Finite Group Databasebeta

← C2×C3×#4759 #11868232

Subgroups — C2×C3×#4759 #11868232

Order 1152.

1061 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic
2C2×C3×#888C211yesmaximal, characteristic, complemented
3C2×C32×#102C211yesmaximal, characteristic
4C2×C3×#832C211yesmaximal, characteristic
5C2×C3×#818C211yesmaximal, characteristic, complemented
6C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C2×C3×#807C211yesmaximal, characteristic, complemented
8C3×#4759C218yesmaximal, direct product factor
9C2×C3×#474—31nomaximal
10C2×#4759C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C3×Dic3—21no—
13C22×C3×Dic6—21no—
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C3×#217—21no—
17C3×D4×Dic3—22no—
18C3×D4×Dic3C2214yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×Dic3⋊C4—21no—
22C2×C3×C4×Dic3C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×#888—22no—
25C3×#888C2214yescomplemented
26C32×#102C2214yes—
27C3×#832C2214yes—
28C3×#818C2214yescomplemented
29C3×Dic3⋊C8C2214yescomplemented
30C3×#807C2214yescomplemented
31C2×C3×C4⋊Q8—31no—
32C2×C3×C4×D4—31no—
33C2×D4×Dic3C2×C311yescharacteristic
34C2×#888C2×C311yescharacteristic, complemented
35C3×#474—38no—
36C2×C3×#107—31no—
37C2×C3×#107—31no—
38C2×C3×C4⋊C8—31no—
39C2×C3×#102—21no—
40C2×C3×#102—31no—
41C2×C3×#102D311yescharacteristic
42C2×C3×#102C2×C311yescharacteristic
43C2×#832C2×C311yescharacteristic
44C2×#818C2×C311yescharacteristic, complemented
45C2×Dic3⋊C8C2×C311yescharacteristic, complemented
46C2×#807C2×C311yescharacteristic, complemented
47#4759C2×C318yesdirect product factor
48C24×C32—21no—
49C2×C32×D4—21no—
50C2×C32×D4—22no—
51C2×C32×D4Q814yescomplemented
52C2×C32×D4C2312yes—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3—21no—
56C22×C3×Dic3—21no—
57C22×C3×Dic3—24no—
58C22×C3×Dic3—44no—
59C22×C3×Dic3D411yescharacteristic
60C22×C3×Dic3D411yescharacteristic
61C2×C3×Dic6—24no—
62C2×C3×Dic6—44no—
63C2×C32×C8—22no—
64C2×C32×C8C2312yes—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2312yes—
67C3×#217—24no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3C2312yes—
71C3×C4⋊Dic3C2312yes—
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3D412yes—
74C3×C4×Dic3C2312yes—
75C2×C3×C3⋊C8—22no—
76C2×C3×C3⋊C8C2312yes—
77C2×#474—31noSylow
78C22×C3×Q8—61no—
79C22×C3×D4—21no—
80C22×C3×D4C2×D311yescharacteristic
81C22×C3×D4C22×C311yescharacteristic
82C23×C3×C4—61no—
83C23×Dic3—21no—
84C22×Dic6—21no—
85C22×C3×C8—21no—
86C22×C3×C8—31no—
87C22×C3×C8C2×D311yescharacteristic
88C22×C3×C8C22×C311yescharacteristic
89C3×C4⋊Q8—34no—
90C3×C4⋊Q8—62no—
91C3×C4×D4—34no—
92C3×C4×D4—62no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C4⋊C4—61no—
97C2×C3×C4⋊C4C2×D311yescharacteristic
98C2×C3×C4⋊C4C22×C311yescharacteristic
99C2×C3×C22⋊C4—61no—
100C2×C3×C42—31no—
101C2×#217—21no—
102D4×Dic3—22no—
103D4×Dic3C22×C314yes—
104C2×C4⋊Dic3C22×C311yescharacteristic
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3C22×C311yescharacteristic
108C22×C3⋊C8C22×C311yescharacteristic
109#888—22no—
110#888C22×C314yescomplemented
111C3×#107—34no—
112C3×#107—34no—
113C3×C4⋊C8—34no—
114C3×#102—24no—
115C3×#102—34no—
116C3×#102C2×D314yes—
117C3×#102C22×C314yes—
118#832C22×C314yes—
119#818C22×C314yescomplemented
120Dic3⋊C8C22×C314yescomplemented
121#807C22×C314yescomplemented
122C23×C32—24no—
123C23×C32—22no—
124C23×C32—42no—
125C23×C32—42no—
126C23×C32C2×D411yescharacteristic
127C32×D4—24no—
128C32×D4—22no—
129C32×D4C2×Q814yescomplemented
130C2×C32×C4—22no—
131C2×C32×C4—42no—
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×Q812yes—
134C2×C32×C4#4112yes—
135C2×C32×C4C2411yescharacteristic
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—42no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—48no—
142C2×C3×Dic3D814yes—
143C2×C3×Dic3C2×D412yes—
144C2×C3×Dic3C2×D412yes—
145C3×Dic6—48no—
146C32×C8—22no—
147C3×C3⋊C8—22no—
148C2×C4⋊Q8—31no—
149C2×C4×D4—31no—
150#474—38no—
151C2×#107—31no—
152C2×#107—31no—
153C2×C4⋊C8—31no—
154C2×#102—31no—
155C2×#102C3×D311yescharacteristic
156C24×C3—21no—
157C24×C3—21no—
158C24×C3—41no—
159C2×C3×Q8—64no—
160C2×C3×Q8—124no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—41no—
168C2×C3×D4—42no—
169C2×C3×D4Dic614yescomplemented
170C2×C3×D4C3×Q814yescomplemented
171C2×C3×D4C22×D312yes—
172C2×C3×D4C23×C312yes—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—41no—
179C22×C3×C4—64no—
180C22×C3×C4—61no—
181C22×C3×C4—61no—
182C22×C3×C4—124no—
183C22×C3×C4C22×D311yescharacteristic
184C22×C3×C4C23×C311yescharacteristic
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—24no—
188C22×Dic3—44no—
189C22×Dic3C3×D411yescharacteristic
190C22×Dic3C3×D411yescharacteristic
191C2×Dic6—24no—
192C2×Dic6—44no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—32no—
197C2×C3×C8—42no—
198C2×C3×C8—62no—
199C2×C3×C8C22×D312yes—
200C2×C3×C8C23×C312yes—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—32no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—64no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4C22×D312yes—
211C3×C4⋊C4C23×C312yes—
212C3×C22⋊C4—64no—
213C3×C42—32no—
214C3×C42—32no—
215#217—24no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3C23×C312yes—
219C4⋊Dic3C23×C312yes—
220Dic3⋊C4—24no—
221C4×Dic3C3×D412yes—
222C4×Dic3C23×C312yes—
223C2×C3⋊C8—22no—
224C2×C3⋊C8C23×C312yes—
225C22×C32—24no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—48no—
229C22×C32C22⋊Q812yes—
230C22×C32C2×D812yes—
231C22×C32#13712yes—
232C22×C32C22×D411yescharacteristic
233C32×C4—42no—
234C32×C4C22⋊Q812yes—
235C32×C4C22×Q811yescharacteristic
236C32×C4C2×#4111yescharacteristic
237C3×Dic3—22no—
238C3×Dic3—42no—
239C3×Dic3—42no—
240C3×Dic3C2×D814yes—
241C22×Q8—61no—
242C22×D4C2×C3×D311yescharacteristic
243C23×C4—61no—
244C22×C8—31no—
245C22×C8C2×C3×D311yescharacteristic
246C4⋊Q8—34no—
247C4⋊Q8—62no—
248C4×D4—34no—
249C4×D4—62no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4C2×C3×D311yescharacteristic
254C2×C22⋊C4—61no—
255C2×C42—31no—
256#107—34no—
257#107—34no—
258C4⋊C8—34no—
259#102—34no—
260#102C2×C3×D314yes—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—21no—
266C23×C3—44no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—82no—
273C23×C3—82no—
274C23×C3D3×D411yescharacteristic
275C23×C3C2×C3×D411yescharacteristic
276C3×Q8—128no—
277C3×D4—24no—
278C3×D4—24no—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—24no—
282C3×D4—44no—
283C3×D4—42no—
284C3×D4C2×Dic614yescomplemented
285C3×D4C2×C3×Q814yescomplemented
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—34no—
293C2×C3×C4—32no—
294C2×C3×C4—32no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—62no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—82no—
302C2×C3×C4—128no—
303C2×C3×C4—122no—
304C2×C3×C4—122no—
305C2×C3×C4C2×Dic612yes—
306C2×C3×C4D3×D411yescharacteristic
307C2×C3×C4#23712yes—
308C2×C3×C4C2×C3×D411yescharacteristic
309C2×C3×C4C2×C3×Q812yes—
310C2×C3×C4C3×#4112yes—
311C2×C3×C4C23×D311yescharacteristic
312C2×C3×C4C24×C311yescharacteristic
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—48no—
319C2×Dic3C3×D814yes—
320C2×Dic3C2×C3×D412yes—
321C2×Dic3C2×C3×D412yes—
322Dic6—48no—
323C3×C8—22no—
324C3×C8—22no—
325C3×C8—42no—
326C3×C8—62no—
327C3⋊C8—22no—
328C2×C32—44no—
329C2×C32#47414yescomplemented
330C2×C32C2×C22⋊Q811yescharacteristic
331C2×C32C22×D811yescharacteristic
332C2×C32C2×#13711yescharacteristic
333C24—21no—
334C2×Q8—64no—
335C2×Q8—124no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4C3×Dic614yescomplemented
339C2×D4C22×C3×D312yes—
340C22×C4—21no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—64no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—124no—
347C22×C4C22×C3×D311yescharacteristic
348C2×C8—22no—
349C2×C8—32no—
350C2×C8—62no—
351C2×C8C22×C3×D312yes—
352C4⋊C4—21no—
353C4⋊C4—32no—
354C4⋊C4—32no—
355C4⋊C4—64no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4C22×C3×D312yes—
359C22⋊C4—64no—
360C42—32no—
361C42—32no—
362C22×C3—24no—
363C22×C3—24no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—44no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—48no—
374C22×C3—48no—
375C22×C3—82no—
376C22×C3—82no—
377C22×C3—88no—
378C22×C3#87812yes—
379C22×C3D3×D812yes—
380C22×C3#91512yes—
381C22×C3C3×C22⋊Q812yes—
382C22×C3C2×C3×D812yes—
383C22×C3C3×#13712yes—
384C22×C3C2×D3×D411yescharacteristic
385C22×C3C22×C3×D411yescharacteristic
386C3×C4—22no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—34no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—62no—
393C3×C4—82no—
394C3×C4—122no—
395C3×C4—122no—
396C3×C4#87812yes—
397C3×C4C3×C22⋊Q812yes—
398C3×C4C22×Dic611yescharacteristic
399C3×C4C2×#23711yescharacteristic
400C3×C4C22×C3×Q811yescharacteristic
401C3×C4C2×C3×#4111yescharacteristic
402Dic3—22no—
403Dic3—42no—
404Dic3—42no—
405Dic3C2×C3×D814yes—
406C32C2×#47411yescharacteristic, complemented, Sylow
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23C3×D3×D411yescharacteristic
412Q8—128no—
413D4—24no—
414D4—22no—
415D4C2×C3×Dic614yescomplemented
416C2×C4—22no—
417C2×C4—34no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—42no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—128no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4C2×C3×Dic612yes—
428C2×C4C3×D3×D411yescharacteristic
429C2×C4C3×#23712yes—
430C2×C4C23×C3×D311yescharacteristic
431C8—22no—
432C8—62no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—24no—
436C2×C3—21no—
437C2×C3—44no—
438C2×C3—44no—
439C2×C3—84no—
440C2×C3#475914yesdirect product factor
441C2×C3C3×#47414yescomplemented
442C2×C3C2×#87811yescharacteristic
443C2×C3C2×D3×D811yescharacteristic
444C2×C3C2×#91511yescharacteristic
445C2×C3C2×C3×C22⋊Q811yescharacteristic
446C2×C3C22×C3×D811yescharacteristic
447C2×C3C2×C3×#13711yescharacteristic
448C22—24no—
449C22—42no—
450C22—42no—
451C22—48no—
452C22C3×#87812yes—
453C22C3×D3×D812yes—
454C22C3×#91512yes—
455C22C2×C3×D3×D411yescharacteristic
456C4—34no—
457C4—42no—
458C4—62no—
459C4—122no—
460C4—122no—
461C4C3×#87812yes—
462C4C22×C3×Dic611yescharacteristic
463C4C2×C3×#23711yescharacteristic
464C3—21no—
465C3C2×#475911yescharacteristic, direct product factor
466C3C2×C3×#47411yescharacteristic, complemented
467C2—44no—
468C2C3×#475914yesdirect product factor
469C2C2×C3×#87811yescharacteristic
470C2C2×C3×D3×D811yescharacteristic
471C2C2×C3×#91511yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×#474
LCSG ▷ C2×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
2. C2×C3×#888C2semidirect
5. C2×C3×#818C2semidirect
6. C2×C3×Dic3⋊C8C2semidirect
7. C2×C3×#807C2semidirect
8. C3×#4759C2direct (N × H)
10. C2×#4759C3direct (N × H)
25. C3×#888C22semidirect
28. C3×#818C22semidirect
29. C3×Dic3⋊C8C22semidirect
30. C3×#807C22semidirect
34. C2×#888C2×C3semidirect
44. C2×#818C2×C3semidirect
45. C2×Dic3⋊C8C2×C3semidirect
46. C2×#807C2×C3semidirect
47. #4759C2×C3direct (N × H)
51. C2×C32×D4Q8semidirect
110. #888C22×C3semidirect
119. #818C22×C3semidirect
120. Dic3⋊C8C22×C3semidirect
121. #807C22×C3semidirect
129. C32×D4C2×Q8semidirect
169. C2×C3×D4Dic6semidirect
170. C2×C3×D4C3×Q8semidirect
284. C3×D4C2×Dic6semidirect
285. C3×D4C2×C3×Q8semidirect
329. C2×C32#474semidirect
338. C2×D4C3×Dic6semidirect
406. C32C2×#474semidirect
415. D4C2×C3×Dic6semidirect
440. C2×C3#4759direct (N × H)
441. C2×C3C3×#474semidirect
465. C3C2×#4759direct (N × H)
466. C3C2×C3×#474semidirect
468. C2C3×#4759direct (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
C2×C3×#8882
C2×C32×#1022
C2×C3×#8322
C2×C3×#8182
C2×C3×Dic3⋊C82
C2×C3×#8072
C3×#47592
C2×C3×#4743
C2×#47593