Finite Group Databasebeta

← C2×C3×#5168 #11874231

Subgroups — C2×C3×#5168 #11874231

Order 1152.

1333 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic, complemented
3C22×C3×#216C211yesmaximal, characteristic, complemented
4C2×C3×#936C211yesmaximal, characteristic, complemented
5C22×C3×#213C211yesmaximal, characteristic
6C22×C3×#204C211yesmaximal, characteristic, complemented
7C3×#5168C218yesmaximal, direct product factor
8C2×C3×C8×Dic3C211yesmaximal, characteristic, complemented
9C2×C3×#495—31nomaximal
10C2×#5168C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×SD16—24no—
16C2×C32×SD16C2214yescomplemented
17C22×C32×C8C2211yescharacteristic
18C2×C3×#217—22no—
19C3×#947—22no—
20C3×#947C2214yescomplemented
21C2×C3×#216—24no—
22C2×C3×#216C2214yescomplemented
23C3×#936—22no—
24C3×#936C2214yescomplemented
25C2×C3×#213—24no—
26C2×C3×#213C2214yes—
27C2×C3×(C2×D3)⋊C4—22no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C2×C3×#204—24no—
31C2×C3×#204C2214yescomplemented
32C3×C8×Dic3C2214yescomplemented
33C22×C3×Q16—31no—
34C22×C3×SD16—21no—
35C22×C3×SD16—31no—
36C22×C3×SD16D311yescharacteristic, complemented
37C22×C3×SD16C2×C311yescharacteristic, complemented
38C22×C3×D8—31no—
39C2×C3×#124—31no—
40C2×C3×#124—31no—
41C2×#947C2×C311yescharacteristic, complemented
42C22×#216C2×C311yescharacteristic, complemented
43C2×#936C2×C311yescharacteristic, complemented
44C22×#213C2×C311yescharacteristic
45C22×#204C2×C311yescharacteristic, complemented
46C3×#495—38no—
47C2×C3×C4×C8—31no—
48#5168C2×C318yesdirect product factor
49C2×C8×Dic3C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C23×C3×D3—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—22no—
55C2×C32×Q8C2312yes—
56C2×C32×D4—22no—
57C2×C32×D4—22no—
58C2×C32×D4—21no—
59C2×C32×D4C2312yes—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C22×C3×Dic3D412yes—
64C2×C3×D12—22no—
65C2×C3×D12—22no—
66C2×C3×D12—21no—
67C2×C3×D12C2312yes—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2312yes—
72C32×SD16—28no—
73C2×C32×C8D414yes—
74C2×C32×C8C2312yes—
75C3×#217—28no—
76C3×#216—28no—
77C3×#213—28no—
78C3×(C2×D3)⋊C4—28no—
79C3×C4×Dic3D412yes—
80C3×C4×Dic3C2312yes—
81C2×C3×C3⋊C8D414yes—
82C2×C3×C3⋊C8C2312yes—
83C3×#204—28no—
84C2×#495—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8—31no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C22×C3×D4—21no—
90C22×C3×D4—31no—
91C22×C3×D4C2×D311yescharacteristic
92C22×C3×D4C22×C311yescharacteristic
93C22×D12C22×C311yescharacteristic
94C22×Dic6C22×C311yescharacteristic
95C2×C3×Q16—34no—
96C2×C3×Q16—64no—
97C2×C3×SD16—28no—
98C2×C3×SD16—24no—
99C2×C3×SD16—24no—
100C2×C3×SD16—24no—
101C2×C3×SD16—34no—
102C2×C3×SD16—64no—
103C2×C3×SD16C2×D314yescomplemented
104C2×C3×SD16C22×C314yescomplemented
105C2×C3×D8—34no—
106C2×C3×D8—64no—
107C22×C3×C8—21no—
108C22×C3×C8—31no—
109C22×C3×C8C2×D311yescharacteristic
110C22×C3×C8C22×C311yescharacteristic
111C3×#124—34no—
112C3×#124—34no—
113C3×#124—62no—
114C3×#124—62no—
115C2×C3×C22⋊C4—62no—
116C2×C3×C22⋊C4—62no—
117C2×C3×C42—31no—
118C2×#217—22no—
119#947—22no—
120#947C22×C314yescomplemented
121C2×#216—24no—
122C2×#216C22×C314yescomplemented
123#936—22no—
124#936C22×C314yescomplemented
125C2×#213—24no—
126C2×#213C22×C314yes—
127C2×(C2×D3)⋊C4—22no—
128C2×C4×Dic3C22×C311yescharacteristic
129C22×C3⋊C8C22×C311yescharacteristic
130C2×#204—24no—
131C2×#204C22×C314yescomplemented
132C3×C4×C8—34no—
133C8×Dic3C22×C314yescomplemented
134C23×C32—22no—
135C23×C32—42no—
136C23×C32—44no—
137C23×C32C2×D411yescharacteristic
138C22×C3×D3—22no—
139C22×C3×D3—42no—
140C22×C3×D3—44no—
141C32×Q8—22no—
142C32×Q8—22no—
143C32×Q8—42no—
144C32×D4—22no—
145C32×D4—22no—
146C32×D4—42no—
147C2×C32×C4—22no—
148C2×C32×C4—42no—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D412yes—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2411yescharacteristic
153C2×C3×Dic3—24no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3C2×D414yes—
157C3×D12—22no—
158C3×D12—22no—
159C3×D12—42no—
160C3×Dic6—22no—
161C3×Dic6—22no—
162C3×Dic6—42no—
163C32×C8C2×D414yes—
164C3×C3⋊C8C2×D414yes—
165C22×Q16—31no—
166C22×SD16—31no—
167C22×SD16C3×D311yescharacteristic, complemented
168C22×D8—31no—
169C2×#124—31no—
170C2×#124—31no—
171#495—38no—
172C2×C4×C8—31no—
173C24×C3—21no—
174C24×C3—21no—
175C24×C3—41no—
176C24×C3—61no—
177C23×D3—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—24no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—32no—
187C2×C3×Q8—41no—
188C2×C3×Q8—42no—
189C2×C3×Q8—61no—
190C2×C3×Q8—62no—
191C2×C3×Q8—62no—
192C2×C3×Q8C22×D312yes—
193C2×C3×Q8C23×C312yes—
194C2×C3×D4—24no—
195C2×C3×D4—22no—
196C2×C3×D4—21no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—21no—
201C2×C3×D4—22no—
202C2×C3×D4—32no—
203C2×C3×D4—42no—
204C2×C3×D4—41no—
205C2×C3×D4—62no—
206C2×C3×D4—61no—
207C2×C3×D4—62no—
208C2×C3×D4C22×D312yes—
209C2×C3×D4C23×C312yes—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—32no—
214C22×C3×C4—41no—
215C22×C3×C4—61no—
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C23×C311yescharacteristic
218C22×Dic3—21no—
219C22×Dic3C3×D412yes—
220C2×D12—22no—
221C2×D12—22no—
222C2×D12—21no—
223C2×D12C23×C312yes—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—21no—
227C2×Dic6C23×C312yes—
228C3×Q16—68no—
229C3×SD16—216no—
230C3×SD16—28no—
231C3×SD16—28no—
232C3×SD16—68no—
233C3×D8—68no—
234C2×C3×C8—24no—
235C2×C3×C8—22no—
236C2×C3×C8—34no—
237C2×C3×C8—32no—
238C2×C3×C8#6714yes—
239C2×C3×C8C3×D414yes—
240C2×C3×C8C22×D312yes—
241C2×C3×C8C23×C312yes—
242C3×C22⋊C4—68no—
243C3×C22⋊C4—68no—
244C3×C42—32no—
245C3×C42—32no—
246#217—28no—
247#216—28no—
248#213—28no—
249(C2×D3)⋊C4—28no—
250C4×Dic3C3×D412yes—
251C4×Dic3C23×C312yes—
252C2×C3⋊C8C3×D414yes—
253C2×C3⋊C8C23×C312yes—
254#204—28no—
255C22×C32—42no—
256C22×C32—44no—
257C22×C32—84no—
258C22×C32#12712yes—
259C22×C32#13514yes—
260C22×C32C22×D411yescharacteristic
261C2×C3×D3—42no—
262C2×C3×D3—44no—
263C2×C3×D3—84no—
264C32×C4—42no—
265C32×C4#12712yes—
266C32×C4C22×D411yescharacteristic
267C32×C4C22×D411yescharacteristic
268C3×Dic3—24no—
269C3×Dic3—42no—
270C22×Q8—31no—
271C22×Q8C2×C3×D311yescharacteristic
272C22×D4—31no—
273C22×D4C2×C3×D311yescharacteristic
274C2×Q16—34no—
275C2×Q16—64no—
276C2×SD16—24no—
277C2×SD16—34no—
278C2×SD16—64no—
279C2×SD16C2×C3×D314yescomplemented
280C2×D8—34no—
281C2×D8—64no—
282C22×C8—31no—
283C22×C8C2×C3×D311yescharacteristic
284#124—34no—
285#124—34no—
286#124—62no—
287#124—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
290C2×C42—31no—
291C4×C8—34no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—42no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—44no—
301C23×C3—62no—
302C23×C3—84no—
303C23×C3—124no—
304C23×C3—122no—
305C23×C3D3×D411yescharacteristic
306C23×C3C2×C3×D411yescharacteristic
307C22×D3—22no—
308C22×D3—42no—
309C22×D3—44no—
310C3×Q8—24no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—22no—
314C3×Q8—22no—
315C3×Q8—42no—
316C3×Q8—44no—
317C3×Q8—42no—
318C3×Q8—42no—
319C3×Q8—62no—
320C3×Q8—62no—
321C3×Q8—122no—
322C3×D4—24no—
323C3×D4—22no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—42no—
328C3×D4—44no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—62no—
332C3×D4—62no—
333C3×D4—122no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—34no—
341C2×C3×C4—42no—
342C2×C3×C4—44no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—64no—
346C2×C3×C4—62no—
347C2×C3×C4—122no—
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D412yes—
350C2×C3×C4C2×#6712yes—
351C2×C3×C4C2×C3×D411yescharacteristic
352C2×C3×C4C2×C3×D412yes—
353C2×C3×C4C2×C3×D412yes—
354C2×C3×C4C23×D311yescharacteristic
355C2×C3×C4C24×C311yescharacteristic
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—42no—
359C2×Dic3C2×C3×D414yes—
360D12—22no—
361D12—22no—
362D12—42no—
363Dic6—22no—
364Dic6—22no—
365Dic6—42no—
366C3×C8—24no—
367C3×C8—34no—
368C3×C8C2×#6714yes—
369C3×C8C2×C3×D414yes—
370C3⋊C8C2×C3×D414yes—
371C2×C32—82no—
372C2×C32#49514yescomplemented
373C2×C32C2×#12711yescharacteristic
374C2×C32C2×#13512yes—
375C3×D3—82no—
376C24—21no—
377C24—61no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—32no—
382C2×Q8—61no—
383C2×Q8—62no—
384C2×Q8—62no—
385C2×Q8C22×C3×D312yes—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—32no—
390C2×D4—62no—
391C2×D4—61no—
392C2×D4—62no—
393C2×D4C22×C3×D312yes—
394C22×C4—21no—
395C22×C4—32no—
396C22×C4—61no—
397C22×C4C22×C3×D311yescharacteristic
398Q16—68no—
399SD16—28no—
400SD16—68no—
401D8—68no—
402C2×C8—32no—
403C2×C8—34no—
404C2×C8C3×#6714yes—
405C2×C8C22×C3×D312yes—
406C22⋊C4—68no—
407C22⋊C4—68no—
408C42—32no—
409C42—32no—
410C22×C3—24no—
411C22×C3—22no—
412C22×C3—21no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—84no—
419C22×C3—88no—
420C22×C3—84no—
421C22×C3—84no—
422C22×C3—124no—
423C22×C3—122no—
424C22×C3—244no—
425C22×C3#91614yes—
426C22×C3#94012yes—
427C22×C3C3×#12712yes—
428C22×C3C3×#13514yes—
429C22×C3C2×D3×D411yescharacteristic
430C22×C3C22×C3×D411yescharacteristic
431C2×D3—44no—
432C2×D3—42no—
433C2×D3—84no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—44no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—64no—
441C3×C4—122no—
442C3×C4#94012yes—
443C3×C4C3×#12712yes—
444C3×C4C2×D3×D411yescharacteristic
445C3×C4C22×#6711yescharacteristic
446C3×C4C22×C3×D411yescharacteristic
447C3×C4C22×C3×D411yescharacteristic
448Dic3—24no—
449Dic3—42no—
450C32C2×#49511yescharacteristic, complemented, Sylow
451C23—22no—
452C23—44no—
453C23—42no—
454C23—62no—
455C23—124no—
456C23—122no—
457C23C3×D3×D411yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—42no—
461Q8—62no—
462Q8—62no—
463Q8—122no—
464D4—22no—
465D4—22no—
466D4—42no—
467D4—62no—
468D4—62no—
469D4—122no—
470C2×C4—22no—
471C2×C4—34no—
472C2×C4—42no—
473C2×C4—64no—
474C2×C4—62no—
475C2×C4—122no—
476C2×C4C3×D3×D411yescharacteristic
477C2×C4C3×D3×D412yes—
478C2×C4C2×C3×#6712yes—
479C2×C4C23×C3×D311yescharacteristic
480C8—34no—
481C8C2×C3×#6714yes—
482C2×C3—24no—
483C2×C3—22no—
484C2×C3—21no—
485C2×C3—84no—
486C2×C3—82no—
487C2×C3—82no—
488C2×C3—242no—
489C2×C3#516814yesdirect product factor
490C2×C3C3×#49514yescomplemented
491C2×C3C2×#91612yes—
492C2×C3C2×#94011yescharacteristic
493C2×C3C2×C3×#12711yescharacteristic
494C2×C3C2×C3×#13512yes—
495D3—82no—
496C22—44no—
497C22—42no—
498C22—84no—
499C22—124no—
500C22—122no—
501C22—244no—
502C22C3×#91614yes—
503C22C3×#94012yes—
504C22C2×C3×D3×D411yescharacteristic
505C4—42no—
506C4—64no—
507C4—122no—
508C4C3×#94012yes—
509C4C2×C3×D3×D411yescharacteristic
510C4C22×C3×#6711yescharacteristic
511C3—21no—
512C3C2×#516811yescharacteristic, direct product factor
513C3C2×C3×#49511yescharacteristic, complemented
514C2—82no—
515C2—242no—
516C2C3×#516814yesdirect product factor
517C2C2×C3×#91612yes—
518C2C2×C3×#94011yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC22×C3×#213C2
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#495
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
1. C22×C32×SD16C2semidirect
2. C2×C3×#947C2semidirect
3. C22×C3×#216C2semidirect
4. C2×C3×#936C2semidirect
6. C22×C3×#204C2semidirect
7. C3×#5168C2direct (N × H)
8. C2×C3×C8×Dic3C2semidirect
10. C2×#5168C3direct (N × H)
16. C2×C32×SD16C22semidirect
20. C3×#947C22semidirect
22. C2×C3×#216C22semidirect
24. C3×#936C22semidirect
31. C2×C3×#204C22semidirect
32. C3×C8×Dic3C22semidirect
36. C22×C3×SD16D3semidirect
37. C22×C3×SD16C2×C3semidirect
41. C2×#947C2×C3semidirect
42. C22×#216C2×C3semidirect
43. C2×#936C2×C3semidirect
45. C22×#204C2×C3semidirect
48. #5168C2×C3direct (N × H)
49. C2×C8×Dic3C2×C3semidirect
103. C2×C3×SD16C2×D3semidirect
104. C2×C3×SD16C22×C3semidirect
120. #947C22×C3semidirect
122. C2×#216C22×C3semidirect
124. #936C22×C3semidirect
131. C2×#204C22×C3semidirect
133. C8×Dic3C22×C3semidirect
167. C22×SD16C3×D3semidirect
279. C2×SD16C2×C3×D3semidirect
372. C2×C32#495semidirect
450. C32C2×#495semidirect
489. C2×C3#5168direct (N × H)
490. C2×C3C3×#495semidirect
512. C3C2×#5168direct (N × H)
513. C3C2×C3×#495semidirect
516. C2C3×#5168direct (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×#9472
C22×C3×#2162
C2×C3×#9362
C22×C3×#2132
C22×C3×#2042
C3×#51682
C2×C3×C8×Dic32
C2×C3×#4953
C2×#51683