Finite Group Databasebeta

← C2×C3×#4861 #11858453

Subgroups — C2×C3×#4861 #11858453

Order 1152.

1137 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
1C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C2×C3×#892C211yesmaximal, characteristic
3C2×C3×D3×C4⋊C4C211yesmaximal, characteristic
4C2×C32×#106C211yesmaximal, characteristic, complemented
5C2×C3×#818C211yesmaximal, characteristic, complemented
6C2×C3×#808C211yesmaximal, characteristic, complemented
7C2×C3×#807C211yesmaximal, characteristic, complemented
8C3×#4861C218yesmaximal, direct product factor
9C2×C3×#428—31nomaximal
10C2×#4861C311yesmaximal, characteristic, direct product factor
11C22×C3×C4×D3—21no—
12C22×C3×C4×D3C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4—21no—
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×Dic3⋊C4—21no—
20C2×C3×C4×Dic3—21no—
21C22×C3×C3⋊C8C2211yescharacteristic
22C2×C3×ΓL(1,𝔽25)C418yescomplemented
23C2×C3×ΓL(1,𝔽25)C2214yes—
24C3×#892—22no—
25C3×#892C2214yes—
26C3×D3×C4⋊C4—22no—
27C3×D3×C4⋊C4C2214yes—
28C32×#106C2214yescomplemented
29C3×#818C2214yescomplemented
30C3×#808C2214yescomplemented
31C3×#807C2214yescomplemented
32C22×C3×M16—31no—
33C2×C3×#117—31no—
34C22×C3×C4⋊C4—31no—
35C22×ΓL(1,𝔽25)C2×C311yescharacteristic
36C2×#892C2×C311yescharacteristic
37C2×D3×C4⋊C4C2×C311yescharacteristic
38C3×#428—38no—
39C2×C3×#107—31no—
40C2×C3×#107—31no—
41C2×C3×#106—21no—
42C2×C3×#106—31no—
43C2×C3×#106D311yescharacteristic, complemented
44C2×C3×#106C2×C311yescharacteristic, complemented
45C2×#818C2×C311yescharacteristic, complemented
46C2×#808C2×C311yescharacteristic, complemented
47C2×#807C2×C311yescharacteristic, complemented
48#4861C2×C318yesdirect product factor
49C23×C3×D3D411yescharacteristic
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3—21no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3D411yescharacteristic
56C2×C3×C4×D3—24no—
57C2×C3×C4×D3—44no—
58C2×C3×C4×D3C2×C414yes—
59C2×C3×C4×D3D412yes—
60C2×C3×C4×D3Q814yes—
61C2×C3×C4×D3C2312yes—
62C2×C32×C8C2×C414yescomplemented
63C2×C32×C8C2312yes—
64C32×C4⋊C4—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2312yes—
67C32×C4⋊C4C2312yes—
68C3×(C2×D3)⋊C4—24no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3C2312yes—
72C3×C4⋊Dic3C2312yes—
73C3×Dic3⋊C4—24no—
74C3×C4×Dic3—24no—
75C2×C3×C3⋊C8C2×C414yescomplemented
76C2×C3×C3⋊C8C2312yes—
77C3×ΓL(1,𝔽25)C2×C4116yescomplemented
78C2×#428—31noSylow
79C23×C3×C4—31no—
80C23×C3×C4—61no—
81C22×C4×D3—21no—
82C22×C4×D3C22×C311yescharacteristic
83C2×C3×M16—38no—
84C2×C3×M16—34no—
85C22×C3×C8—21no—
86C22×C3×C8—31no—
87C22×C3×C8C2×D311yescharacteristic
88C22×C3×C8C22×C311yescharacteristic
89C3×#117—34no—
90C3×#117—62no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—31no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—34no—
96C2×C3×C4⋊C4—62no—
97C2×C3×C4⋊C4—61no—
98C2×C3×C4⋊C4C2×D311yescharacteristic
99C2×C3×C4⋊C4C2×D311yescharacteristic
100C2×C3×C4⋊C4C22×C311yescharacteristic
101C2×C3×C4⋊C4C22×C311yescharacteristic
102C2×C3×C22⋊C4—61no—
103C2×C3×C42—61no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×C4⋊Dic3C22×C311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C22×C311yescharacteristic
110C2×ΓL(1,𝔽25)C3×C418yescomplemented
111C2×ΓL(1,𝔽25)C22×C314yes—
112#892—22no—
113#892C22×C314yes—
114D3×C4⋊C4—22no—
115D3×C4⋊C4C22×C314yes—
116C3×#107—34no—
117C3×#107—34no—
118C3×#106—24no—
119C3×#106—34no—
120C3×#106C2×D314yescomplemented
121C3×#106C22×C314yescomplemented
122#818C22×C314yescomplemented
123#808C22×C314yescomplemented
124#807C22×C314yescomplemented
125C23×C32C2×D411yescharacteristic
126C22×C3×D3—22no—
127C22×C3×D3—22no—
128C22×C3×D3C4⋊C414yes—
129C22×C3×D3C2×D412yes—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—42no—
133C2×C32×C4—42no—
134C2×C32×C4C22×C412yes—
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×Q812yes—
137C2×C32×C4C2411yescharacteristic
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3C4⋊C414yes—
143C2×C3×Dic3C2×D412yes—
144C3×C4×D3—48no—
145C3×C4×D3C4⋊C418yes—
146C3×C4×D3C22×C414yes—
147C3×C4×D3C2×Q814yes—
148C32×C8C22×C414yescomplemented
149C3×C3⋊C8C22×C414yescomplemented
150C22×M16—31no—
151C2×#117—31no—
152C22×C4⋊C4—31no—
153#428—38no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#106—31no—
157C2×#106C3×D311yescharacteristic, complemented
158C24×C3—31no—
159C23×D3C3×D411yescharacteristic
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—31no—
166C22×C3×C4—34no—
167C22×C3×C4—32no—
168C22×C3×C4—34no—
169C22×C3×C4—32no—
170C22×C3×C4—41no—
171C22×C3×C4—41no—
172C22×C3×C4—64no—
173C22×C3×C4—61no—
174C22×C3×C4—61no—
175C22×C3×C4—124no—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C23×C311yescharacteristic
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3C3×D411yescharacteristic
181C2×C4×D3—24no—
182C2×C4×D3—44no—
183C2×C4×D3C2×C3×C414yes—
184C2×C4×D3C3×D412yes—
185C2×C4×D3C3×Q814yes—
186C2×C4×D3C23×C312yes—
187C3×M16—316no—
188C2×C3×C8—24no—
189C2×C3×C8—22no—
190C2×C3×C8—34no—
191C2×C3×C8—32no—
192C2×C3×C8C4×D314yescomplemented
193C2×C3×C8C2×C3×C414yescomplemented
194C2×C3×C8C22×D312yes—
195C2×C3×C8C23×C312yes—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—32no—
203C3×C4⋊C4—32no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—64no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4C22×D312yes—
210C3×C4⋊C4C22×D312yes—
211C3×C4⋊C4C23×C312yes—
212C3×C4⋊C4C23×C312yes—
213C3×C22⋊C4—64no—
214C3×C42—64no—
215(C2×D3)⋊C4—24no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3C23×C312yes—
219C4⋊Dic3C23×C312yes—
220Dic3⋊C4—24no—
221C4×Dic3—24no—
222C2×C3⋊C8C2×C3×C414yescomplemented
223C2×C3⋊C8C23×C312yes—
224ΓL(1,𝔽25)C2×C3×C4116yescomplemented
225C22×C32C2×C4⋊C412yes—
226C22×C32AGL(1,ℤ/8)12yes—
227C22×C32#13712yes—
228C22×C32C22×D411yescharacteristic
229C2×C3×D3—22no—
230C2×C3×D3—22no—
231C2×C3×D3—28no—
232C2×C3×D3C2×C4⋊C414yes—
233C32×C4—42no—
234C32×C4—42no—
235C32×C4C2×C4⋊C412yes—
236C32×C4C23×C411yescharacteristic
237C32×C4C22×Q811yescharacteristic
238C3×Dic3—42no—
239C3×Dic3—42no—
240C3×Dic3C2×C4⋊C414yes—
241C23×C4—31no—
242C23×C4—61no—
243C2×M16—38no—
244C2×M16—34no—
245C22×C8—31no—
246C22×C8C2×C3×D311yescharacteristic
247#117—34no—
248#117—62no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—34no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4C2×C3×D311yescharacteristic
255C2×C4⋊C4C2×C3×D311yescharacteristic
256C2×C22⋊C4—61no—
257C2×C42—61no—
258#107—34no—
259#107—34no—
260#106—34no—
261#106C2×C3×D314yescomplemented
262C23×C3—21no—
263C23×C3—34no—
264C23×C3—32no—
265C23×C3—62no—
266C23×C3—62no—
267C23×C3D3×D411yescharacteristic
268C23×C3C2×C3×D411yescharacteristic
269C22×D3—22no—
270C22×D3—22no—
271C22×D3C3×C4⋊C414yes—
272C22×D3C2×C3×D412yes—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—34no—
282C2×C3×C4—32no—
283C2×C3×C4—38no—
284C2×C3×C4—34no—
285C2×C3×C4—34no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—82no—
295C2×C3×C4—82no—
296C2×C3×C4—128no—
297C2×C3×C4—122no—
298C2×C3×C4—122no—
299C2×C3×C4C2×C4×D312yes—
300C2×C3×C4D3×D411yescharacteristic
301C2×C3×C4D3×Q812yes—
302C2×C3×C4C22×C3×C412yes—
303C2×C3×C4C2×C3×D411yescharacteristic
304C2×C3×C4C2×C3×Q812yes—
305C2×C3×C4C23×D311yescharacteristic
306C2×C3×C4C24×C311yescharacteristic
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3C3×C4⋊C414yes—
312C2×Dic3C2×C3×D412yes—
313C4×D3—48no—
314C4×D3C3×C4⋊C418yes—
315C4×D3C22×C3×C414yes—
316C4×D3C2×C3×Q814yes—
317C3×C8—24no—
318C3×C8—34no—
319C3×C8C2×C4×D314yescomplemented
320C3×C8C22×C3×C414yescomplemented
321C3⋊C8C22×C3×C414yescomplemented
322C2×C32#42814yescomplemented
323C2×C32C22×C4⋊C411yescharacteristic
324C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
325C2×C32C2×#13711yescharacteristic
326C3×D3—24no—
327C24—31no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—31no—
331C22×C4—32no—
332C22×C4—34no—
333C22×C4—32no—
334C22×C4—34no—
335C22×C4—64no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—124no—
339C22×C4C22×C3×D311yescharacteristic
340M16—316no—
341C2×C8—32no—
342C2×C8—34no—
343C2×C8C3×C4×D314yescomplemented
344C2×C8C22×C3×D312yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—32no—
348C4⋊C4—32no—
349C4⋊C4—64no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4C22×C3×D312yes—
353C4⋊C4C22×C3×D312yes—
354C22⋊C4—64no—
355C42—64no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—21no—
360C22×C3—34no—
361C22×C3—62no—
362C22×C3—62no—
363C22×C3—68no—
364C22×C3D3×C4⋊C412yes—
365C22×C3#91412yes—
366C22×C3#91512yes—
367C22×C3C2×C3×C4⋊C412yes—
368C22×C3C3×AGL(1,ℤ/8)12yes—
369C22×C3C3×#13712yes—
370C22×C3C2×D3×D411yescharacteristic
371C22×C3C22×C3×D411yescharacteristic
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—28no—
375C2×D3C2×C3×C4⋊C414yes—
376C3×C4—22no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—34no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—42no—
384C3×C4—82no—
385C3×C4—82no—
386C3×C4—122no—
387C3×C4—122no—
388C3×C4D3×C4⋊C412yes—
389C3×C4C2×C3×C4⋊C412yes—
390C3×C4C22×C4×D311yescharacteristic
391C3×C4C2×D3×Q811yescharacteristic
392C3×C4C23×C3×C411yescharacteristic
393C3×C4C22×C3×Q811yescharacteristic
394Dic3—42no—
395Dic3—42no—
396Dic3C2×C3×C4⋊C414yes—
397C32C2×#42811yescharacteristic, complemented, Sylow
398C23—34no—
399C23—32no—
400C23—62no—
401C23—62no—
402C23C3×D3×D411yescharacteristic
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—34no—
406C2×C4—32no—
407C2×C4—38no—
408C2×C4—34no—
409C2×C4—34no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—128no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4C2×C3×C4×D312yes—
418C2×C4C3×D3×D411yescharacteristic
419C2×C4C3×D3×Q812yes—
420C2×C4C23×C3×D311yescharacteristic
421C8—34no—
422C8C2×C3×C4×D314yescomplemented
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—24no—
427C2×C3—64no—
428C2×C3#486114yesdirect product factor
429C2×C3C3×#42814yescomplemented
430C2×C3C2×D3×C4⋊C411yescharacteristic
431C2×C3C2×#91411yescharacteristic
432C2×C3C2×#91511yescharacteristic
433C2×C3C22×C3×C4⋊C411yescharacteristic
434C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
435C2×C3C2×C3×#13711yescharacteristic
436D3—24no—
437C22—34no—
438C22—62no—
439C22—62no—
440C22—68no—
441C22C3×D3×C4⋊C412yes—
442C22C3×#91412yes—
443C22C3×#91512yes—
444C22C2×C3×D3×D411yescharacteristic
445C4—34no—
446C4—42no—
447C4—42no—
448C4—122no—
449C4—122no—
450C4C3×D3×C4⋊C412yes—
451C4C22×C3×C4×D311yescharacteristic
452C4C2×C3×D3×Q811yescharacteristic
453C3—21no—
454C3C2×#486111yescharacteristic, direct product factor
455C3C2×C3×#42811yescharacteristic, complemented
456C2—64no—
457C2C3×#486114yesdirect product factor
458C2C2×C3×D3×C4⋊C411yescharacteristic
459C2C2×C3×#91411yescharacteristic
460C2C2×C3×#91511yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#106C2
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#106D3
hypocenterC3C2×C3×#428
LCSG ▷ 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
4. C2×C32×#106C2semidirect
5. C2×C3×#818C2semidirect
6. C2×C3×#808C2semidirect
7. C2×C3×#807C2semidirect
8. C3×#4861C2direct (N × H)
10. C2×#4861C3direct (N × H)
22. C2×C3×ΓL(1,𝔽25)C4semidirect
28. C32×#106C22semidirect
29. C3×#818C22semidirect
30. C3×#808C22semidirect
31. C3×#807C22semidirect
43. C2×C3×#106D3semidirect
44. C2×C3×#106C2×C3semidirect
45. C2×#818C2×C3semidirect
46. C2×#808C2×C3semidirect
47. C2×#807C2×C3semidirect
48. #4861C2×C3direct (N × H)
62. C2×C32×C8C2×C4semidirect
75. C2×C3×C3⋊C8C2×C4semidirect
77. C3×ΓL(1,𝔽25)C2×C4semidirect
110. C2×ΓL(1,𝔽25)C3×C4semidirect
120. C3×#106C2×D3semidirect
121. C3×#106C22×C3semidirect
122. #818C22×C3semidirect
123. #808C22×C3semidirect
124. #807C22×C3semidirect
148. C32×C8C22×C4semidirect
149. C3×C3⋊C8C22×C4semidirect
157. C2×#106C3×D3semidirect
192. C2×C3×C8C4×D3semidirect
193. C2×C3×C8C2×C3×C4semidirect
222. C2×C3⋊C8C2×C3×C4semidirect
224. ΓL(1,𝔽25)C2×C3×C4semidirect
261. #106C2×C3×D3semidirect
319. C3×C8C2×C4×D3semidirect
320. C3×C8C22×C3×C4semidirect
321. C3⋊C8C22×C3×C4semidirect
322. C2×C32#428semidirect
343. C2×C8C3×C4×D3semidirect
397. C32C2×#428semidirect
422. C8C2×C3×C4×D3semidirect
428. C2×C3#4861direct (N × H)
429. C2×C3C3×#428semidirect
454. C3C2×#4861direct (N × H)
455. C3C2×C3×#428semidirect
457. C2C3×#4861direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽25)2
C2×C3×#8922
C2×C3×D3×C4⋊C42
C2×C32×#1062
C2×C3×#8182
C2×C3×#8082
C2×C3×#8072
C3×#48612
C2×C3×#4283
C2×#48613