Finite Group Databasebeta

← C2×C3×#4758 #11859874

Subgroups — C2×C3×#4758 #11859874

Order 1152.

1205 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic
3C2×C3×#887C211yesmaximal, characteristic, complemented
4C2×C32×#102C211yesmaximal, characteristic
5C2×C3×#816C211yesmaximal, characteristic, complemented
6C2×C3×C8×Dic3C211yesmaximal, characteristic, complemented
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C3×#4758C218yesmaximal, direct product factor
9C2×C3×C4×SD16—31nomaximal
10C2×#4758C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C3×Dic3—21no—
13C22×C3×Dic6C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C3×#217—21no—
17C3×D4×Dic3—22no—
18C3×D4×Dic3C2214yes—
19C2×C3×#214C418yescomplemented
20C2×C3×#214C2214yes—
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×Dic3⋊C4—21no—
23C2×C3×C4×Dic3—21no—
24C2×C3×C4×Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#887—22no—
27C3×#887C2214yescomplemented
28C32×#102C2214yes—
29C3×#816C2214yescomplemented
30C3×C8×Dic3C2214yescomplemented
31C3×#808C2214yescomplemented
32C22×C3×SD16—31no—
33C2×C3×C4×Q8—31no—
34C2×C3×C4×D4—31no—
35C2×D4×Dic3C2×C311yescharacteristic
36C22×#214C2×C311yescharacteristic
37C2×#887C2×C311yescharacteristic, complemented
38C3×C4×SD16—38no—
39C2×C3×#106—31no—
40C2×C3×#103—31no—
41C2×C3×#102—21no—
42C2×C3×#102D311yescharacteristic
43C2×C3×#102C2×C311yescharacteristic
44C2×C3×C4×C8—31no—
45C2×#816C2×C311yescharacteristic, complemented
46C2×C8×Dic3C2×C311yescharacteristic, complemented
47C2×#808C2×C311yescharacteristic, complemented
48#4758C2×C318yesdirect product factor
49C24×C32—21no—
50C2×C32×D4—21no—
51C2×C32×D4—22no—
52C2×C32×D4C2×C414yes—
53C2×C32×D4C2312yes—
54C22×C32×C4—21no—
55C22×C32×C4C2311yescharacteristic
56C22×C3×Dic3—21no—
57C22×C3×Dic3—21no—
58C22×C3×Dic3—24no—
59C22×C3×Dic3—44no—
60C22×C3×Dic3D411yescharacteristic
61C22×C3×Dic3D411yescharacteristic
62C2×C3×Dic6—21no—
63C2×C3×Dic6—22no—
64C2×C3×Dic6C2×C414yescomplemented
65C2×C3×Dic6C2312yes—
66C2×C32×C8—22no—
67C2×C32×C8C2312yes—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2312yes—
70C3×#217—24no—
71C3×#214C2×C4116yescomplemented
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3C2312yes—
74C3×Dic3⋊C4—24no—
75C3×C4×Dic3—24no—
76C3×C4×Dic3D412yes—
77C3×C4×Dic3C2312yes—
78C2×C3×C3⋊C8C2×C414yescomplemented
79C2×C3×C3⋊C8C2312yes—
80C2×C4×SD16—31noSylow
81C22×C3×Q8—31no—
82C22×C3×D4—21no—
83C22×C3×D4C2×D311yescharacteristic
84C22×C3×D4C22×C311yescharacteristic
85C23×C3×C4—61no—
86C23×Dic3—21no—
87C22×Dic6C22×C311yescharacteristic
88C2×C3×SD16—38no—
89C2×C3×SD16—34no—
90C22×C3×C8—21no—
91C22×C3×C8—31no—
92C22×C3×C8C2×D311yescharacteristic
93C22×C3×C8C22×C311yescharacteristic
94C3×C4×Q8—34no—
95C3×C4×Q8—62no—
96C3×C4×D4—34no—
97C3×C4×D4—62no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—61no—
101C2×C3×C4⋊C4C2×D311yescharacteristic
102C2×C3×C4⋊C4C22×C311yescharacteristic
103C2×C3×C22⋊C4—61no—
104C2×C3×C42—31no—
105C2×C3×C42—61no—
106C2×#217—21no—
107D4×Dic3—22no—
108D4×Dic3C22×C314yes—
109C2×#214C3×C418yescomplemented
110C2×#214C22×C314yes—
111C2×C4⋊Dic3C22×C311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C22×C311yescharacteristic
115C22×C3⋊C8C22×C311yescharacteristic
116#887—22no—
117#887C22×C314yescomplemented
118C3×#106—34no—
119C3×#103—34no—
120C3×#102—24no—
121C3×#102C2×D314yes—
122C3×#102C22×C314yes—
123C3×C4×C8—34no—
124#816C22×C314yescomplemented
125C8×Dic3C22×C314yescomplemented
126#808C22×C314yescomplemented
127C23×C32—24no—
128C23×C32—22no—
129C23×C32—42no—
130C23×C32—42no—
131C23×C32C2×D411yescharacteristic
132C32×D4—24no—
133C32×D4—22no—
134C32×D4C22×C414yes—
135C2×C32×C4—22no—
136C2×C32×C4—42no—
137C2×C32×C4C22×C412yes—
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4#4112yes—
140C2×C32×C4C2411yescharacteristic
141C2×C3×Dic3—24no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3—48no—
147C2×C3×Dic3SD1614yescomplemented
148C2×C3×Dic3C2×D412yes—
149C2×C3×Dic3C2×D412yes—
150C3×Dic6—24no—
151C3×Dic6—22no—
152C3×Dic6C22×C414yescomplemented
153C32×C8—22no—
154C3×C3⋊C8C22×C414yescomplemented
155C22×SD16—31no—
156C2×C4×Q8—31no—
157C2×C4×D4—31no—
158C4×SD16—38no—
159C2×#106—31no—
160C2×#103—31no—
161C2×#102C3×D311yescharacteristic
162C2×C4×C8—31no—
163C24×C3—21no—
164C24×C3—21no—
165C24×C3—41no—
166C2×C3×Q8—34no—
167C2×C3×Q8—32no—
168C2×C3×Q8—61no—
169C2×C3×Q8—62no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—41no—
177C2×C3×D4—42no—
178C2×C3×D4C4×D314yes—
179C2×C3×D4C2×C3×C414yes—
180C2×C3×D4C22×D312yes—
181C2×C3×D4C23×C312yes—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—41no—
188C22×C3×C4—64no—
189C22×C3×C4—61no—
190C22×C3×C4—61no—
191C22×C3×C4—124no—
192C22×C3×C4C22×D311yescharacteristic
193C22×C3×C4C23×C311yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—24no—
197C22×Dic3—44no—
198C22×Dic3C3×D411yescharacteristic
199C22×Dic3C3×D411yescharacteristic
200C2×Dic6—21no—
201C2×Dic6—22no—
202C2×Dic6C2×C3×C414yescomplemented
203C2×Dic6C23×C312yes—
204C3×SD16—316no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—34no—
209C2×C3×C8—32no—
210C2×C3×C8—42no—
211C2×C3×C8C22×D312yes—
212C2×C3×C8C23×C312yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—32no—
217C3×C4⋊C4—41no—
218C3×C4⋊C4—64no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4C22×D312yes—
221C3×C4⋊C4C23×C312yes—
222C3×C22⋊C4—64no—
223C3×C42—32no—
224C3×C42—32no—
225C3×C42—64no—
226#217—24no—
227#214C2×C3×C4116yescomplemented
228C4⋊Dic3—21no—
229C4⋊Dic3C23×C312yes—
230Dic3⋊C4—24no—
231C4×Dic3—24no—
232C4×Dic3C3×D412yes—
233C4×Dic3C23×C312yes—
234C2×C3⋊C8C2×C3×C414yescomplemented
235C2×C3⋊C8C23×C312yes—
236C22×C32—24no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—48no—
240C22×C32C4×D412yes—
241C22×C32C2×SD1612yes—
242C22×C32#13512yes—
243C22×C32C22×D411yescharacteristic
244C32×C4—42no—
245C32×C4C4×D412yes—
246C32×C4C23×C411yescharacteristic
247C32×C4C2×#4111yescharacteristic
248C3×Dic3—24no—
249C3×Dic3—22no—
250C3×Dic3—42no—
251C3×Dic3C2×SD1614yescomplemented
252C22×Q8—31no—
253C22×D4C2×C3×D311yescharacteristic
254C23×C4—61no—
255C2×SD16—38no—
256C2×SD16—34no—
257C22×C8—31no—
258C22×C8C2×C3×D311yescharacteristic
259C4×Q8—34no—
260C4×Q8—62no—
261C4×D4—34no—
262C4×D4—62no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4C2×C3×D311yescharacteristic
266C2×C22⋊C4—61no—
267C2×C42—31no—
268C2×C42—61no—
269#106—34no—
270#103—34no—
271#102C2×C3×D314yes—
272C4×C8—34no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—44no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—82no—
285C23×C3—82no—
286C23×C3D3×D411yescharacteristic
287C23×C3C2×C3×D411yescharacteristic
288C3×Q8—34no—
289C3×Q8—64no—
290C3×Q8—62no—
291C3×D4—24no—
292C3×D4—24no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—24no—
296C3×D4—44no—
297C3×D4—42no—
298C3×D4C2×C4×D314yes—
299C3×D4C22×C3×C414yes—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—34no—
307C2×C3×C4—32no—
308C2×C3×C4—32no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—64no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—82no—
317C2×C3×C4—128no—
318C2×C3×C4—122no—
319C2×C3×C4C2×C4×D312yes—
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4#23712yes—
322C2×C3×C4C22×C3×C412yes—
323C2×C3×C4C2×C3×D411yescharacteristic
324C2×C3×C4C3×#4112yes—
325C2×C3×C4C23×D311yescharacteristic
326C2×C3×C4C24×C311yescharacteristic
327C2×Dic3—24no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—42no—
332C2×Dic3—48no—
333C2×Dic3C3×SD1614yescomplemented
334C2×Dic3C2×C3×D412yes—
335C2×Dic3C2×C3×D412yes—
336Dic6—24no—
337Dic6—22no—
338Dic6C22×C3×C414yescomplemented
339C3×C8—22no—
340C3×C8—22no—
341C3×C8—34no—
342C3×C8—42no—
343C3⋊C8C22×C3×C414yescomplemented
344C2×C32—44no—
345C2×C32C4×SD1614yescomplemented
346C2×C32C2×C4×D411yescharacteristic
347C2×C32C22×SD1611yescharacteristic
348C2×C32C2×#13511yescharacteristic
349C24—21no—
350C2×Q8—32no—
351C2×Q8—34no—
352C2×Q8—61no—
353C2×Q8—62no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4C3×C4×D314yes—
357C2×D4C22×C3×D312yes—
358C22×C4—21no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—64no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—124no—
365C22×C4C22×C3×D311yescharacteristic
366SD16—316no—
367C2×C8—22no—
368C2×C8—32no—
369C2×C8—34no—
370C2×C8C22×C3×D312yes—
371C4⋊C4—21no—
372C4⋊C4—32no—
373C4⋊C4—64no—
374C4⋊C4—61no—
375C4⋊C4C22×C3×D312yes—
376C22⋊C4—64no—
377C42—32no—
378C42—32no—
379C42—64no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—21no—
386C22×C3—44no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—48no—
392C22×C3—48no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3—88no—
396C22×C3#67⋊C412yes—
397C22×C3#91212yes—
398C22×C3D3×SD1612yes—
399C22×C3C3×C4×D412yes—
400C22×C3C2×C3×SD1612yes—
401C22×C3C3×#13512yes—
402C22×C3C2×D3×D411yescharacteristic
403C22×C3C22×C3×D411yescharacteristic
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—34no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—64no—
411C3×C4—62no—
412C3×C4—82no—
413C3×C4—122no—
414C3×C4#67⋊C412yes—
415C3×C4C3×C4×D412yes—
416C3×C4C22×C4×D311yescharacteristic
417C3×C4C2×#23711yescharacteristic
418C3×C4C23×C3×C411yescharacteristic
419C3×C4C2×C3×#4111yescharacteristic
420Dic3—22no—
421Dic3—24no—
422Dic3—42no—
423Dic3C2×C3×SD1614yescomplemented
424C32C2×C4×SD1611yescharacteristic, complemented, Sylow
425C23—24no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23C3×D3×D411yescharacteristic
430Q8—34no—
431Q8—64no—
432Q8—62no—
433D4—24no—
434D4—22no—
435D4C2×C3×C4×D314yes—
436C2×C4—22no—
437C2×C4—34no—
438C2×C4—32no—
439C2×C4—32no—
440C2×C4—42no—
441C2×C4—64no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—128no—
446C2×C4—122no—
447C2×C4C2×C3×C4×D312yes—
448C2×C4C3×D3×D411yescharacteristic
449C2×C4C3×#23712yes—
450C2×C4C23×C3×D311yescharacteristic
451C8—22no—
452C8—34no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—24no—
456C2×C3—21no—
457C2×C3—44no—
458C2×C3—44no—
459C2×C3—84no—
460C2×C3#475814yesdirect product factor
461C2×C3C3×C4×SD1614yescomplemented
462C2×C3C2×#67⋊C411yescharacteristic
463C2×C3C2×#91211yescharacteristic
464C2×C3C2×D3×SD1611yescharacteristic
465C2×C3C2×C3×C4×D411yescharacteristic
466C2×C3C22×C3×SD1611yescharacteristic
467C2×C3C2×C3×#13511yescharacteristic
468C22—24no—
469C22—42no—
470C22—42no—
471C22—48no—
472C22C3×#67⋊C412yes—
473C22C3×#91212yes—
474C22C3×D3×SD1612yes—
475C22C2×C3×D3×D411yescharacteristic
476C4—34no—
477C4—42no—
478C4—62no—
479C4—64no—
480C4—122no—
481C4C3×#67⋊C412yes—
482C4C22×C3×C4×D311yescharacteristic
483C4C2×C3×#23711yescharacteristic
484C3—21no—
485C3C2×#475811yescharacteristic, direct product factor
486C3C2×C3×C4×SD1611yescharacteristic, complemented
487C2—44no—
488C2C3×#475814yesdirect product factor
489C2C2×C3×#67⋊C411yescharacteristic
490C2C2×C3×#91211yescharacteristic
491C2C2×C3×D3×SD1611yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×C4×SD16
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
3. C2×C3×#887C2semidirect
5. C2×C3×#816C2semidirect
6. C2×C3×C8×Dic3C2semidirect
7. C2×C3×#808C2semidirect
8. C3×#4758C2direct (N × H)
10. C2×#4758C3direct (N × H)
19. C2×C3×#214C4semidirect
27. C3×#887C22semidirect
29. C3×#816C22semidirect
30. C3×C8×Dic3C22semidirect
31. C3×#808C22semidirect
37. C2×#887C2×C3semidirect
45. C2×#816C2×C3semidirect
46. C2×C8×Dic3C2×C3semidirect
47. C2×#808C2×C3semidirect
48. #4758C2×C3direct (N × H)
64. C2×C3×Dic6C2×C4semidirect
71. C3×#214C2×C4semidirect
78. C2×C3×C3⋊C8C2×C4semidirect
109. C2×#214C3×C4semidirect
117. #887C22×C3semidirect
124. #816C22×C3semidirect
125. C8×Dic3C22×C3semidirect
126. #808C22×C3semidirect
147. C2×C3×Dic3SD16semidirect
152. C3×Dic6C22×C4semidirect
154. C3×C3⋊C8C22×C4semidirect
202. C2×Dic6C2×C3×C4semidirect
227. #214C2×C3×C4semidirect
234. C2×C3⋊C8C2×C3×C4semidirect
251. C3×Dic3C2×SD16semidirect
333. C2×Dic3C3×SD16semidirect
338. Dic6C22×C3×C4semidirect
343. C3⋊C8C22×C3×C4semidirect
345. C2×C32C4×SD16semidirect
423. Dic3C2×C3×SD16semidirect
424. C32C2×C4×SD16semidirect
460. C2×C3#4758direct (N × H)
461. C2×C3C3×C4×SD16semidirect
485. C3C2×#4758direct (N × H)
486. C3C2×C3×C4×SD16semidirect
488. C2C3×#4758direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic32
C22×C3×#2142
C2×C3×#8872
C2×C32×#1022
C2×C3×#8162
C2×C3×C8×Dic32
C2×C3×#8082
C3×#47582
C2×C3×C4×SD163
C2×#47583