Finite Group Databasebeta

← C2×C3×#5162 #11865470

Subgroups — C2×C3×#5162 #11865470

Order 1152.

1237 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic
2C2×C3×#944C211yesmaximal, characteristic, complemented
3C2×C3×D4×Dic3C211yesmaximal, characteristic
4C22×C3×#214C211yesmaximal, characteristic
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C3×#5162C218yesmaximal, direct product factor
7C2×C3×#816C211yesmaximal, characteristic, complemented
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×#461—31nomaximal
10C2×#5162C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C23×C3×Dic3—21no—
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×SD16—24no—
16C2×C32×SD16C2214yes—
17C22×C32×C8C2211yescharacteristic
18C2×C3×#217—21no—
19C3×#944—22no—
20C3×#944C2214yescomplemented
21C3×D4×Dic3—22no—
22C3×D4×Dic3C2214yes—
23C2×C3×#214—24no—
24C2×C3×#214C2214yes—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×Dic3⋊C4—21no—
27C2×C3×Dic3⋊C4—21no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#835C2214yescomplemented
31C3×#816C2214yescomplemented
32C3×Dic3⋊C8C2214yescomplemented
33C22×C3×SD16—21no—
34C22×C3×SD16—31no—
35C22×C3×SD16D311yescharacteristic
36C22×C3×SD16C2×C311yescharacteristic
37C2×C3×C4⋊Q8—31no—
38C2×C3×C4×D4—31no—
39C2×#944C2×C311yescharacteristic, complemented
40C2×D4×Dic3C2×C311yescharacteristic
41C22×#214C2×C311yescharacteristic
42C3×#461—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#103—31no—
46C2×#835C2×C311yescharacteristic, complemented
47#5162C2×C318yesdirect product factor
48C2×#816C2×C311yescharacteristic, complemented
49C2×Dic3⋊C8C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—22no—
54C2×C32×Q8C2312yes—
55C2×C32×D4—21no—
56C2×C32×D4—22no—
57C2×C32×D4D414yes—
58C2×C32×D4C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C22×C3×Dic3—24no—
64C22×C3×Dic3—44no—
65C22×C3×Dic3D411yescharacteristic
66C22×C3×Dic3D411yescharacteristic
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C32×SD16—28no—
72C2×C32×C8—22no—
73C2×C32×C8C2312yes—
74C3×#217—24no—
75C3×#214—28no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3C2312yes—
78C3×Dic3⋊C4—24no—
79C3×Dic3⋊C4—24no—
80C3×C4×Dic3D412yes—
81C3×C4×Dic3C2312yes—
82C2×C3×C3⋊C8—22no—
83C2×C3×C3⋊C8C2312yes—
84C2×#461—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8—31no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C22×C3×D4—21no—
90C22×C3×D4C2×D311yescharacteristic
91C22×C3×D4C22×C311yescharacteristic
92C23×C3×C4—61no—
93C23×Dic3—21no—
94C22×Dic6C22×C311yescharacteristic
95C2×C3×SD16—28no—
96C2×C3×SD16—24no—
97C2×C3×SD16—24no—
98C2×C3×SD16—24no—
99C2×C3×SD16—34no—
100C2×C3×SD16—64no—
101C2×C3×SD16C2×D314yes—
102C2×C3×SD16C22×C314yes—
103C22×C3×C8—21no—
104C22×C3×C8—31no—
105C22×C3×C8C2×D311yescharacteristic
106C22×C3×C8C22×C311yescharacteristic
107C3×C4⋊Q8—34no—
108C3×C4⋊Q8—62no—
109C3×C4×D4—34no—
110C3×C4×D4—62no—
111C2×C3×C4⋊C4—31no—
112C2×C3×C4⋊C4—61no—
113C2×C3×C4⋊C4—61no—
114C2×C3×C22⋊C4—61no—
115C2×C3×C42—31no—
116C2×#217—21no—
117#944—22no—
118#944C22×C314yescomplemented
119D4×Dic3—22no—
120D4×Dic3C22×C314yes—
121C2×#214—24no—
122C2×#214C22×C314yes—
123C2×C4⋊Dic3C22×C311yescharacteristic
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×C4×Dic3C22×C311yescharacteristic
127C22×C3⋊C8C22×C311yescharacteristic
128C3×C4⋊C8—34no—
129C3×#103—34no—
130C3×#103—34no—
131#835C22×C314yescomplemented
132#816C22×C314yescomplemented
133Dic3⋊C8C22×C314yescomplemented
134C23×C32—24no—
135C23×C32—22no—
136C23×C32—42no—
137C23×C32—42no—
138C23×C32C2×D411yescharacteristic
139C32×Q8—22no—
140C32×Q8—22no—
141C32×Q8—42no—
142C32×D4—24no—
143C32×D4—22no—
144C32×D4C2×D414yes—
145C2×C32×C4—22no—
146C2×C32×C4—42no—
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D412yes—
149C2×C32×C4#4112yes—
150C2×C32×C4C2411yescharacteristic
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—42no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3—48no—
157C2×C3×Dic3SD1614yescomplemented
158C2×C3×Dic3C2×D412yes—
159C2×C3×Dic3C2×D412yes—
160C3×Dic6—22no—
161C3×Dic6—22no—
162C3×Dic6—42no—
163C32×C8—22no—
164C3×C3⋊C8—22no—
165C22×SD16—31no—
166C22×SD16C3×D311yescharacteristic
167C2×C4⋊Q8—31no—
168C2×C4×D4—31no—
169#461—38no—
170C2×C4⋊C8—31no—
171C2×#103—31no—
172C2×#103—31no—
173C24×C3—21no—
174C24×C3—21no—
175C24×C3—41no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—24no—
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—41no—
186C2×C3×Q8—42no—
187C2×C3×Q8—61no—
188C2×C3×Q8—62no—
189C2×C3×Q8—62no—
190C2×C3×Q8C22×D312yes—
191C2×C3×Q8C23×C312yes—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—24no—
197C2×C3×D4—22no—
198C2×C3×D4—41no—
199C2×C3×D4—42no—
200C2×C3×D4#6714yes—
201C2×C3×D4C3×D414yes—
202C2×C3×D4C22×D312yes—
203C2×C3×D4C23×C312yes—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—41no—
210C22×C3×C4—64no—
211C22×C3×C4—61no—
212C22×C3×C4—61no—
213C22×C3×C4—124no—
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C23×C311yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—24no—
219C22×Dic3—44no—
220C22×Dic3C3×D411yescharacteristic
221C22×Dic3C3×D411yescharacteristic
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—21no—
225C2×Dic6C23×C312yes—
226C3×SD16—216no—
227C3×SD16—28no—
228C3×SD16—28no—
229C3×SD16—68no—
230C2×C3×C8—24no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—32no—
235C2×C3×C8—62no—
236C2×C3×C8C22×D312yes—
237C2×C3×C8C23×C312yes—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—64no—
240C3×C4⋊C4—64no—
241C3×C4⋊C4—61no—
242C3×C22⋊C4—64no—
243C3×C42—32no—
244C3×C42—32no—
245#217—24no—
246#214—28no—
247C4⋊Dic3—21no—
248C4⋊Dic3C23×C312yes—
249Dic3⋊C4—24no—
250Dic3⋊C4—24no—
251C4×Dic3C3×D412yes—
252C4×Dic3C23×C312yes—
253C2×C3⋊C8—22no—
254C2×C3⋊C8C23×C312yes—
255C22×C32—24no—
256C22×C32—42no—
257C22×C32—42no—
258C22×C32—48no—
259C22×C32#12112yes—
260C22×C32C2×SD1612yes—
261C22×C32#13712yes—
262C22×C32C22×D411yescharacteristic
263C32×C4—42no—
264C32×C4#12112yes—
265C32×C4C22×D411yescharacteristic
266C32×C4C2×#4111yescharacteristic
267C3×Dic3—22no—
268C3×Dic3—42no—
269C3×Dic3—42no—
270C3×Dic3C2×SD1614yescomplemented
271C22×Q8—31no—
272C22×Q8C2×C3×D311yescharacteristic
273C22×D4C2×C3×D311yescharacteristic
274C23×C4—61no—
275C2×SD16—24no—
276C2×SD16—34no—
277C2×SD16—64no—
278C2×SD16C2×C3×D314yes—
279C22×C8—31no—
280C22×C8C2×C3×D311yescharacteristic
281C4⋊Q8—34no—
282C4⋊Q8—62no—
283C4×D4—34no—
284C4×D4—62no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C42—31no—
290C4⋊C8—34no—
291#103—34no—
292#103—34no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—44no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—82no—
305C23×C3—82no—
306C23×C3D3×D411yescharacteristic
307C23×C3C2×C3×D411yescharacteristic
308C3×Q8—24no—
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—42no—
314C3×Q8—44no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8—62no—
318C3×Q8—62no—
319C3×Q8—122no—
320C3×D4—24no—
321C3×D4—24no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—24no—
325C3×D4—44no—
326C3×D4—42no—
327C3×D4C2×#6714yes—
328C3×D4C2×C3×D414yes—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—34no—
336C2×C3×C4—32no—
337C2×C3×C4—32no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—62no—
345C2×C3×C4—128no—
346C2×C3×C4—122no—
347C2×C3×C4—122no—
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4#23712yes—
350C2×C3×C4C2×#6712yes—
351C2×C3×C4C2×C3×D411yescharacteristic
352C2×C3×C4C2×C3×D412yes—
353C2×C3×C4C3×#4112yes—
354C2×C3×C4C23×D311yescharacteristic
355C2×C3×C4C24×C311yescharacteristic
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—48no—
362C2×Dic3C3×SD1614yescomplemented
363C2×Dic3C2×C3×D412yes—
364C2×Dic3C2×C3×D412yes—
365Dic6—22no—
366Dic6—22no—
367Dic6—42no—
368C3×C8—24no—
369C3×C8—22no—
370C3×C8—22no—
371C3×C8—62no—
372C3⋊C8—22no—
373C2×C32—44no—
374C2×C32#46114yescomplemented
375C2×C32C2×#12111yescharacteristic
376C2×C32C22×SD1611yescharacteristic
377C2×C32C2×#13711yescharacteristic
378C24—21no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—32no—
383C2×Q8—61no—
384C2×Q8—62no—
385C2×Q8—62no—
386C2×Q8C22×C3×D312yes—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4C3×#6714yes—
390C2×D4C22×C3×D312yes—
391C22×C4—21no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—64no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—124no—
398C22×C4C22×C3×D311yescharacteristic
399SD16—28no—
400SD16—68no—
401C2×C8—22no—
402C2×C8—32no—
403C2×C8—62no—
404C2×C8C22×C3×D312yes—
405C4⋊C4—32no—
406C4⋊C4—64no—
407C4⋊C4—64no—
408C4⋊C4—61no—
409C22⋊C4—64no—
410C42—32no—
411C42—32no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—44no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—48no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—48no—
425C22×C3—88no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3D3×SD1612yes—
429C22×C3#91512yes—
430C22×C3#93912yes—
431C22×C3C3×#12112yes—
432C22×C3C2×C3×SD1612yes—
433C22×C3C3×#13712yes—
434C22×C3C2×D3×D411yescharacteristic
435C22×C3C22×C3×D411yescharacteristic
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—34no—
440C3×C4—44no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—62no—
444C3×C4—122no—
445C3×C4—122no—
446C3×C4#93912yes—
447C3×C4C3×#12112yes—
448C3×C4C2×#23711yescharacteristic
449C3×C4C22×#6711yescharacteristic
450C3×C4C22×C3×D411yescharacteristic
451C3×C4C2×C3×#4111yescharacteristic
452Dic3—22no—
453Dic3—42no—
454Dic3—42no—
455Dic3C2×C3×SD1614yescomplemented
456C32C2×#46111yescharacteristic, complemented, Sylow
457C23—24no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23C3×D3×D411yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—42no—
465Q8—62no—
466Q8—62no—
467Q8—122no—
468D4—24no—
469D4—22no—
470D4C2×C3×#6714yes—
471C2×C4—22no—
472C2×C4—34no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—42no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—128no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4C3×D3×D411yescharacteristic
483C2×C4C3×#23712yes—
484C2×C4C2×C3×#6712yes—
485C2×C4C23×C3×D311yescharacteristic
486C8—22no—
487C8—62no—
488C2×C3—24no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—44no—
493C2×C3—44no—
494C2×C3—84no—
495C2×C3#516214yesdirect product factor
496C2×C3C3×#46114yescomplemented
497C2×C3C2×D3×SD1611yescharacteristic
498C2×C3C2×#91511yescharacteristic
499C2×C3C2×#93911yescharacteristic
500C2×C3C2×C3×#12111yescharacteristic
501C2×C3C22×C3×SD1611yescharacteristic
502C2×C3C2×C3×#13711yescharacteristic
503C22—24no—
504C22—48no—
505C22—42no—
506C22—42no—
507C22C3×D3×SD1612yes—
508C22C3×#91512yes—
509C22C3×#93912yes—
510C22C2×C3×D3×D411yescharacteristic
511C4—34no—
512C4—42no—
513C4—62no—
514C4—122no—
515C4—122no—
516C4C3×#93912yes—
517C4C2×C3×#23711yescharacteristic
518C4C22×C3×#6711yescharacteristic
519C3—21no—
520C3C2×#516211yescharacteristic, direct product factor
521C3C2×C3×#46111yescharacteristic, complemented
522C2—44no—
523C2C3×#516214yesdirect product factor
524C2C2×C3×D3×SD1611yescharacteristic
525C2C2×C3×#91511yescharacteristic
526C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#461
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
2. C2×C3×#944C2semidirect
5. C2×C3×#835C2semidirect
6. C3×#5162C2direct (N × H)
7. C2×C3×#816C2semidirect
8. C2×C3×Dic3⋊C8C2semidirect
10. C2×#5162C3direct (N × H)
20. C3×#944C22semidirect
30. C3×#835C22semidirect
31. C3×#816C22semidirect
32. C3×Dic3⋊C8C22semidirect
39. C2×#944C2×C3semidirect
46. C2×#835C2×C3semidirect
47. #5162C2×C3direct (N × H)
48. C2×#816C2×C3semidirect
49. C2×Dic3⋊C8C2×C3semidirect
118. #944C22×C3semidirect
131. #835C22×C3semidirect
132. #816C22×C3semidirect
133. Dic3⋊C8C22×C3semidirect
157. C2×C3×Dic3SD16semidirect
270. C3×Dic3C2×SD16semidirect
362. C2×Dic3C3×SD16semidirect
374. C2×C32#461semidirect
455. Dic3C2×C3×SD16semidirect
456. C32C2×#461semidirect
495. C2×C3#5162direct (N × H)
496. C2×C3C3×#461semidirect
520. C3C2×#5162direct (N × H)
521. C3C2×C3×#461semidirect
523. C2C3×#5162direct (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×#9442
C2×C3×D4×Dic32
C22×C3×#2142
C2×C3×#8352
C3×#51622
C2×C3×#8162
C2×C3×Dic3⋊C82
C2×C3×#4613
C2×#51623