Finite Group Databasebeta

← C2×C3×#5134 #11866895

Subgroups — C2×C3×#5134 #11866895

Order 1152.

1237 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×M16C211yesmaximal, characteristic, complemented
2C2×C3×#930C211yesmaximal, characteristic
3C2×C3×#924C211yesmaximal, characteristic, complemented
4C22×C3×#204C211yesmaximal, characteristic, complemented
5C3×#5134C218yesmaximal, direct product factor
6C2×C3×#821C211yesmaximal, characteristic, complemented
7C2×C3×#818C211yesmaximal, characteristic, complemented
8C2×C3×#816C211yesmaximal, characteristic, complemented
9C2×C3×#468—31nomaximal
10C2×#5134C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×D12C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×M16—24no—
16C2×C32×M16C2214yescomplemented
17C22×C32×C8C2211yescharacteristic
18C22×C32×C8C2211yescharacteristic
19C2×C3×#217—21no—
20C3×#930—22no—
21C3×#930C2214yes—
22C2×C3×(C2×D3)⋊C4—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C3×#924—22no—
26C3×#924C2214yescomplemented
27C2×C3×Dic3⋊C4—21no—
28C2×C3×#204—24no—
29C2×C3×#204C2214yescomplemented
30C3×#821C2214yescomplemented
31C3×#818C2214yescomplemented
32C3×#816C2214yescomplemented
33C22×C3×SD16—31no—
34C22×C3×M16—21no—
35C22×C3×M16D311yescharacteristic, complemented
36C22×C3×M16C2×C311yescharacteristic, complemented
37C2×C3×C22⋊Q8—31no—
38C2×C3×#121—31no—
39C2×#930C2×C311yescharacteristic
40C2×#924C2×C311yescharacteristic, complemented
41C22×#204C2×C311yescharacteristic, complemented
42C3×#468—38no—
43C2×C3×#107—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46#5134C2×C318yesdirect product factor
47C2×#821C2×C311yescharacteristic, complemented
48C2×#818C2×C311yescharacteristic, complemented
49C2×#816C2×C311yescharacteristic, complemented
50C24×C32D411yescharacteristic
51C23×C3×D3—21no—
52C22×C32×C4—22no—
53C22×C32×C4—22no—
54C22×C32×C4D411yescharacteristic
55C22×C32×C4D412yes—
56C22×C32×C4C2312yes—
57C22×C32×C4C2311yescharacteristic
58C2×C3×#67—24no—
59C2×C3×#67—44no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C2×C3×D12—22no—
64C2×C3×D12—22no—
65C2×C3×D12—21no—
66C2×C3×D12C2312yes—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C32×M16—28no—
72C2×C32×C8—22no—
73C2×C32×C8D414yescomplemented
74C2×C32×C8C2312yes—
75C2×C32×C8C2312yes—
76C3×#217—24no—
77C3×(C2×D3)⋊C4—24no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3C2312yes—
81C3×C4⋊Dic3C2312yes—
82C3×Dic3⋊C4—24no—
83C3×#204—28no—
84C2×#468—31noSylow
85C22×C3×Q8—31no—
86C22×C3×D4—31no—
87C22×C3×D4—61no—
88C23×C3×C4—21no—
89C23×C3×C4C2×D311yescharacteristic
90C23×C3×C4C22×C311yescharacteristic
91C22×#67—21no—
92C22×D12C22×C311yescharacteristic
93C22×Dic6C22×C311yescharacteristic
94C2×C3×SD16—34no—
95C2×C3×SD16—64no—
96C2×C3×M16—28no—
97C2×C3×M16—24no—
98C2×C3×M16—24no—
99C2×C3×M16—24no—
100C2×C3×M16C2×D314yescomplemented
101C2×C3×M16C22×C314yescomplemented
102C22×C3×C8—21no—
103C22×C3×C8—21no—
104C22×C3×C8C2×D311yescharacteristic
105C22×C3×C8C2×D311yescharacteristic
106C22×C3×C8C22×C311yescharacteristic
107C22×C3×C8C22×C311yescharacteristic
108C3×C22⋊Q8—34no—
109C3×C22⋊Q8—62no—
110C3×#121—34no—
111C3×#121—62no—
112C2×C3×C4⋊C4—31no—
113C2×C3×C4⋊C4—31no—
114C2×C3×C4⋊C4—61no—
115C2×C3×C22⋊C4—61no—
116C2×C3×C22⋊C4—61no—
117C2×#217—21no—
118#930—22no—
119#930C22×C314yes—
120C2×(C2×D3)⋊C4—21no—
121C2×C4⋊Dic3C22×C311yescharacteristic
122C2×C4⋊Dic3C22×C311yescharacteristic
123#924—22no—
124#924C22×C314yescomplemented
125C2×Dic3⋊C4—21no—
126C2×#204—24no—
127C2×#204C22×C314yescomplemented
128C3×#107—34no—
129C3×#103—34no—
130C3×#102—34no—
131#821C22×C314yescomplemented
132#818C22×C314yescomplemented
133#816C22×C314yescomplemented
134C23×C32—22no—
135C23×C32—22no—
136C23×C32—22no—
137C23×C32C2×D412yes—
138C23×C32C2×D411yescharacteristic
139C22×C3×D3—22no—
140C22×C3×D3—42no—
141C22×C3×D3—42no—
142C22×C3×D3—42no—
143C2×C32×C4—22no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D412yes—
150C2×C32×C4#4112yes—
151C2×C32×C4C2411yescharacteristic
152C3×#67—48no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—42no—
157C2×C3×Dic3—42no—
158C2×C3×Dic3—42no—
159C3×D12—22no—
160C3×D12—22no—
161C3×D12—42no—
162C3×Dic6—22no—
163C3×Dic6—22no—
164C3×Dic6—42no—
165C32×C8—22no—
166C32×C8C2×D414yescomplemented
167C22×SD16—31no—
168C22×M16C3×D311yescharacteristic, complemented
169C2×C22⋊Q8—31no—
170C2×#121—31no—
171#468—38no—
172C2×#107—31no—
173C2×#103—31no—
174C2×#102—31no—
175C24×C3—21no—
176C24×C3—61no—
177C24×C3D1211yescharacteristic
178C24×C3C3×D411yescharacteristic
179C23×D3—21no—
180C2×C3×Q8—32no—
181C2×C3×Q8—61no—
182C2×C3×Q8—62no—
183C2×C3×Q8—62no—
184C2×C3×D4—32no—
185C2×C3×D4—64no—
186C2×C3×D4—61no—
187C2×C3×D4—62no—
188C2×C3×D4—62no—
189C2×C3×D4—124no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—61no—
201C22×C3×C4—61no—
202C22×C3×C4—61no—
203C22×C3×C4D1212yes—
204C22×C3×C4D1211yescharacteristic
205C22×C3×C4C3×D411yescharacteristic
206C22×C3×C4C3×D412yes—
207C22×C3×C4C22×D312yes—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C23×C312yes—
210C22×C3×C4C23×C311yescharacteristic
211C2×#67—24no—
212C2×#67—44no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C2×D12—22no—
217C2×D12—22no—
218C2×D12—21no—
219C2×D12C23×C312yes—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6C23×C312yes—
224C3×SD16—68no—
225C3×M16—216no—
226C3×M16—28no—
227C3×M16—28no—
228C2×C3×C8—24no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—24no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8#6714yescomplemented
235C2×C3×C8C3×D414yescomplemented
236C2×C3×C8C22×D312yes—
237C2×C3×C8C22×D312yes—
238C2×C3×C8C23×C312yes—
239C2×C3×C8C23×C312yes—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—64no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—61no—
245C3×C22⋊C4—64no—
246C3×C22⋊C4—64no—
247#217—24no—
248(C2×D3)⋊C4—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3C23×C312yes—
252C4⋊Dic3C23×C312yes—
253Dic3⋊C4—24no—
254#204—28no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—44no—
259C22×C32#12112yes—
260C22×C32AGL(1,ℤ/8)12yes—
261C22×C32#13712yes—
262C22×C32C22×D411yescharacteristic
263C2×C3×D3—42no—
264C2×C3×D3—42no—
265C2×C3×D3—42no—
266C2×C3×D3—84no—
267C32×C4—22no—
268C32×C4#12112yes—
269C32×C4C22×D411yescharacteristic
270C32×C4C2×#4111yescharacteristic
271C3×Dic3—42no—
272C3×Dic3—42no—
273C3×Dic3—42no—
274C22×Q8—31no—
275C22×D4—31no—
276C22×D4—61no—
277C23×C4C2×C3×D311yescharacteristic
278C2×SD16—34no—
279C2×SD16—64no—
280C2×M16—24no—
281C2×M16C2×C3×D314yescomplemented
282C22×C8C2×C3×D311yescharacteristic
283C22×C8C2×C3×D311yescharacteristic
284C22⋊Q8—34no—
285C22⋊Q8—62no—
286#121—34no—
287#121—62no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293#107—34no—
294#103—34no—
295#102—34no—
296C23×C3—24no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—22no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—62no—
308C23×C3—122no—
309C23×C3—122no—
310C23×C3—122no—
311C23×C3C2×D1212yes—
312C23×C3C2×D1211yescharacteristic
313C23×C3C2×C3×D412yes—
314C23×C3C2×C3×D411yescharacteristic
315C22×D3—22no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C3×Q8—62no—
320C3×Q8—62no—
321C3×Q8—122no—
322C3×D4—62no—
323C3×D4—62no—
324C3×D4—128no—
325C3×D4—122no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—28no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—62no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—122no—
347C2×C3×C4—122no—
348C2×C3×C4—122no—
349C2×C3×C4C2×D1212yes—
350C2×C3×C4C2×D1211yescharacteristic
351C2×C3×C4#23512yes—
352C2×C3×C4C2×#6712yes—
353C2×C3×C4C2×C3×D412yes—
354C2×C3×C4C2×C3×D411yescharacteristic
355C2×C3×C4C2×C3×D412yes—
356C2×C3×C4C3×#4112yes—
357C2×C3×C4C23×D311yescharacteristic
358C2×C3×C4C24×C311yescharacteristic
359#67—48no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366D12—22no—
367D12—22no—
368D12—42no—
369Dic6—22no—
370Dic6—22no—
371Dic6—42no—
372C3×C8—24no—
373C3×C8—22no—
374C3×C8—22no—
375C3×C8—24no—
376C3×C8C2×#6714yescomplemented
377C3×C8C2×C3×D414yescomplemented
378C2×C32—42no—
379C2×C32#46814yescomplemented
380C2×C32C2×#12111yescharacteristic
381C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
382C2×C32C2×#13711yescharacteristic
383C3×D3—82no—
384C24—61no—
385C24C3×D1211yescharacteristic
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×D4—32no—
391C2×D4—64no—
392C2×D4—61no—
393C2×D4—62no—
394C2×D4—62no—
395C2×D4—124no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4C3×D1211yescharacteristic
402C22×C4C3×D1212yes—
403C22×C4C22×C3×D312yes—
404C22×C4C22×C3×D311yescharacteristic
405SD16—68no—
406M16—28no—
407C2×C8—22no—
408C2×C8C3×#6714yescomplemented
409C2×C8C22×C3×D312yes—
410C2×C8C22×C3×D312yes—
411C4⋊C4—32no—
412C4⋊C4—32no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22×C3—24no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—48no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—122no—
435C22×C3—122no—
436C22×C3—122no—
437C22×C3—244no—
438C22×C3#90812yes—
439C22×C3#90912yes—
440C22×C3#93012yes—
441C22×C3C3×#12112yes—
442C22×C3C3×AGL(1,ℤ/8)12yes—
443C22×C3C3×#13712yes—
444C22×C3C22×D1211yescharacteristic
445C22×C3C22×C3×D411yescharacteristic
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—84no—
450C3×C4—24no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—122no—
457C3×C4—122no—
458C3×C4—122no—
459C3×C4#93012yes—
460C3×C4C3×#12112yes—
461C3×C4C2×#23511yescharacteristic
462C3×C4C22×#6711yescharacteristic
463C3×C4C22×C3×D411yescharacteristic
464C3×C4C2×C3×#4111yescharacteristic
465Dic3—42no—
466Dic3—42no—
467Dic3—42no—
468C32C2×#46811yescharacteristic, complemented, Sylow
469C23—22no—
470C23—22no—
471C23—22no—
472C23—62no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23C2×C3×D1212yes—
477C23C2×C3×D1211yescharacteristic
478Q8—62no—
479Q8—62no—
480Q8—122no—
481D4—62no—
482D4—62no—
483D4—128no—
484D4—122no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4C2×C3×D1212yes—
496C2×C4C2×C3×D1211yescharacteristic
497C2×C4C3×#23512yes—
498C2×C4C2×C3×#6712yes—
499C2×C4C23×C3×D311yescharacteristic
500C8—22no—
501C8C2×C3×#6714yescomplemented
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—24no—
506C2×C3—44no—
507C2×C3—42no—
508C2×C3—42no—
509C2×C3—242no—
510C2×C3#513414yesdirect product factor
511C2×C3C3×#46814yescomplemented
512C2×C3C2×#90811yescharacteristic
513C2×C3C2×#90911yescharacteristic
514C2×C3C2×#93011yescharacteristic
515C2×C3C2×C3×#12111yescharacteristic
516C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
517C2×C3C2×C3×#13711yescharacteristic
518D3—82no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—44no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22—244no—
527C22C3×#90812yes—
528C22C3×#90912yes—
529C22C3×#93012yes—
530C22C22×C3×D1211yescharacteristic
531C4—22no—
532C4—122no—
533C4—122no—
534C4—122no—
535C4C3×#93012yes—
536C4C2×C3×#23511yescharacteristic
537C4C22×C3×#6711yescharacteristic
538C3—21no—
539C3C2×#513411yescharacteristic, direct product factor
540C3C2×C3×#46811yescharacteristic, complemented
541C2—42no—
542C2—242no—
543C2C3×#513414yesdirect product factor
544C2C2×C3×#90811yescharacteristic
545C2C2×C3×#90911yescharacteristic
546C2C2×C3×#93011yescharacteristic
subgroupG/H
centerC23×C3C2×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×M16C2
socleC23×C32C2×D4
prime radicalC2×C3×#930C2
hypercenterC22×C3×M16D3
hypocenterC3C2×C3×#468
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C22×C3×M16

Complements

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

NH ≅ G/Ntype
1. C22×C32×M16C2semidirect
3. C2×C3×#924C2semidirect
4. C22×C3×#204C2semidirect
5. C3×#5134C2direct (N × H)
6. C2×C3×#821C2semidirect
7. C2×C3×#818C2semidirect
8. C2×C3×#816C2semidirect
10. C2×#5134C3direct (N × H)
16. C2×C32×M16C22semidirect
26. C3×#924C22semidirect
29. C2×C3×#204C22semidirect
30. C3×#821C22semidirect
31. C3×#818C22semidirect
32. C3×#816C22semidirect
35. C22×C3×M16D3semidirect
36. C22×C3×M16C2×C3semidirect
40. C2×#924C2×C3semidirect
41. C22×#204C2×C3semidirect
46. #5134C2×C3direct (N × H)
47. C2×#821C2×C3semidirect
48. C2×#818C2×C3semidirect
49. C2×#816C2×C3semidirect
73. C2×C32×C8D4semidirect
100. C2×C3×M16C2×D3semidirect
101. C2×C3×M16C22×C3semidirect
124. #924C22×C3semidirect
127. C2×#204C22×C3semidirect
131. #821C22×C3semidirect
132. #818C22×C3semidirect
133. #816C22×C3semidirect
166. C32×C8C2×D4semidirect
168. C22×M16C3×D3semidirect
234. C2×C3×C8#67semidirect
235. C2×C3×C8C3×D4semidirect
281. C2×M16C2×C3×D3semidirect
376. C3×C8C2×#67semidirect
377. C3×C8C2×C3×D4semidirect
379. C2×C32#468semidirect
408. C2×C8C3×#67semidirect
468. C32C2×#468semidirect
501. C8C2×C3×#67semidirect
510. C2×C3#5134direct (N × H)
511. C2×C3C3×#468semidirect
539. C3C2×#5134direct (N × H)
540. C3C2×C3×#468semidirect
543. C2C3×#5134direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×M162
C2×C3×#9302
C2×C3×#9242
C22×C3×#2042
C3×#51342
C2×C3×#8212
C2×C3×#8182
C2×C3×#8162
C2×C3×#4683
C2×#51343