Finite Group Databasebeta

← C2×C3×#5153 #11865516

Subgroups — C2×C3×#5153 #11865516

Order 1152.

1297 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D8C211yesmaximal, characteristic
2C2×C3×#936C211yesmaximal, characteristic, complemented
3C2×C3×D4×Dic3C211yesmaximal, characteristic, complemented
4C22×C3×#214C211yesmaximal, characteristic, complemented
5C2×C3×#832C211yesmaximal, characteristic, complemented
6C3×#5153C218yesmaximal, direct product factor
7C2×C3×#816C211yesmaximal, characteristic, complemented
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×#463—31nomaximal
10C2×#5153C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C23×C3×Dic3—21no—
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×D8—24no—
16C2×C32×D8C2214yes—
17C22×C32×C8C2211yescharacteristic
18C2×C3×#217—21no—
19C2×C3×#217—21no—
20C2×C3×#217—21no—
21C3×#936—22no—
22C3×#936C2214yescomplemented
23C3×D4×Dic3—22no—
24C3×D4×Dic3C2214yescomplemented
25C2×C3×#214—24no—
26C2×C3×#214C2214yescomplemented
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#832C2214yescomplemented
31C3×#816C2214yescomplemented
32C3×Dic3⋊C8C2214yescomplemented
33C22×C3×SD16—31no—
34C22×C3×D8—21no—
35C22×C3×D8D311yescharacteristic
36C22×C3×D8C2×C311yescharacteristic
37C2×C3×#124—31no—
38C2×C3×C4×D4—31no—
39C2×#936C2×C311yescharacteristic, complemented
40C2×D4×Dic3C2×C311yescharacteristic, complemented
41C22×#214C2×C311yescharacteristic, complemented
42C3×#463—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46C2×#832C2×C311yescharacteristic, complemented
47#5153C2×C318yesdirect product factor
48C2×#816C2×C311yescharacteristic, complemented
49C2×Dic3⋊C8C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C24×C32—21no—
52C2×C32×D4—21no—
53C2×C32×D4—22no—
54C2×C32×D4—21no—
55C2×C32×D4—22no—
56C2×C32×D4—22no—
57C2×C32×D4D414yes—
58C2×C32×D4C2312yes—
59C2×C32×D4C2312yes—
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×D8—28no—
72C2×C32×C8—22no—
73C2×C32×C8C2312yes—
74C3×#217—24no—
75C3×#217—24no—
76C3×#217—24no—
77C3×#214—28no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3C2312yes—
80C3×C4×Dic3D412yes—
81C3×C4×Dic3C2312yes—
82C2×C3×C3⋊C8—22no—
83C2×C3×C3⋊C8C2312yes—
84C2×#463—31noSylow
85C22×C3×Q8—31no—
86C22×C3×D4—21no—
87C22×C3×D4—21no—
88C22×C3×D4C2×D311yescharacteristic
89C22×C3×D4C2×D311yescharacteristic
90C22×C3×D4C22×C311yescharacteristic
91C22×C3×D4C22×C311yescharacteristic
92C23×C3×C4—61no—
93C23×Dic3—21no—
94C22×Dic6C22×C311yescharacteristic
95C2×C3×SD16—34no—
96C2×C3×SD16—64no—
97C2×C3×D8—28no—
98C2×C3×D8—24no—
99C2×C3×D8—24no—
100C2×C3×D8—24no—
101C2×C3×D8C2×D314yes—
102C2×C3×D8C22×C314yes—
103C22×C3×C8—21no—
104C22×C3×C8—31no—
105C22×C3×C8C2×D311yescharacteristic
106C22×C3×C8C22×C311yescharacteristic
107C3×#124—34no—
108C3×#124—62no—
109C3×C4×D4—34no—
110C3×C4×D4—62no—
111C2×C3×C4⋊C4—31no—
112C2×C3×C22⋊C4—61no—
113C2×C3×C22⋊C4—61no—
114C2×C3×C22⋊C4—61no—
115C2×C3×C42—31no—
116C2×#217—21no—
117C2×#217—21no—
118C2×#217—21no—
119#936—22no—
120#936C22×C314yescomplemented
121D4×Dic3—22no—
122D4×Dic3C22×C314yescomplemented
123C2×#214—24no—
124C2×#214C22×C314yescomplemented
125C2×C4⋊Dic3C22×C311yescharacteristic
126C2×C4×Dic3C22×C311yescharacteristic
127C22×C3⋊C8C22×C311yescharacteristic
128C3×C4⋊C8—34no—
129C3×#103—34no—
130C3×#102—34no—
131#832C22×C314yescomplemented
132#816C22×C314yescomplemented
133Dic3⋊C8C22×C314yescomplemented
134C23×C32—24no—
135C23×C32—22no—
136C23×C32—22no—
137C23×C32—42no—
138C23×C32—42no—
139C23×C32—42no—
140C23×C32—42no—
141C23×C32—42no—
142C23×C32C2×D411yescharacteristic
143C32×D4—22no—
144C32×D4—24no—
145C32×D4—22no—
146C32×D4—22no—
147C32×D4—42no—
148C32×D4C2×D414yes—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D412yes—
151C2×C32×C4#4112yes—
152C2×C32×C4C2411yescharacteristic
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—48no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3C2×D412yes—
161C2×C3×Dic3C2×D412yes—
162C3×Dic6—22no—
163C3×Dic6—22no—
164C3×Dic6—42no—
165C32×C8—22no—
166C3×C3⋊C8—22no—
167C22×SD16—31no—
168C22×D8C3×D311yescharacteristic
169C2×#124—31no—
170C2×C4×D4—31no—
171#463—38no—
172C2×C4⋊C8—31no—
173C2×#103—31no—
174C2×#102—31no—
175C24×C3—21no—
176C24×C3—21no—
177C24×C3—21no—
178C24×C3—21no—
179C24×C3—41no—
180C24×C3—41no—
181C2×C3×Q8—32no—
182C2×C3×Q8—61no—
183C2×C3×Q8—62no—
184C2×C3×Q8—62no—
185C2×C3×D4—24no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—24no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4—42no—
202C2×C3×D4—42no—
203C2×C3×D4#6714yes—
204C2×C3×D4C3×D414yes—
205C2×C3×D4C22×D312yes—
206C2×C3×D4C22×D312yes—
207C2×C3×D4C23×C312yes—
208C2×C3×D4C23×C312yes—
209C22×C3×C4—21no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—64no—
213C22×C3×C4—61no—
214C22×C3×C4—61no—
215C22×C3×C4—124no—
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C23×C311yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—24no—
221C22×Dic3—44no—
222C22×Dic3C3×D411yescharacteristic
223C22×Dic3C3×D411yescharacteristic
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—21no—
227C2×Dic6C23×C312yes—
228C3×SD16—68no—
229C3×D8—216no—
230C3×D8—28no—
231C3×D8—28no—
232C2×C3×C8—24no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—32no—
237C2×C3×C8—62no—
238C2×C3×C8C22×D312yes—
239C2×C3×C8C23×C312yes—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—61no—
242C3×C22⋊C4—64no—
243C3×C22⋊C4—64no—
244C3×C22⋊C4—64no—
245C3×C42—32no—
246C3×C42—32no—
247#217—24no—
248#217—24no—
249#217—24no—
250#214—28no—
251C4⋊Dic3—21no—
252C4⋊Dic3C23×C312yes—
253C4×Dic3C3×D412yes—
254C4×Dic3C23×C312yes—
255C2×C3⋊C8—22no—
256C2×C3⋊C8C23×C312yes—
257C22×C32—24no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32—42no—
261C22×C32—48no—
262C22×C32—42no—
263C22×C32—42no—
264C22×C32—84no—
265C22×C32#12112yes—
266C22×C32#13512yes—
267C22×C32AGL(1,ℤ/8)12yes—
268C22×C32C22×D411yescharacteristic
269C32×C4#12112yes—
270C32×C4C22×D411yescharacteristic
271C32×C4C2×#4111yescharacteristic
272C3×Dic3—22no—
273C3×Dic3—22no—
274C3×Dic3—42no—
275C3×Dic3—42no—
276C22×Q8—31no—
277C22×D4C2×C3×D311yescharacteristic
278C22×D4C2×C3×D311yescharacteristic
279C23×C4—61no—
280C2×SD16—34no—
281C2×SD16—64no—
282C2×D8—24no—
283C2×D8C2×C3×D314yes—
284C22×C8—31no—
285C22×C8C2×C3×D311yescharacteristic
286#124—34no—
287#124—62no—
288C4×D4—34no—
289C4×D4—62no—
290C2×C4⋊C4—31no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—31no—
295C4⋊C8—34no—
296#103—34no—
297#102—34no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—44no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—44no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—82no—
320C23×C3—82no—
321C23×C3—82no—
322C23×C3—82no—
323C23×C3D3×D411yescharacteristic
324C23×C3C2×C3×D411yescharacteristic
325C3×Q8—62no—
326C3×Q8—62no—
327C3×Q8—122no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—24no—
338C3×D4—44no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—44no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4C2×#6714yes—
345C3×D4C2×C3×D414yes—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—62no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—128no—
357C2×C3×C4—122no—
358C2×C3×C4—122no—
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4#23712yes—
361C2×C3×C4C2×#6712yes—
362C2×C3×C4C2×C3×D411yescharacteristic
363C2×C3×C4C2×C3×D412yes—
364C2×C3×C4C3×#4112yes—
365C2×C3×C4C23×D311yescharacteristic
366C2×C3×C4C24×C311yescharacteristic
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—48no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3C2×C3×D412yes—
375C2×Dic3C2×C3×D412yes—
376Dic6—22no—
377Dic6—22no—
378Dic6—42no—
379C3×C8—24no—
380C3×C8—22no—
381C3×C8—22no—
382C3×C8—62no—
383C3⋊C8—22no—
384C2×C32—44no—
385C2×C32—82no—
386C2×C32#46314yescomplemented
387C2×C32C2×#12111yescharacteristic
388C2×C32C2×#13511yescharacteristic
389C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
390C24—21no—
391C24—21no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—62no—
395C2×Q8—62no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4C3×#6714yes—
402C2×D4C22×C3×D312yes—
403C2×D4C22×C3×D312yes—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—64no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—124no—
410C22×C4C22×C3×D311yescharacteristic
411SD16—68no—
412D8—28no—
413C2×C8—22no—
414C2×C8—32no—
415C2×C8—62no—
416C2×C8C22×C3×D312yes—
417C4⋊C4—32no—
418C4⋊C4—61no—
419C22⋊C4—64no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C42—32no—
423C42—32no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—21no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—48no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—48no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—88no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—88no—
450C22×C3—84no—
451C22×C3—84no—
452C22×C3#91112yes—
453C22×C3#91212yes—
454C22×C3#93912yes—
455C22×C3C3×#12112yes—
456C22×C3C3×#13512yes—
457C22×C3C3×AGL(1,ℤ/8)12yes—
458C22×C3C2×D3×D411yescharacteristic
459C22×C3C22×C3×D411yescharacteristic
460C3×C4—22no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—62no—
464C3×C4—62no—
465C3×C4—122no—
466C3×C4—122no—
467C3×C4#93912yes—
468C3×C4C3×#12112yes—
469C3×C4C2×#23711yescharacteristic
470C3×C4C22×#6711yescharacteristic
471C3×C4C22×C3×D411yescharacteristic
472C3×C4C2×C3×#4111yescharacteristic
473Dic3—22no—
474Dic3—22no—
475Dic3—42no—
476Dic3—42no—
477C32C2×#46311yescharacteristic, complemented, Sylow
478C23—24no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C3×D3×D411yescharacteristic
487Q8—62no—
488Q8—62no—
489Q8—122no—
490D4—24no—
491D4—22no—
492D4—22no—
493D4—22no—
494D4—42no—
495D4C2×C3×#6714yes—
496C2×C4—32no—
497C2×C4—32no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—128no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4C3×D3×D411yescharacteristic
506C2×C4C3×#23712yes—
507C2×C4C2×C3×#6712yes—
508C2×C4C23×C3×D311yescharacteristic
509C8—22no—
510C8—62no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—24no—
515C2×C3—44no—
516C2×C3—44no—
517C2×C3—84no—
518C2×C3—84no—
519C2×C3—82no—
520C2×C3—82no—
521C2×C3#515314yesdirect product factor
522C2×C3C3×#46314yescomplemented
523C2×C3C2×#91111yescharacteristic
524C2×C3C2×#91211yescharacteristic
525C2×C3C2×#93911yescharacteristic
526C2×C3C2×C3×#12111yescharacteristic
527C2×C3C2×C3×#13511yescharacteristic
528C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
529C22—24no—
530C22—42no—
531C22—42no—
532C22—48no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22C3×#91112yes—
538C22C3×#91212yes—
539C22C3×#93912yes—
540C22C2×C3×D3×D411yescharacteristic
541C4—62no—
542C4—62no—
543C4—122no—
544C4—122no—
545C4C3×#93912yes—
546C4C2×C3×#23711yescharacteristic
547C4C22×C3×#6711yescharacteristic
548C3—21no—
549C3C2×#515311yescharacteristic, direct product factor
550C3C2×C3×#46311yescharacteristic, complemented
551C2—44no—
552C2—82no—
553C2C3×#515314yesdirect product factor
554C2C2×C3×#91111yescharacteristic
555C2C2×C3×#91211yescharacteristic
556C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC22×C32×D8C2
socleC23×C32C2×D4
hypercenterC22×C3×D8D3
hypocenterC3C2×C3×#463
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×D8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#936C2semidirect
3. C2×C3×D4×Dic3C2semidirect
4. C22×C3×#214C2semidirect
5. C2×C3×#832C2semidirect
6. C3×#5153C2direct (N × H)
7. C2×C3×#816C2semidirect
8. C2×C3×Dic3⋊C8C2semidirect
10. C2×#5153C3direct (N × H)
22. C3×#936C22semidirect
24. C3×D4×Dic3C22semidirect
26. C2×C3×#214C22semidirect
30. C3×#832C22semidirect
31. C3×#816C22semidirect
32. C3×Dic3⋊C8C22semidirect
39. C2×#936C2×C3semidirect
40. C2×D4×Dic3C2×C3semidirect
41. C22×#214C2×C3semidirect
46. C2×#832C2×C3semidirect
47. #5153C2×C3direct (N × H)
48. C2×#816C2×C3semidirect
49. C2×Dic3⋊C8C2×C3semidirect
120. #936C22×C3semidirect
122. D4×Dic3C22×C3semidirect
124. C2×#214C22×C3semidirect
131. #832C22×C3semidirect
132. #816C22×C3semidirect
133. Dic3⋊C8C22×C3semidirect
386. C2×C32#463semidirect
477. C32C2×#463semidirect
521. C2×C3#5153direct (N × H)
522. C2×C3C3×#463semidirect
549. C3C2×#5153direct (N × H)
550. C3C2×C3×#463semidirect
553. C2C3×#5153direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×D82
C2×C3×#9362
C2×C3×D4×Dic32
C22×C3×#2142
C2×C3×#8322
C3×#51532
C2×C3×#8162
C2×C3×Dic3⋊C82
C2×C3×#4633
C2×#51533