Finite Group Databasebeta

← C2×C3×#4746 #11864817

Subgroups — C2×C3×#4746 #11864817

Order 1152.

1613 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1004C211yesmaximal, characteristic
2C2×C3×#936C211yesmaximal, characteristic, complemented
3C2×C3×#909C212yesmaximal, complemented
4C2×C32×#100C211yesmaximal, characteristic, complemented
5C2×C3×#825C212yesmaximal, complemented
6C3×#4746C218yesmaximal, direct product factor
7C2×C3×#455—31nomaximal
8C2×#4746C311yesmaximal, characteristic, direct product factor
9C22×C32×D4C2211yescharacteristic
10C22×C3×#67—22no—
11C3×#1004—22no—
12C3×#1004—24no—
13C3×#1004C2214yes—
14C2×C3×#237—22no—
15C2×C3×D3×D4—22no—
16C2×C3×#235C2212yes—
17C22×C3×Dic6C2211yescharacteristic, complemented
18C2×C32×M16C2212yescomplemented
19C2×C3×#217—22no—
20C3×#936—22no—
21C3×#936C2214yescomplemented
22C2×C3×C4×Dic3C2211yescharacteristic, complemented
23C3×#909—24no—
24C3×#909C2218yescomplemented
25C2×C3×Dic12—22no—
26C2×C3×#204—22no—
27C32×#100C2214yescomplemented
28C3×#825C2218yescomplemented
29C2×C3×H(5,𝔽2)—31no—
30C2×#1004C2×C311yescharacteristic
31C2×C3×#137—32no—
32C2×C3×#124—31no—
33C2×#936C2×C311yescharacteristic, complemented
34C2×#909C2×C312yescomplemented
35C3×#455—38no—
36C2×C3×#104—32no—
37C2×C3×#100—21no—
38C2×C3×#100D311yescharacteristic, complemented
39C2×C3×#100C2×C311yescharacteristic, complemented
40C2×#825C2×C312yescomplemented
41#4746C2×C318yesdirect product factor
42C24×C32D412yes—
43C23×C3×D3—22no—
44C2×C32×D4—22no—
45C2×C32×D4—22no—
46C2×C32×D4D412yes—
47C2×C32×D4C2312yes—
48C22×C32×C4C2311yescharacteristic
49C2×C3×#67—24no—
50C2×C3×#67—28no—
51C2×C3×#67—22no—
52C2×C3×#67—28no—
53C22×C3×Dic3—22no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3—21no—
56C3×#237—28no—
57C3×#237—28no—
58C3×D3×D4—28no—
59C3×D3×D4—28no—
60C3×#235—22no—
61C3×#235—22no—
62C3×#235D414yes—
63C3×#235C2314yes—
64C2×C3×D12D412yes—
65C2×C3×C4×D3—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6D412yes—
70C2×C3×Dic6C2312yescomplemented
71C32×M16—22no—
72C32×M16C2314yescomplemented
73C2×C32×C8—22no—
74C3×#217—28no—
75C3×C4×Dic3—21no—
76C3×C4×Dic3C2312yescomplemented
77C3×Dic12—28no—
78C3×#204—28no—
79C2×#455—31noSylow
80C3×H(5,𝔽2)—34no—
81C3×H(5,𝔽2)—64no—
82C3×H(5,𝔽2)—62no—
83C2×C3×#41—32no—
84C2×C3×#41—62no—
85C22×C3×Q8—31no—
86C22×C3×D4—21no—
87C22×C3×D4—62no—
88C22×C3×D4—62no—
89C22×C3×D4C2×D311yescharacteristic
90C22×C3×D4C22×C311yescharacteristic
91C22×#67—22no—
92#1004—22no—
93#1004—24no—
94#1004C22×C314yes—
95C2×#237—22no—
96C2×D3×D4—22no—
97C2×#235C22×C312yes—
98C22×Dic6C22×C311yescharacteristic, complemented
99C3×#137—38no—
100C3×#137—64no—
101C2×C3×Q16—62no—
102C2×C3×SD16—62no—
103C2×C3×M16—22no—
104C2×C3×M16C2×D312yescomplemented
105C2×C3×M16C22×C312yescomplemented
106C3×#124—34no—
107C3×#124—62no—
108C2×C3×C22⋊C4—62no—
109C2×C3×C42—31no—
110C2×#217—22no—
111#936—22no—
112#936C22×C314yescomplemented
113C2×C4×Dic3C22×C311yescharacteristic, complemented
114#909—24no—
115#909C22×C318yescomplemented
116C2×Dic12—22no—
117C2×#204—22no—
118C3×#104—38no—
119C3×#100—24no—
120C3×#100C2×D314yescomplemented
121C3×#100C22×C314yescomplemented
122#825C22×C318yescomplemented
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—44no—
126C23×C32C2×D411yescharacteristic
127C23×C32C2×D414yes—
128C22×C3×D3—22no—
129C22×C3×D3—28no—
130C22×C3×D3—84no—
131C32×D4—24no—
132C32×D4—24no—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D412yes—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2411yescharacteristic
137C3×#67—216no—
138C3×#67—24no—
139C3×#67—24no—
140C2×C3×Dic3—28no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C3×D12—22no—
148C3×D12C2×D414yes—
149C3×C4×D3—24no—
150C3×C4×D3—24no—
151C3×Dic6—22no—
152C3×Dic6—24no—
153C3×Dic6—42no—
154C3×Dic6C2×D414yes—
155C32×C8—24no—
156C2×H(5,𝔽2)—31no—
157C2×#137—32no—
158C2×#124—31no—
159#455—38no—
160C2×#104—32no—
161C2×#100C3×D311yescharacteristic, complemented
162C24×C3—22no—
163C24×C3—62no—
164C24×C3D1212yes—
165C24×C3C3×D412yes—
166C23×D3—22no—
167C3×#41—34no—
168C3×#41—34no—
169C3×#41—68no—
170C3×#41—68no—
171C3×#41—62no—
172C3×#41—62no—
173C2×C3×Q8—32no—
174C2×C3×Q8—32no—
175C2×C3×Q8—61no—
176C2×C3×Q8—62no—
177C2×C3×Q8—62no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—32no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—68no—
188C2×C3×D4—68no—
189C2×C3×D4—64no—
190C2×C3×D4—68no—
191C2×C3×D4—62no—
192C2×C3×D4—68no—
193C2×C3×D4D1212yes—
194C2×C3×D4C3×D412yes—
195C2×C3×D4C22×D312yes—
196C2×C3×D4C23×C312yes—
197C22×C3×C4—21no—
198C22×C3×C4—62no—
199C22×C3×C4—62no—
200C22×C3×C4—61no—
201C22×C3×C4—61no—
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C23×C311yescharacteristic
204C2×#67—24no—
205C2×#67—28no—
206C2×#67—22no—
207C2×#67—28no—
208C22×Dic3—22no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211#237—28no—
212#237—28no—
213D3×D4—28no—
214D3×D4—28no—
215#235—22no—
216#235—22no—
217#235C3×D414yes—
218#235C23×C314yes—
219C2×D12C3×D412yes—
220C2×C4×D3—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—21no—
224C2×Dic6C3×D412yes—
225C2×Dic6C23×C312yescomplemented
226C3×Q16—68no—
227C3×SD16—68no—
228C3×M16—22no—
229C3×M16—22no—
230C3×M16—24no—
231C3×M16—42no—
232C3×M16C22×D314yescomplemented
233C3×M16C23×C314yescomplemented
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—42no—
237C3×C22⋊C4—68no—
238C3×C42—32no—
239C3×C42—61no—
240#217—28no—
241C4×Dic3—21no—
242C4×Dic3C23×C312yescomplemented
243Dic12—28no—
244#204—28no—
245C22×C32—22no—
246C22×C32—24no—
247C22×C32—44no—
248C22×C32—44no—
249C22×C32—82no—
250C22×C32#12011yescharacteristic
251C22×C32#12011yescharacteristic
252C22×C32C22×D411yescharacteristic
253C2×C3×D3—24no—
254C2×C3×D3—44no—
255C2×C3×D3—88no—
256C32×C4#12012yes—
257C32×C4C22×D412yes—
258C3×Dic3—24no—
259C3×Dic3—42no—
260C3×Dic3—42no—
261H(5,𝔽2)—34no—
262H(5,𝔽2)—64no—
263H(5,𝔽2)—62no—
264C2×#41—32no—
265C2×#41—62no—
266C22×Q8—31no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4C2×C3×D311yescharacteristic
270#137—38no—
271#137—64no—
272C2×Q16—62no—
273C2×SD16—62no—
274C2×M16C2×C3×D312yescomplemented
275#124—34no—
276#124—62no—
277C2×C22⋊C4—62no—
278C2×C42—31no—
279#104—38no—
280#100C2×C3×D314yescomplemented
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—21no—
286C23×C3—24no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—68no—
292C23×C3—62no—
293C23×C3—84no—
294C23×C3—244no—
295C23×C3C2×D1211yescharacteristic
296C23×C3C2×D1214yes—
297C23×C3C2×C3×D411yescharacteristic
298C23×C3C2×C3×D414yes—
299C22×D3—22no—
300C22×D3—28no—
301C22×D3—84no—
302C3×Q8—34no—
303C3×Q8—62no—
304C3×Q8—64no—
305C3×Q8—122no—
306C3×D4—24no—
307C3×D4—24no—
308C3×D4—24no—
309C3×D4—24no—
310C3×D4—34no—
311C3×D4—44no—
312C3×D4—44no—
313C3×D4—616no—
314C3×D4—64no—
315C3×D4—64no—
316C3×D4—62no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—68no—
322C2×C3×C4—62no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—122no—
328C2×C3×C4—121no—
329C2×C3×C4—121no—
330C2×C3×C4C2×D1211yescharacteristic
331C2×C3×C4D3×D412yes—
332C2×C3×C4D3×D412yes—
333C2×C3×C4C2×C3×D411yescharacteristic
334C2×C3×C4C2×C3×D412yes—
335C2×C3×C4C2×C3×D412yes—
336C2×C3×C4C23×D311yescharacteristic
337C2×C3×C4C24×C311yescharacteristic
338#67—216no—
339#67—24no—
340#67—24no—
341C2×Dic3—28no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—42no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348D12—22no—
349D12C2×C3×D414yes—
350C4×D3—24no—
351C4×D3—24no—
352Dic6—22no—
353Dic6—24no—
354Dic6—42no—
355Dic6C2×C3×D414yes—
356C3×C8—24no—
357C3×C8—24no—
358C3×C8—44no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—44no—
362C2×C32#45512yescomplemented
363C2×C32C2×#12011yescharacteristic
364C3×D3—44no—
365C24—62no—
366C24C3×D1212yes—
367#41—34no—
368#41—34no—
369#41—68no—
370#41—68no—
371#41—62no—
372#41—62no—
373C2×Q8—32no—
374C2×Q8—32no—
375C2×Q8—61no—
376C2×Q8—62no—
377C2×Q8—62no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—32no—
381C2×D4—68no—
382C2×D4—68no—
383C2×D4—64no—
384C2×D4—68no—
385C2×D4—62no—
386C2×D4—68no—
387C2×D4C3×D1212yes—
388C2×D4C22×C3×D312yes—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4C22×C3×D311yescharacteristic
394Q16—68no—
395SD16—68no—
396M16—22no—
397M16C22×C3×D314yescomplemented
398C2×C8—22no—
399C22⋊C4—68no—
400C42—32no—
401C42—61no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—64no—
416C22×C3—84no—
417C22×C3—84no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—84no—
421C22×C3—124no—
422C22×C3—248no—
423C22×C3#88211yescharacteristic
424C22×C3#88211yescharacteristic
425C22×C3C3×#12011yescharacteristic
426C22×C3C3×#12011yescharacteristic
427C22×C3C22×D1211yescharacteristic
428C22×C3C22×C3×D411yescharacteristic
429C2×D3—24no—
430C2×D3—44no—
431C2×D3—88no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—64no—
435C3×C4—122no—
436C3×C4—122no—
437C3×C4#88212yes—
438C3×C4C3×#12012yes—
439C3×C4C2×D3×D412yes—
440C3×C4C22×C3×D412yes—
441Dic3—24no—
442Dic3—42no—
443Dic3—42no—
444C32C2×#45511yescharacteristic, complemented, Sylow
445C23—22no—
446C23—22no—
447C23—44no—
448C23—68no—
449C23—62no—
450C23—244no—
451C23C2×C3×D1211yescharacteristic
452C23C2×C3×D1214yes—
453Q8—34no—
454Q8—62no—
455Q8—64no—
456Q8—122no—
457D4—24no—
458D4—24no—
459D4—34no—
460D4—616no—
461D4—64no—
462D4—64no—
463D4—62no—
464C2×C4—68no—
465C2×C4—62no—
466C2×C4—64no—
467C2×C4—64no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4C2×C3×D1211yescharacteristic
474C2×C4C3×D3×D412yes—
475C2×C4C3×D3×D412yes—
476C2×C4C23×C3×D311yescharacteristic
477C8—24no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—44no—
487C2×C3—44no—
488C2×C3—84no—
489C2×C3—124no—
490C2×C3#474612yesdirect product factor
491C2×C3C3×#45512yescomplemented
492C2×C3C2×#88211yescharacteristic
493C2×C3C2×C3×#12011yescharacteristic
494D3—44no—
495C22—22no—
496C22—24no—
497C22—44no—
498C22—44no—
499C22—64no—
500C22—82no—
501C22—124no—
502C22—248no—
503C22C3×#88211yescharacteristic
504C22C3×#88211yescharacteristic
505C22C22×C3×D1211yescharacteristic
506C4—64no—
507C4—122no—
508C4—122no—
509C4C3×#88212yes—
510C4C2×C3×D3×D412yes—
511C3—21no—
512C3C2×#474611yescharacteristic, direct product factor
513C3C2×C3×#45511yescharacteristic, complemented
514C2—21no—
515C2—21no—
516C2—44no—
517C2—124no—
518C2C3×#474612yesdirect product factor
519C2C2×C3×#88211yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#100C2
socleC22×C32#120
prime radicalC2×C3×#1004C2
hypercenterC2×C3×#100D3
hypocenterC3C2×C3×#455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#100

Complements

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

NH ≅ G/Ntype
2. C2×C3×#936C2semidirect
3. C2×C3×#909C2semidirect
4. C2×C32×#100C2semidirect
5. C2×C3×#825C2semidirect
6. C3×#4746C2direct (N × H)
8. C2×#4746C3direct (N × H)
17. C22×C3×Dic6C22semidirect
18. C2×C32×M16C22semidirect
21. C3×#936C22semidirect
22. C2×C3×C4×Dic3C22semidirect
24. C3×#909C22semidirect
27. C32×#100C22semidirect
28. C3×#825C22semidirect
33. C2×#936C2×C3semidirect
34. C2×#909C2×C3semidirect
38. C2×C3×#100D3semidirect
39. C2×C3×#100C2×C3semidirect
40. C2×#825C2×C3semidirect
41. #4746C2×C3direct (N × H)
70. C2×C3×Dic6C23semidirect
72. C32×M16C23semidirect
76. C3×C4×Dic3C23semidirect
98. C22×Dic6C22×C3semidirect
104. C2×C3×M16C2×D3semidirect
105. C2×C3×M16C22×C3semidirect
112. #936C22×C3semidirect
113. C2×C4×Dic3C22×C3semidirect
115. #909C22×C3semidirect
120. C3×#100C2×D3semidirect
121. C3×#100C22×C3semidirect
122. #825C22×C3semidirect
161. C2×#100C3×D3semidirect
225. C2×Dic6C23×C3semidirect
232. C3×M16C22×D3semidirect
233. C3×M16C23×C3semidirect
242. C4×Dic3C23×C3semidirect
274. C2×M16C2×C3×D3semidirect
280. #100C2×C3×D3semidirect
362. C2×C32#455semidirect
397. M16C22×C3×D3semidirect
444. C32C2×#455semidirect
490. C2×C3#4746direct (N × H)
491. C2×C3C3×#455semidirect
512. C3C2×#4746direct (N × H)
513. C3C2×C3×#455semidirect
518. C2C3×#4746direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10042
C2×C3×#9362
C2×C3×#9092
C2×C32×#1002
C2×C3×#8252
C3×#47462
C2×C3×#4553
C2×#47463