Finite Group Databasebeta

← C2×C3×#4870 #11868280

Subgroups — C2×C3×#4870 #11868280

Order 1152.

1025 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#893C211yesmaximal, characteristic
2C2×C3×#888C211yesmaximal, characteristic, complemented
3C2×C32×#106C211yesmaximal, characteristic, complemented
4C2×C3×#821C211yesmaximal, characteristic
5C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×#809C211yesmaximal, characteristic
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C3×#4870C218yesmaximal, direct product factor
9C2×C3×#476—31nomaximal
10C2×#4870C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3—21no—
13C22×C3×Dic6—21no—
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C3×(C2×D3)⋊C4—21no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×Dic3⋊C4—21no—
20C2×C3×C4×Dic3C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C3×#893—22no—
23C3×#893C2214yes—
24C3×#888—22no—
25C3×#888C2214yescomplemented
26C32×#106C2214yescomplemented
27C3×#821C2214yes—
28C3×Dic3⋊C8C2214yescomplemented
29C3×#809C2214yes—
30C3×#808C2214yescomplemented
31C2×C3×C4⋊Q8—31no—
32C2×C3×C4×D4—31no—
33C2×#893C2×C311yescharacteristic
34C2×#888C2×C311yescharacteristic, complemented
35C3×#476—38no—
36C2×C3×#106—21no—
37C2×C3×#106—31no—
38C2×C3×#106D311yescharacteristic, complemented
39C2×C3×#106C2×C311yescharacteristic, complemented
40C2×C3×C4⋊C8—31no—
41C2×C3×#102—31no—
42C2×C3×#102—31no—
43C2×#821C2×C311yescharacteristic
44C2×Dic3⋊C8C2×C311yescharacteristic, complemented
45C2×#809C2×C311yescharacteristic
46C2×#808C2×C311yescharacteristic, complemented
47#4870C2×C318yesdirect product factor
48C23×C3×D3—21no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3—21no—
53C22×C3×Dic3D411yescharacteristic
54C22×C3×Dic3D411yescharacteristic
55C2×C3×D12—21no—
56C2×C3×D12—22no—
57C2×C3×D12Q814yescomplemented
58C2×C3×D12C2312yes—
59C2×C3×C4×D3—24no—
60C2×C3×C4×D3—44no—
61C2×C3×Dic6—24no—
62C2×C3×Dic6—44no—
63C2×C32×C8—22no—
64C2×C32×C8C2312yes—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2312yes—
68C32×C4⋊C4C2312yes—
69C3×(C2×D3)⋊C4—24no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3C2312yes—
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3D412yes—
74C3×C4×Dic3C2312yes—
75C2×C3×C3⋊C8—22no—
76C2×C3×C3⋊C8C2312yes—
77C2×#476—31noSylow
78C22×C3×Q8—61no—
79C22×C3×D4—31no—
80C23×C3×C4—61no—
81C22×D12C22×C311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6—21no—
84C22×C3×C8—21no—
85C22×C3×C8—31no—
86C22×C3×C8C2×D311yescharacteristic
87C22×C3×C8C22×C311yescharacteristic
88C3×C4⋊Q8—34no—
89C3×C4⋊Q8—62no—
90C3×C4×D4—34no—
91C3×C4×D4—62no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—61no—
96C2×C3×C4⋊C4C2×D311yescharacteristic
97C2×C3×C4⋊C4C2×D311yescharacteristic
98C2×C3×C4⋊C4C22×C311yescharacteristic
99C2×C3×C4⋊C4C22×C311yescharacteristic
100C2×C3×C22⋊C4—61no—
101C2×C3×C42—31no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3C22×C311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3C22×C311yescharacteristic
106C22×C3⋊C8C22×C311yescharacteristic
107#893—22no—
108#893C22×C314yes—
109#888—22no—
110#888C22×C314yescomplemented
111C3×#106—24no—
112C3×#106—34no—
113C3×#106C2×D314yescomplemented
114C3×#106C22×C314yescomplemented
115C3×C4⋊C8—34no—
116C3×#102—34no—
117C3×#102—34no—
118#821C22×C314yes—
119Dic3⋊C8C22×C314yescomplemented
120#809C22×C314yes—
121#808C22×C314yescomplemented
122C23×C32C2×D411yescharacteristic
123C22×C3×D3—24no—
124C22×C3×D3—22no—
125C22×C3×D3—42no—
126C22×C3×D3—42no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—42no—
130C2×C32×C4—42no—
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×Q812yes—
133C2×C32×C4#4112yes—
134C2×C32×C4C2411yescharacteristic
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—42no—
138C2×C3×Dic3SD1614yes—
139C2×C3×Dic3C2×D412yes—
140C2×C3×Dic3C2×D412yes—
141C3×D12—24no—
142C3×D12—22no—
143C3×D12C2×Q814yescomplemented
144C3×C4×D3—48no—
145C3×Dic6—48no—
146C32×C8—22no—
147C3×C3⋊C8—22no—
148C2×C4⋊Q8—31no—
149C2×C4×D4—31no—
150#476—38no—
151C2×#106—31no—
152C2×#106C3×D311yescharacteristic, complemented
153C2×C4⋊C8—31no—
154C2×#102—31no—
155C2×#102—31no—
156C24×C3—61no—
157C23×D3—21no—
158C2×C3×Q8—64no—
159C2×C3×Q8—124no—
160C2×C3×D4—34no—
161C2×C3×D4—32no—
162C2×C3×D4—62no—
163C2×C3×D4—61no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4—41no—
172C22×C3×C4—41no—
173C22×C3×C4—64no—
174C22×C3×C4—61no—
175C22×C3×C4—124no—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C23×C311yescharacteristic
178C22×Dic3—21no—
179C22×Dic3C3×D411yescharacteristic
180C22×Dic3C3×D411yescharacteristic
181C2×D12—21no—
182C2×D12—22no—
183C2×D12C3×Q814yescomplemented
184C2×D12C23×C312yes—
185C2×C4×D3—24no—
186C2×C4×D3—44no—
187C2×Dic6—24no—
188C2×Dic6—44no—
189C2×C3×C8—24no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—32no—
194C2×C3×C8—62no—
195C2×C3×C8C22×D312yes—
196C2×C3×C8C23×C312yes—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—32no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—64no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4C22×D312yes—
209C3×C4⋊C4C22×D312yes—
210C3×C4⋊C4C23×C312yes—
211C3×C4⋊C4C23×C312yes—
212C3×C22⋊C4—64no—
213C3×C42—32no—
214C3×C42—32no—
215(C2×D3)⋊C4—24no—
216C4⋊Dic3—21no—
217C4⋊Dic3C23×C312yes—
218Dic3⋊C4—24no—
219C4×Dic3C3×D412yes—
220C4×Dic3C23×C312yes—
221C2×C3⋊C8—22no—
222C2×C3⋊C8C23×C312yes—
223C22×C32C22⋊Q812yes—
224C22×C32C2×SD1612yes—
225C22×C32AGL(1,ℤ/8)12yes—
226C22×C32C22×D411yescharacteristic
227C2×C3×D3—24no—
228C2×C3×D3—42no—
229C2×C3×D3—42no—
230C2×C3×D3—48no—
231C32×C4—42no—
232C32×C4—42no—
233C32×C4C22⋊Q812yes—
234C32×C4C22×Q811yescharacteristic
235C32×C4C2×#4111yescharacteristic
236C3×Dic3—22no—
237C3×Dic3—42no—
238C3×Dic3C2×SD1614yes—
239C22×Q8—61no—
240C22×D4—31no—
241C23×C4—61no—
242C22×C8—31no—
243C22×C8C2×C3×D311yescharacteristic
244C4⋊Q8—34no—
245C4⋊Q8—62no—
246C4×D4—34no—
247C4×D4—62no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4C2×C3×D311yescharacteristic
251C2×C4⋊C4C2×C3×D311yescharacteristic
252C2×C22⋊C4—61no—
253C2×C42—31no—
254#106—34no—
255#106C2×C3×D314yescomplemented
256C4⋊C8—34no—
257#102—34no—
258#102—34no—
259C23×C3—21no—
260C23×C3—64no—
261C23×C3—62no—
262C23×C3—122no—
263C23×C3—122no—
264C23×C3D3×D411yescharacteristic
265C23×C3C2×C3×D411yescharacteristic
266C22×D3—24no—
267C22×D3—22no—
268C22×D3—42no—
269C22×D3—42no—
270C3×Q8—128no—
271C3×D4—34no—
272C3×D4—64no—
273C3×D4—62no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—34no—
283C2×C3×C4—32no—
284C2×C3×C4—32no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—82no—
294C2×C3×C4—82no—
295C2×C3×C4—128no—
296C2×C3×C4—122no—
297C2×C3×C4#23512yes—
298C2×C3×C4D3×D411yescharacteristic
299C2×C3×C4D3×Q812yes—
300C2×C3×C4C2×C3×D411yescharacteristic
301C2×C3×C4C2×C3×Q812yes—
302C2×C3×C4C3×#4112yes—
303C2×C3×C4C23×D311yescharacteristic
304C2×C3×C4C24×C311yescharacteristic
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—42no—
308C2×Dic3C3×SD1614yes—
309C2×Dic3C2×C3×D412yes—
310C2×Dic3C2×C3×D412yes—
311D12—24no—
312D12—22no—
313D12C2×C3×Q814yescomplemented
314C4×D3—48no—
315Dic6—48no—
316C3×C8—24no—
317C3×C8—22no—
318C3×C8—22no—
319C3×C8—62no—
320C3⋊C8—22no—
321C2×C32#47614yescomplemented
322C2×C32C2×C22⋊Q811yescharacteristic
323C2×C32C22×SD1611yescharacteristic
324C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
325C3×D3—44no—
326C24—61no—
327C2×Q8—64no—
328C2×Q8—124no—
329C2×D4—32no—
330C2×D4—34no—
331C2×D4—62no—
332C2×D4—61no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—64no—
338C22×C4—61no—
339C22×C4—124no—
340C22×C4C22×C3×D311yescharacteristic
341C2×C8—22no—
342C2×C8—32no—
343C2×C8—62no—
344C2×C8C22×C3×D312yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—32no—
348C4⋊C4—64no—
349C4⋊C4—61no—
350C4⋊C4C22×C3×D312yes—
351C4⋊C4C22×C3×D312yes—
352C22⋊C4—64no—
353C42—32no—
354C42—32no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—21no—
359C22×C3—64no—
360C22×C3—128no—
361C22×C3—122no—
362C22×C3—122no—
363C22×C3#89612yes—
364C22×C3D3×SD1612yes—
365C22×C3#91412yes—
366C22×C3C3×C22⋊Q812yes—
367C22×C3C2×C3×SD1612yes—
368C22×C3C3×AGL(1,ℤ/8)12yes—
369C22×C3C2×D3×D411yescharacteristic
370C22×C3C22×C3×D411yescharacteristic
371C2×D3—24no—
372C2×D3—42no—
373C2×D3—42no—
374C2×D3—48no—
375C3×C4—22no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—34no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—62no—
384C3×C4—82no—
385C3×C4—82no—
386C3×C4—122no—
387C3×C4#89612yes—
388C3×C4C3×C22⋊Q812yes—
389C3×C4C2×#23511yescharacteristic
390C3×C4C2×D3×Q811yescharacteristic
391C3×C4C22×C3×Q811yescharacteristic
392C3×C4C2×C3×#4111yescharacteristic
393Dic3—22no—
394Dic3—42no—
395Dic3C2×C3×SD1614yes—
396C32C2×#47611yescharacteristic, complemented, Sylow
397C23—64no—
398C23—62no—
399C23—122no—
400C23—122no—
401C23C3×D3×D411yescharacteristic
402Q8—128no—
403D4—34no—
404D4—64no—
405D4—62no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—34no—
409C2×C4—32no—
410C2×C4—32no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—128no—
416C2×C4—122no—
417C2×C4C3×#23512yes—
418C2×C4C3×D3×D411yescharacteristic
419C2×C4C3×D3×Q812yes—
420C2×C4C23×C3×D311yescharacteristic
421C8—22no—
422C8—62no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—24no—
427C2×C3—124no—
428C2×C3#487014yesdirect product factor
429C2×C3C3×#47614yescomplemented
430C2×C3C2×#89611yescharacteristic
431C2×C3C2×D3×SD1611yescharacteristic
432C2×C3C2×#91411yescharacteristic
433C2×C3C2×C3×C22⋊Q811yescharacteristic
434C2×C3C22×C3×SD1611yescharacteristic
435C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
436D3—44no—
437C22—64no—
438C22—122no—
439C22—122no—
440C22—128no—
441C22C3×#89612yes—
442C22C3×D3×SD1612yes—
443C22C3×#91412yes—
444C22C2×C3×D3×D411yescharacteristic
445C4—34no—
446C4—42no—
447C4—42no—
448C4—62no—
449C4—122no—
450C4C3×#89612yes—
451C4C2×C3×#23511yescharacteristic
452C4C2×C3×D3×Q811yescharacteristic
453C3—21no—
454C3C2×#487011yescharacteristic, direct product factor
455C3C2×C3×#47611yescharacteristic, complemented
456C2—124no—
457C2C3×#487014yesdirect product factor
458C2C2×C3×#89611yescharacteristic
459C2C2×C3×D3×SD1611yescharacteristic
460C2C2×C3×#91411yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#106C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
hypercenterC2×C3×#106D3
hypocenterC3C2×C3×#476
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#106

Complements

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

NH ≅ G/Ntype
2. C2×C3×#888C2semidirect
3. C2×C32×#106C2semidirect
5. C2×C3×Dic3⋊C8C2semidirect
7. C2×C3×#808C2semidirect
8. C3×#4870C2direct (N × H)
10. C2×#4870C3direct (N × H)
25. C3×#888C22semidirect
26. C32×#106C22semidirect
28. C3×Dic3⋊C8C22semidirect
30. C3×#808C22semidirect
34. C2×#888C2×C3semidirect
38. C2×C3×#106D3semidirect
39. C2×C3×#106C2×C3semidirect
44. C2×Dic3⋊C8C2×C3semidirect
46. C2×#808C2×C3semidirect
47. #4870C2×C3direct (N × H)
57. C2×C3×D12Q8semidirect
110. #888C22×C3semidirect
113. C3×#106C2×D3semidirect
114. C3×#106C22×C3semidirect
119. Dic3⋊C8C22×C3semidirect
121. #808C22×C3semidirect
143. C3×D12C2×Q8semidirect
152. C2×#106C3×D3semidirect
183. C2×D12C3×Q8semidirect
255. #106C2×C3×D3semidirect
313. D12C2×C3×Q8semidirect
321. C2×C32#476semidirect
396. C32C2×#476semidirect
428. C2×C3#4870direct (N × H)
429. C2×C3C3×#476semidirect
454. C3C2×#4870direct (N × H)
455. C3C2×C3×#476semidirect
457. C2C3×#4870direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#8932
C2×C3×#8882
C2×C32×#1062
C2×C3×#8212
C2×C3×Dic3⋊C82
C2×C3×#8092
C2×C3×#8082
C3×#48702
C2×C3×#4763
C2×#48703