Finite Group Databasebeta

← C2×C3×#4780 #11866892

Subgroups — C2×C3×#4780 #11866892

Order 1152.

1189 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#938C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic, complemented
3C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C2×C3×#897C211yesmaximal, characteristic, complemented
5C2×C32×#102C211yesmaximal, characteristic, complemented
6C2×C3×#816C211yesmaximal, characteristic, complemented
7C2×C3×#807C211yesmaximal, characteristic, complemented
8C3×#4780C218yesmaximal, direct product factor
9C2×C3×#468—31nomaximal
10C2×#4780C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×C4×D3C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C3×#217—21no—
18C3×#938—22no—
19C3×#938C2214yes—
20C2×C3×#214—24no—
21C2×C3×#214C2214yescomplemented
22C2×C3×(C2×D3)⋊C4—21no—
23C2×C3×C4⋊Dic3—21no—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C2×C3×ΓL(1,𝔽25)—24no—
27C2×C3×ΓL(1,𝔽25)C2214yescomplemented
28C3×#897—22no—
29C3×#897C2214yescomplemented
30C32×#102C2214yescomplemented
31C3×#816C2214yescomplemented
32C3×#807C2214yescomplemented
33C22×C3×SD16—31no—
34C22×C3×M16—31no—
35C2×C3×C22⋊Q8—31no—
36C2×C3×#121—31no—
37C2×#938C2×C311yescharacteristic
38C22×#214C2×C311yescharacteristic, complemented
39C22×ΓL(1,𝔽25)C2×C311yescharacteristic, complemented
40C2×#897C2×C311yescharacteristic, complemented
41C3×#468—38no—
42C2×C3×#107—31no—
43C2×C3×#103—31no—
44C2×C3×#102—21no—
45C2×C3×#102D311yescharacteristic, complemented
46C2×C3×#102C2×C311yescharacteristic, complemented
47C2×#816C2×C311yescharacteristic, complemented
48C2×#807C2×C311yescharacteristic, complemented
49#4780C2×C318yesdirect product factor
50C24×C32—21no—
51C23×C3×D3D411yescharacteristic
52C2×C32×D4—22no—
53C2×C32×D4—21no—
54C2×C32×D4—22no—
55C2×C32×D4C2312yes—
56C22×C32×C4—21no—
57C22×C32×C4C2311yescharacteristic
58C2×C3×#67—24no—
59C2×C3×#67—44no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3D411yescharacteristic
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3—22no—
65C2×C3×C4×D3D412yes—
66C2×C3×C4×D3C2312yes—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C2×C32×C8—22no—
72C2×C32×C8C2312yes—
73C32×C4⋊C4—21no—
74C32×C4⋊C4C2312yes—
75C3×#217—24no—
76C3×#214—28no—
77C3×(C2×D3)⋊C4—24no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—24no—
80C3×C4⋊Dic3C2312yes—
81C2×C3×C3⋊C8D414yescomplemented
82C2×C3×C3⋊C8C2312yes—
83C3×ΓL(1,𝔽25)—28no—
84C2×#468—31noSylow
85C22×C3×Q8—31no—
86C22×C3×D4—21no—
87C22×C3×D4—61no—
88C22×C3×D4C2×D311yescharacteristic
89C22×C3×D4C22×C311yescharacteristic
90C23×C3×C4—31no—
91C22×#67—21no—
92C22×C4×D3C22×C311yescharacteristic
93C22×Dic6C22×C311yescharacteristic
94C2×C3×SD16—34no—
95C2×C3×SD16—64no—
96C2×C3×M16—34no—
97C2×C3×M16—64no—
98C22×C3×C8—21no—
99C22×C3×C8—31no—
100C22×C3×C8C2×D311yescharacteristic
101C22×C3×C8C22×C311yescharacteristic
102C3×C22⋊Q8—34no—
103C3×C22⋊Q8—62no—
104C3×#121—34no—
105C3×#121—62no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—61no—
109C2×C3×C4⋊C4C2×D311yescharacteristic
110C2×C3×C4⋊C4C22×C311yescharacteristic
111C2×C3×C22⋊C4—61no—
112C2×C3×C22⋊C4—61no—
113C2×#217—21no—
114#938—22no—
115#938C22×C314yes—
116C2×#214—24no—
117C2×#214C22×C314yescomplemented
118C2×(C2×D3)⋊C4—21no—
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3C22×C311yescharacteristic
121C22×C3⋊C8C22×C311yescharacteristic
122C2×ΓL(1,𝔽25)—24no—
123C2×ΓL(1,𝔽25)C22×C314yescomplemented
124#897—22no—
125#897C22×C314yescomplemented
126C3×#107—34no—
127C3×#103—34no—
128C3×#102—24no—
129C3×#102C2×D314yescomplemented
130C3×#102C22×C314yescomplemented
131#816C22×C314yescomplemented
132#807C22×C314yescomplemented
133C23×C32—22no—
134C23×C32—42no—
135C23×C32—42no—
136C23×C32—42no—
137C23×C32C2×D411yescharacteristic
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3C2×D412yes—
142C32×D4—22no—
143C32×D4—22no—
144C32×D4—42no—
145C2×C32×C4—22no—
146C2×C32×C4—42no—
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D412yes—
149C2×C32×C4#4112yes—
150C2×C32×C4C2411yescharacteristic
151C3×#67—48no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3—42no—
157C2×C3×Dic3C2×D412yes—
158C3×C4×D3—24no—
159C3×C4×D3—22no—
160C3×C4×D3—22no—
161C3×Dic6—22no—
162C3×Dic6—22no—
163C3×Dic6—42no—
164C32×C8—22no—
165C3×C3⋊C8C2×D414yescomplemented
166C22×SD16—31no—
167C22×M16—31no—
168C2×C22⋊Q8—31no—
169C2×#121—31no—
170#468—38no—
171C2×#107—31no—
172C2×#103—31no—
173C2×#102C3×D311yescharacteristic, complemented
174C24×C3—21no—
175C24×C3—21no—
176C24×C3—31no—
177C24×C3—41no—
178C23×D3C3×D411yescharacteristic
179C2×C3×Q8—32no—
180C2×C3×Q8—61no—
181C2×C3×Q8—62no—
182C2×C3×Q8—62no—
183C2×C3×D4—24no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—42no—
192C2×C3×D4—41no—
193C2×C3×D4—64no—
194C2×C3×D4—124no—
195C2×C3×D4C22×D312yes—
196C2×C3×D4C23×C312yes—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—31no—
201C22×C3×C4—32no—
202C22×C3×C4—32no—
203C22×C3×C4—41no—
204C22×C3×C4—62no—
205C22×C3×C4—62no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C23×C311yescharacteristic
210C2×#67—24no—
211C2×#67—44no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C3×D411yescharacteristic
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3C3×D412yes—
218C2×C4×D3C23×C312yes—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—21no—
222C2×Dic6C23×C312yes—
223C3×SD16—68no—
224C3×M16—68no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—34no—
229C2×C3×C8—32no—
230C2×C3×C8—42no—
231C2×C3×C8C22×D312yes—
232C2×C3×C8C23×C312yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—64no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4C22×D312yes—
241C3×C4⋊C4C23×C312yes—
242C3×C22⋊C4—64no—
243C3×C22⋊C4—64no—
244#217—24no—
245#214—28no—
246(C2×D3)⋊C4—24no—
247C4⋊Dic3—21no—
248C4⋊Dic3—24no—
249C4⋊Dic3C23×C312yes—
250C2×C3⋊C8C3×D414yescomplemented
251C2×C3⋊C8C23×C312yes—
252ΓL(1,𝔽25)—28no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—84no—
257C22×C32#12112yes—
258C22×C32AGL(1,ℤ/8)12yes—
259C22×C32#13712yes—
260C22×C32C22×D411yescharacteristic
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C2×C3×D3—44no—
265C32×C4—42no—
266C32×C4#12112yes—
267C32×C4C22×D411yescharacteristic
268C32×C4C2×#4111yescharacteristic
269C3×Dic3—22no—
270C3×Dic3—42no—
271C3×Dic3—42no—
272C22×Q8—31no—
273C22×D4—61no—
274C22×D4C2×C3×D311yescharacteristic
275C23×C4—31no—
276C2×SD16—34no—
277C2×SD16—64no—
278C2×M16—34no—
279C2×M16—64no—
280C22×C8—31no—
281C22×C8C2×C3×D311yescharacteristic
282C22⋊Q8—34no—
283C22⋊Q8—62no—
284#121—34no—
285#121—62no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C2×C3×D311yescharacteristic
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291#107—34no—
292#103—34no—
293#102C2×C3×D314yescomplemented
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—32no—
298C23×C3—42no—
299C23×C3—44no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—62no—
307C23×C3—62no—
308C23×C3—62no—
309C23×C3—82no—
310C23×C3—82no—
311C23×C3D3×D411yescharacteristic
312C23×C3C2×C3×D411yescharacteristic
313C22×D3—22no—
314C22×D3—22no—
315C22×D3—22no—
316C22×D3C2×C3×D412yes—
317C3×Q8—62no—
318C3×Q8—62no—
319C3×Q8—122no—
320C3×D4—24no—
321C3×D4—22no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—22no—
325C3×D4—42no—
326C3×D4—44no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—128no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—32no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—62no—
341C2×C3×C4—64no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—82no—
347C2×C3×C4—122no—
348C2×C3×C4—122no—
349C2×C3×C4#23512yes—
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4D3×D412yes—
352C2×C3×C4C2×C3×D411yescharacteristic
353C2×C3×C4C2×C3×D412yes—
354C2×C3×C4C3×#4112yes—
355C2×C3×C4C23×D311yescharacteristic
356C2×C3×C4C24×C311yescharacteristic
357#67—48no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3C2×C3×D412yes—
364C4×D3—24no—
365C4×D3—22no—
366C4×D3—22no—
367Dic6—22no—
368Dic6—22no—
369Dic6—42no—
370C3×C8—22no—
371C3×C8—22no—
372C3×C8—34no—
373C3×C8—42no—
374C3⋊C8C2×C3×D414yescomplemented
375C2×C32—82no—
376C2×C32#46814yescomplemented
377C2×C32C2×#12111yescharacteristic
378C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
379C2×C32C2×#13711yescharacteristic
380C3×D3—42no—
381C24—21no—
382C24—31no—
383C2×Q8—32no—
384C2×Q8—61no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—64no—
391C2×D4—124no—
392C2×D4C22×C3×D312yes—
393C22×C4—21no—
394C22×C4—31no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4C22×C3×D311yescharacteristic
402SD16—68no—
403M16—68no—
404C2×C8—22no—
405C2×C8—32no—
406C2×C8—34no—
407C2×C8C22×C3×D312yes—
408C4⋊C4—21no—
409C4⋊C4—32no—
410C4⋊C4—64no—
411C4⋊C4—61no—
412C4⋊C4C22×C3×D312yes—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—88no—
432C22×C3—84no—
433C22×C3—84no—
434C22×C3—124no—
435C22×C3#88412yes—
436C22×C3#91112yes—
437C22×C3#91512yes—
438C22×C3C3×#12112yes—
439C22×C3C3×AGL(1,ℤ/8)12yes—
440C22×C3C3×#13712yes—
441C22×C3C2×D3×D411yescharacteristic
442C22×C3C22×C3×D411yescharacteristic
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—22no—
446C2×D3—44no—
447C3×C4—22no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—62no—
453C3×C4—82no—
454C3×C4—122no—
455C3×C4—122no—
456C3×C4#88412yes—
457C3×C4C3×#12112yes—
458C3×C4C2×#23511yescharacteristic
459C3×C4C2×D3×D411yescharacteristic
460C3×C4C22×C3×D411yescharacteristic
461C3×C4C2×C3×#4111yescharacteristic
462Dic3—22no—
463Dic3—42no—
464Dic3—42no—
465C32C2×#46811yescharacteristic, complemented, Sylow
466C23—22no—
467C23—32no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—62no—
472C23—62no—
473C23—62no—
474C23C3×D3×D411yescharacteristic
475Q8—62no—
476Q8—62no—
477Q8—122no—
478D4—22no—
479D4—22no—
480D4—42no—
481D4—128no—
482C2×C4—22no—
483C2×C4—32no—
484C2×C4—42no—
485C2×C4—62no—
486C2×C4—64no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4C3×#23512yes—
494C2×C4C3×D3×D411yescharacteristic
495C2×C4C3×D3×D412yes—
496C2×C4C23×C3×D311yescharacteristic
497C8—22no—
498C8—34no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—24no—
503C2×C3—84no—
504C2×C3—82no—
505C2×C3—82no—
506C2×C3—122no—
507C2×C3#478014yesdirect product factor
508C2×C3C3×#46814yescomplemented
509C2×C3C2×#88411yescharacteristic
510C2×C3C2×#91111yescharacteristic
511C2×C3C2×#91511yescharacteristic
512C2×C3C2×C3×#12111yescharacteristic
513C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
514C2×C3C2×C3×#13711yescharacteristic
515D3—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—62no—
520C22—62no—
521C22—62no—
522C22—84no—
523C22—124no—
524C22C3×#88412yes—
525C22C3×#91112yes—
526C22C3×#91512yes—
527C22C2×C3×D3×D411yescharacteristic
528C4—42no—
529C4—62no—
530C4—122no—
531C4—122no—
532C4C3×#88412yes—
533C4C2×C3×#23511yescharacteristic
534C4C2×C3×D3×D411yescharacteristic
535C3—21no—
536C3C2×#478011yescharacteristic, direct product factor
537C3C2×C3×#46811yescharacteristic, complemented
538C2—82no—
539C2—122no—
540C2C3×#478014yesdirect product factor
541C2C2×C3×#88411yescharacteristic
542C2C2×C3×#91111yescharacteristic
543C2C2×C3×#91511yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC2×C3×#938C2
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×#468
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
2. C22×C3×#214C2semidirect
3. C22×C3×ΓL(1,𝔽25)C2semidirect
4. C2×C3×#897C2semidirect
5. C2×C32×#102C2semidirect
6. C2×C3×#816C2semidirect
7. C2×C3×#807C2semidirect
8. C3×#4780C2direct (N × H)
10. C2×#4780C3direct (N × H)
21. C2×C3×#214C22semidirect
27. C2×C3×ΓL(1,𝔽25)C22semidirect
29. C3×#897C22semidirect
30. C32×#102C22semidirect
31. C3×#816C22semidirect
32. C3×#807C22semidirect
38. C22×#214C2×C3semidirect
39. C22×ΓL(1,𝔽25)C2×C3semidirect
40. C2×#897C2×C3semidirect
45. C2×C3×#102D3semidirect
46. C2×C3×#102C2×C3semidirect
47. C2×#816C2×C3semidirect
48. C2×#807C2×C3semidirect
49. #4780C2×C3direct (N × H)
81. C2×C3×C3⋊C8D4semidirect
117. C2×#214C22×C3semidirect
123. C2×ΓL(1,𝔽25)C22×C3semidirect
125. #897C22×C3semidirect
129. C3×#102C2×D3semidirect
130. C3×#102C22×C3semidirect
131. #816C22×C3semidirect
132. #807C22×C3semidirect
165. C3×C3⋊C8C2×D4semidirect
173. C2×#102C3×D3semidirect
250. C2×C3⋊C8C3×D4semidirect
293. #102C2×C3×D3semidirect
374. C3⋊C8C2×C3×D4semidirect
376. C2×C32#468semidirect
465. C32C2×#468semidirect
507. C2×C3#4780direct (N × H)
508. C2×C3C3×#468semidirect
536. C3C2×#4780direct (N × H)
537. C3C2×C3×#468semidirect
540. C2C3×#4780direct (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×#2142
C22×C3×ΓL(1,𝔽25)2
C2×C3×#8972
C2×C32×#1022
C2×C3×#8162
C2×C3×#8072
C3×#47802
C2×C3×#4683
C2×#47803