Finite Group Databasebeta

← C2×C3×#5233 #11857780

Subgroups — C2×C3×#5233 #11857780

Order 1152.

2029 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#41C211yesmaximal, characteristic
2C2×C3×#926C211yesmaximal, characteristic, complemented
3C23×C3×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#5233C218yesmaximal, direct product factor
5C2×C3×#835C212yesmaximal, complemented
6C2×C3×#832C212yesmaximal, complemented
7C2×C3×#416—31nomaximal
8C2×#5233C311yesmaximal, characteristic, direct product factor
9C2×C32×#41—24no—
10C2×C32×#41—22no—
11C2×C32×#41C418yescomplemented
12C2×C32×#41C2214yes—
13C22×C32×Q8C2211yescharacteristic
14C22×C32×D4—21no—
15C22×C32×D4C2211yescharacteristic
16C23×C32×C4—21no—
17C23×C32×C4C2211yescharacteristic
18C2×C3×#217—21no—
19C3×#926—22no—
20C3×#926C2214yescomplemented
21C2×C3×C4⋊Dic3C2212yes—
22C2×C3×C4×Dic3—21no—
23C22×C3×C3⋊C8—24no—
24C22×C3×C3⋊C8C2212yes—
25C22×C3×C3⋊C8C2214yescomplemented
26C3×#835C2218yescomplemented
27C3×#832C2218yescomplemented
28C22×C3×#41—21no—
29C22×C3×#41D311yescharacteristic
30C22×C3×#41C2×C311yescharacteristic
31C23×C3×C8—31no—
32C2×C3×#117—31no—
33C2×#926C2×C311yescharacteristic, complemented
34C23×C3⋊C8C2×C311yescharacteristic, complemented
35C3×#416—38no—
36C2×C3×#103—32no—
37C2×C3×#102—32no—
38#5233C2×C318yesdirect product factor
39C2×#835C2×C312yescomplemented
40C2×#832C2×C312yescomplemented
41C24×C32—21no—
42C24×C32D411yescharacteristic
43C32×#41—216no—
44C32×#41—28no—
45C32×#41C2×C4116yescomplemented
46C2×C32×Q8—22no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×C414yes—
49C2×C32×Q8C2312yes—
50C2×C32×D4—28no—
51C2×C32×D4—24no—
52C2×C32×D4—21no—
53C2×C32×D4—22no—
54C2×C32×D4C2×C414yes—
55C2×C32×D4C2312yes—
56C22×C32×C4—21no—
57C22×C32×C4—28no—
58C22×C32×C4—24no—
59C22×C32×C4C2×C414yes—
60C22×C32×C4D412yes—
61C22×C32×C4D411yescharacteristic
62C22×C32×C4D414yes—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2312yes—
65C22×C3×Dic3—22no—
66C3×#217—24no—
67C3×C4⋊Dic3—22no—
68C3×C4⋊Dic3C2314yes—
69C3×C4×Dic3—24no—
70C2×C3×C3⋊C8—24no—
71C2×C3×C3⋊C8—28no—
72C2×C3×C3⋊C8C2314yes—
73C2×#416—31noSylow
74C2×C3×#41—28no—
75C2×C3×#41—24no—
76C2×C3×#41—24no—
77C2×C3×#41—22no—
78C2×C3×#41—24no—
79C2×C3×#41—22no—
80C2×C3×#41—28no—
81C2×C3×#41—24no—
82C2×C3×#41Dic318yescomplemented
83C2×C3×#41C3×C418yescomplemented
84C2×C3×#41C2×D314yes—
85C2×C3×#41C22×C314yes—
86C22×C3×Q8—21no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C22×C3×D4—22no—
90C22×C3×D4—21no—
91C22×C3×D4—21no—
92C22×C3×D4—21no—
93C22×C3×D4C2×D311yescharacteristic
94C22×C3×D4C22×C311yescharacteristic
95C23×C3×C4—22no—
96C23×C3×C4—21no—
97C23×C3×C4—21no—
98C23×C3×C4—21no—
99C23×C3×C4C2×D311yescharacteristic
100C23×C3×C4C22×C311yescharacteristic
101C22×C3×C8—32no—
102C22×C3×C8—34no—
103C22×C3×C8—64no—
104C3×#117—34no—
105C3×#117—62no—
106C2×C3×C4⋊C4—32no—
107C2×C3×C22⋊C4—61no—
108C2×C3×C42—61no—
109C2×#217—21no—
110#926—22no—
111#926C22×C314yescomplemented
112C2×C4⋊Dic3C22×C312yes—
113C2×C4×Dic3—21no—
114C22×C3⋊C8—24no—
115C22×C3⋊C8C22×C312yes—
116C22×C3⋊C8C22×C314yescomplemented
117C3×#103—38no—
118C3×#102—38no—
119#835C22×C318yescomplemented
120#832C22×C318yescomplemented
121C23×C32—24no—
122C23×C32—22no—
123C23×C32—24no—
124C23×C32—44no—
125C23×C32C22⋊C414yes—
126C23×C32C2×D412yes—
127C23×C32C2×D411yescharacteristic
128C32×Q8—24no—
129C32×Q8—22no—
130C32×Q8C22×C414yes—
131C32×D4—216no—
132C32×D4—24no—
133C32×D4—22no—
134C32×D4C22×C414yes—
135C2×C32×C4—24no—
136C2×C32×C4—22no—
137C2×C32×C4—216no—
138C2×C32×C4C22⋊C414yes—
139C2×C32×C4C22⋊C418yes—
140C2×C32×C4C22×C412yes—
141C2×C32×C4C22×C414yes—
142C2×C32×C4C2×D412yes—
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D412yes—
145C2×C32×C4C2×D414yes—
146C2×C32×C4C2411yescharacteristic
147C2×C3×Dic3—24no—
148C2×C3×Dic3—44no—
149C3×C3⋊C8—24no—
150C22×#41C3×D311yescharacteristic
151C23×C8—31no—
152C2×#117—31no—
153#416—38no—
154C2×#103—32no—
155C2×#102—32no—
156C24×C3—22no—
157C24×C3—21no—
158C24×C3—21no—
159C24×C3—21no—
160C24×C3#6711yescharacteristic
161C24×C3C3×D411yescharacteristic
162C3×#41—232no—
163C3×#41—216no—
164C3×#41—216no—
165C3×#41—216no—
166C3×#41—28no—
167C3×#41—28no—
168C3×#41—216no—
169C3×#41C2×Dic3116yescomplemented
170C3×#41C2×C3×C4116yescomplemented
171C2×C3×Q8—24no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×Q8—24no—
178C2×C3×Q8—22no—
179C2×C3×Q8C2×Dic314yes—
180C2×C3×Q8C2×C3×C414yes—
181C2×C3×Q8C22×D312yes—
182C2×C3×Q8C23×C312yes—
183C2×C3×D4—216no—
184C2×C3×D4—28no—
185C2×C3×D4—22no—
186C2×C3×D4—24no—
187C2×C3×D4—28no—
188C2×C3×D4—28no—
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4C2×Dic314yes—
198C2×C3×D4C2×C3×C414yes—
199C2×C3×D4C22×D312yes—
200C2×C3×D4C23×C312yes—
201C22×C3×C4—22no—
202C22×C3×C4—216no—
203C22×C3×C4—28no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—28no—
207C22×C3×C4—28no—
208C22×C3×C4—24no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—21no—
215C22×C3×C4—22no—
216C22×C3×C4—62no—
217C22×C3×C4C2×Dic314yes—
218C22×C3×C4#6712yes—
219C22×C3×C4#6714yes—
220C22×C3×C4#6711yescharacteristic
221C22×C3×C4C2×C3×C414yes—
222C22×C3×C4C3×D411yescharacteristic
223C22×C3×C4C3×D412yes—
224C22×C3×C4C3×D414yes—
225C22×C3×C4C22×D311yescharacteristic
226C22×C3×C4C22×D312yes—
227C22×C3×C4C23×C311yescharacteristic
228C22×C3×C4C23×C312yes—
229C22×Dic3—22no—
230C2×C3×C8—34no—
231C2×C3×C8—64no—
232C2×C3×C8—68no—
233C3×C4⋊C4—34no—
234C3×C4⋊C4—62no—
235C3×C22⋊C4—64no—
236C3×C42—64no—
237#217—24no—
238C4⋊Dic3—22no—
239C4⋊Dic3C23×C314yes—
240C4×Dic3—24no—
241C2×C3⋊C8—24no—
242C2×C3⋊C8—28no—
243C2×C3⋊C8C23×C314yes—
244C22×C32—24no—
245C22×C32—24no—
246C22×C32—28no—
247C22×C32—44no—
248C22×C32—48no—
249C22×C32C2×C22⋊C414yes—
250C22×C32C2×C22⋊C412yes—
251C22×C32#13514yes—
252C22×C32C22×D411yescharacteristic
253C32×C4—24no—
254C32×C4C2×C22⋊C414yes—
255C32×C4C2×C22⋊C412yes—
256C32×C4C23×C411yescharacteristic
257C32×C4C22×D411yescharacteristic
258C3×Dic3—44no—
259C2×#41—24no—
260C2×#41—22no—
261C2×#41C3×Dic318yescomplemented
262C2×#41C2×C3×D314yes—
263C22×Q8C2×C3×D311yescharacteristic
264C22×D4—21no—
265C22×D4C2×C3×D311yescharacteristic
266C23×C4—21no—
267C23×C4C2×C3×D311yescharacteristic
268C22×C8—32no—
269C22×C8—34no—
270C22×C8—64no—
271#117—34no—
272#117—62no—
273C2×C4⋊C4—32no—
274C2×C22⋊C4—61no—
275C2×C42—61no—
276#103—38no—
277#102—38no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—28no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—24no—
287C23×C3—21no—
288C23×C3—22no—
289C23×C3—44no—
290C23×C3—48no—
291C23×C3—44no—
292C23×C3—44no—
293C23×C3#21714yes—
294C23×C3C3×C22⋊C414yes—
295C23×C3C2×#6712yes—
296C23×C3C2×#6711yescharacteristic
297C23×C3C2×C3×D412yes—
298C23×C3C2×C3×D411yescharacteristic
299C3×Q8—28no—
300C3×Q8—24no—
301C3×Q8—24no—
302C3×Q8—24no—
303C3×Q8—22no—
304C3×Q8—22no—
305C3×Q8—24no—
306C3×Q8C22×Dic314yes—
307C3×Q8C22×C3×C414yes—
308C3×D4—232no—
309C3×D4—28no—
310C3×D4—24no—
311C3×D4—216no—
312C3×D4—216no—
313C3×D4—24no—
314C3×D4—24no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—24no—
318C3×D4C22×Dic314yes—
319C3×D4C22×C3×C414yes—
320C2×C3×C4—28no—
321C2×C3×C4—24no—
322C2×C3×C4—232no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—216no—
328C2×C3×C4—216no—
329C2×C3×C4—21no—
330C2×C3×C4—24no—
331C2×C3×C4—28no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—64no—
339C2×C3×C4—124no—
340C2×C3×C4#21714yes—
341C2×C3×C4#21718yes—
342C2×C3×C4C3×C22⋊C414yes—
343C2×C3×C4C3×C22⋊C418yes—
344C2×C3×C4C22×Dic312yes—
345C2×C3×C4C22×Dic314yes—
346C2×C3×C4C2×#6712yes—
347C2×C3×C4C2×#6711yescharacteristic
348C2×C3×C4C2×#6712yes—
349C2×C3×C4C2×#6714yes—
350C2×C3×C4C22×C3×C412yes—
351C2×C3×C4C22×C3×C414yes—
352C2×C3×C4C2×C3×D412yes—
353C2×C3×C4C2×C3×D411yescharacteristic
354C2×C3×C4C2×C3×D412yes—
355C2×C3×C4C2×C3×D414yes—
356C2×C3×C4C23×D311yescharacteristic
357C2×C3×C4C24×C311yescharacteristic
358C2×Dic3—24no—
359C2×Dic3—44no—
360C3×C8—64no—
361C3⋊C8—24no—
362C2×C32—24no—
363C2×C32—44no—
364C2×C32#41614yescomplemented
365C2×C32C22×C22⋊C411yescharacteristic
366C2×C32C2×#13512yes—
367C24—21no—
368C24C3×#6711yescharacteristic
369#41—216no—
370#41—28no—
371#41C2×C3×Dic3116yescomplemented
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8C2×C3×Dic314yes—
375C2×Q8C22×C3×D312yes—
376C2×D4—28no—
377C2×D4—24no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4C2×C3×Dic314yes—
381C2×D4C22×C3×D312yes—
382C22×C4—21no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—62no—
386C22×C4C2×C3×Dic314yes—
387C22×C4C3×#6711yescharacteristic
388C22×C4C3×#6712yes—
389C22×C4C3×#6714yes—
390C22×C4C22×C3×D311yescharacteristic
391C22×C4C22×C3×D312yes—
392C2×C8—34no—
393C2×C8—64no—
394C2×C8—68no—
395C4⋊C4—34no—
396C4⋊C4—62no—
397C22⋊C4—64no—
398C42—64no—
399C22×C3—24no—
400C22×C3—28no—
401C22×C3—28no—
402C22×C3—24no—
403C22×C3—28no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—22no—
408C22×C3—24no—
409C22×C3—21no—
410C22×C3—48no—
411C22×C3—44no—
412C22×C3—416no—
413C22×C3—48no—
414C22×C3—44no—
415C22×C3—48no—
416C22×C3—48no—
417C22×C3—44no—
418C22×C3#95014yes—
419C22×C3C2×#21714yes—
420C22×C3C2×#21712yes—
421C22×C3C2×C3×C22⋊C414yes—
422C22×C3C2×C3×C22⋊C412yes—
423C22×C3C3×#13514yes—
424C22×C3C22×#6711yescharacteristic
425C22×C3C22×C3×D411yescharacteristic
426C3×C4—28no—
427C3×C4—24no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—22no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—124no—
434C3×C4C2×#21714yes—
435C3×C4C2×#21712yes—
436C3×C4C2×C3×C22⋊C414yes—
437C3×C4C2×C3×C22⋊C412yes—
438C3×C4C23×Dic311yescharacteristic
439C3×C4C22×#6711yescharacteristic
440C3×C4C23×C3×C411yescharacteristic
441C3×C4C22×C3×D411yescharacteristic
442Dic3—44no—
443C32C2×#41611yescharacteristic, complemented, Sylow
444C23—24no—
445C23—24no—
446C23—22no—
447C23—44no—
448C23C3×#21714yes—
449C23C2×C3×#6712yes—
450C23C2×C3×#6711yescharacteristic
451Q8—24no—
452Q8—22no—
453Q8C22×C3×Dic314yes—
454D4—216no—
455D4—24no—
456D4—22no—
457D4C22×C3×Dic314yes—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—216no—
461C2×C4—64no—
462C2×C4—124no—
463C2×C4C3×#21714yes—
464C2×C4C3×#21718yes—
465C2×C4C22×C3×Dic312yes—
466C2×C4C22×C3×Dic314yes—
467C2×C4C2×C3×#6712yes—
468C2×C4C2×C3×#6711yescharacteristic
469C2×C4C2×C3×#6712yes—
470C2×C4C2×C3×#6714yes—
471C2×C4C23×C3×D311yescharacteristic
472C8—64no—
473C2×C3—24no—
474C2×C3—24no—
475C2×C3—24no—
476C2×C3—22no—
477C2×C3—21no—
478C2×C3—44no—
479C2×C3—48no—
480C2×C3—44no—
481C2×C3—44no—
482C2×C3#523314yesdirect product factor
483C2×C3C3×#41614yescomplemented
484C2×C3C2×#95012yes—
485C2×C3C22×#21711yescharacteristic
486C2×C3C22×C3×C22⋊C411yescharacteristic
487C2×C3C2×C3×#13512yes—
488C22—28no—
489C22—24no—
490C22—24no—
491C22—48no—
492C22—44no—
493C22C3×#95014yes—
494C22C2×C3×#21714yes—
495C22C2×C3×#21712yes—
496C22C22×C3×#6711yescharacteristic
497C4—24no—
498C4—124no—
499C4C2×C3×#21714yes—
500C4C2×C3×#21712yes—
501C4C23×C3×Dic311yescharacteristic
502C4C22×C3×#6711yescharacteristic
503C3—21no—
504C3C2×#523311yescharacteristic, direct product factor
505C3C2×C3×#41611yescharacteristic, complemented
506C2—24no—
507C2—44no—
508C2C3×#523314yesdirect product factor
509C2C2×C3×#95012yes—
510C2C22×C3×#21711yescharacteristic
subgroupG/H
centerC22×C3×C4#67
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC22×C32×#41C2
socleC23×C32C2×D4
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×#41D3
hypocenterC3C2×C3×#416
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ C22×C3×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#926C2semidirect
3. C23×C3×C3⋊C8C2semidirect
4. C3×#5233C2direct (N × H)
5. C2×C3×#835C2semidirect
6. C2×C3×#832C2semidirect
8. C2×#5233C3direct (N × H)
11. C2×C32×#41C4semidirect
20. C3×#926C22semidirect
25. C22×C3×C3⋊C8C22semidirect
26. C3×#835C22semidirect
27. C3×#832C22semidirect
33. C2×#926C2×C3semidirect
34. C23×C3⋊C8C2×C3semidirect
38. #5233C2×C3direct (N × H)
39. C2×#835C2×C3semidirect
40. C2×#832C2×C3semidirect
45. C32×#41C2×C4semidirect
82. C2×C3×#41Dic3semidirect
83. C2×C3×#41C3×C4semidirect
111. #926C22×C3semidirect
116. C22×C3⋊C8C22×C3semidirect
119. #835C22×C3semidirect
120. #832C22×C3semidirect
169. C3×#41C2×Dic3semidirect
170. C3×#41C2×C3×C4semidirect
261. C2×#41C3×Dic3semidirect
364. C2×C32#416semidirect
371. #41C2×C3×Dic3semidirect
443. C32C2×#416semidirect
482. C2×C3#5233direct (N × H)
483. C2×C3C3×#416semidirect
504. C3C2×#5233direct (N × H)
505. C3C2×C3×#416semidirect
508. C2C3×#5233direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×#412
C2×C3×#9262
C23×C3×C3⋊C82
C3×#52332
C2×C3×#8352
C2×C3×#8322
C2×C3×#4163
C2×#52333