleC22×C33C24
hypercenterC2×C3×#41C3⋊D3
hypocenterC32C22×C3×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C3×#41×C3⋊D3C2direct (N × H)
3. C2×C3×#3832C2semidirect
4. C2×C3×Q8×C3⋊D3C2semidirect
5. C2×C3×#3830C2semidirect
6. C2×C3×D4×C3⋊D3C2semidirect
7. C2×C3×#3828C2semidirect
8. C22×C3×C4×C3⋊D3C2semidirect
9. C2×#41×C3⋊D3C3direct (N × H)
12. C33×#41C22semidirect
13. C2×C33×Q8C22semidirect
14. C2×C33×D4C22semidirect
15. C22×C33×C4C22semidirect
16. C2×C3×#638C22semidirect
17. C22×C3×C3⋊Dic3C22semidirect
18. C3×#3832C22semidirect
19. C3×Q8×C3⋊D3C22semidirect
20. C3×#3830C22semidirect
21. C3×D4×C3⋊D3C22semidirect
22. C3×#3828C22semidirect
23. C2×C3×C4⋊(C3⋊D3)C22semidirect
26. C2×C3×C4×C3⋊D3C22semidirect
27. C2×C3×C4×C3⋊D3C22semidirect
28. C2×C3×C4×C3⋊D3C22semidirect
29. C2×C3×#634C22semidirect
31. C2×C32×#41D3semidirect
32. C2×C32×#41C2×C3semidirect
33. #41×C3⋊D3C2×C3direct (N × H)
34. C2×#3832C2×C3semidirect
35. C2×Q8×C3⋊D3C2×C3semidirect
36. C2×#3830C2×C3semidirect
37. C2×D4×C3⋊D3C2×C3semidirect
38. C2×#3828C2×C3semidirect
39. C22×C4×C3⋊D3C2×C3semidirect
52. C33×Q8C23semidirect
53. C33×D4C23semidirect
56. C2×C33×C4C23semidirect
57. C3×#638C23semidirect
60. C2×C3×C3⋊Dic3C23semidirect
61. C3×C4⋊(C3⋊D3)C23semidirect
64. C3×C4×C3⋊D3C23semidirect
65. C3×C4×C3⋊D3C23semidirect
66. C3×#634C23semidirect
72. C32×#41C2×D3semidirect
73. C32×#41C22×C3semidirect
75. C2×C32×Q8C2×D3semidirect
76. C2×C32×Q8C22×C3semidirect
78. C2×C32×D4C2×D3semidirect
79. C2×C32×D4C22×C3semidirect
81. C22×C32×C4C2×D3semidirect
82. C22×C32×C4C22×C3semidirect
83. C2×#638C22×C3semidirect
84. C22×C3⋊Dic3C22×C3semidirect
85. #3832C22×C3semidirect
86. Q8×C3⋊D3C22×C3semidirect
87. #3830C22×C3semidirect
88. D4×C3⋊D3C22×C3semidirect
89. #3828C22×C3semidirect
90. C2×C4⋊(C3⋊D3)C22×C3semidirect
93. C2×C4×C3⋊D3C22×C3semidirect
94. C2×C4×C3⋊D3C22×C3semidirect
95. C2×C4×C3⋊D3C22×C3semidirect
96. C2×#634C22×C3semidirect
116. C2×C3×C3⋊D3#41direct (N × H)
128. C2×C3×#41C3×D3semidirect
129. C2×C3×#41C3⋊D3direct (N × H)
152. C32×Q8C22×D3semidirect
153. C32×Q8C23×C3semidirect
155. C32×D4C22×D3semidirect
156. C32×D4C23×C3semidirect
162. C2×C32×C4C22×D3semidirect
165. C2×C32×C4C23×C3semidirect
166. #638C23×C3semidirect
169. C2×C3⋊Dic3C23×C3semidirect
170. C4⋊(C3⋊D3)C23×C3semidirect
173. C4×C3⋊D3C23×C3semidirect
174. C4×C3⋊D3C23×C3semidirect
175. #634C23×C3semidirect
188. C2×C33C2×#41semidirect
191. C3×C3⋊D3C2×#41direct (N × H)
198. C3×#41C2×C3×D3semidirect
199. C3×#41C2×C3⋊D3direct (N × H)
203. C2×C3×Q8C2×C3×D3semidirect
204. C2×C3×Q8C2×C3⋊D3semidirect
209. C2×C3×D4C2×C3×D3semidirect
210. C2×C3×D4C2×C3⋊D3semidirect
218. C22×C3×C4C2×C3×D3semidirect
219. C22×C3×C4C2×C3⋊D3semidirect
245. C2×C3⋊D3C3×#41direct (N × H)
267. C2×#41C3×C3⋊D3direct (N × H)
271. C33C22×#41semidirect
285. C3×Q8C22×C3×D3semidirect
286. C3×Q8C22×C3⋊D3semidirect
290. C3×D4C22×C3×D3semidirect
291. C3×D4C22×C3⋊D3semidirect
304. C2×C3×C4C22×C3×D3semidirect
307. C2×C3×C4C22×C3⋊D3semidirect
323. C2×C32D3×#41semidirect
324. C2×C32C2×C3×#41semidirect
328. C3⋊D3C2×C3×#41direct (N × H)
335. #41C2×C3×C3⋊D3direct (N × H)
338. C2×Q8C2×C3×C3⋊D3semidirect
342. C2×D4C2×C3×C3⋊D3semidirect
349. C22×C4C2×C3×C3⋊D3semidirect
380. C32C2×D3×#41semidirect
381. C32C22×C3×#41semidirect
388. Q8C22×C3×C3⋊D3semidirect
391. D4C22×C3×C3⋊D3semidirect
401. C2×C4C22×C3×C3⋊D3semidirect
409. C2×C3C3×D3×#41semidirect
410. C2×C3#41×C3⋊D3direct (N × H)
428. C3C2×C3×D3×#41semidirect
429. C3C2×#41×C3⋊D3direct (N × H)
433. C2C3×#41×C3⋊D3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
C3×#41×C3⋊D32
C2×C3×#38322
C2×C3×Q8×C3⋊D32
C2×C3×#38302
C2×C3×D4×C3⋊D32
C2×C3×#38282
C22×C3×C4×C3⋊D32
C2×#41×C3⋊D33
C2×C3×D3×#413