Finite Group Databasebeta

← C2×C3×#4737 #11869426

Subgroups — C2×C3×#4737 #11869426

Order 1152.

1037 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#930C211yesmaximal, characteristic
2C2×C3×#926C211yesmaximal, characteristic, complemented
3C2×C32×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×C3×#821C211yesmaximal, characteristic, complemented
5C2×C3×#821C211yesmaximal, characteristic, complemented
6C2×C3×#818C211yesmaximal, characteristic, complemented
7C2×C3×#817C211yesmaximal, characteristic, complemented
8C3×#4737C218yesmaximal, direct product factor
9C2×C3×#482—31nomaximal
10C2×#4737C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×D12C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×C3×#217—21no—
17C3×#930—22no—
18C3×#930C2214yes—
19C2×C3×(C2×D3)⋊C4—21no—
20C3×#926—22no—
21C3×#926C2214yescomplemented
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4×Dic3—21no—
26C32×C22⋊C8C2214yescomplemented
27C3×#821C2214yescomplemented
28C3×#821C2214yescomplemented
29C3×#818C2214yescomplemented
30C3×#817C2214yescomplemented
31C2×C3×#121—31no—
32C2×C3×#117—31no—
33C2×#930C2×C311yescharacteristic
34C2×#926C2×C311yescharacteristic, complemented
35C3×#482—38no—
36C2×C3×#107—31no—
37C2×C3×#106—31no—
38C2×C3×#102—31no—
39C2×C3×#102—31no—
40C2×C3×C22⋊C8—21no—
41C2×C3×C22⋊C8D311yescharacteristic, complemented
42C2×C3×C22⋊C8C2×C311yescharacteristic, complemented
43C2×#821C2×C311yescharacteristic, complemented
44C2×#821C2×C311yescharacteristic, complemented
45C2×#818C2×C311yescharacteristic, complemented
46C2×#817C2×C311yescharacteristic, complemented
47#4737C2×C318yesdirect product factor
48C24×C32D411yescharacteristic
49C23×C3×D3—21no—
50C22×C32×C4—22no—
51C22×C32×C4—22no—
52C22×C32×C4D411yescharacteristic
53C22×C32×C4D412yes—
54C22×C32×C4C2312yes—
55C22×C32×C4C2311yescharacteristic
56C2×C3×#67—24no—
57C2×C3×#67—44no—
58C22×C3×Dic3—21no—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—21no—
61C2×C3×D12—22no—
62C2×C3×D12—22no—
63C2×C3×D12—21no—
64C2×C3×D12C2312yes—
65C2×C32×C8—22no—
66C2×C32×C8—22no—
67C2×C32×C8C2312yes—
68C2×C32×C8C2312yes—
69C3×#217—24no—
70C3×(C2×D3)⋊C4—24no—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3C2312yes—
75C3×C4⋊Dic3C2312yes—
76C3×C4⋊Dic3C2312yes—
77C3×C4×Dic3—24no—
78C2×#482—31noSylow
79C22×C3×D4—31no—
80C22×C3×D4—61no—
81C23×C3×C4—21no—
82C23×C3×C4C2×D311yescharacteristic
83C23×C3×C4C22×C311yescharacteristic
84C22×#67—21no—
85C22×D12C22×C311yescharacteristic
86C22×C3×C8—21no—
87C22×C3×C8—21no—
88C22×C3×C8C2×D311yescharacteristic
89C22×C3×C8C2×D311yescharacteristic
90C22×C3×C8C22×C311yescharacteristic
91C22×C3×C8C22×C311yescharacteristic
92C3×#121—34no—
93C3×#121—62no—
94C3×#117—34no—
95C3×#117—62no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C4⋊C4—31no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C22⋊C4—61no—
100C2×C3×C22⋊C4—61no—
101C2×C3×C42—61no—
102C2×#217—21no—
103#930—22no—
104#930C22×C314yes—
105C2×(C2×D3)⋊C4—21no—
106#926—22no—
107#926C22×C314yescomplemented
108C2×C4⋊Dic3C22×C311yescharacteristic
109C2×C4⋊Dic3C22×C311yescharacteristic
110C2×C4⋊Dic3C22×C311yescharacteristic
111C2×C4×Dic3—21no—
112C3×#107—34no—
113C3×#106—34no—
114C3×#102—34no—
115C3×#102—34no—
116C3×C22⋊C8—24no—
117C3×C22⋊C8C2×D314yescomplemented
118C3×C22⋊C8C22×C314yescomplemented
119#821C22×C314yescomplemented
120#821C22×C314yescomplemented
121#818C22×C314yescomplemented
122#817C22×C314yescomplemented
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32C2×D412yes—
127C23×C32C2×D411yescharacteristic
128C22×C3×D3—22no—
129C22×C3×D3—42no—
130C22×C3×D3—42no—
131C22×C3×D3—42no—
132C2×C32×C4—22no—
133C2×C32×C4—24no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4C2×D412yes—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4#4112yes—
139C2×C32×C4#4112yes—
140C2×C32×C4C2411yescharacteristic
141C3×#67—48no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3—42no—
147C2×C3×Dic3—42no—
148C3×D12—22no—
149C3×D12—22no—
150C3×D12—42no—
151C32×C8—22no—
152C32×C8—22no—
153C2×#121—31no—
154C2×#117—31no—
155#482—38no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×C22⋊C8C3×D311yescharacteristic, complemented
161C24×C3—21no—
162C24×C3—61no—
163C24×C3D1211yescharacteristic
164C24×C3C3×D411yescharacteristic
165C23×D3—21no—
166C2×C3×D4—32no—
167C2×C3×D4—64no—
168C2×C3×D4—61no—
169C2×C3×D4—62no—
170C2×C3×D4—62no—
171C2×C3×D4—124no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4—61no—
185C22×C3×C4D1212yes—
186C22×C3×C4D1211yescharacteristic
187C22×C3×C4C3×D411yescharacteristic
188C22×C3×C4C3×D412yes—
189C22×C3×C4C22×D312yes—
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C23×C312yes—
192C22×C3×C4C23×C311yescharacteristic
193C2×#67—24no—
194C2×#67—44no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—21no—
201C2×D12C23×C312yes—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—42no—
209C2×C3×C8—42no—
210C2×C3×C8C22×D312yes—
211C2×C3×C8C22×D312yes—
212C2×C3×C8C23×C312yes—
213C2×C3×C8C23×C312yes—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—32no—
216C3×C4⋊C4—32no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—61no—
220C3×C22⋊C4—64no—
221C3×C22⋊C4—64no—
222C3×C42—64no—
223#217—24no—
224(C2×D3)⋊C4—24no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3C23×C312yes—
229C4⋊Dic3C23×C312yes—
230C4⋊Dic3C23×C312yes—
231C4×Dic3—24no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—44no—
236C22×C32#12312yes—
237C22×C32#13512yes—
238C22×C32AGL(1,ℤ/8)12yes—
239C22×C32C22×D411yescharacteristic
240C2×C3×D3—42no—
241C2×C3×D3—42no—
242C2×C3×D3—42no—
243C2×C3×D3—84no—
244C32×C4—22no—
245C32×C4#12312yes—
246C32×C4C2×#4111yescharacteristic
247C32×C4C2×#4111yescharacteristic
248C3×Dic3—42no—
249C3×Dic3—42no—
250C3×Dic3—42no—
251C22×D4—31no—
252C22×D4—61no—
253C23×C4C2×C3×D311yescharacteristic
254C22×C8C2×C3×D311yescharacteristic
255C22×C8C2×C3×D311yescharacteristic
256#121—34no—
257#121—62no—
258#117—34no—
259#117—62no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C42—61no—
266#107—34no—
267#106—34no—
268#102—34no—
269#102—34no—
270C22⋊C8C2×C3×D314yescomplemented
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—21no—
279C23×C3—22no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—62no—
283C23×C3—122no—
284C23×C3—122no—
285C23×C3—122no—
286C23×C3C2×D1212yes—
287C23×C3C2×D1211yescharacteristic
288C23×C3C2×C3×D412yes—
289C23×C3C2×C3×D411yescharacteristic
290C22×D3—22no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C3×D4—62no—
295C3×D4—62no—
296C3×D4—128no—
297C3×D4—122no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—44no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—122no—
319C2×C3×C4—122no—
320C2×C3×C4—122no—
321C2×C3×C4C2×D1212yes—
322C2×C3×C4C2×D1211yescharacteristic
323C2×C3×C4#23712yes—
324C2×C3×C4#23712yes—
325C2×C3×C4C2×C3×D412yes—
326C2×C3×C4C2×C3×D411yescharacteristic
327C2×C3×C4C3×#4112yes—
328C2×C3×C4C3×#4112yes—
329C2×C3×C4C23×D311yescharacteristic
330C2×C3×C4C24×C311yescharacteristic
331#67—48no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338D12—22no—
339D12—22no—
340D12—42no—
341C3×C8—22no—
342C3×C8—22no—
343C3×C8—22no—
344C3×C8—22no—
345C3×C8—42no—
346C3×C8—42no—
347C2×C32—42no—
348C2×C32#48214yescomplemented
349C2×C32C2×#12311yescharacteristic
350C2×C32C2×#13511yescharacteristic
351C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
352C3×D3—82no—
353C24—61no—
354C24C3×D1211yescharacteristic
355C2×D4—32no—
356C2×D4—64no—
357C2×D4—61no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—124no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4C3×D1211yescharacteristic
367C22×C4C3×D1212yes—
368C22×C4C22×C3×D312yes—
369C22×C4C22×C3×D311yescharacteristic
370C2×C8—22no—
371C2×C8—22no—
372C2×C8C22×C3×D312yes—
373C2×C8C22×C3×D312yes—
374C4⋊C4—32no—
375C4⋊C4—32no—
376C4⋊C4—32no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C22⋊C4—64no—
381C22⋊C4—64no—
382C42—64no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—21no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—48no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—122no—
400C22×C3—122no—
401C22×C3—122no—
402C22×C3—244no—
403C22×C3#88612yes—
404C22×C3#90412yes—
405C22×C3#90812yes—
406C22×C3C3×#12312yes—
407C22×C3C3×#13512yes—
408C22×C3C3×AGL(1,ℤ/8)12yes—
409C22×C3C22×D1211yescharacteristic
410C22×C3C22×C3×D411yescharacteristic
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—84no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—42no—
421C3×C4—122no—
422C3×C4—122no—
423C3×C4—122no—
424C3×C4#88612yes—
425C3×C4C3×#12312yes—
426C3×C4C2×#23711yescharacteristic
427C3×C4C2×#23711yescharacteristic
428C3×C4C2×C3×#4111yescharacteristic
429C3×C4C2×C3×#4111yescharacteristic
430Dic3—42no—
431Dic3—42no—
432Dic3—42no—
433C32C2×#48211yescharacteristic, complemented, Sylow
434C23—22no—
435C23—22no—
436C23—22no—
437C23—62no—
438C23—122no—
439C23—122no—
440C23—122no—
441C23C2×C3×D1212yes—
442C23C2×C3×D1211yescharacteristic
443D4—62no—
444D4—62no—
445D4—128no—
446D4—122no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4C2×C3×D1212yes—
458C2×C4C2×C3×D1211yescharacteristic
459C2×C4C3×#23712yes—
460C2×C4C3×#23712yes—
461C2×C4C23×C3×D311yescharacteristic
462C8—22no—
463C8—22no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—24no—
467C2×C3—21no—
468C2×C3—44no—
469C2×C3—42no—
470C2×C3—42no—
471C2×C3—242no—
472C2×C3#473714yesdirect product factor
473C2×C3C3×#48214yescomplemented
474C2×C3C2×#88611yescharacteristic
475C2×C3C2×#90411yescharacteristic
476C2×C3C2×#90811yescharacteristic
477C2×C3C2×C3×#12311yescharacteristic
478C2×C3C2×C3×#13511yescharacteristic
479C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
480D3—82no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—44no—
485C22—122no—
486C22—122no—
487C22—122no—
488C22—244no—
489C22C3×#88612yes—
490C22C3×#90412yes—
491C22C3×#90812yes—
492C22C22×C3×D1211yescharacteristic
493C4—22no—
494C4—122no—
495C4—122no—
496C4—122no—
497C4C3×#88612yes—
498C4C2×C3×#23711yescharacteristic
499C4C2×C3×#23711yescharacteristic
500C3—21no—
501C3C2×#473711yescharacteristic, direct product factor
502C3C2×C3×#48211yescharacteristic, complemented
503C2—42no—
504C2—242no—
505C2C3×#473714yesdirect product factor
506C2C2×C3×#88611yescharacteristic
507C2C2×C3×#90411yescharacteristic
508C2C2×C3×#90811yescharacteristic
subgroupG/H
centerC23×C3C2×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C22⋊C8C2
socleC23×C32C2×D4
prime radicalC2×C3×#930C2
hypercenterC2×C3×C22⋊C8D3
hypocenterC3C2×C3×#482
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#926C2semidirect
3. C2×C32×C22⋊C8C2semidirect
4. C2×C3×#821C2semidirect
5. C2×C3×#821C2semidirect
6. C2×C3×#818C2semidirect
7. C2×C3×#817C2semidirect
8. C3×#4737C2direct (N × H)
10. C2×#4737C3direct (N × H)
21. C3×#926C22semidirect
26. C32×C22⋊C8C22semidirect
27. C3×#821C22semidirect
28. C3×#821C22semidirect
29. C3×#818C22semidirect
30. C3×#817C22semidirect
34. C2×#926C2×C3semidirect
41. C2×C3×C22⋊C8D3semidirect
42. C2×C3×C22⋊C8C2×C3semidirect
43. C2×#821C2×C3semidirect
44. C2×#821C2×C3semidirect
45. C2×#818C2×C3semidirect
46. C2×#817C2×C3semidirect
47. #4737C2×C3direct (N × H)
107. #926C22×C3semidirect
117. C3×C22⋊C8C2×D3semidirect
118. C3×C22⋊C8C22×C3semidirect
119. #821C22×C3semidirect
120. #821C22×C3semidirect
121. #818C22×C3semidirect
122. #817C22×C3semidirect
160. C2×C22⋊C8C3×D3semidirect
270. C22⋊C8C2×C3×D3semidirect
348. C2×C32#482semidirect
433. C32C2×#482semidirect
472. C2×C3#4737direct (N × H)
473. C2×C3C3×#482semidirect
501. C3C2×#4737direct (N × H)
502. C3C2×C3×#482semidirect
505. C2C3×#4737direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9302
C2×C3×#9262
C2×C32×C22⋊C82
C2×C3×#8212
C2×C3×#8182
C2×C3×#8172
C3×#47372
C2×C3×#4823
C2×#47373