Finite Group Databasebeta

← C2×C3×#4782 #11866955

Subgroups — C2×C3×#4782 #11866955

Order 1152.

1293 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#938C211yesmaximal, characteristic, complemented
2C22×C3×#213C211yesmaximal, characteristic, complemented
3C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C2×C3×#895C211yesmaximal, characteristic, complemented
5C2×C32×#102C211yesmaximal, characteristic, complemented
6C2×C3×#821C211yesmaximal, characteristic, complemented
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C3×#4782C218yesmaximal, direct product factor
9C2×C3×#469—31nomaximal
10C2×#4782C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×D12—21no—
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C22×C32×C8C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C3×#217—21no—
19C3×#938—22no—
20C3×#938C2214yescomplemented
21C2×C3×#213—24no—
22C2×C3×#213C2214yescomplemented
23C2×C3×(C2×D3)⋊C4—21no—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C2×C3×ΓL(1,𝔽25)—24no—
27C2×C3×ΓL(1,𝔽25)C2214yescomplemented
28C3×#895—22no—
29C3×#895C2214yescomplemented
30C32×#102C2214yescomplemented
31C3×#821C2214yescomplemented
32C3×#808C2214yescomplemented
33C22×C3×D8—31no—
34C22×C3×M16—31no—
35C2×C3×#121—31no—
36C2×C3×#121—31no—
37C2×#938C2×C311yescharacteristic, complemented
38C22×#213C2×C311yescharacteristic, complemented
39C22×ΓL(1,𝔽25)C2×C311yescharacteristic, complemented
40C2×#895C2×C311yescharacteristic, complemented
41C3×#469—38no—
42C2×C3×#106—31no—
43C2×C3×#102—21no—
44C2×C3×#102—31no—
45C2×C3×#102D311yescharacteristic, complemented
46C2×C3×#102C2×C311yescharacteristic, complemented
47C2×#821C2×C311yescharacteristic, complemented
48C2×#808C2×C311yescharacteristic, complemented
49#4782C2×C318yesdirect product factor
50C24×C32—21no—
51C23×C3×D3—21no—
52C23×C3×D3D411yescharacteristic
53C2×C32×D4—22no—
54C2×C32×D4—21no—
55C2×C32×D4—22no—
56C2×C32×D4C2312yes—
57C22×C32×C4—21no—
58C22×C32×C4C2311yescharacteristic
59C2×C3×#67—24no—
60C2×C3×#67—44no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3D411yescharacteristic
63C2×C3×D12—22no—
64C2×C3×D12—22no—
65C2×C3×D12—24no—
66C2×C3×D12—21no—
67C2×C3×D12—44no—
68C2×C3×D12C2312yes—
69C2×C3×C4×D3—22no—
70C2×C3×C4×D3—22no—
71C2×C3×C4×D3D412yes—
72C2×C3×C4×D3C2312yes—
73C2×C32×C8—22no—
74C2×C32×C8C2312yes—
75C32×C4⋊C4—21no—
76C32×C4⋊C4C2312yes—
77C3×#217—24no—
78C3×#213—28no—
79C3×(C2×D3)⋊C4—24no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3C2312yes—
82C2×C3×C3⋊C8D414yescomplemented
83C2×C3×C3⋊C8C2312yes—
84C3×ΓL(1,𝔽25)—28no—
85C2×#469—31noSylow
86C22×C3×D4—21no—
87C22×C3×D4—31no—
88C22×C3×D4—61no—
89C22×C3×D4—61no—
90C22×C3×D4C2×D311yescharacteristic
91C22×C3×D4C22×C311yescharacteristic
92C23×C3×C4—31no—
93C22×#67—21no—
94C22×D12—21no—
95C22×D12C22×C311yescharacteristic
96C22×C4×D3C22×C311yescharacteristic
97C2×C3×D8—34no—
98C2×C3×D8—64no—
99C2×C3×M16—34no—
100C2×C3×M16—64no—
101C22×C3×C8—21no—
102C22×C3×C8—31no—
103C22×C3×C8C2×D311yescharacteristic
104C22×C3×C8C22×C311yescharacteristic
105C3×#121—34no—
106C3×#121—34no—
107C3×#121—62no—
108C3×#121—62no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4C2×D311yescharacteristic
112C2×C3×C4⋊C4C22×C311yescharacteristic
113C2×C3×C22⋊C4—61no—
114C2×C3×C22⋊C4—61no—
115C2×#217—21no—
116#938—22no—
117#938C22×C314yescomplemented
118C2×#213—24no—
119C2×#213C22×C314yescomplemented
120C2×(C2×D3)⋊C4—21no—
121C2×C4⋊Dic3C22×C311yescharacteristic
122C22×C3⋊C8C22×C311yescharacteristic
123C2×ΓL(1,𝔽25)—24no—
124C2×ΓL(1,𝔽25)C22×C314yescomplemented
125#895—22no—
126#895C22×C314yescomplemented
127C3×#106—34no—
128C3×#102—24no—
129C3×#102—34no—
130C3×#102C2×D314yescomplemented
131C3×#102C22×C314yescomplemented
132#821C22×C314yescomplemented
133#808C22×C314yescomplemented
134C23×C32—22no—
135C23×C32—42no—
136C23×C32—42no—
137C23×C32—42no—
138C23×C32C2×D411yescharacteristic
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—42no—
144C22×C3×D3—42no—
145C22×C3×D3—42no—
146C22×C3×D3C2×D412yes—
147C32×D4—22no—
148C32×D4—22no—
149C32×D4—42no—
150C2×C32×C4—22no—
151C2×C32×C4—42no—
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4C2×D412yes—
154C2×C32×C4#4112yes—
155C2×C32×C4C2411yescharacteristic
156C3×#67—48no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3C2×D412yes—
161C3×D12—22no—
162C3×D12—22no—
163C3×D12—48no—
164C3×D12—42no—
165C3×C4×D3—24no—
166C3×C4×D3—22no—
167C3×C4×D3—22no—
168C32×C8—22no—
169C3×C3⋊C8C2×D414yescomplemented
170C22×D8—31no—
171C22×M16—31no—
172C2×#121—31no—
173C2×#121—31no—
174#469—38no—
175C2×#106—31no—
176C2×#102—31no—
177C2×#102C3×D311yescharacteristic, complemented
178C24×C3—21no—
179C24×C3—21no—
180C24×C3—31no—
181C24×C3—41no—
182C24×C3—61no—
183C23×D3—21no—
184C23×D3C3×D411yescharacteristic
185C2×C3×D4—24no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—32no—
194C2×C3×D4—42no—
195C2×C3×D4—41no—
196C2×C3×D4—64no—
197C2×C3×D4—64no—
198C2×C3×D4—62no—
199C2×C3×D4—62no—
200C2×C3×D4—61no—
201C2×C3×D4—124no—
202C2×C3×D4—124no—
203C2×C3×D4C22×D312yes—
204C2×C3×D4C23×C312yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—31no—
209C22×C3×C4—32no—
210C22×C3×C4—32no—
211C22×C3×C4—41no—
212C22×C3×C4—62no—
213C22×C3×C4—62no—
214C22×C3×C4—61no—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C23×C311yescharacteristic
217C2×#67—24no—
218C2×#67—44no—
219C22×Dic3—21no—
220C22×Dic3C3×D411yescharacteristic
221C2×D12—22no—
222C2×D12—22no—
223C2×D12—24no—
224C2×D12—21no—
225C2×D12—44no—
226C2×D12C23×C312yes—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3C3×D412yes—
230C2×C4×D3C23×C312yes—
231C3×D8—68no—
232C3×M16—68no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—34no—
237C2×C3×C8—32no—
238C2×C3×C8—42no—
239C2×C3×C8C22×D312yes—
240C2×C3×C8C23×C312yes—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4C22×D312yes—
248C3×C4⋊C4C23×C312yes—
249C3×C22⋊C4—64no—
250C3×C22⋊C4—64no—
251#217—24no—
252#213—28no—
253(C2×D3)⋊C4—24no—
254C4⋊Dic3—21no—
255C4⋊Dic3C23×C312yes—
256C2×C3⋊C8C3×D414yescomplemented
257C2×C3⋊C8C23×C312yes—
258ΓL(1,𝔽25)—28no—
259C22×C32—42no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—84no—
263C22×C32#12112yes—
264C22×C32AGL(1,ℤ/8)12yes—
265C22×C32AGL(1,ℤ/8)12yes—
266C22×C32C22×D411yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—22no—
269C2×C3×D3—22no—
270C2×C3×D3—44no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3—42no—
274C2×C3×D3—84no—
275C32×C4—42no—
276C32×C4#12112yes—
277C32×C4C22×D411yescharacteristic
278C32×C4C2×#4111yescharacteristic
279C3×Dic3—22no—
280C3×Dic3—42no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C22×D4C2×C3×D311yescharacteristic
285C23×C4—31no—
286C2×D8—34no—
287C2×D8—64no—
288C2×M16—34no—
289C2×M16—64no—
290C22×C8—31no—
291C22×C8C2×C3×D311yescharacteristic
292#121—34no—
293#121—34no—
294#121—62no—
295#121—62no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4C2×C3×D311yescharacteristic
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300#106—34no—
301#102—34no—
302#102C2×C3×D314yescomplemented
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—21no—
306C23×C3—32no—
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—62no—
316C23×C3—62no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—82no—
320C23×C3—82no—
321C23×C3—122no—
322C23×C3—122no—
323C23×C3—122no—
324C23×C3D3×D411yescharacteristic
325C23×C3C2×C3×D411yescharacteristic
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3C2×C3×D412yes—
334C3×D4—24no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—42no—
340C3×D4—44no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—62no—
344C3×D4—62no—
345C3×D4—128no—
346C3×D4—128no—
347C3×D4—122no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—32no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—62no—
359C2×C3×C4—64no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—82no—
364C2×C3×C4—122no—
365C2×C3×C4#23512yes—
366C2×C3×C4D3×D411yescharacteristic
367C2×C3×C4D3×D412yes—
368C2×C3×C4C2×C3×D411yescharacteristic
369C2×C3×C4C2×C3×D412yes—
370C2×C3×C4C3×#4112yes—
371C2×C3×C4C23×D311yescharacteristic
372C2×C3×C4C24×C311yescharacteristic
373#67—48no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—42no—
377C2×Dic3C2×C3×D412yes—
378D12—22no—
379D12—22no—
380D12—48no—
381D12—42no—
382C4×D3—24no—
383C4×D3—22no—
384C4×D3—22no—
385C3×C8—22no—
386C3×C8—22no—
387C3×C8—34no—
388C3×C8—42no—
389C3⋊C8C2×C3×D414yescomplemented
390C2×C32—82no—
391C2×C32#46914yescomplemented
392C2×C32C2×#12111yescharacteristic
393C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
394C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
395C3×D3—42no—
396C3×D3—82no—
397C24—21no—
398C24—31no—
399C24—61no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—32no—
404C2×D4—64no—
405C2×D4—64no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—61no—
409C2×D4—124no—
410C2×D4—124no—
411C2×D4C22×C3×D312yes—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4C22×C3×D311yescharacteristic
420D8—68no—
421M16—68no—
422C2×C8—22no—
423C2×C8—32no—
424C2×C8—34no—
425C2×C8C22×C3×D312yes—
426C4⋊C4—21no—
427C4⋊C4—32no—
428C4⋊C4—61no—
429C4⋊C4C22×C3×D312yes—
430C22⋊C4—64no—
431C22⋊C4—64no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—44no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—62no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—88no—
449C22×C3—84no—
450C22×C3—84no—
451C22×C3—124no—
452C22×C3—122no—
453C22×C3—122no—
454C22×C3—122no—
455C22×C3—244no—
456C22×C3#88412yes—
457C22×C3#91112yes—
458C22×C3#91412yes—
459C22×C3C3×#12112yes—
460C22×C3C3×AGL(1,ℤ/8)12yes—
461C22×C3C3×AGL(1,ℤ/8)12yes—
462C22×C3C2×D3×D411yescharacteristic
463C22×C3C22×C3×D411yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—44no—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—84no—
472C3×C4—22no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—62no—
478C3×C4—82no—
479C3×C4—122no—
480C3×C4#88412yes—
481C3×C4C3×#12112yes—
482C3×C4C2×#23511yescharacteristic
483C3×C4C2×D3×D411yescharacteristic
484C3×C4C22×C3×D411yescharacteristic
485C3×C4C2×C3×#4111yescharacteristic
486Dic3—22no—
487Dic3—42no—
488C32C2×#46911yescharacteristic, complemented, Sylow
489C23—22no—
490C23—32no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—62no—
495C23—62no—
496C23—62no—
497C23—62no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23C3×D3×D411yescharacteristic
502D4—22no—
503D4—22no—
504D4—42no—
505D4—62no—
506D4—62no—
507D4—128no—
508D4—128no—
509D4—122no—
510C2×C4—22no—
511C2×C4—32no—
512C2×C4—42no—
513C2×C4—62no—
514C2×C4—64no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—122no—
519C2×C4C3×#23512yes—
520C2×C4C3×D3×D411yescharacteristic
521C2×C4C3×D3×D412yes—
522C2×C4C23×C3×D311yescharacteristic
523C8—22no—
524C8—34no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—24no—
529C2×C3—84no—
530C2×C3—82no—
531C2×C3—82no—
532C2×C3—122no—
533C2×C3—242no—
534C2×C3#478214yesdirect product factor
535C2×C3C3×#46914yescomplemented
536C2×C3C2×#88411yescharacteristic
537C2×C3C2×#91111yescharacteristic
538C2×C3C2×#91411yescharacteristic
539C2×C3C2×C3×#12111yescharacteristic
540C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
541C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
542D3—42no—
543D3—82no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—62no—
548C22—62no—
549C22—62no—
550C22—84no—
551C22—124no—
552C22—122no—
553C22—122no—
554C22—122no—
555C22—244no—
556C22C3×#88412yes—
557C22C3×#91112yes—
558C22C3×#91412yes—
559C22C2×C3×D3×D411yescharacteristic
560C4—42no—
561C4—62no—
562C4—122no—
563C4C3×#88412yes—
564C4C2×C3×#23511yescharacteristic
565C4C2×C3×D3×D411yescharacteristic
566C3—21no—
567C3C2×#478211yescharacteristic, direct product factor
568C3C2×C3×#46911yescharacteristic, complemented
569C2—82no—
570C2—122no—
571C2—242no—
572C2C3×#478214yesdirect product factor
573C2C2×C3×#88411yescharacteristic
574C2C2×C3×#91111yescharacteristic
575C2C2×C3×#91411yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×#469
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×C3×#938C2semidirect
2. C22×C3×#213C2semidirect
3. C22×C3×ΓL(1,𝔽25)C2semidirect
4. C2×C3×#895C2semidirect
5. C2×C32×#102C2semidirect
6. C2×C3×#821C2semidirect
7. C2×C3×#808C2semidirect
8. C3×#4782C2direct (N × H)
10. C2×#4782C3direct (N × H)
20. C3×#938C22semidirect
22. C2×C3×#213C22semidirect
27. C2×C3×ΓL(1,𝔽25)C22semidirect
29. C3×#895C22semidirect
30. C32×#102C22semidirect
31. C3×#821C22semidirect
32. C3×#808C22semidirect
37. C2×#938C2×C3semidirect
38. C22×#213C2×C3semidirect
39. C22×ΓL(1,𝔽25)C2×C3semidirect
40. C2×#895C2×C3semidirect
45. C2×C3×#102D3semidirect
46. C2×C3×#102C2×C3semidirect
47. C2×#821C2×C3semidirect
48. C2×#808C2×C3semidirect
49. #4782C2×C3direct (N × H)
82. C2×C3×C3⋊C8D4semidirect
117. #938C22×C3semidirect
119. C2×#213C22×C3semidirect
124. C2×ΓL(1,𝔽25)C22×C3semidirect
126. #895C22×C3semidirect
130. C3×#102C2×D3semidirect
131. C3×#102C22×C3semidirect
132. #821C22×C3semidirect
133. #808C22×C3semidirect
169. C3×C3⋊C8C2×D4semidirect
177. C2×#102C3×D3semidirect
256. C2×C3⋊C8C3×D4semidirect
302. #102C2×C3×D3semidirect
389. C3⋊C8C2×C3×D4semidirect
391. C2×C32#469semidirect
488. C32C2×#469semidirect
534. C2×C3#4782direct (N × H)
535. C2×C3C3×#469semidirect
567. C3C2×#4782direct (N × H)
568. C3C2×C3×#469semidirect
572. C2C3×#4782direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9382
C22×C3×#2132
C22×C3×ΓL(1,𝔽25)2
C2×C3×#8952
C2×C32×#1022
C2×C3×#8212
C2×C3×#8082
C3×#47822
C2×C3×#4693
C2×#47823