Finite Group Databasebeta

← C2×C3×#4751 #11857631

Subgroups — C2×C3×#4751 #11857631

Order 1152.

1585 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#239C211yesmaximal, characteristic
2C2×C3×D3×M16C212yesmaximal, complemented
3C2×C32×#101C211yesmaximal, characteristic, complemented
4C2×C3×#836C211yesmaximal, characteristic, complemented
5C2×C3×#823C212yesmaximal, complemented
6C3×#4751C218yesmaximal, direct product factor
7C2×C3×#413—31nomaximal
8C2×#4751C311yesmaximal, characteristic, direct product factor
9C22×C32×Q8C2211yescharacteristic
10C2×C3×#239—22no—
11C2×C3×#239—22no—
12C2×C3×#239—24no—
13C2×C3×#239C414yes—
14C2×C3×#239C2214yes—
15C22×C3×D12C412yes—
16C22×C3×D12C2211yescharacteristic
17C22×C3×C4×D3C412yes—
18C22×C3×C4×D3C2211yescharacteristic
19C2×C32×M16C2212yes—
20C2×C3×#208C2212yes—
21C3×D3×M16—24no—
22C3×D3×M16C2218yescomplemented
23C2×C3×ΓL(1,𝔽25)—22no—
24C2×C3×D3×C8—22no—
25C32×#101C2214yescomplemented
26C3×#836C2214yescomplemented
27C3×#823C2218yescomplemented
28C22×C3×#41—31no—
29C22×#239C2×C311yescharacteristic
30C22×C3×M16—32no—
31C2×D3×M16C2×C312yescomplemented
32C3×#413—38no—
33C2×C3×#101—21no—
34C2×C3×#101—31no—
35C2×C3×#101D311yescharacteristic, complemented
36C2×C3×#101C2×C311yescharacteristic, complemented
37C2×C3×#100—32no—
38C2×#836C2×C311yescharacteristic, complemented
39C2×#823C2×C312yescomplemented
40#4751C2×C318yesdirect product factor
41C23×C3×D3C2×C412yes—
42C23×C3×D3C2×C411yescharacteristic
43C2×C32×Q8—22no—
44C2×C32×Q8—22no—
45C2×C32×Q8C2×C412yes—
46C2×C32×Q8C2312yes—
47C22×C32×C4C2×C412yes—
48C22×C32×C4C2311yescharacteristic
49C22×C3×Dic3C2×C411yescharacteristic
50C3×#239—28no—
51C3×#239—28no—
52C3×#239—28no—
53C3×#239—28no—
54C2×C3×D12—24no—
55C2×C3×D12—24no—
56C2×C3×D12—22no—
57C2×C3×D12—22no—
58C2×C3×D12C2×C418yes—
59C2×C3×D12C2×C412yes—
60C2×C3×D12C2312yes—
61C2×C3×C4×D3—24no—
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3C2×C418yes—
65C2×C3×C4×D3C2×C412yes—
66C2×C3×C4×D3D412yes—
67C2×C3×C4×D3D412yes—
68C2×C3×C4×D3D414yes—
69C2×C3×C4×D3C2312yes—
70C32×M16—22no—
71C32×M16C2314yes—
72C2×C32×C8—22no—
73C3×#208—22no—
74C3×#208C2314yes—
75C2×C3×C3⋊C8—22no—
76C3×ΓL(1,𝔽25)—28no—
77C3×D3×C8—28no—
78C2×#413—31noSylow
79C2×C3×#41—34no—
80C2×C3×#41—34no—
81C2×C3×#41—62no—
82C2×C3×#41—64no—
83C2×C3×#41—62no—
84C22×C3×Q8—21no—
85C22×C3×Q8C2×D311yescharacteristic
86C22×C3×Q8C22×C311yescharacteristic
87C22×C3×D4—32no—
88C22×C3×D4—31no—
89C23×C3×C4—31no—
90C23×C3×C4—32no—
91C2×#239—22no—
92C2×#239—22no—
93C2×#239—24no—
94C2×#239C3×C414yes—
95C2×#239C22×C314yes—
96C22×D12C3×C412yes—
97C22×D12C22×C311yescharacteristic
98C22×C4×D3C3×C412yes—
99C22×C4×D3C22×C311yescharacteristic
100C2×C3×M16—22no—
101C2×C3×M16—32no—
102C2×C3×M16—38no—
103C2×C3×M16—62no—
104C2×C3×M16—64no—
105C2×C3×M16C2×D312yes—
106C2×C3×M16C22×C312yes—
107C22×C3×C8—62no—
108C2×#208C22×C312yes—
109D3×M16—24no—
110D3×M16C22×C318yescomplemented
111C2×ΓL(1,𝔽25)—22no—
112C2×D3×C8—22no—
113C3×#101—24no—
114C3×#101—34no—
115C3×#101C2×D314yescomplemented
116C3×#101C22×C314yescomplemented
117C3×#100—38no—
118#836C22×C314yescomplemented
119#823C22×C318yescomplemented
120C23×C32C22×C411yescharacteristic
121C22×C3×D3—22no—
122C22×C3×D3—22no—
123C22×C3×D3—42no—
124C22×C3×D3—44no—
125C22×C3×D3C22⋊C412yes—
126C22×C3×D3C22⋊C412yes—
127C22×C3×D3C22×C414yes—
128C22×C3×D3C22×C412yes—
129C32×Q8—24no—
130C32×Q8—24no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4C22×C414yes—
134C2×C32×C4C22×C411yescharacteristic
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×D412yes—
137C2×C32×C4C2411yescharacteristic
138C2×C3×Dic3C22⋊C412yes—
139C2×C3×Dic3C22⋊C411yescharacteristic
140C2×C3×Dic3C22⋊C411yescharacteristic
141C2×C3×Dic3C22×C412yes—
142C3×D12—216no—
143C3×D12—24no—
144C3×D12—24no—
145C3×C4×D3—28no—
146C3×C4×D3—28no—
147C3×C4×D3C22⋊C414yes—
148C3×C4×D3C22⋊C414yes—
149C3×C4×D3C2×D414yes—
150C3×C4×D3C2×D414yes—
151C32×C8—24no—
152C3×C3⋊C8—24no—
153C22×#41—31no—
154C22×M16—32no—
155#413—38no—
156C2×#101—31no—
157C2×#101C3×D311yescharacteristic, complemented
158C2×#100—32no—
159C24×C3—31no—
160C24×C3—32no—
161C23×D3C2×C3×C412yes—
162C23×D3C2×C3×C411yescharacteristic
163C3×#41—68no—
164C3×#41—68no—
165C3×#41—68no—
166C3×#41—68no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—42no—
174C2×C3×Q8—42no—
175C2×C3×Q8C4×D312yes—
176C2×C3×Q8C2×C3×C412yes—
177C2×C3×Q8C22×D312yes—
178C2×C3×Q8C23×C312yes—
179C2×C3×D4—38no—
180C2×C3×D4—32no—
181C2×C3×D4—32no—
182C2×C3×D4—64no—
183C2×C3×D4—64no—
184C2×C3×D4—62no—
185C2×C3×D4—62no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—31no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—34no—
192C22×C3×C4—32no—
193C22×C3×C4—32no—
194C22×C3×C4—38no—
195C22×C3×C4—64no—
196C22×C3×C4—62no—
197C22×C3×C4—62no—
198C22×C3×C4C4×D312yes—
199C22×C3×C4C2×C3×C412yes—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C23×C311yescharacteristic
202C22×Dic3C2×C3×C411yescharacteristic
203#239—28no—
204#239—28no—
205#239—28no—
206#239—28no—
207C2×D12—24no—
208C2×D12—24no—
209C2×D12—22no—
210C2×D12—22no—
211C2×D12C2×C3×C418yes—
212C2×D12C2×C3×C412yes—
213C2×D12C23×C312yes—
214C2×C4×D3—24no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3C2×C3×C418yes—
218C2×C4×D3C2×C3×C412yes—
219C2×C4×D3C3×D412yes—
220C2×C4×D3C3×D412yes—
221C2×C4×D3C3×D414yes—
222C2×C4×D3C23×C312yes—
223C3×M16—22no—
224C3×M16—22no—
225C3×M16—24no—
226C3×M16—34no—
227C3×M16—42no—
228C3×M16—68no—
229C3×M16—62no—
230C3×M16C22×D314yes—
231C3×M16C23×C314yes—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—42no—
235C2×C3×C8—68no—
236C2×C3×C8—62no—
237#208—22no—
238#208C23×C314yes—
239C2×C3⋊C8—22no—
240ΓL(1,𝔽25)—28no—
241D3×C8—28no—
242C22×C32—22no—
243C22×C32C2×C22⋊C411yescharacteristic
244C22×C32C2×C22⋊C411yescharacteristic
245C22×C32C23×C411yescharacteristic
246C2×C3×D3—24no—
247C2×C3×D3—24no—
248C2×C3×D3—42no—
249C2×C3×D3—42no—
250C2×C3×D3—44no—
251C2×C3×D3—44no—
252C2×C3×D3—44no—
253C2×C3×D3C2×C22⋊C414yes—
254C32×C4—24no—
255C32×C4C2×C22⋊C412yes—
256C32×C4C22×D412yes—
257C3×Dic3C2×C22⋊C412yes—
258C3×Dic3C2×C22⋊C412yes—
259C2×#41—34no—
260C2×#41—34no—
261C2×#41—62no—
262C2×#41—64no—
263C2×#41—62no—
264C22×Q8C2×C3×D311yescharacteristic
265C22×D4—32no—
266C22×D4—31no—
267C23×C4—31no—
268C23×C4—32no—
269C2×M16—32no—
270C2×M16—38no—
271C2×M16—62no—
272C2×M16—64no—
273C2×M16C2×C3×D312yes—
274C22×C8—62no—
275#101—34no—
276#101C2×C3×D314yescomplemented
277#100—38no—
278C23×C3—21no—
279C23×C3—32no—
280C23×C3—32no—
281C23×C3—32no—
282C23×C3—34no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—122no—
286C23×C3—124no—
287C23×C3C2×C4×D311yescharacteristic
288C23×C3C22×C3×C411yescharacteristic
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—42no—
292C22×D3—44no—
293C22×D3C3×C22⋊C412yes—
294C22×D3C3×C22⋊C412yes—
295C22×D3C22×C3×C414yes—
296C22×D3C22×C3×C412yes—
297C3×Q8—24no—
298C3×Q8—24no—
299C3×Q8—24no—
300C3×Q8—24no—
301C3×Q8—44no—
302C3×Q8—44no—
303C3×D4—616no—
304C3×D4—64no—
305C3×D4—64no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—32no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—32no—
319C2×C3×C4—34no—
320C2×C3×C4—34no—
321C2×C3×C4—34no—
322C2×C3×C4—34no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—68no—
326C2×C3×C4—68no—
327C2×C3×C4C2×C4×D314yes—
328C2×C3×C4C2×C4×D311yescharacteristic
329C2×C3×C4D3×D412yes—
330C2×C3×C4D3×D412yes—
331C2×C3×C4C22×C3×C414yes—
332C2×C3×C4C22×C3×C411yescharacteristic
333C2×C3×C4C2×C3×D412yes—
334C2×C3×C4C2×C3×D412yes—
335C2×C3×C4C23×D311yescharacteristic
336C2×C3×C4C24×C311yescharacteristic
337C2×Dic3C3×C22⋊C412yes—
338C2×Dic3C3×C22⋊C411yescharacteristic
339C2×Dic3C3×C22⋊C411yescharacteristic
340C2×Dic3C22×C3×C412yes—
341D12—216no—
342D12—24no—
343D12—24no—
344C4×D3—28no—
345C4×D3—28no—
346C4×D3C3×C22⋊C414yes—
347C4×D3C3×C22⋊C414yes—
348C4×D3C2×C3×D414yes—
349C4×D3C2×C3×D414yes—
350C3×C8—24no—
351C3×C8—24no—
352C3×C8—44no—
353C3×C8—64no—
354C3⋊C8—24no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32#41312yescomplemented
358C2×C32C22×C22⋊C411yescharacteristic
359C3×D3—24no—
360C3×D3—44no—
361C24—31no—
362C24—32no—
363#41—68no—
364#41—68no—
365#41—68no—
366#41—68no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8C3×C4×D312yes—
370C2×Q8C22×C3×D312yes—
371C2×D4—32no—
372C2×D4—38no—
373C2×D4—32no—
374C2×D4—64no—
375C2×D4—64no—
376C2×D4—62no—
377C2×D4—62no—
378C22×C4—31no—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—34no—
383C22×C4—32no—
384C22×C4—38no—
385C22×C4—64no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4C3×C4×D312yes—
389C22×C4C22×C3×D311yescharacteristic
390M16—22no—
391M16—34no—
392M16—68no—
393M16—62no—
394M16C22×C3×D314yes—
395C2×C8—22no—
396C2×C8—68no—
397C2×C8—62no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—34no—
404C22×C3—42no—
405C22×C3—64no—
406C22×C3—64no—
407C22×C3—122no—
408C22×C3—122no—
409C22×C3—124no—
410C22×C3—124no—
411C22×C3—124no—
412C22×C3D3×C22⋊C411yescharacteristic
413C22×C3D3×C22⋊C411yescharacteristic
414C22×C3C2×C3×C22⋊C411yescharacteristic
415C22×C3C2×C3×C22⋊C411yescharacteristic
416C22×C3C22×C4×D311yescharacteristic
417C22×C3C23×C3×C411yescharacteristic
418C2×D3—24no—
419C2×D3—24no—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—44no—
423C2×D3—44no—
424C2×D3—44no—
425C2×D3C2×C3×C22⋊C414yes—
426C3×C4—24no—
427C3×C4—24no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—32no—
431C3×C4—32no—
432C3×C4—44no—
433C3×C4D3×C22⋊C412yes—
434C3×C4C2×C3×C22⋊C412yes—
435C3×C4C2×D3×D412yes—
436C3×C4C22×C3×D412yes—
437Dic3C2×C3×C22⋊C412yes—
438Dic3C2×C3×C22⋊C412yes—
439C32C2×#41311yescharacteristic, complemented, Sylow
440C23—32no—
441C23—32no—
442C23—32no—
443C23—34no—
444C23—62no—
445C23—62no—
446C23—122no—
447C23—124no—
448C23C2×C3×C4×D311yescharacteristic
449Q8—24no—
450Q8—24no—
451D4—616no—
452D4—64no—
453D4—64no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—32no—
457C2×C4—31no—
458C2×C4—31no—
459C2×C4—32no—
460C2×C4—34no—
461C2×C4—34no—
462C2×C4—34no—
463C2×C4—34no—
464C2×C4—68no—
465C2×C4—68no—
466C2×C4C2×C3×C4×D314yes—
467C2×C4C2×C3×C4×D311yescharacteristic
468C2×C4C3×D3×D412yes—
469C2×C4C3×D3×D412yes—
470C2×C4C23×C3×D311yescharacteristic
471C8—24no—
472C8—64no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—22no—
478C2×C3—21no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—64no—
482C2×C3—124no—
483C2×C3#475112yesdirect product factor
484C2×C3C3×#41312yescomplemented
485C2×C3C2×D3×C22⋊C411yescharacteristic
486C2×C3C22×C3×C22⋊C411yescharacteristic
487D3—24no—
488D3—44no—
489C22—22no—
490C22—34no—
491C22—64no—
492C22—64no—
493C22—122no—
494C22—122no—
495C22—124no—
496C22—124no—
497C22—124no—
498C22C3×D3×C22⋊C411yescharacteristic
499C22C3×D3×C22⋊C411yescharacteristic
500C22C22×C3×C4×D311yescharacteristic
501C4—24no—
502C4—32no—
503C4—32no—
504C4C3×D3×C22⋊C412yes—
505C4C2×C3×D3×D412yes—
506C3—21no—
507C3C2×#475111yescharacteristic, direct product factor
508C3C2×C3×#41311yescharacteristic, complemented
509C2—21no—
510C2—21no—
511C2—64no—
512C2—124no—
513C2C3×#475112yesdirect product factor
514C2C2×C3×D3×C22⋊C411yescharacteristic
subgroupG/H
centerC22×C3D3×C22⋊C4
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#101C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C3×#239C2
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#101D3
hypocenterC3C2×C3×#413
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
2. C2×C3×D3×M16C2semidirect
3. C2×C32×#101C2semidirect
4. C2×C3×#836C2semidirect
5. C2×C3×#823C2semidirect
6. C3×#4751C2direct (N × H)
8. C2×#4751C3direct (N × H)
22. C3×D3×M16C22semidirect
25. C32×#101C22semidirect
26. C3×#836C22semidirect
27. C3×#823C22semidirect
31. C2×D3×M16C2×C3semidirect
35. C2×C3×#101D3semidirect
36. C2×C3×#101C2×C3semidirect
38. C2×#836C2×C3semidirect
39. C2×#823C2×C3semidirect
40. #4751C2×C3direct (N × H)
110. D3×M16C22×C3semidirect
115. C3×#101C2×D3semidirect
116. C3×#101C22×C3semidirect
118. #836C22×C3semidirect
119. #823C22×C3semidirect
157. C2×#101C3×D3semidirect
276. #101C2×C3×D3semidirect
357. C2×C32#413semidirect
439. C32C2×#413semidirect
483. C2×C3#4751direct (N × H)
484. C2×C3C3×#413semidirect
507. C3C2×#4751direct (N × H)
508. C3C2×C3×#413semidirect
513. C2C3×#4751direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C2×C3×D3×M162
C2×C32×#1012
C2×C3×#8362
C2×C3×#8232
C3×#47512
C2×C3×#4133
C2×#47513