Finite Group Databasebeta

← C2×C3×#5176 #11874330

Subgroups — C2×C3×#5176 #11874330

Order 1152.

1485 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic, complemented
3C22×C3×#215C211yesmaximal, characteristic, complemented
4C2×C3×#940C211yesmaximal, characteristic, complemented
5C22×C3×#213C211yesmaximal, characteristic, complemented
6C22×C3×D24C211yesmaximal, characteristic, complemented
7C3×#5176C218yesmaximal, direct product factor
8C2×C3×#815C211yesmaximal, characteristic, complemented
9C2×C3×#496—31nomaximal
10C2×#5176C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67—22no—
14C22×C3×D12C2211yescharacteristic
15C22×C3×D12C2211yescharacteristic
16C2×C32×SD16—24no—
17C2×C32×SD16C2214yescomplemented
18C22×C32×C8C2211yescharacteristic
19C3×#947—22no—
20C3×#947C2214yescomplemented
21C2×C3×#215—24no—
22C2×C3×#215C2214yescomplemented
23C3×#940—22no—
24C3×#940C2214yescomplemented
25C2×C3×#213—24no—
26C2×C3×#213C2214yescomplemented
27C2×C3×(C2×D3)⋊C4—22no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C2×C3×D24—24no—
31C2×C3×D24C2214yescomplemented
32C3×#815C2214yescomplemented
33C22×C3×SD16—21no—
34C22×C3×SD16—31no—
35C22×C3×SD16D311yescharacteristic, complemented
36C22×C3×SD16C2×C311yescharacteristic, complemented
37C22×C3×D8—31no—
38C22×C3×D8—31no—
39C2×C3×#127—31no—
40C2×C3×#124—31no—
41C2×#947C2×C311yescharacteristic, complemented
42C22×#215C2×C311yescharacteristic, complemented
43C2×#940C2×C311yescharacteristic, complemented
44C22×#213C2×C311yescharacteristic, complemented
45C22×D24C2×C311yescharacteristic, complemented
46C3×#496—38no—
47C2×C3×#97—31no—
48#5176C2×C318yesdirect product factor
49C2×#815C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C23×C3×D3—21no—
52C23×C3×D3—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8—22no—
55C2×C32×Q8—22no—
56C2×C32×Q8C2312yes—
57C2×C32×D4—22no—
58C2×C32×D4—22no—
59C2×C32×D4—21no—
60C2×C32×D4C2312yes—
61C22×C32×C4—21no—
62C22×C32×C4C2311yescharacteristic
63C2×C3×#67—28no—
64C2×C3×#67—48no—
65C22×C3×Dic3D412yes—
66C2×C3×D12—22no—
67C2×C3×D12—22no—
68C2×C3×D12—22no—
69C2×C3×D12—22no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12C2312yes—
73C2×C3×D12C2312yes—
74C32×SD16—28no—
75C2×C32×C8D414yescomplemented
76C2×C32×C8C2312yes—
77C3×#215—28no—
78C3×#213—28no—
79C3×(C2×D3)⋊C4—28no—
80C3×C4×Dic3D412yes—
81C3×C4×Dic3C2312yes—
82C2×C3×C3⋊C8D414yescomplemented
83C2×C3×C3⋊C8C2312yes—
84C3×D24—28no—
85C2×#496—31noSylow
86C22×C3×Q8—21no—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C22×C311yescharacteristic
89C22×C3×D4—21no—
90C22×C3×D4—31no—
91C22×C3×D4—31no—
92C22×C3×D4—62no—
93C22×C3×D4C2×D311yescharacteristic
94C22×C3×D4C22×C311yescharacteristic
95C22×#67—22no—
96C22×D12C22×C311yescharacteristic
97C22×D12C22×C311yescharacteristic
98C2×C3×SD16—28no—
99C2×C3×SD16—24no—
100C2×C3×SD16—24no—
101C2×C3×SD16—24no—
102C2×C3×SD16—34no—
103C2×C3×SD16—64no—
104C2×C3×SD16C2×D314yescomplemented
105C2×C3×SD16C22×C314yescomplemented
106C2×C3×D8—34no—
107C2×C3×D8—34no—
108C2×C3×D8—64no—
109C2×C3×D8—64no—
110C22×C3×C8—21no—
111C22×C3×C8—31no—
112C22×C3×C8C2×D311yescharacteristic
113C22×C3×C8C22×C311yescharacteristic
114C3×#127—34no—
115C3×#127—62no—
116C3×#124—34no—
117C3×#124—62no—
118C2×C3×C22⋊C4—62no—
119C2×C3×C42—31no—
120#947—22no—
121#947C22×C314yescomplemented
122C2×#215—24no—
123C2×#215C22×C314yescomplemented
124#940—22no—
125#940C22×C314yescomplemented
126C2×#213—24no—
127C2×#213C22×C314yescomplemented
128C2×(C2×D3)⋊C4—22no—
129C2×C4×Dic3C22×C311yescharacteristic
130C22×C3⋊C8C22×C311yescharacteristic
131C2×D24—24no—
132C2×D24C22×C314yescomplemented
133C3×#97—34no—
134#815C22×C314yescomplemented
135C23×C32—22no—
136C23×C32—42no—
137C23×C32—44no—
138C23×C32C2×D411yescharacteristic
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—42no—
142C22×C3×D3—42no—
143C22×C3×D3—44no—
144C22×C3×D3—44no—
145C32×Q8—22no—
146C32×Q8—22no—
147C32×Q8—42no—
148C32×D4—22no—
149C32×D4—22no—
150C32×D4—42no—
151C2×C32×C4—22no—
152C2×C32×C4—42no—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2×D412yes—
156C2×C32×C4C2411yescharacteristic
157C3×#67—416no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3C2×D414yes—
160C3×D12—22no—
161C3×D12—22no—
162C3×D12—22no—
163C3×D12—22no—
164C3×D12—42no—
165C3×D12—42no—
166C32×C8C2×D414yescomplemented
167C3×C3⋊C8C2×D414yescomplemented
168C22×SD16—31no—
169C22×SD16C3×D311yescharacteristic, complemented
170C22×D8—31no—
171C22×D8—31no—
172C2×#127—31no—
173C2×#124—31no—
174#496—38no—
175C2×#97—31no—
176C24×C3—21no—
177C24×C3—21no—
178C24×C3—41no—
179C24×C3—61no—
180C24×C3—61no—
181C23×D3—21no—
182C23×D3—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—24no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—41no—
192C2×C3×Q8—42no—
193C2×C3×Q8C22×D312yes—
194C2×C3×Q8C23×C312yes—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—21no—
202C2×C3×D4—22no—
203C2×C3×D4—32no—
204C2×C3×D4—32no—
205C2×C3×D4—42no—
206C2×C3×D4—41no—
207C2×C3×D4—68no—
208C2×C3×D4—62no—
209C2×C3×D4—62no—
210C2×C3×D4—61no—
211C2×C3×D4—61no—
212C2×C3×D4—62no—
213C2×C3×D4—62no—
214C2×C3×D4—128no—
215C2×C3×D4C22×D312yes—
216C2×C3×D4C23×C312yes—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—32no—
221C22×C3×C4—41no—
222C22×C3×C4C22×D311yescharacteristic
223C22×C3×C4C23×C311yescharacteristic
224C2×#67—28no—
225C2×#67—48no—
226C22×Dic3C3×D412yes—
227C2×D12—22no—
228C2×D12—22no—
229C2×D12—22no—
230C2×D12—22no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12C23×C312yes—
234C2×D12C23×C312yes—
235C3×SD16—216no—
236C3×SD16—28no—
237C3×SD16—28no—
238C3×SD16—68no—
239C3×D8—68no—
240C3×D8—68no—
241C2×C3×C8—24no—
242C2×C3×C8—22no—
243C2×C3×C8—34no—
244C2×C3×C8—32no—
245C2×C3×C8#6714yescomplemented
246C2×C3×C8C3×D414yescomplemented
247C2×C3×C8C22×D312yes—
248C2×C3×C8C23×C312yes—
249C3×C22⋊C4—68no—
250C3×C42—32no—
251C3×C42—32no—
252#215—28no—
253#213—28no—
254(C2×D3)⋊C4—28no—
255C4×Dic3C3×D412yes—
256C4×Dic3C23×C312yes—
257C2×C3⋊C8C3×D414yescomplemented
258C2×C3⋊C8C23×C312yes—
259D24—28no—
260C22×C32—42no—
261C22×C32—44no—
262C22×C32—84no—
263C22×C32#12712yes—
264C22×C32AGL(1,ℤ/8)14yes—
265C22×C32C22×D411yescharacteristic
266C2×C3×D3—42no—
267C2×C3×D3—42no—
268C2×C3×D3—44no—
269C2×C3×D3—44no—
270C2×C3×D3—84no—
271C2×C3×D3—84no—
272C32×C4—42no—
273C32×C4#12712yes—
274C32×C4C22×D411yescharacteristic
275C32×C4C22×D411yescharacteristic
276C3×Dic3—24no—
277C22×Q8C2×C3×D311yescharacteristic
278C22×D4—31no—
279C22×D4—31no—
280C22×D4—62no—
281C22×D4C2×C3×D311yescharacteristic
282C2×SD16—24no—
283C2×SD16—34no—
284C2×SD16—64no—
285C2×SD16C2×C3×D314yescomplemented
286C2×D8—34no—
287C2×D8—34no—
288C2×D8—64no—
289C2×D8—64no—
290C22×C8—31no—
291C22×C8C2×C3×D311yescharacteristic
292#127—34no—
293#127—62no—
294#124—34no—
295#124—62no—
296C2×C22⋊C4—62no—
297C2×C42—31no—
298#97—34no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—42no—
303C23×C3—44no—
304C23×C3—44no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—44no—
308C23×C3—62no—
309C23×C3—62no—
310C23×C3—84no—
311C23×C3—124no—
312C23×C3—124no—
313C23×C3—122no—
314C23×C3—122no—
315C23×C3D3×D411yescharacteristic
316C23×C3C2×C3×D411yescharacteristic
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—44no—
322C22×D3—44no—
323C3×Q8—24no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—42no—
329C3×Q8—44no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×D4—24no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—42no—
338C3×D4—44no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—62no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—62no—
345C3×D4—1216no—
346C3×D4—122no—
347C3×D4—122no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—34no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—64no—
360C2×C3×C4D3×D411yescharacteristic
361C2×C3×C4D3×D412yes—
362C2×C3×C4C2×#6712yes—
363C2×C3×C4C2×C3×D411yescharacteristic
364C2×C3×C4C2×C3×D412yes—
365C2×C3×C4C2×C3×D412yes—
366C2×C3×C4C23×D311yescharacteristic
367C2×C3×C4C24×C311yescharacteristic
368#67—416no—
369C2×Dic3—24no—
370C2×Dic3C2×C3×D414yes—
371D12—22no—
372D12—22no—
373D12—22no—
374D12—22no—
375D12—42no—
376D12—42no—
377C3×C8—24no—
378C3×C8—34no—
379C3×C8C2×#6714yescomplemented
380C3×C8C2×C3×D414yescomplemented
381C3⋊C8C2×C3×D414yescomplemented
382C2×C32—82no—
383C2×C32#49614yescomplemented
384C2×C32C2×#12711yescharacteristic
385C2×C32C2×AGL(1,ℤ/8)12yes—
386C3×D3—82no—
387C3×D3—82no—
388C24—21no—
389C24—61no—
390C24—61no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8C22×C3×D312yes—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—32no—
399C2×D4—32no—
400C2×D4—68no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—128no—
408C2×D4C22×C3×D312yes—
409C22×C4—21no—
410C22×C4—32no—
411C22×C4C22×C3×D311yescharacteristic
412SD16—28no—
413SD16—68no—
414D8—68no—
415D8—68no—
416C2×C8—32no—
417C2×C8—34no—
418C2×C8C3×#6714yescomplemented
419C2×C8C22×C3×D312yes—
420C22⋊C4—68no—
421C42—32no—
422C42—32no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—21no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—84no—
432C22×C3—88no—
433C22×C3—84no—
434C22×C3—84no—
435C22×C3—124no—
436C22×C3—124no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—244no—
440C22×C3—244no—
441C22×C3#91414yes—
442C22×C3#94012yes—
443C22×C3C3×#12712yes—
444C22×C3C3×AGL(1,ℤ/8)14yes—
445C22×C3C2×D3×D411yescharacteristic
446C22×C3C22×C3×D411yescharacteristic
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—84no—
452C2×D3—84no—
453C3×C4—22no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—44no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—64no—
460C3×C4#94012yes—
461C3×C4C3×#12712yes—
462C3×C4C2×D3×D411yescharacteristic
463C3×C4C22×#6711yescharacteristic
464C3×C4C22×C3×D411yescharacteristic
465C3×C4C22×C3×D411yescharacteristic
466Dic3—24no—
467C32C2×#49611yescharacteristic, complemented, Sylow
468C23—22no—
469C23—44no—
470C23—42no—
471C23—62no—
472C23—62no—
473C23—124no—
474C23—124no—
475C23—122no—
476C23—122no—
477C23C3×D3×D411yescharacteristic
478Q8—22no—
479Q8—22no—
480Q8—42no—
481D4—22no—
482D4—22no—
483D4—42no—
484D4—62no—
485D4—62no—
486D4—62no—
487D4—62no—
488D4—1216no—
489D4—122no—
490D4—122no—
491C2×C4—22no—
492C2×C4—34no—
493C2×C4—42no—
494C2×C4—64no—
495C2×C4C3×D3×D411yescharacteristic
496C2×C4C3×D3×D412yes—
497C2×C4C2×C3×#6712yes—
498C2×C4C23×C3×D311yescharacteristic
499C8—34no—
500C8C2×C3×#6714yescomplemented
501C2×C3—22no—
502C2×C3—24no—
503C2×C3—21no—
504C2×C3—84no—
505C2×C3—82no—
506C2×C3—82no—
507C2×C3—242no—
508C2×C3—242no—
509C2×C3#517614yesdirect product factor
510C2×C3C3×#49614yescomplemented
511C2×C3C2×#91412yes—
512C2×C3C2×#94011yescharacteristic
513C2×C3C2×C3×#12711yescharacteristic
514C2×C3C2×C3×AGL(1,ℤ/8)12yes—
515D3—82no—
516D3—82no—
517C22—44no—
518C22—42no—
519C22—84no—
520C22—124no—
521C22—124no—
522C22—122no—
523C22—122no—
524C22—244no—
525C22—244no—
526C22C3×#91414yes—
527C22C3×#94012yes—
528C22C2×C3×D3×D411yescharacteristic
529C4—42no—
530C4—64no—
531C4C3×#94012yes—
532C4C2×C3×D3×D411yescharacteristic
533C4C22×C3×#6711yescharacteristic
534C3—21no—
535C3C2×#517611yescharacteristic, direct product factor
536C3C2×C3×#49611yescharacteristic, complemented
537C2—82no—
538C2—242no—
539C2—242no—
540C2C3×#517614yesdirect product factor
541C2C2×C3×#91412yes—
542C2C2×C3×#94011yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#496
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
1. C22×C32×SD16C2semidirect
2. C2×C3×#947C2semidirect
3. C22×C3×#215C2semidirect
4. C2×C3×#940C2semidirect
5. C22×C3×#213C2semidirect
6. C22×C3×D24C2semidirect
7. C3×#5176C2direct (N × H)
8. C2×C3×#815C2semidirect
10. C2×#5176C3direct (N × H)
17. C2×C32×SD16C22semidirect
20. C3×#947C22semidirect
22. C2×C3×#215C22semidirect
24. C3×#940C22semidirect
26. C2×C3×#213C22semidirect
31. C2×C3×D24C22semidirect
32. C3×#815C22semidirect
35. C22×C3×SD16D3semidirect
36. C22×C3×SD16C2×C3semidirect
41. C2×#947C2×C3semidirect
42. C22×#215C2×C3semidirect
43. C2×#940C2×C3semidirect
44. C22×#213C2×C3semidirect
45. C22×D24C2×C3semidirect
48. #5176C2×C3direct (N × H)
49. C2×#815C2×C3semidirect
75. C2×C32×C8D4semidirect
82. C2×C3×C3⋊C8D4semidirect
104. C2×C3×SD16C2×D3semidirect
105. C2×C3×SD16C22×C3semidirect
121. #947C22×C3semidirect
123. C2×#215C22×C3semidirect
125. #940C22×C3semidirect
127. C2×#213C22×C3semidirect
132. C2×D24C22×C3semidirect
134. #815C22×C3semidirect
166. C32×C8C2×D4semidirect
167. C3×C3⋊C8C2×D4semidirect
169. C22×SD16C3×D3semidirect
245. C2×C3×C8#67semidirect
246. C2×C3×C8C3×D4semidirect
257. C2×C3⋊C8C3×D4semidirect
285. C2×SD16C2×C3×D3semidirect
379. C3×C8C2×#67semidirect
380. C3×C8C2×C3×D4semidirect
381. C3⋊C8C2×C3×D4semidirect
383. C2×C32#496semidirect
418. C2×C8C3×#67semidirect
467. C32C2×#496semidirect
500. C8C2×C3×#67semidirect
509. C2×C3#5176direct (N × H)
510. C2×C3C3×#496semidirect
535. C3C2×#5176direct (N × H)
536. C3C2×C3×#496semidirect
540. C2C3×#5176direct (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×#9472
C22×C3×#2152
C2×C3×#9402
C22×C3×#2132
C22×C3×D242
C3×#51762
C2×C3×#8152
C2×C3×#4963
C2×#51763