Finite Group Databasebeta

← C2×D3×#3713 #13170522

Subgroups — C2×D3×#3713 #13170522

Order 1728.

1602 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×D12C212yesmaximal, complemented
2C2×#88810C211yesmaximal, characteristic, complemented
3C2×#88789C212yesmaximal, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C2×C3×#3713C211yesmaximal, characteristic, complemented
6D3×#3713C218yesmaximal, direct product factor
7C2×D3×#213—32nomaximal
8C22×#3713—31nomaximal
9C22×C3×D32—22no—
10C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
11C2×C32×D12C2212yescomplemented
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×#626—22no—
14C3×D3×D12—24no—
15C3×D3×D12C2218yescomplemented
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
18#88810C2214yescomplemented
19#88789C2218yescomplemented
20D3×C3⋊(C3⋊C8)C2214yescomplemented
21C3×#3713C2214yescomplemented
22C2×C3×D3×D4—32no—
23C22×C3×D12—32no—
24C2×D3×D12D312yescomplemented
25C2×#3753—32no—
26C22×C3⋊(C3⋊C8)—31no—
27C2×C3×#213—32no—
28D3×#213—316no—
29C2×#3714—32no—
30C2×#3713—32no—
31C2×#3713—32no—
32C2×#3713—31no—
33C2×#3713—38no—
34C2×#3713—62no—
35C2×#3713D311yescharacteristic, direct product factor
36C2×D3×C3⋊C8—32no—
37C2×D3×C3⋊C8—61no—
38C22×C3×C3⋊D3—22no—
39C22×C32×D3—22no—
40C22×C32×D3D411yescharacteristic
41C2×C3×D32—28no—
42C2×C3×D32—48no—
43C2×C33×C4C2311yescharacteristic
44C3×C4⋊(C3⋊D3)—22no—
45C3×C4⋊(C3⋊D3)C2314yescomplemented
46C2×C32×Dic3D411yescharacteristic
47C32×D12—22no—
48C32×D12C2314yescomplemented
49C32×C4×D3D412yes—
50C32×C4×D3C2312yes—
51C3×#626—28no—
52C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
53C3×C3⋊(C3⋊C8)C2312yescomplemented
54C22×#213—92no—
55C2×D3×D8—91no—
56C23×C3×D3—62no—
57C23×C3×D3—62no—
58C22×D32—22no—
59C2×C32×D4—32no—
60C22×C32×C4—31no—
61C2×C4⋊(C3⋊D3)C2×D312yescomplemented
62C2×C3×#67—62no—
63C3×D3×D4—38no—
64C3×D3×D4—64no—
65C2×C3×D12—22no—
66C2×C3×D12—32no—
67C2×C3×D12—32no—
68C2×C3×D12—38no—
69C2×C3×D12—64no—
70C2×C3×D12—62no—
71C2×C3×D12—62no—
72C2×C3×D12C2×D312yescomplemented
73C2×C3×D12C2×D312yescomplemented
74C2×C3×C4×D3—21no—
75C2×C3×C4×D3C2×D312yes—
76C2×#626—22no—
77D3×D12—24no—
78D3×D12C2×D318yescomplemented
79#3753—38no—
80C2×C3⋊(C3⋊C8)—32no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—34no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—62no—
85C2×C3⋊(C3⋊C8)—121no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
87C3×#213—38no—
88C2×C3×C3⋊C8—32no—
89C2×C3×C3⋊C8—61no—
90#3714—38no—
91#3713—38no—
92#3713—38no—
93#3713—34no—
94#3713—68no—
95#3713C2×D314yesdirect product factor
96D3×C3⋊C8—38no—
97D3×C3⋊C8—64no—
98C22×C33C2×D411yescharacteristic
99C2×C3×C3⋊D3—24no—
100C2×C3×C3⋊D3—44no—
101C2×C32×D3—24no—
102C2×C32×D3—44no—
103C2×C32×D3D814yescomplemented
104C2×C32×D3C2×D412yes—
105C3×D32—416no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2411yescharacteristic
108C32×Dic3C2×D412yes—
109C22×C3×D4—92no—
110C2×D3×D4—32no—
111C22×D12—32no—
112C2×C3×D8—91no—
113C2×#213—32no—
114C2×#213—92no—
115C2×#213—916no—
116C2×#213—92no—
117C2×#213—92no—
118C2×#213—92no—
119C2×#213—182no—
120C22×C3⋊C8—92no—
121C22×C3⋊C8—181no—
122D3×D8—98no—
123C2×D24—91no—
124C2×D3×C8—91no—
125C23×C32—31no—
126C23×C32—62no—
127C22×C3⋊D3—22no—
128C22×C3×D3—22no—
129C22×C3×D3—22no—
130C22×C3×D3—21no—
131C22×C3×D3—42no—
132C22×C3×D3—68no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—68no—
136C22×C3×D3—122no—
137C22×C3×D3—128no—
138C22×C3×D3—128no—
139C22×C3×D3#6712yes—
140C2×D32—28no—
141C2×D32—48no—
142C32×D4—34no—
143C32×D4—62no—
144C2×C32×C4—21no—
145C2×C32×C4—22no—
146C2×C32×C4—31no—
147C2×C32×C4—32no—
148C2×C32×C4—32no—
149C2×C32×C4—41no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D312yes—
152C4⋊(C3⋊D3)—22no—
153C4⋊(C3⋊D3)C22×D314yescomplemented
154C3×#67—68no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3#6712yes—
157C3×D12—22no—
158C3×D12—22no—
159C3×D12—24no—
160C3×D12—34no—
161C3×D12—34no—
162C3×D12—42no—
163C3×D12—64no—
164C3×D12—68no—
165C3×D12—62no—
166C3×D12—62no—
167C3×D12—122no—
168C3×D12C22×D314yescomplemented
169C3×D12C22×D314yescomplemented
170C3×C4×D3—22no—
171C3×C4×D3—22no—
172C3×C4×D3#6714yes—
173C3×C4×D3C22×D314yes—
174#626—28no—
175C3⋊(C3⋊C8)—34no—
176C3⋊(C3⋊C8)—32no—
177C3⋊(C3⋊C8)—62no—
178C3⋊(C3⋊C8)—64no—
179C3⋊(C3⋊C8)—122no—
180C3⋊(C3⋊C8)C22×D312yescomplemented
181C3×C3⋊C8—34no—
182C3×C3⋊C8—62no—
183C22×D8—271noSylow
184C2×C33C2×D812yescomplemented
185C2×C33C22×D411yescharacteristic
186C3×C3⋊D3—44no—
187C32×D3—44no—
188C32×D3C2×D814yescomplemented
189C24×C3—182no—
190C23×D3—62no—
191C23×D3—62no—
192C2×C3×D4—32no—
193C2×C3×D4—32no—
194C2×C3×D4—62no—
195C2×C3×D4—98no—
196C2×C3×D4—92no—
197C2×C3×D4—184no—
198C2×C3×D4—182no—
199C22×C3×C4—32no—
200C22×C3×C4—61no—
201C2×#67—62no—
202D3×D4—38no—
203D3×D4—64no—
204C2×D12—32no—
205C2×D12—32no—
206C2×D12—38no—
207C2×D12—62no—
208C2×D12—64no—
209C2×D12—62no—
210C2×D12D3212yescomplemented
211C2×C4×D3D3211yescharacteristic
212C3×D8—94no—
213C2×C3×C8—91no—
214#213—38no—
215#213—98no—
216#213—98no—
217#213—98no—
218#213—98no—
219#213—188no—
220C2×C3⋊C8—32no—
221C2×C3⋊C8—61no—
222C2×C3⋊C8—98no—
223C2×C3⋊C8—91no—
224C2×C3⋊C8—92no—
225C2×C3⋊C8—184no—
226C2×C3⋊C8—181no—
227C2×C3⋊C8—182no—
228C2×C3⋊C8—361no—
229D24—94no—
230D3×C8—94no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32—34no—
234C22×C32—32no—
235C22×C32—41no—
236C22×C32—64no—
237C22×C32—124no—
238C22×C32D3×D411yescharacteristic
239C22×C32C2×#6712yes—
240C2×C3⋊D3—24no—
241C2×C3⋊D3—44no—
242C2×C3×D3—24no—
243C2×C3×D3—24no—
244C2×C3×D3—24no—
245C2×C3×D3—22no—
246C2×C3×D3—44no—
247C2×C3×D3—44no—
248C2×C3×D3—44no—
249C2×C3×D3—64no—
250C2×C3×D3—64no—
251C2×C3×D3—84no—
252C2×C3×D3—124no—
253C2×C3×D3—1216no—
254C2×C3×D3—124no—
255C2×C3×D3—124no—
256C2×C3×D3—1216no—
257C2×C3×D3—244no—
258C2×C3×D3#21318yescomplemented
259C2×C3×D3C2×#6714yes—
260D32—416no—
261C32×C4—21no—
262C32×C4—22no—
263C32×C4—21no—
264C32×C4—22no—
265C32×C4—32no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4D3×D411yescharacteristic
269C32×C4C2×#6712yes—
270C32×C4C23×D311yescharacteristic
271C32×C4C23×D312yes—
272C3×Dic3—22no—
273C3×Dic3C2×#6714yes—
274C22×D4—92no—
275C2×D8—91no—
276C2×D8—271no—
277C2×D8—278no—
278C2×D8—272no—
279C22×C8—271no—
280C33C22×D811yescharacteristic, complemented, Sylow
281C23×C3—32no—
282C23×C3—61no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—122no—
286C23×C3—188no—
287C23×C3—182no—
288C23×C3—368no—
289C22×D3—22no—
290C22×D3—68no—
291C22×D3—62no—
292C22×D3—62no—
293C22×D3—68no—
294C22×D3—122no—
295C22×D3—128no—
296C22×D3—128no—
297C22×D3#62511yescharacteristic
298C3×D4—34no—
299C3×D4—34no—
300C3×D4—64no—
301C3×D4—62no—
302C3×D4—62no—
303C3×D4—94no—
304C3×D4—122no—
305C3×D4—188no—
306C3×D4—182no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—32no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—41no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—61no—
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3212yes—
318#67—68no—
319C2×Dic3#62511yescharacteristic
320D12—22no—
321D12—34no—
322D12—34no—
323D12—64no—
324D12—68no—
325D12—62no—
326D12—62no—
327D12—122no—
328D12C2×D3214yescomplemented
329C4×D3#62512yes—
330C4×D3C2×D3212yes—
331C3×C8—92no—
332C3⋊C8—34no—
333C3⋊C8—62no—
334C3⋊C8—92no—
335C3⋊C8—94no—
336C3⋊C8—182no—
337C3⋊C8—184no—
338C3⋊C8—362no—
339C2×C32—21no—
340C2×C32—22no—
341C2×C32—22no—
342C2×C32—24no—
343C2×C32—34no—
344C2×C32—42no—
345C2×C32—41no—
346C2×C32—124no—
347C2×C32D3×D812yescomplemented
348C2×C32C2×#21314yescomplemented
349C2×C32C2×D3×D411yescharacteristic
350C2×C32C22×#6712yes—
351C3⋊D3—44no—
352C3×D3—24no—
353C3×D3—44no—
354C3×D3—44no—
355C3×D3—84no—
356C3×D3—124no—
357C3×D3—124no—
358C3×D3—244no—
359C3×D3C2×#21318yescomplemented
360C24—182no—
361C2×D4—32no—
362C2×D4—98no—
363C2×D4—92no—
364C2×D4—184no—
365C2×D4—182no—
366C22×C4—31no—
367D8—94no—
368D8—274no—
369D8—278no—
370C2×C8—91no—
371C2×C8—274no—
372C2×C8—271no—
373C22×C3—21no—
374C22×C3—22no—
375C22×C3—34no—
376C22×C3—38no—
377C22×C3—41no—
378C22×C3—62no—
379C22×C3—64no—
380C22×C3—64no—
381C22×C3—64no—
382C22×C3—124no—
383C22×C3—124no—
384C22×C3—124no—
385C22×C3—184no—
386C22×C3—244no—
387C22×C3—3616no—
388C22×C3—364no—
389C22×C3C2×#62511yescharacteristic
390C22×C3D3×#6712yes—
391C2×D3—24no—
392C2×D3—44no—
393C2×D3—64no—
394C2×D3—64no—
395C2×D3—124no—
396C2×D3—1216no—
397C2×D3—124no—
398C2×D3—124no—
399C2×D3—1216no—
400C2×D3—244no—
401C2×D3#371314yesdirect product factor
402C2×D3C2×#62512yes—
403C3×C4—21no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—21no—
407C3×C4—34no—
408C3×C4—41no—
409C3×C4—41no—
410C3×C4—62no—
411C3×C4C2×#62511yescharacteristic
412C3×C4D3×#6712yes—
413C3×C4C22×D3211yescharacteristic
414C3×C4C22×D3212yes—
415Dic3C2×#62512yes—
416C32—22no—
417C32—21no—
418C32—41no—
419C32C2×D3×D811yescharacteristic, complemented
420C32C22×#21312yescomplemented
421C23—31no—
422C23—62no—
423C23—188no—
424C23—182no—
425C23—368no—
426D4—34no—
427D4—62no—
428D4—94no—
429D4—188no—
430D4—182no—
431C2×C4—31no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4D3311yescharacteristic
435C8—92no—
436C8—272no—
437C2×C3—24no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3—22no—
441C2×C3—38no—
442C2×C3—42no—
443C2×C3—41no—
444C2×C3—64no—
445C2×C3—124no—
446C2×C3—124no—
447C2×C3—244no—
448C2×C3—364no—
449C2×C3C2×#371312yescomplemented
450C2×C3D3×#21314yescomplemented
451C2×C3C22×#62511yescharacteristic
452C2×C3C2×D3×#6712yes—
453D3—44no—
454D3—124no—
455D3—124no—
456D3—244no—
457D3C2×#371314yesdirect product factor
458C22—34no—
459C22—32no—
460C22—64no—
461C22—124no—
462C22—184no—
463C22—3616no—
464C22—364no—
465C22D3×#62511yescharacteristic
466C4—32no—
467C4D3×#62511yescharacteristic
468C4C2×D3311yescharacteristic
469C3—22no—
470C3—21no—
471C3—41no—
472C3C22×#371311yescharacteristic, complemented
473C3C2×D3×#21312yescomplemented
474C2—34no—
475C2—124no—
476C2—364no—
477C2D3×#371312yesdirect product factor
478C2C2×D3×#62511yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
hypercenterC2×C4D33
hypocenterC33C22×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×D12C2semidirect
2. C2×#88810C2semidirect
3. C2×#88789C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. C2×C3×#3713C2semidirect
6. D3×#3713C2direct (N × H)
10. C2×C3×C4⋊(C3⋊D3)C22semidirect
11. C2×C32×D12C22semidirect
15. C3×D3×D12C22semidirect
16. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
17. C2×C3×C3⋊(C3⋊C8)C22semidirect
18. #88810C22semidirect
19. #88789C22semidirect
20. D3×C3⋊(C3⋊C8)C22semidirect
21. C3×#3713C22semidirect
24. C2×D3×D12D3semidirect
35. C2×#3713D3direct (N × H)
45. C3×C4⋊(C3⋊D3)C23semidirect
48. C32×D12C23semidirect
52. C3⋊(C3⋊(C3⋊C8))C23semidirect
53. C3×C3⋊(C3⋊C8)C23semidirect
61. C2×C4⋊(C3⋊D3)C2×D3semidirect
72. C2×C3×D12C2×D3semidirect
73. C2×C3×D12C2×D3semidirect
78. D3×D12C2×D3semidirect
86. C2×C3⋊(C3⋊C8)C2×D3semidirect
95. #3713C2×D3direct (N × H)
103. C2×C32×D3D8semidirect
153. C4⋊(C3⋊D3)C22×D3semidirect
168. C3×D12C22×D3semidirect
169. C3×D12C22×D3semidirect
180. C3⋊(C3⋊C8)C22×D3semidirect
184. C2×C33C2×D8semidirect
188. C32×D3C2×D8semidirect
210. C2×D12D32semidirect
258. C2×C3×D3#213semidirect
280. C33C22×D8semidirect
328. D12C2×D32semidirect
347. C2×C32D3×D8semidirect
348. C2×C32C2×#213semidirect
359. C3×D3C2×#213semidirect
401. C2×D3#3713direct (N × H)
419. C32C2×D3×D8semidirect
420. C32C22×#213semidirect
449. C2×C3C2×#3713semidirect
450. C2×C3D3×#213semidirect
457. D3C2×#3713direct (N × H)
472. C3C22×#3713semidirect
473. C3C2×D3×#213semidirect
477. C2D3×#3713direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D3×D122
C2×#888102
C2×#887892
C2×D3×C3⋊(C3⋊C8)2
C2×C3×#37132
D3×#37132
C2×D3×#2133
C22×#37133