Finite Group Databasebeta

← C2×C3×#63633 #13162993

Subgroups — C2×C3×#63633 #13162993

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C2×C33×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#3757C211yesmaximal, characteristic, complemented
4C3×#63633C214yesmaximal, direct product factor
5C2×#63633C311yesmaximal, characteristic, direct product factor
6C2×C3×#806—34nomaximal
7C24×C33C2211yescharacteristic
8C23×C3×C3⋊D3C411yescharacteristic, complemented
9C22×C33×C4—21no—
10C2×C3×#638—21no—
11C2×C3×#638—21no—
12C2×C3×#638—21no—
13C2×C3×#638—21no—
14C2×C3×#638C412yescomplemented
15C2×C3×#638C2212yes—
16C22×C3×C3⋊Dic3—21no—
17C22×C3×C3⋊Dic3C411yescharacteristic, complemented
18C33×C22⋊C4—21no—
19C33×C22⋊C4C2212yescomplemented
20C3×#3757—21no—
21C3×#3757C2212yescomplemented
22C22×#638C2×C311yescharacteristic
23C22×C3×#67—34no—
24C2×C32×C22⋊C4—24no—
25C2×C32×C22⋊C4D314yescomplemented
26C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
27C2×#3757C2×C311yescharacteristic, complemented
28C2×C3×#217—34no—
29#63633C2×C314yesdirect product factor
30C3×#806—316no—
31C23×C33—42no—
32C23×C33C2×C411yescharacteristic
33C23×C33C2×C411yescharacteristic
34C23×C33D411yescharacteristic
35C23×C33D411yescharacteristic
36C23×C33D411yescharacteristic
37C23×C33D411yescharacteristic
38C23×C33C2311yescharacteristic
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—42no—
42C22×C3×C3⋊D3C2×C412yescomplemented
43C2×C33×C4—22no—
44C2×C33×C4—42no—
45C3×#638—22no—
46C3×#638—22no—
47C3×#638—22no—
48C3×#638—22no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—22no—
52C2×C3×C3⋊Dic3—42no—
53C2×C3×C3⋊Dic3C2×C412yescomplemented
54C2×C3×#99—91no—
55C2×#806—34no—
56C24×C32—24no—
57C24×C32C2×D314yes—
58C24×C32C22×C311yescharacteristic
59C23×C3⋊D3C3×C411yescharacteristic, complemented
60C23×C3×D3—34no—
61C22×C32×C4—28no—
62C22×C32×C4—24no—
63C22×C32×C4—21no—
64C2×#638—21no—
65C2×#638—21no—
66C2×#638—21no—
67C2×#638—21no—
68C2×#638C3×C412yescomplemented
69C2×#638C22×C312yes—
70C22×C3⋊Dic3—21no—
71C22×C3⋊Dic3C3×C411yescharacteristic, complemented
72C2×C3×#67—38no—
73C2×C3×#67—38no—
74C2×C3×#67—64no—
75C2×C3×#67—64no—
76C2×C3×#67—64no—
77C2×C3×#67—64no—
78C22×C3×Dic3—34no—
79C22×C3×Dic3—64no—
80C32×C22⋊C4—28no—
81C32×C22⋊C4—28no—
82C32×C22⋊C4—24no—
83C32×C22⋊C4—21no—
84C32×C22⋊C4C2×D318yescomplemented
85C32×C22⋊C4C22×C312yescomplemented
86#3757—21no—
87#3757C22×C312yescomplemented
88C3×#217—38no—
89C3×#217—64no—
90C22×C33—22no—
91C22×C33—22no—
92C22×C33—22no—
93C22×C33—41no—
94C22×C33—41no—
95C22×C33—41no—
96C22×C33—41no—
97C22×C33C22⋊C411yescharacteristic
98C22×C33C22⋊C411yescharacteristic
99C22×C33C22⋊C411yescharacteristic
100C22×C33C22⋊C411yescharacteristic
101C22×C33C22×C411yescharacteristic
102C22×C33C2×D411yescharacteristic
103C22×C33C2×D411yescharacteristic
104C2×C3×C3⋊D3—22no—
105C2×C3×C3⋊D3—42no—
106C2×C3×C3⋊D3—42no—
107C2×C3×C3⋊D3—42no—
108C33×C4—42no—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—42no—
111C22×C3×D4—91no—
112C22×#67—34no—
113C2×C3×C22⋊C4—24no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4C3×D314yescomplemented
116C2×C3×C22⋊C4C3⋊D311yescharacteristic, complemented
117C2×#217—34no—
118C3×#99—94no—
119#806—316no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—24no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32—48no—
128C23×C32—42no—
129C23×C32—416no—
130C23×C32C4×D314yes—
131C23×C32C4×D314yes—
132C23×C32D1214yes—
133C23×C32D1214yes—
134C23×C32#6714yes—
135C23×C32#6714yes—
136C23×C32C2×C3×C411yescharacteristic
137C23×C32C2×C3×C411yescharacteristic
138C23×C32C3×D411yescharacteristic
139C23×C32C3×D411yescharacteristic
140C23×C32C3×D411yescharacteristic
141C23×C32C3×D411yescharacteristic
142C23×C32C22×D314yes—
143C23×C32C23×C311yescharacteristic
144C22×C3⋊D3—21no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3—42no—
147C22×C3⋊D3C2×C3×C412yescomplemented
148C22×C3×D3—38no—
149C22×C3×D3—64no—
150C22×C3×D3—64no—
151C22×C3×D3—128no—
152C2×C32×C4—216no—
153C2×C32×C4—28no—
154C2×C32×C4—22no—
155C2×C32×C4—416no—
156C2×C32×C4—48no—
157C2×C32×C4—42no—
158#638—22no—
159#638—22no—
160#638—22no—
161#638—22no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—42no—
166C2×C3⋊Dic3C2×C3×C412yescomplemented
167C3×#67—68no—
168C3×#67—68no—
169C3×#67—68no—
170C3×#67—68no—
171C2×C3×Dic3—38no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—68no—
175C2×C3×Dic3—128no—
176C2×#99—91noSylow
177C2×C33—21no—
178C2×C33—21no—
179C2×C33—21no—
180C2×C33—21no—
181C2×C33—21no—
182C2×C33—21no—
183C2×C33#9912yescomplemented
184C2×C33C2×C22⋊C411yescharacteristic
185C3×C3⋊D3—42no—
186C24×C3—24no—
187C24×C3—91no—
188C24×C3C2×C3×D314yes—
189C24×C3C2×C3⋊D311yescharacteristic
190C23×D3—34no—
191C2×C3×D4—92no—
192C2×C3×D4—92no—
193C2×C3×D4—181no—
194C2×C3×D4—181no—
195C2×C3×D4—181no—
196C2×C3×D4—181no—
197C22×C3×C4—28no—
198C22×C3×C4—24no—
199C22×C3×C4—21no—
200C22×C3×C4—91no—
201C22×C3×C4—181no—
202C2×#67—38no—
203C2×#67—38no—
204C2×#67—64no—
205C2×#67—64no—
206C2×#67—64no—
207C2×#67—64no—
208C22×Dic3—34no—
209C22×Dic3—64no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—21no—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—181no—
216C3×C22⋊C4C2×C3×D318yescomplemented
217C3×C22⋊C4C2×C3⋊D312yescomplemented
218#217—38no—
219#217—64no—
220C22×C32—216no—
221C22×C32—28no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—28no—
233C22×C32—28no—
234C22×C32—48no—
235C22×C32—48no—
236C22×C32—44no—
237C22×C32—44no—
238C22×C32—44no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—44no—
243C22×C32—41no—
244C22×C32—48no—
245C22×C32—48no—
246C22×C32—48no—
247C22×C32—48no—
248C22×C32(C2×D3)⋊C414yes—
249C22×C32(C2×D3)⋊C414yes—
250C22×C32(C2×D3)⋊C414yes—
251C22×C32(C2×D3)⋊C414yes—
252C22×C32C3×C22⋊C411yescharacteristic
253C22×C32C3×C22⋊C411yescharacteristic
254C22×C32C3×C22⋊C411yescharacteristic
255C22×C32C3×C22⋊C411yescharacteristic
256C22×C32C2×C4×D314yes—
257C22×C32C2×D1214yes—
258C22×C32C2×#6714yes—
259C22×C32C22×C3×C411yescharacteristic
260C22×C32C2×C3×D411yescharacteristic
261C22×C32C2×C3×D411yescharacteristic
262C2×C3⋊D3—22no—
263C2×C3⋊D3—42no—
264C2×C3⋊D3—42no—
265C2×C3⋊D3—42no—
266C2×C3×D3—68no—
267C2×C3×D3—128no—
268C2×C3×D3—128no—
269C2×C3×D3—128no—
270C32×C4—416no—
271C32×C4—48no—
272C32×C4—42no—
273C3⋊Dic3—22no—
274C3⋊Dic3—42no—
275C3×Dic3—68no—
276C3×Dic3—128no—
277C22×D4—91no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4C3×C3⋊D311yescharacteristic, complemented
280#99—94no—
281C33C2×#9911yescharacteristic, complemented, Sylow
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—48no—
290C23×C3—42no—
291C23×C3—416no—
292C23×C3—92no—
293C23×C3—181no—
294C23×C3—181no—
295C23×C3—362no—
296C23×C3C3×C4×D314yes—
297C23×C3C3×C4×D314yes—
298C23×C3C3×D1214yes—
299C23×C3C3×D1214yes—
300C23×C3C3×#6714yes—
301C23×C3C3×#6714yes—
302C23×C3C4×C3⋊D311yescharacteristic
303C23×C3C4×C3⋊D311yescharacteristic
304C23×C3C4⋊(C3⋊D3)11yescharacteristic
305C23×C3C4⋊(C3⋊D3)11yescharacteristic
306C23×C3#63811yescharacteristic
307C23×C3#63811yescharacteristic
308C23×C3C22×C3×D314yes—
309C23×C3C22×C3⋊D311yescharacteristic
310C22×D3—38no—
311C22×D3—64no—
312C22×D3—64no—
313C22×D3—128no—
314C3×D4—182no—
315C3×D4—182no—
316C3×D4—182no—
317C3×D4—182no—
318C2×C3×C4—216no—
319C2×C3×C4—28no—
320C2×C3×C4—22no—
321C2×C3×C4—416no—
322C2×C3×C4—48no—
323C2×C3×C4—42no—
324C2×C3×C4—92no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4—182no—
328C2×C3×C4—362no—
329#67—68no—
330#67—68no—
331#67—68no—
332#67—68no—
333C2×Dic3—38no—
334C2×Dic3—64no—
335C2×Dic3—64no—
336C2×Dic3—68no—
337C2×Dic3—128no—
338C2×C32—28no—
339C2×C32—28no—
340C2×C32—24no—
341C2×C32—24no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—24no—
349C2×C32—28no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32—44no—
355C2×C32—44no—
356C2×C32—44no—
357C2×C32—44no—
358C2×C32#80618yescomplemented
359C2×C32C3×#9912yescomplemented
360C2×C32C2×(C2×D3)⋊C414yes—
361C2×C32C2×C3×C22⋊C411yescharacteristic
362C3⋊D3—42no—
363C3×D3—128no—
364C24—91no—
365C24C2×C3×C3⋊D311yescharacteristic
366C2×D4—92no—
367C2×D4—92no—
368C2×D4—181no—
369C2×D4—181no—
370C2×D4—181no—
371C2×D4—181no—
372C22×C4—21no—
373C22×C4—91no—
374C22×C4—181no—
375C22⋊C4—21no—
376C22⋊C4—92no—
377C22⋊C4—181no—
378C22⋊C4C2×C3×C3⋊D312yescomplemented
379C22×C3—216no—
380C22×C3—28no—
381C22×C3—28no—
382C22×C3—28no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—24no—
390C22×C3—24no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—48no—
394C22×C3—48no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—44no—
402C22×C3—41no—
403C22×C3—48no—
404C22×C3—48no—
405C22×C3—48no—
406C22×C3—48no—
407C22×C3—182no—
408C22×C3—362no—
409C22×C3—362no—
410C22×C3—362no—
411C22×C3C3×(C2×D3)⋊C414yes—
412C22×C3C3×(C2×D3)⋊C414yes—
413C22×C3C3×(C2×D3)⋊C414yes—
414C22×C3C3×(C2×D3)⋊C414yes—
415C22×C3#375211yescharacteristic
416C22×C3#375211yescharacteristic
417C22×C3#375211yescharacteristic
418C22×C3#375211yescharacteristic
419C22×C3C2×C3×C4×D314yes—
420C22×C3C2×C3×D1214yes—
421C22×C3C2×C3×#6714yes—
422C22×C3C2×C4×C3⋊D311yescharacteristic
423C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
424C22×C3C2×#63811yescharacteristic
425C2×D3—68no—
426C2×D3—128no—
427C2×D3—128no—
428C2×D3—128no—
429C3×C4—416no—
430C3×C4—48no—
431C3×C4—42no—
432C3×C4—182no—
433C3×C4—362no—
434Dic3—68no—
435Dic3—128no—
436C32—24no—
437C32C2×#80614yescomplemented
438C32C2×C3×#9911yescharacteristic, complemented
439C23—42no—
440C23—92no—
441C23—181no—
442C23—181no—
443C23—362no—
444C23C3×C4×C3⋊D311yescharacteristic
445C23C3×C4×C3⋊D311yescharacteristic
446C23C3×C4⋊(C3⋊D3)11yescharacteristic
447C23C3×C4⋊(C3⋊D3)11yescharacteristic
448C23C3×#63811yescharacteristic
449C23C3×#63811yescharacteristic
450C23C22×C3×C3⋊D311yescharacteristic
451D4—182no—
452D4—182no—
453D4—182no—
454D4—182no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—92no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—182no—
461C2×C4—362no—
462C2×C3—28no—
463C2×C3—28no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—24no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—24no—
477C2×C3—28no—
478C2×C3—44no—
479C2×C3—44no—
480C2×C3—44no—
481C2×C3—44no—
482C2×C3—362no—
483C2×C3C3×#80618yescomplemented
484C2×C3#6363312yesdirect product factor
485C2×C3C2×C3×(C2×D3)⋊C414yes—
486C2×C3C2×#375211yescharacteristic
487D3—128no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—182no—
496C22—362no—
497C22—362no—
498C22—362no—
499C22C3×#375211yescharacteristic
500C22C3×#375211yescharacteristic
501C22C3×#375211yescharacteristic
502C22C3×#375211yescharacteristic
503C22C2×C3×C4×C3⋊D311yescharacteristic
504C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
505C22C2×C3×#63811yescharacteristic
506C4—42no—
507C4—182no—
508C4—362no—
509C3—24no—
510C3C2×C3×#80614yescomplemented
511C3C2×#6363311yescharacteristic, direct product factor
512C2—21no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—21no—
518C2—362no—
519C2C3×#6363312yesdirect product factor
520C2C2×C3×#375211yescharacteristic
subgroupG/H
centerC22×C3#3752
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C22⋊C4C2
socleC22×C33C22⋊C4
prime radicalC22×C3×#638C2
cosocleC23×C32C23×C3
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterC32C2×C3×#99
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C33×C22⋊C4C2semidirect
3. C2×C3×#3757C2semidirect
4. C3×#63633C2direct (N × H)
5. C2×#63633C3direct (N × H)
8. C23×C3×C3⋊D3C4semidirect
14. C2×C3×#638C4semidirect
17. C22×C3×C3⋊Dic3C4semidirect
19. C33×C22⋊C4C22semidirect
21. C3×#3757C22semidirect
25. C2×C32×C22⋊C4D3semidirect
26. C2×C32×C22⋊C4C2×C3semidirect
27. C2×#3757C2×C3semidirect
29. #63633C2×C3direct (N × H)
42. C22×C3×C3⋊D3C2×C4semidirect
53. C2×C3×C3⋊Dic3C2×C4semidirect
59. C23×C3⋊D3C3×C4semidirect
68. C2×#638C3×C4semidirect
71. C22×C3⋊Dic3C3×C4semidirect
84. C32×C22⋊C4C2×D3semidirect
85. C32×C22⋊C4C22×C3semidirect
87. #3757C22×C3semidirect
115. C2×C3×C22⋊C4C3×D3semidirect
116. C2×C3×C22⋊C4C3⋊D3semidirect
147. C22×C3⋊D3C2×C3×C4semidirect
166. C2×C3⋊Dic3C2×C3×C4semidirect
183. C2×C33#99semidirect
216. C3×C22⋊C4C2×C3×D3semidirect
217. C3×C22⋊C4C2×C3⋊D3semidirect
279. C2×C22⋊C4C3×C3⋊D3semidirect
281. C33C2×#99semidirect
358. C2×C32#806semidirect
359. C2×C32C3×#99semidirect
378. C22⋊C4C2×C3×C3⋊D3semidirect
437. C32C2×#806semidirect
438. C32C2×C3×#99semidirect
483. C2×C3C3×#806semidirect
484. C2×C3#63633direct (N × H)
510. C3C2×C3×#806semidirect
511. C3C2×#63633direct (N × H)
519. C2C3×#63633direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×C33×C22⋊C42
C2×C3×#37572
C3×#636332
C2×#636333
C2×C3×#8063