Finite Group Databasebeta

← C2×C3×#5052 #11866997

Subgroups — C2×C3×#5052 #11866997

Order 1152.

1145 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C22⋊Q8C211yesmaximal, characteristic
2C22×C3×#216C211yesmaximal, characteristic, complemented
3C2×C3×#924C211yesmaximal, characteristic
4C22×C3×#208C211yesmaximal, characteristic
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C3×#5052C218yesmaximal, direct product factor
7C2×C3×#810C211yesmaximal, characteristic, complemented
8C2×C3×#808C211yesmaximal, characteristic, complemented
9C2×C3×#470—31nomaximal
10C2×#5052C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C23×C32×C4C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C32×C22⋊Q8—22no—
15C32×C22⋊Q8C2214yes—
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4—21no—
19C2×C3×#217—21no—
20C2×C3×#216—24no—
21C2×C3×#216C2214yescomplemented
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C3×#924—22no—
24C3×#924C2214yes—
25C2×C3×Dic3⋊C4—21no—
26C2×C3×#208—24no—
27C2×C3×#208C2214yes—
28C22×C3×C3⋊C8C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#835C2214yescomplemented
31C3×#810C2214yescomplemented
32C3×#808C2214yescomplemented
33C22×C3×Q16—31no—
34C22×C3×M16—31no—
35C2×C3×C22⋊Q8—21no—
36C2×C3×C22⋊Q8—31no—
37C2×C3×C22⋊Q8D311yescharacteristic
38C2×C3×C22⋊Q8C2×C311yescharacteristic
39C22×#216C2×C311yescharacteristic, complemented
40C2×#924C2×C311yescharacteristic
41C22×#208C2×C311yescharacteristic
42C3×#470—38no—
43C2×C3×#106—31no—
44C2×C3×#103—31no—
45C2×C3×#103—31no—
46C2×#835C2×C311yescharacteristic, complemented
47#5052C2×C318yesdirect product factor
48C2×#810C2×C311yescharacteristic, complemented
49C2×#808C2×C311yescharacteristic, complemented
50C24×C32D411yescharacteristic
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—22no—
54C2×C32×Q8C2312yes—
55C22×C32×C4—22no—
56C22×C32×C4—22no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4D411yescharacteristic
60C22×C32×C4D412yes—
61C22×C32×C4C2312yes—
62C22×C32×C4C2311yescharacteristic
63C22×C3×Dic3—21no—
64C22×C3×Dic3—21no—
65C2×C3×Dic6—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2312yes—
69C32×C4⋊C4—24no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4C2312yes—
72C32×C22⋊C4—24no—
73C3×#217—24no—
74C3×#216—28no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3C2312yes—
77C3×Dic3⋊C4—24no—
78C3×#208—28no—
79C2×C3×C3⋊C8—22no—
80C2×C3×C3⋊C8D414yes—
81C2×C3×C3⋊C8C2312yes—
82C2×C3×C3⋊C8C2312yes—
83C2×#470—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—31no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C23×C3×C4—21no—
89C23×C3×C4C2×D311yescharacteristic
90C23×C3×C4C22×C311yescharacteristic
91C22×Dic6C22×C311yescharacteristic
92C2×C3×Q16—34no—
93C2×C3×Q16—64no—
94C2×C3×M16—34no—
95C2×C3×M16—64no—
96C22×C3×C8—31no—
97C22×C3×C8—31no—
98C3×C22⋊Q8—24no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—24no—
102C3×C22⋊Q8—34no—
103C3×C22⋊Q8—62no—
104C3×C22⋊Q8C2×D314yes—
105C3×C22⋊Q8C22×C314yes—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4—61no—
112C2×C3×C4⋊C4C2×D311yescharacteristic
113C2×C3×C4⋊C4C22×C311yescharacteristic
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—61no—
118C2×#217—21no—
119C2×#216—24no—
120C2×#216C22×C314yescomplemented
121C2×C4⋊Dic3C22×C311yescharacteristic
122#924—22no—
123#924C22×C314yes—
124C2×Dic3⋊C4—21no—
125C2×#208—24no—
126C2×#208C22×C314yes—
127C22×C3⋊C8C22×C311yescharacteristic
128C22×C3⋊C8C22×C311yescharacteristic
129C3×#106—34no—
130C3×#103—34no—
131C3×#103—34no—
132#835C22×C314yescomplemented
133#810C22×C314yescomplemented
134#808C22×C314yescomplemented
135C23×C32—22no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32C2×D412yes—
139C23×C32C2×D411yescharacteristic
140C32×Q8—22no—
141C32×Q8—22no—
142C32×Q8—42no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—24no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—42no—
150C2×C32×C4—42no—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D412yes—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4#4112yes—
155C2×C32×C4C2411yescharacteristic
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—42no—
160C3×Dic6—22no—
161C3×Dic6—22no—
162C3×Dic6—42no—
163C3×C3⋊C8—22no—
164C3×C3⋊C8C2×D414yes—
165C22×Q16—31no—
166C22×M16—31no—
167C2×C22⋊Q8—31no—
168C2×C22⋊Q8C3×D311yescharacteristic
169#470—38no—
170C2×#106—31no—
171C2×#103—31no—
172C2×#103—31no—
173C24×C3—21no—
174C24×C3#6711yescharacteristic
175C24×C3C3×D411yescharacteristic
176C2×C3×Q8—22no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—32no—
185C2×C3×Q8—42no—
186C2×C3×Q8—42no—
187C2×C3×Q8—61no—
188C2×C3×Q8—62no—
189C2×C3×Q8—62no—
190C2×C3×Q8C22×D312yes—
191C2×C3×Q8C23×C312yes—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—61no—
209C22×C3×C4—61no—
210C22×C3×C4#6712yes—
211C22×C3×C4#6711yescharacteristic
212C22×C3×C4C3×D411yescharacteristic
213C22×C3×C4C3×D412yes—
214C22×C3×C4C22×D312yes—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C23×C312yes—
217C22×C3×C4C23×C311yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6C23×C312yes—
224C3×Q16—68no—
225C3×M16—68no—
226C2×C3×C8—34no—
227C2×C3×C8—32no—
228C2×C3×C8—32no—
229C2×C3×C8—62no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—64no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4C22×D312yes—
241C3×C4⋊C4C23×C312yes—
242C3×C22⋊C4—28no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—64no—
246#217—24no—
247#216—28no—
248C4⋊Dic3—21no—
249C4⋊Dic3C23×C312yes—
250Dic3⋊C4—24no—
251#208—28no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8C3×D414yes—
254C2×C3⋊C8C23×C312yes—
255C2×C3⋊C8C23×C312yes—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—44no—
260C22×C32#12112yes—
261C22×C32#13712yes—
262C22×C32#13712yes—
263C22×C32C22×D411yescharacteristic
264C32×C4—22no—
265C32×C4—42no—
266C32×C4—42no—
267C32×C4#12112yes—
268C32×C4C22×D411yescharacteristic
269C32×C4C2×#4111yescharacteristic
270C3×Dic3—42no—
271C3×Dic3—42no—
272C22×Q8—31no—
273C22×Q8C2×C3×D311yescharacteristic
274C23×C4C2×C3×D311yescharacteristic
275C2×Q16—34no—
276C2×Q16—64no—
277C2×M16—34no—
278C2×M16—64no—
279C22×C8—31no—
280C22×C8—31no—
281C22⋊Q8—22no—
282C22⋊Q8—34no—
283C22⋊Q8—62no—
284C22⋊Q8C2×C3×D314yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C2×C3×D311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—61no—
291#106—34no—
292#103—34no—
293#103—34no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3C2×#6712yes—
306C23×C3C2×#6711yescharacteristic
307C23×C3C2×C3×D412yes—
308C23×C3C2×C3×D411yescharacteristic
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—44no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—62no—
319C3×Q8—62no—
320C3×Q8—122no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—62no—
351C2×C3×C4—62no—
352C2×C3×C4—122no—
353C2×C3×C4—122no—
354C2×C3×C4D3×D412yes—
355C2×C3×C4#23712yes—
356C2×C3×C4C2×#6712yes—
357C2×C3×C4C2×#6711yescharacteristic
358C2×C3×C4C2×C3×D412yes—
359C2×C3×C4C2×C3×D412yes—
360C2×C3×C4C2×C3×D411yescharacteristic
361C2×C3×C4C3×#4112yes—
362C2×C3×C4C23×D311yescharacteristic
363C2×C3×C4C24×C311yescharacteristic
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368Dic6—22no—
369Dic6—22no—
370Dic6—42no—
371C3×C8—34no—
372C3×C8—62no—
373C3⋊C8—22no—
374C3⋊C8C2×C3×D414yes—
375C2×C32—42no—
376C2×C32#47014yescomplemented
377C2×C32C2×#12111yescharacteristic
378C2×C32C2×#13711yescharacteristic
379C2×C32C2×#13711yescharacteristic
380C24C3×#6711yescharacteristic
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—32no—
385C2×Q8—61no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8C22×C3×D312yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4C3×#6711yescharacteristic
396C22×C4C3×#6712yes—
397C22×C4C22×C3×D312yes—
398C22×C4C22×C3×D311yescharacteristic
399Q16—68no—
400M16—68no—
401C2×C8—32no—
402C2×C8—34no—
403C2×C8—32no—
404C2×C8—62no—
405C4⋊C4—24no—
406C4⋊C4—21no—
407C4⋊C4—32no—
408C4⋊C4—61no—
409C4⋊C4—64no—
410C4⋊C4C22×C3×D312yes—
411C22⋊C4—24no—
412C22⋊C4—64no—
413C22×C3—24no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—21no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—48no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3#93912yes—
430C22×C3#94212yes—
431C22×C3#95112yes—
432C22×C3C3×#12112yes—
433C22×C3C3×#13712yes—
434C22×C3C3×#13712yes—
435C22×C3C22×#6711yescharacteristic
436C22×C3C22×C3×D411yescharacteristic
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—42no—
443C3×C4—44no—
444C3×C4—44no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4#93912yes—
452C3×C4C3×#12112yes—
453C3×C4C2×D3×D411yescharacteristic
454C3×C4C2×#23711yescharacteristic
455C3×C4C22×C3×D411yescharacteristic
456C3×C4C2×C3×#4111yescharacteristic
457Dic3—42no—
458Dic3—42no—
459C32C2×#47011yescharacteristic, complemented, Sylow
460C23—22no—
461C23—22no—
462C23—22no—
463C23C2×C3×#6712yes—
464C23C2×C3×#6711yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—42no—
468Q8—62no—
469Q8—62no—
470Q8—122no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4C3×D3×D412yes—
484C2×C4C3×#23712yes—
485C2×C4C2×C3×#6712yes—
486C2×C4C2×C3×#6711yescharacteristic
487C2×C4C23×C3×D311yescharacteristic
488C8—34no—
489C8—62no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—24no—
494C2×C3—44no—
495C2×C3—42no—
496C2×C3—42no—
497C2×C3#505214yesdirect product factor
498C2×C3C3×#47014yescomplemented
499C2×C3C2×#93911yescharacteristic
500C2×C3C2×#94211yescharacteristic
501C2×C3C2×#95111yescharacteristic
502C2×C3C2×C3×#12111yescharacteristic
503C2×C3C2×C3×#13711yescharacteristic
504C2×C3C2×C3×#13711yescharacteristic
505C22—22no—
506C22—22no—
507C22—22no—
508C22—44no—
509C22C3×#93912yes—
510C22C3×#94212yes—
511C22C3×#95112yes—
512C22C22×C3×#6711yescharacteristic
513C4—22no—
514C4—42no—
515C4—42no—
516C4—122no—
517C4—122no—
518C4C3×#93912yes—
519C4C2×C3×D3×D411yescharacteristic
520C4C2×C3×#23711yescharacteristic
521C3—21no—
522C3C2×#505211yescharacteristic, direct product factor
523C3C2×C3×#47011yescharacteristic, complemented
524C2—42no—
525C2C3×#505214yesdirect product factor
526C2C2×C3×#93911yescharacteristic
527C2C2×C3×#94211yescharacteristic
528C2C2×C3×#95111yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C22⋊Q8C2
socleC23×C32C2×D4
prime radicalC24×C32D4
hypercenterC2×C3×C22⋊Q8D3
hypocenterC3C2×C3×#470
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×C3×#216C2semidirect
5. C2×C3×#835C2semidirect
6. C3×#5052C2direct (N × H)
7. C2×C3×#810C2semidirect
8. C2×C3×#808C2semidirect
10. C2×#5052C3direct (N × H)
21. C2×C3×#216C22semidirect
30. C3×#835C22semidirect
31. C3×#810C22semidirect
32. C3×#808C22semidirect
39. C22×#216C2×C3semidirect
46. C2×#835C2×C3semidirect
47. #5052C2×C3direct (N × H)
48. C2×#810C2×C3semidirect
49. C2×#808C2×C3semidirect
120. C2×#216C22×C3semidirect
132. #835C22×C3semidirect
133. #810C22×C3semidirect
134. #808C22×C3semidirect
376. C2×C32#470semidirect
459. C32C2×#470semidirect
497. C2×C3#5052direct (N × H)
498. C2×C3C3×#470semidirect
522. C3C2×#5052direct (N × H)
523. C3C2×C3×#470semidirect
525. C2C3×#5052direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊Q82
C22×C3×#2162
C2×C3×#9242
C22×C3×#2082
C2×C3×#8352
C3×#50522
C2×C3×#8102
C2×C3×#8082
C2×C3×#4703
C2×#50523