Finite Group Databasebeta

← C2×C3×#5167 #11874374

Subgroups — C2×C3×#5167 #11874374

Order 1152.

1277 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic
2C2×C3×#944C211yesmaximal, characteristic, complemented
3C22×C3×#216C211yesmaximal, characteristic, complemented
4C2×C3×#936C211yesmaximal, characteristic
5C22×C3×#214C211yesmaximal, characteristic
6C22×C3×Dic12C211yesmaximal, characteristic, complemented
7C3×#5167C218yesmaximal, direct product factor
8C2×C3×#815C211yesmaximal, characteristic, complemented
9C2×C3×#497—31nomaximal
10C2×#5167C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×SD16—24no—
16C2×C32×SD16C2214yes—
17C22×C32×C8C2211yescharacteristic
18C2×C3×#217—22no—
19C3×#944—22no—
20C3×#944C2214yescomplemented
21C2×C3×#216—24no—
22C2×C3×#216C2214yescomplemented
23C3×#936—22no—
24C3×#936C2214yes—
25C2×C3×#214—24no—
26C2×C3×#214C2214yes—
27C2×C3×Dic3⋊C4—22no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C2×C3×Dic12—24no—
31C2×C3×Dic12C2214yescomplemented
32C3×#815C2214yescomplemented
33C22×C3×Q16—31no—
34C22×C3×Q16—31no—
35C22×C3×SD16—21no—
36C22×C3×SD16—31no—
37C22×C3×SD16D311yescharacteristic
38C22×C3×SD16C2×C311yescharacteristic
39C2×C3×C4⋊Q8—31no—
40C2×C3×#124—31no—
41C2×#944C2×C311yescharacteristic, complemented
42C22×#216C2×C311yescharacteristic, complemented
43C2×#936C2×C311yescharacteristic
44C22×#214C2×C311yescharacteristic
45C22×Dic12C2×C311yescharacteristic, complemented
46C3×#497—38no—
47C2×C3×#97—31no—
48#5167C2×C318yesdirect product factor
49C2×#815C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—22no—
54C2×C32×Q8C2312yes—
55C2×C32×D4—22no—
56C2×C32×D4—22no—
57C2×C32×D4—21no—
58C2×C32×D4C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C22×C3×Dic3D412yes—
64C2×C3×Dic6—22no—
65C2×C3×Dic6—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C2×C3×Dic6C2312yes—
72C32×SD16—28no—
73C2×C32×C8D414yes—
74C2×C32×C8C2312yes—
75C3×#217—28no—
76C3×#216—28no—
77C3×#214—28no—
78C3×Dic3⋊C4—28no—
79C3×C4×Dic3D412yes—
80C3×C4×Dic3C2312yes—
81C2×C3×C3⋊C8D414yes—
82C2×C3×C3⋊C8C2312yes—
83C3×Dic12—28no—
84C2×#497—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8—31no—
87C22×C3×Q8—31no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—21no—
91C22×C3×D4C2×D311yescharacteristic
92C22×C3×D4C22×C311yescharacteristic
93C22×Dic6C22×C311yescharacteristic
94C22×Dic6C22×C311yescharacteristic
95C2×C3×Q16—34no—
96C2×C3×Q16—34no—
97C2×C3×Q16—64no—
98C2×C3×Q16—64no—
99C2×C3×SD16—28no—
100C2×C3×SD16—24no—
101C2×C3×SD16—24no—
102C2×C3×SD16—24no—
103C2×C3×SD16—34no—
104C2×C3×SD16—64no—
105C2×C3×SD16C2×D314yes—
106C2×C3×SD16C22×C314yes—
107C22×C3×C8—21no—
108C22×C3×C8—31no—
109C22×C3×C8C2×D311yescharacteristic
110C22×C3×C8C22×C311yescharacteristic
111C3×C4⋊Q8—34no—
112C3×C4⋊Q8—62no—
113C3×#124—34no—
114C3×#124—62no—
115C2×C3×C4⋊C4—62no—
116C2×C3×C22⋊C4—62no—
117C2×C3×C42—31no—
118C2×#217—22no—
119#944—22no—
120#944C22×C314yescomplemented
121C2×#216—24no—
122C2×#216C22×C314yescomplemented
123#936—22no—
124#936C22×C314yes—
125C2×#214—24no—
126C2×#214C22×C314yes—
127C2×Dic3⋊C4—22no—
128C2×C4×Dic3C22×C311yescharacteristic
129C22×C3⋊C8C22×C311yescharacteristic
130C2×Dic12—24no—
131C2×Dic12C22×C314yescomplemented
132C3×#97—34no—
133#815C22×C314yescomplemented
134C23×C32—22no—
135C23×C32—42no—
136C23×C32—44no—
137C23×C32C2×D411yescharacteristic
138C32×Q8—22no—
139C32×Q8—22no—
140C32×Q8—42no—
141C32×D4—22no—
142C32×D4—22no—
143C32×D4—42no—
144C2×C32×C4—22no—
145C2×C32×C4—42no—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D412yes—
149C2×C32×C4C2411yescharacteristic
150C2×C3×Dic3—24no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—42no—
154C2×C3×Dic3—42no—
155C2×C3×Dic3C2×D414yes—
156C3×Dic6—22no—
157C3×Dic6—22no—
158C3×Dic6—22no—
159C3×Dic6—22no—
160C3×Dic6—42no—
161C3×Dic6—42no—
162C32×C8C2×D414yes—
163C3×C3⋊C8C2×D414yes—
164C22×Q16—31no—
165C22×Q16—31no—
166C22×SD16—31no—
167C22×SD16C3×D311yescharacteristic
168C2×C4⋊Q8—31no—
169C2×#124—31no—
170#497—38no—
171C2×#97—31no—
172C24×C3—21no—
173C24×C3—21no—
174C24×C3—41no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—24no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—32no—
184C2×C3×Q8—32no—
185C2×C3×Q8—41no—
186C2×C3×Q8—42no—
187C2×C3×Q8—61no—
188C2×C3×Q8—61no—
189C2×C3×Q8—62no—
190C2×C3×Q8—62no—
191C2×C3×Q8—62no—
192C2×C3×Q8—62no—
193C2×C3×Q8C22×D312yes—
194C2×C3×Q8C23×C312yes—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—21no—
202C2×C3×D4—22no—
203C2×C3×D4—42no—
204C2×C3×D4—41no—
205C2×C3×D4C22×D312yes—
206C2×C3×D4C23×C312yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—32no—
211C22×C3×C4—41no—
212C22×C3×C4—61no—
213C22×C3×C4—61no—
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C23×C311yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3C3×D412yes—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C23×C312yes—
226C2×Dic6C23×C312yes—
227C3×Q16—68no—
228C3×Q16—68no—
229C3×SD16—216no—
230C3×SD16—28no—
231C3×SD16—28no—
232C3×SD16—68no—
233C2×C3×C8—24no—
234C2×C3×C8—22no—
235C2×C3×C8—34no—
236C2×C3×C8—32no—
237C2×C3×C8#6714yes—
238C2×C3×C8C3×D414yes—
239C2×C3×C8C22×D312yes—
240C2×C3×C8C23×C312yes—
241C3×C4⋊C4—68no—
242C3×C22⋊C4—68no—
243C3×C42—32no—
244C3×C42—32no—
245#217—28no—
246#216—28no—
247#214—28no—
248Dic3⋊C4—28no—
249C4×Dic3C3×D412yes—
250C4×Dic3C23×C312yes—
251C2×C3⋊C8C3×D414yes—
252C2×C3⋊C8C23×C312yes—
253Dic12—28no—
254C22×C32—42no—
255C22×C32—44no—
256C22×C32—84no—
257C22×C32#12712yes—
258C22×C32#13714yes—
259C22×C32C22×D411yescharacteristic
260C32×C4—42no—
261C32×C4#12712yes—
262C32×C4C22×D411yescharacteristic
263C32×C4C22×D411yescharacteristic
264C3×Dic3—24no—
265C3×Dic3—42no—
266C3×Dic3—42no—
267C22×Q8—31no—
268C22×Q8—31no—
269C22×Q8C2×C3×D311yescharacteristic
270C22×D4C2×C3×D311yescharacteristic
271C2×Q16—34no—
272C2×Q16—34no—
273C2×Q16—64no—
274C2×Q16—64no—
275C2×SD16—24no—
276C2×SD16—34no—
277C2×SD16—64no—
278C2×SD16C2×C3×D314yes—
279C22×C8—31no—
280C22×C8C2×C3×D311yescharacteristic
281C4⋊Q8—34no—
282C4⋊Q8—62no—
283#124—34no—
284#124—62no—
285C2×C4⋊C4—62no—
286C2×C22⋊C4—62no—
287C2×C42—31no—
288#97—34no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—21no—
292C23×C3—42no—
293C23×C3—44no—
294C23×C3—44no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—44no—
298C23×C3—84no—
299C23×C3D3×D411yescharacteristic
300C23×C3C2×C3×D411yescharacteristic
301C3×Q8—24no—
302C3×Q8—22no—
303C3×Q8—22no—
304C3×Q8—22no—
305C3×Q8—22no—
306C3×Q8—42no—
307C3×Q8—44no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—62no—
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—62no—
314C3×Q8—122no—
315C3×Q8—122no—
316C3×D4—24no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—22no—
321C3×D4—42no—
322C3×D4—44no—
323C3×D4—42no—
324C3×D4—42no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—34no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—64no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—122no—
340C2×C3×C4—122no—
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4D3×D412yes—
343C2×C3×C4C2×#6712yes—
344C2×C3×C4C2×C3×D411yescharacteristic
345C2×C3×C4C2×C3×D412yes—
346C2×C3×C4C2×C3×D412yes—
347C2×C3×C4C23×D311yescharacteristic
348C2×C3×C4C24×C311yescharacteristic
349C2×Dic3—24no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3C2×C3×D414yes—
355Dic6—22no—
356Dic6—22no—
357Dic6—22no—
358Dic6—22no—
359Dic6—42no—
360Dic6—42no—
361C3×C8—24no—
362C3×C8—34no—
363C3×C8C2×#6714yes—
364C3×C8C2×C3×D414yes—
365C3⋊C8C2×C3×D414yes—
366C2×C32—82no—
367C2×C32#49714yescomplemented
368C2×C32C2×#12711yescharacteristic
369C2×C32C2×#13712yes—
370C24—21no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—32no—
375C2×Q8—32no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—62no—
382C2×Q8C22×C3×D312yes—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4C22×C3×D312yes—
387C22×C4—21no—
388C22×C4—32no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4C22×C3×D311yescharacteristic
392Q16—68no—
393Q16—68no—
394SD16—28no—
395SD16—68no—
396C2×C8—32no—
397C2×C8—34no—
398C2×C8C3×#6714yes—
399C2×C8C22×C3×D312yes—
400C4⋊C4—68no—
401C22⋊C4—68no—
402C42—32no—
403C42—32no—
404C22×C3—24no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—44no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—84no—
413C22×C3—88no—
414C22×C3—84no—
415C22×C3—84no—
416C22×C3#91514yes—
417C22×C3#94012yes—
418C22×C3C3×#12712yes—
419C22×C3C3×#13714yes—
420C22×C3C2×D3×D411yescharacteristic
421C22×C3C22×C3×D411yescharacteristic
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—44no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—64no—
429C3×C4—122no—
430C3×C4—122no—
431C3×C4#94012yes—
432C3×C4C3×#12712yes—
433C3×C4C2×D3×D411yescharacteristic
434C3×C4C22×#6711yescharacteristic
435C3×C4C22×C3×D411yescharacteristic
436C3×C4C22×C3×D411yescharacteristic
437Dic3—24no—
438Dic3—42no—
439Dic3—42no—
440C32C2×#49711yescharacteristic, complemented, Sylow
441C23—22no—
442C23—44no—
443C23—42no—
444C23C3×D3×D411yescharacteristic
445Q8—22no—
446Q8—22no—
447Q8—42no—
448Q8—62no—
449Q8—62no—
450Q8—62no—
451Q8—62no—
452Q8—122no—
453Q8—122no—
454D4—22no—
455D4—22no—
456D4—42no—
457C2×C4—22no—
458C2×C4—34no—
459C2×C4—42no—
460C2×C4—64no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4C3×D3×D411yescharacteristic
466C2×C4C3×D3×D412yes—
467C2×C4C2×C3×#6712yes—
468C2×C4C23×C3×D311yescharacteristic
469C8—34no—
470C8C2×C3×#6714yes—
471C2×C3—24no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—84no—
475C2×C3—82no—
476C2×C3—82no—
477C2×C3#516714yesdirect product factor
478C2×C3C3×#49714yescomplemented
479C2×C3C2×#91512yes—
480C2×C3C2×#94011yescharacteristic
481C2×C3C2×C3×#12711yescharacteristic
482C2×C3C2×C3×#13712yes—
483C22—44no—
484C22—42no—
485C22—84no—
486C22C3×#91514yes—
487C22C3×#94012yes—
488C22C2×C3×D3×D411yescharacteristic
489C4—42no—
490C4—64no—
491C4—122no—
492C4—122no—
493C4C3×#94012yes—
494C4C2×C3×D3×D411yescharacteristic
495C4C22×C3×#6711yescharacteristic
496C3—21no—
497C3C2×#516711yescharacteristic, direct product factor
498C3C2×C3×#49711yescharacteristic, complemented
499C2—82no—
500C2C3×#516714yesdirect product factor
501C2C2×C3×#91512yes—
502C2C2×C3×#94011yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#497
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C3×#944C2semidirect
3. C22×C3×#216C2semidirect
6. C22×C3×Dic12C2semidirect
7. C3×#5167C2direct (N × H)
8. C2×C3×#815C2semidirect
10. C2×#5167C3direct (N × H)
20. C3×#944C22semidirect
22. C2×C3×#216C22semidirect
31. C2×C3×Dic12C22semidirect
32. C3×#815C22semidirect
41. C2×#944C2×C3semidirect
42. C22×#216C2×C3semidirect
45. C22×Dic12C2×C3semidirect
48. #5167C2×C3direct (N × H)
49. C2×#815C2×C3semidirect
120. #944C22×C3semidirect
122. C2×#216C22×C3semidirect
131. C2×Dic12C22×C3semidirect
133. #815C22×C3semidirect
367. C2×C32#497semidirect
440. C32C2×#497semidirect
477. C2×C3#5167direct (N × H)
478. C2×C3C3×#497semidirect
497. C3C2×#5167direct (N × H)
498. C3C2×C3×#497semidirect
500. C2C3×#5167direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SD162
C2×C3×#9442
C22×C3×#2162
C2×C3×#9362
C22×C3×#2142
C22×C3×Dic122
C3×#51672
C2×C3×#8152
C2×C3×#4973
C2×#51673