Finite Group Databasebeta

← C2×C3×C4×#215 #11859865

Subgroups — C2×C3×C4×#215 #11859865

Order 1152.

1253 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C22×C3×#215C211yesmaximal, characteristic
3C2×C3×C4×D12C211yesmaximal, characteristic
4C2×C3×#835C211yesmaximal, characteristic, complemented
5C3×C4×#215C218yesmaximal, direct product factor
6C2×C3×#809C211yesmaximal, characteristic
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C2×C3×C4×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×C4×SD16—31nomaximal
10C2×C4×#215C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C22×C3×C4×D3—21no—
14C32×C4×Q8—22no—
15C32×C4×Q8C2214yescomplemented
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C42—21no—
19C2×C32×C42C2211yescharacteristic
20C2×C3×#215C418yesdirect product factor
21C2×C3×#215C2214yes—
22C2×C3×(C2×D3)⋊C4—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×C4×D12—22no—
27C3×C4×D12C2214yes—
28C3×#835C2214yescomplemented
29C3×#809C2214yes—
30C3×#808C2214yescomplemented
31C3×C4×C3⋊C8C2214yescomplemented
32C22×C3×SD16—31no—
33C2×C3×C4×Q8—21no—
34C2×C3×C4×Q8D311yescharacteristic, complemented
35C2×C3×C4×Q8C2×C311yescharacteristic, complemented
36C2×C3×C4×D4—31no—
37C22×#215C2×C311yescharacteristic
38C2×C4×D12C2×C311yescharacteristic
39C3×C4×SD16—38no—
40C2×C3×#106—31no—
41C2×C3×#103—31no—
42C2×C3×#102—31no—
43C2×C3×C4×C8—31no—
44C2×#835C2×C311yescharacteristic, complemented
45C4×#215C2×C318yesdirect product factor
46C2×#809C2×C311yescharacteristic
47C2×#808C2×C311yescharacteristic, complemented
48C2×C4×C3⋊C8C2×C311yescharacteristic, complemented
49C23×C3×D3—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—22no—
52C2×C32×Q8C2×C414yescomplemented
53C2×C32×Q8C2312yes—
54C22×C32×C4—21no—
55C22×C32×C4—21no—
56C22×C32×C4D411yescharacteristic
57C22×C32×C4D411yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C22×C3×Dic3—21no—
60C2×C3×D12—21no—
61C2×C3×D12—22no—
62C2×C3×D12C2×C414yes—
63C2×C3×D12C2312yes—
64C2×C3×C4×D3—24no—
65C2×C3×C4×D3—44no—
66C32×C4⋊C4—24no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2312yes—
69C32×C42—24no—
70C32×C42D412yes—
71C32×C42C2312yes—
72C3×#215C2×C4116yesdirect product factor
73C3×(C2×D3)⋊C4—24no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3C2312yes—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8C2×C414yescomplemented
78C2×C3×C3⋊C8C2312yes—
79C2×C3×C3⋊C8C2312yes—
80C2×C4×SD16—31noSylow
81C22×C3×Q8—21no—
82C22×C3×Q8C2×D311yescharacteristic
83C22×C3×Q8C22×C311yescharacteristic
84C22×C3×D4—31no—
85C23×C3×C4—61no—
86C22×D12C22×C311yescharacteristic
87C22×C4×D3—21no—
88C2×C3×SD16—38no—
89C2×C3×SD16—34no—
90C22×C3×C8—31no—
91C22×C3×C8—31no—
92C3×C4×Q8—24no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8—24no—
96C3×C4×Q8C2×D314yescomplemented
97C3×C4×Q8C22×C314yescomplemented
98C3×C4×D4—34no—
99C3×C4×D4—62no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4C2×D311yescharacteristic
106C2×C3×C4⋊C4C22×C311yescharacteristic
107C2×C3×C22⋊C4—61no—
108C2×C3×C42—22no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2×D311yescharacteristic
113C2×C3×C42C22×C311yescharacteristic
114C2×#215C3×C418yesdirect product factor
115C2×#215C22×C314yes—
116C2×(C2×D3)⋊C4—21no—
117C2×C4⋊Dic3C22×C311yescharacteristic
118C22×C3⋊C8C22×C311yescharacteristic
119C22×C3⋊C8C22×C311yescharacteristic
120C4×D12—22no—
121C4×D12C22×C314yes—
122C3×#106—34no—
123C3×#103—34no—
124C3×#102—34no—
125C3×C4×C8—34no—
126#835C22×C314yescomplemented
127#809C22×C314yes—
128#808C22×C314yescomplemented
129C4×C3⋊C8C22×C314yescomplemented
130C23×C32C2×D411yescharacteristic
131C22×C3×D3—24no—
132C22×C3×D3—22no—
133C22×C3×D3—42no—
134C22×C3×D3—42no—
135C32×Q8—24no—
136C32×Q8—22no—
137C32×Q8C22×C414yescomplemented
138C2×C32×C4—24no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—42no—
143C2×C32×C4SD1614yescomplemented
144C2×C32×C4C22×C412yes—
145C2×C32×C4C2×D412yes—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D412yes—
148C2×C32×C4#4112yes—
149C2×C32×C4C2411yescharacteristic
150C2×C3×Dic3—22no—
151C2×C3×Dic3—42no—
152C3×D12—24no—
153C3×D12—22no—
154C3×D12C22×C414yes—
155C3×C4×D3—48no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8C22×C414yescomplemented
158C22×SD16—31no—
159C2×C4×Q8C3×D311yescharacteristic, complemented
160C2×C4×D4—31no—
161C4×SD16—38no—
162C2×#106—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×C4×C8—31no—
166C24×C3—61no—
167C23×D3—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—24no—
170C2×C3×Q8—22no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—22no—
174C2×C3×Q8—24no—
175C2×C3×Q8—22no—
176C2×C3×Q8C4×D314yescomplemented
177C2×C3×Q8C2×C3×C414yescomplemented
178C2×C3×Q8C22×D312yes—
179C2×C3×Q8C23×C312yes—
180C2×C3×D4—34no—
181C2×C3×D4—32no—
182C2×C3×D4—62no—
183C2×C3×D4—61no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—61no—
194C22×C3×C4—64no—
195C22×C3×C4—124no—
196C22×C3×C4#6711yescharacteristic
197C22×C3×C4#6711yescharacteristic
198C22×C3×C4C3×D411yescharacteristic
199C22×C3×C4C3×D411yescharacteristic
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C23×C311yescharacteristic
202C22×Dic3—21no—
203C2×D12—21no—
204C2×D12—22no—
205C2×D12C2×C3×C414yes—
206C2×D12C23×C312yes—
207C2×C4×D3—24no—
208C2×C4×D3—44no—
209C3×SD16—316no—
210C2×C3×C8—34no—
211C2×C3×C8—32no—
212C2×C3×C8—32no—
213C2×C3×C8—62no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4C22×D312yes—
224C3×C4⋊C4C23×C312yes—
225C3×C22⋊C4—64no—
226C3×C42—28no—
227C3×C42—24no—
228C3×C42—24no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42#6712yes—
232C3×C42C3×D412yes—
233C3×C42C22×D312yes—
234C3×C42C23×C312yes—
235#215C2×C3×C4116yesdirect product factor
236(C2×D3)⋊C4—24no—
237C4⋊Dic3—21no—
238C4⋊Dic3C23×C312yes—
239C2×C3⋊C8—22no—
240C2×C3⋊C8C2×C3×C414yescomplemented
241C2×C3⋊C8C23×C312yes—
242C2×C3⋊C8C23×C312yes—
243C22×C32C4×D412yes—
244C22×C32C2×SD1612yes—
245C22×C32#13512yes—
246C22×C32C22×D411yescharacteristic
247C2×C3×D3—24no—
248C2×C3×D3—42no—
249C2×C3×D3—42no—
250C2×C3×D3—48no—
251C32×C4—24no—
252C32×C4—22no—
253C32×C4—42no—
254C32×C4C4×D412yes—
255C32×C4C2×SD1614yescomplemented
256C32×C4C23×C411yescharacteristic
257C32×C4C2×#4111yescharacteristic
258C3×Dic3—42no—
259C22×Q8C2×C3×D311yescharacteristic
260C22×D4—31no—
261C23×C4—61no—
262C2×SD16—38no—
263C2×SD16—34no—
264C22×C8—31no—
265C22×C8—31no—
266C4×Q8—22no—
267C4×Q8C2×C3×D314yescomplemented
268C4×D4—34no—
269C4×D4—62no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4C2×C3×D311yescharacteristic
273C2×C22⋊C4—61no—
274C2×C42—21no—
275C2×C42C2×C3×D311yescharacteristic
276#106—34no—
277#103—34no—
278#102—34no—
279C4×C8—34no—
280C23×C3—21no—
281C23×C3—64no—
282C23×C3—62no—
283C23×C3—122no—
284C23×C3—122no—
285C23×C3C2×#6711yescharacteristic
286C23×C3C2×C3×D411yescharacteristic
287C22×D3—24no—
288C22×D3—22no—
289C22×D3—42no—
290C22×D3—42no—
291C3×Q8—28no—
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—24no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—24no—
298C3×Q8C2×C4×D314yescomplemented
299C3×Q8C22×C3×C414yescomplemented
300C3×D4—34no—
301C3×D4—64no—
302C3×D4—62no—
303C2×C3×C4—24no—
304C2×C3×C4—28no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—44no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—62no—
326C2×C3×C4—122no—
327C2×C3×C4—128no—
328C2×C3×C4#21514yesdirect product factor
329C2×C3×C4C3×SD1614yescomplemented
330C2×C3×C4C2×C4×D312yes—
331C2×C3×C4#23512yes—
332C2×C3×C4C2×#6712yes—
333C2×C3×C4C2×#6711yescharacteristic
334C2×C3×C4C2×#6712yes—
335C2×C3×C4C22×C3×C412yes—
336C2×C3×C4C2×C3×D412yes—
337C2×C3×C4C2×C3×D411yescharacteristic
338C2×C3×C4C2×C3×D412yes—
339C2×C3×C4C3×#4112yes—
340C2×C3×C4C23×D311yescharacteristic
341C2×C3×C4C24×C311yescharacteristic
342C2×Dic3—22no—
343C2×Dic3—42no—
344D12—24no—
345D12—22no—
346D12C22×C3×C414yes—
347C4×D3—48no—
348C3×C8—34no—
349C3×C8—62no—
350C3⋊C8—22no—
351C3⋊C8C22×C3×C414yescomplemented
352C2×C32C4×SD1614yescomplemented
353C2×C32C2×C4×D411yescharacteristic
354C2×C32C22×SD1611yescharacteristic
355C2×C32C2×#13511yescharacteristic
356C3×D3—44no—
357C24—61no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8C3×C4×D314yescomplemented
361C2×Q8C22×C3×D312yes—
362C2×D4—32no—
363C2×D4—34no—
364C2×D4—62no—
365C2×D4—61no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—61no—
369C22×C4—64no—
370C22×C4—124no—
371C22×C4C3×#6711yescharacteristic
372C22×C4C3×#6711yescharacteristic
373C22×C4C22×C3×D311yescharacteristic
374SD16—316no—
375C2×C8—32no—
376C2×C8—34no—
377C2×C8—32no—
378C2×C8—62no—
379C4⋊C4—24no—
380C4⋊C4—21no—
381C4⋊C4—32no—
382C4⋊C4—61no—
383C4⋊C4C22×C3×D312yes—
384C22⋊C4—64no—
385C42—24no—
386C42C3×#6712yes—
387C42C22×C3×D312yes—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—64no—
393C22×C3—128no—
394C22×C3—122no—
395C22×C3—122no—
396C22×C3C4×#6712yes—
397C22×C3C2×#21512yes—
398C22×C3#95012yes—
399C22×C3C3×C4×D412yes—
400C22×C3C2×C3×SD1612yes—
401C22×C3C3×#13512yes—
402C22×C3C22×#6711yescharacteristic
403C22×C3C22×C3×D411yescharacteristic
404C2×D3—24no—
405C2×D3—42no—
406C2×D3—42no—
407C2×D3—48no—
408C3×C4—28no—
409C3×C4—22no—
410C3×C4—24no—
411C3×C4—22no—
412C3×C4—24no—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—42no—
418C3×C4—44no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—122no—
422C3×C4C4×#6712yes—
423C3×C4C2×#21514yesdirect product factor
424C3×C4C3×C4×D412yes—
425C3×C4C2×C3×SD1614yescomplemented
426C3×C4C22×C4×D311yescharacteristic
427C3×C4C2×#23511yescharacteristic
428C3×C4C23×C3×C411yescharacteristic
429C3×C4C2×C3×#4111yescharacteristic
430Dic3—42no—
431C32C2×C4×SD1611yescharacteristic, complemented, Sylow
432C23—64no—
433C23—62no—
434C23—122no—
435C23—122no—
436C23C2×C3×#6711yescharacteristic
437Q8—24no—
438Q8—22no—
439Q8C2×C3×C4×D314yescomplemented
440D4—34no—
441D4—64no—
442D4—62no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—62no—
449C2×C4—122no—
450C2×C4—128no—
451C2×C4C3×#21514yesdirect product factor
452C2×C4C2×C3×C4×D312yes—
453C2×C4C3×#23512yes—
454C2×C4C2×C3×#6712yes—
455C2×C4C2×C3×#6711yescharacteristic
456C2×C4C2×C3×#6712yes—
457C2×C4C23×C3×D311yescharacteristic
458C8—34no—
459C8—62no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—24no—
464C2×C3—124no—
465C2×C3C4×#21514yesdirect product factor
466C2×C3C3×C4×SD1614yescomplemented
467C2×C3C2×C4×#6711yescharacteristic
468C2×C3C22×#21511yescharacteristic
469C2×C3C2×#95011yescharacteristic
470C2×C3C2×C3×C4×D411yescharacteristic
471C2×C3C22×C3×SD1611yescharacteristic
472C2×C3C2×C3×#13511yescharacteristic
473D3—44no—
474C22—64no—
475C22—122no—
476C22—122no—
477C22—128no—
478C22C3×C4×#6712yes—
479C22C2×C3×#21512yes—
480C22C3×#95012yes—
481C22C22×C3×#6711yescharacteristic
482C4—24no—
483C4—22no—
484C4—42no—
485C4—122no—
486C4C3×C4×#6712yes—
487C4C2×C3×#21514yesdirect product factor
488C4C22×C3×C4×D311yescharacteristic
489C4C2×C3×#23511yescharacteristic
490C3—21no—
491C3C2×C4×#21511yescharacteristic, direct product factor
492C3C2×C3×C4×SD1611yescharacteristic, complemented
493C2—124no—
494C2C3×C4×#21514yesdirect product factor
495C2C2×C3×C4×#6711yescharacteristic
496C2C22×C3×#21511yescharacteristic
497C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC22×C3×C4#67
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×Q8C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×C4×SD16
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
4. C2×C3×#835C2semidirect
5. C3×C4×#215C2direct (N × H)
7. C2×C3×#808C2semidirect
8. C2×C3×C4×C3⋊C8C2semidirect
10. C2×C4×#215C3direct (N × H)
15. C32×C4×Q8C22semidirect
20. C2×C3×#215C4direct (N × H)
28. C3×#835C22semidirect
30. C3×#808C22semidirect
31. C3×C4×C3⋊C8C22semidirect
34. C2×C3×C4×Q8D3semidirect
35. C2×C3×C4×Q8C2×C3semidirect
44. C2×#835C2×C3semidirect
45. C4×#215C2×C3direct (N × H)
47. C2×#808C2×C3semidirect
48. C2×C4×C3⋊C8C2×C3semidirect
52. C2×C32×Q8C2×C4semidirect
72. C3×#215C2×C4direct (N × H)
77. C2×C3×C3⋊C8C2×C4semidirect
96. C3×C4×Q8C2×D3semidirect
97. C3×C4×Q8C22×C3semidirect
114. C2×#215C3×C4direct (N × H)
126. #835C22×C3semidirect
128. #808C22×C3semidirect
129. C4×C3⋊C8C22×C3semidirect
137. C32×Q8C22×C4semidirect
143. C2×C32×C4SD16semidirect
157. C3×C3⋊C8C22×C4semidirect
159. C2×C4×Q8C3×D3semidirect
176. C2×C3×Q8C4×D3semidirect
177. C2×C3×Q8C2×C3×C4semidirect
235. #215C2×C3×C4direct (N × H)
240. C2×C3⋊C8C2×C3×C4semidirect
255. C32×C4C2×SD16semidirect
267. C4×Q8C2×C3×D3semidirect
298. C3×Q8C2×C4×D3semidirect
299. C3×Q8C22×C3×C4semidirect
328. C2×C3×C4#215direct (N × H)
329. C2×C3×C4C3×SD16semidirect
351. C3⋊C8C22×C3×C4semidirect
352. C2×C32C4×SD16semidirect
360. C2×Q8C3×C4×D3semidirect
423. C3×C4C2×#215direct (N × H)
425. C3×C4C2×C3×SD16semidirect
431. C32C2×C4×SD16semidirect
439. Q8C2×C3×C4×D3semidirect
451. C2×C4C3×#215direct (N × H)
465. C2×C3C4×#215direct (N × H)
466. C2×C3C3×C4×SD16semidirect
487. C4C2×C3×#215direct (N × H)
491. C3C2×C4×#215direct (N × H)
492. C3C2×C3×C4×SD16semidirect
494. C2C3×C4×#215direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C22×C3×#2152
C2×C3×C4×D122
C2×C3×#8352
C3×C4×#2152
C2×C3×#8092
C2×C3×#8082
C2×C3×C4×C3⋊C82
C2×C3×C4×SD163
C2×C4×#2153