Finite Group Databasebeta

← C2×C3×#5175 #11874141

Subgroups — C2×C3×#5175 #11874141

Order 1152.

1461 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#944C211yesmaximal, characteristic, complemented
3C22×C3×#215C211yesmaximal, characteristic, complemented
4C2×C3×#940C211yesmaximal, characteristic
5C22×C3×#214C211yesmaximal, characteristic, complemented
6C22×C3×#204C211yesmaximal, characteristic, complemented
7C3×#5175C218yesmaximal, direct product factor
8C2×C3×C8×Dic3C211yesmaximal, characteristic, complemented
9C2×C3×#492—31nomaximal
10C2×#5175C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67—22no—
14C22×C3×D12C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C32×SD16—24no—
17C2×C32×SD16C2214yescomplemented
18C22×C32×C8C2211yescharacteristic
19C3×#944—22no—
20C3×#944C2214yescomplemented
21C2×C3×#215—24no—
22C2×C3×#215C2214yescomplemented
23C3×#940—22no—
24C3×#940C2214yes—
25C2×C3×#214—24no—
26C2×C3×#214C2214yescomplemented
27C2×C3×Dic3⋊C4—22no—
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C2×C3×#204—24no—
31C2×C3×#204C2214yescomplemented
32C3×C8×Dic3C2214yescomplemented
33C22×C3×SD16—21no—
34C22×C3×SD16—31no—
35C22×C3×SD16—31no—
36C22×C3×SD16—31no—
37C22×C3×SD16D311yescharacteristic, complemented
38C22×C3×SD16C2×C311yescharacteristic, complemented
39C2×C3×C4⋊Q8—31no—
40C2×C3×#127—31no—
41C2×#944C2×C311yescharacteristic, complemented
42C22×#215C2×C311yescharacteristic, complemented
43C2×#940C2×C311yescharacteristic
44C22×#214C2×C311yescharacteristic, complemented
45C22×#204C2×C311yescharacteristic, complemented
46C3×#492—38no—
47C2×C3×C4×C8—31no—
48#5175C2×C318yesdirect product factor
49C2×C8×Dic3C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C23×C3×D3—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—22no—
55C2×C32×Q8C2312yes—
56C2×C32×D4—22no—
57C2×C32×D4—22no—
58C2×C32×D4—21no—
59C2×C32×D4C2312yes—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67—28no—
63C2×C3×#67—48no—
64C22×C3×Dic3—21no—
65C22×C3×Dic3D412yes—
66C2×C3×D12—22no—
67C2×C3×D12—22no—
68C2×C3×D12—21no—
69C2×C3×D12C2312yes—
70C2×C3×Dic6—22no—
71C2×C3×Dic6—22no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6C2312yes—
74C32×SD16—28no—
75C2×C32×C8D414yescomplemented
76C2×C32×C8C2312yes—
77C3×#215—28no—
78C3×#214—28no—
79C3×Dic3⋊C4—28no—
80C3×C4×Dic3D412yes—
81C3×C4×Dic3C2312yes—
82C2×C3×C3⋊C8D414yescomplemented
83C2×C3×C3⋊C8C2312yes—
84C3×#204—28no—
85C2×#492—31noSylow
86C22×C3×Q8—21no—
87C22×C3×Q8—31no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—21no—
91C22×C3×D4—31no—
92C22×C3×D4—62no—
93C22×C3×D4C2×D311yescharacteristic
94C22×C3×D4C22×C311yescharacteristic
95C22×#67—22no—
96C22×D12C22×C311yescharacteristic
97C22×Dic6C22×C311yescharacteristic
98C2×C3×SD16—28no—
99C2×C3×SD16—24no—
100C2×C3×SD16—24no—
101C2×C3×SD16—24no—
102C2×C3×SD16—34no—
103C2×C3×SD16—34no—
104C2×C3×SD16—34no—
105C2×C3×SD16—64no—
106C2×C3×SD16—64no—
107C2×C3×SD16—64no—
108C2×C3×SD16C2×D314yescomplemented
109C2×C3×SD16C22×C314yescomplemented
110C22×C3×C8—21no—
111C22×C3×C8—31no—
112C22×C3×C8C2×D311yescharacteristic
113C22×C3×C8C22×C311yescharacteristic
114C3×C4⋊Q8—34no—
115C3×C4⋊Q8—62no—
116C3×#127—34no—
117C3×#127—62no—
118C2×C3×C4⋊C4—62no—
119C2×C3×C42—31no—
120#944—22no—
121#944C22×C314yescomplemented
122C2×#215—24no—
123C2×#215C22×C314yescomplemented
124#940—22no—
125#940C22×C314yes—
126C2×#214—24no—
127C2×#214C22×C314yescomplemented
128C2×Dic3⋊C4—22no—
129C2×C4×Dic3C22×C311yescharacteristic
130C22×C3⋊C8C22×C311yescharacteristic
131C2×#204—24no—
132C2×#204C22×C314yescomplemented
133C3×C4×C8—34no—
134C8×Dic3C22×C314yescomplemented
135C23×C32—22no—
136C23×C32—42no—
137C23×C32—44no—
138C23×C32C2×D411yescharacteristic
139C22×C3×D3—22no—
140C22×C3×D3—42no—
141C22×C3×D3—44no—
142C32×Q8—22no—
143C32×Q8—22no—
144C32×Q8—42no—
145C32×D4—22no—
146C32×D4—22no—
147C32×D4—42no—
148C2×C32×C4—22no—
149C2×C32×C4—42no—
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D412yes—
153C2×C32×C4C2411yescharacteristic
154C3×#67—416no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—42no—
157C2×C3×Dic3SD1618yescomplemented
158C2×C3×Dic3C2×D414yes—
159C3×D12—22no—
160C3×D12—22no—
161C3×D12—42no—
162C3×Dic6—22no—
163C3×Dic6—22no—
164C3×Dic6—42no—
165C32×C8C2×D414yescomplemented
166C3×C3⋊C8C2×D414yescomplemented
167C22×SD16—31no—
168C22×SD16—31no—
169C22×SD16—31no—
170C22×SD16C3×D311yescharacteristic, complemented
171C2×C4⋊Q8—31no—
172C2×#127—31no—
173#492—38no—
174C2×C4×C8—31no—
175C24×C3—21no—
176C24×C3—21no—
177C24×C3—41no—
178C24×C3—61no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—32no—
189C2×C3×Q8—41no—
190C2×C3×Q8—42no—
191C2×C3×Q8—61no—
192C2×C3×Q8—62no—
193C2×C3×Q8—62no—
194C2×C3×Q8C22×D312yes—
195C2×C3×Q8C23×C312yes—
196C2×C3×D4—24no—
197C2×C3×D4—22no—
198C2×C3×D4—21no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4—32no—
205C2×C3×D4—42no—
206C2×C3×D4—41no—
207C2×C3×D4—68no—
208C2×C3×D4—62no—
209C2×C3×D4—61no—
210C2×C3×D4—62no—
211C2×C3×D4—128no—
212C2×C3×D4C22×D312yes—
213C2×C3×D4C23×C312yes—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—32no—
218C22×C3×C4—41no—
219C22×C3×C4—61no—
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C23×C311yescharacteristic
222C2×#67—28no—
223C2×#67—48no—
224C22×Dic3—21no—
225C22×Dic3C3×D412yes—
226C2×D12—22no—
227C2×D12—22no—
228C2×D12—21no—
229C2×D12C23×C312yes—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—21no—
233C2×Dic6C23×C312yes—
234C3×SD16—216no—
235C3×SD16—28no—
236C3×SD16—28no—
237C3×SD16—68no—
238C3×SD16—68no—
239C3×SD16—68no—
240C2×C3×C8—24no—
241C2×C3×C8—22no—
242C2×C3×C8—34no—
243C2×C3×C8—32no—
244C2×C3×C8#6714yescomplemented
245C2×C3×C8C3×D414yescomplemented
246C2×C3×C8C22×D312yes—
247C2×C3×C8C23×C312yes—
248C3×C4⋊C4—68no—
249C3×C42—32no—
250C3×C42—32no—
251#215—28no—
252#214—28no—
253Dic3⋊C4—28no—
254C4×Dic3C3×D412yes—
255C4×Dic3C23×C312yes—
256C2×C3⋊C8C3×D414yescomplemented
257C2×C3⋊C8C23×C312yes—
258#204—28no—
259C22×C32—42no—
260C22×C32—44no—
261C22×C32—84no—
262C22×C32#12712yes—
263C22×C32C2×SD1614yes—
264C22×C32C22×D411yescharacteristic
265C2×C3×D3—42no—
266C2×C3×D3—44no—
267C2×C3×D3—84no—
268C32×C4—42no—
269C32×C4#12712yes—
270C32×C4C22×D411yescharacteristic
271C32×C4C22×D411yescharacteristic
272C3×Dic3—42no—
273C3×Dic3C2×SD1618yescomplemented
274C22×Q8—31no—
275C22×Q8C2×C3×D311yescharacteristic
276C22×D4—31no—
277C22×D4—62no—
278C22×D4C2×C3×D311yescharacteristic
279C2×SD16—24no—
280C2×SD16—34no—
281C2×SD16—34no—
282C2×SD16—34no—
283C2×SD16—64no—
284C2×SD16—64no—
285C2×SD16—64no—
286C2×SD16C2×C3×D314yescomplemented
287C22×C8—31no—
288C22×C8C2×C3×D311yescharacteristic
289C4⋊Q8—34no—
290C4⋊Q8—62no—
291#127—34no—
292#127—62no—
293C2×C4⋊C4—62no—
294C2×C42—31no—
295C4×C8—34no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—42no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—44no—
305C23×C3—62no—
306C23×C3—84no—
307C23×C3—124no—
308C23×C3—122no—
309C23×C3D3×D411yescharacteristic
310C23×C3C2×C3×D411yescharacteristic
311C22×D3—22no—
312C22×D3—42no—
313C22×D3—44no—
314C3×Q8—24no—
315C3×Q8—22no—
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—22no—
319C3×Q8—42no—
320C3×Q8—44no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×Q8—62no—
324C3×Q8—62no—
325C3×Q8—122no—
326C3×D4—24no—
327C3×D4—22no—
328C3×D4—22no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—42no—
332C3×D4—44no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—62no—
336C3×D4—62no—
337C3×D4—1216no—
338C3×D4—122no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—38no—
346C2×C3×C4—34no—
347C2×C3×C4—42no—
348C2×C3×C4—44no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—62no—
352C2×C3×C4—122no—
353C2×C3×C4D3×D411yescharacteristic
354C2×C3×C4D3×D412yes—
355C2×C3×C4C2×#6712yes—
356C2×C3×C4C2×C3×D411yescharacteristic
357C2×C3×C4C2×C3×D412yes—
358C2×C3×C4C2×C3×D412yes—
359C2×C3×C4C23×D311yescharacteristic
360C2×C3×C4C24×C311yescharacteristic
361#67—416no—
362C2×Dic3—22no—
363C2×Dic3—42no—
364C2×Dic3C3×SD1618yescomplemented
365C2×Dic3C2×C3×D414yes—
366D12—22no—
367D12—22no—
368D12—42no—
369Dic6—22no—
370Dic6—22no—
371Dic6—42no—
372C3×C8—24no—
373C3×C8—34no—
374C3×C8C2×#6714yescomplemented
375C3×C8C2×C3×D414yescomplemented
376C3⋊C8C2×C3×D414yescomplemented
377C2×C32—82no—
378C2×C32#49214yescomplemented
379C2×C32C2×#12711yescharacteristic
380C2×C32C22×SD1612yes—
381C3×D3—82no—
382C24—21no—
383C24—61no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—32no—
388C2×Q8—61no—
389C2×Q8—62no—
390C2×Q8—62no—
391C2×Q8C22×C3×D312yes—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—32no—
396C2×D4—68no—
397C2×D4—62no—
398C2×D4—61no—
399C2×D4—62no—
400C2×D4—128no—
401C2×D4C22×C3×D312yes—
402C22×C4—21no—
403C22×C4—32no—
404C22×C4—61no—
405C22×C4C22×C3×D311yescharacteristic
406SD16—28no—
407SD16—68no—
408SD16—68no—
409SD16—68no—
410C2×C8—32no—
411C2×C8—34no—
412C2×C8C3×#6714yescomplemented
413C2×C8C22×C3×D312yes—
414C4⋊C4—68no—
415C42—32no—
416C42—32no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—84no—
426C22×C3—88no—
427C22×C3—84no—
428C22×C3—84no—
429C22×C3—124no—
430C22×C3—122no—
431C22×C3—244no—
432C22×C3D3×SD1614yes—
433C22×C3#94012yes—
434C22×C3C3×#12712yes—
435C22×C3C2×C3×SD1614yes—
436C22×C3C2×D3×D411yescharacteristic
437C22×C3C22×C3×D411yescharacteristic
438C2×D3—44no—
439C2×D3—42no—
440C2×D3—84no—
441C3×C4—22no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—38no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—122no—
449C3×C4#94012yes—
450C3×C4C3×#12712yes—
451C3×C4C2×D3×D411yescharacteristic
452C3×C4C22×#6711yescharacteristic
453C3×C4C22×C3×D411yescharacteristic
454C3×C4C22×C3×D411yescharacteristic
455Dic3—42no—
456Dic3C2×C3×SD1618yescomplemented
457C32C2×#49211yescharacteristic, complemented, Sylow
458C23—22no—
459C23—44no—
460C23—42no—
461C23—62no—
462C23—124no—
463C23—122no—
464C23C3×D3×D411yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—42no—
468Q8—62no—
469Q8—62no—
470Q8—122no—
471D4—22no—
472D4—22no—
473D4—42no—
474D4—62no—
475D4—62no—
476D4—1216no—
477D4—122no—
478C2×C4—22no—
479C2×C4—38no—
480C2×C4—34no—
481C2×C4—42no—
482C2×C4—62no—
483C2×C4—122no—
484C2×C4C3×D3×D411yescharacteristic
485C2×C4C3×D3×D412yes—
486C2×C4C2×C3×#6712yes—
487C2×C4C23×C3×D311yescharacteristic
488C8—34no—
489C8C2×C3×#6714yescomplemented
490C2×C3—22no—
491C2×C3—24no—
492C2×C3—21no—
493C2×C3—84no—
494C2×C3—82no—
495C2×C3—82no—
496C2×C3—242no—
497C2×C3#517514yesdirect product factor
498C2×C3C3×#49214yescomplemented
499C2×C3C2×D3×SD1612yes—
500C2×C3C2×#94011yescharacteristic
501C2×C3C2×C3×#12711yescharacteristic
502C2×C3C22×C3×SD1612yes—
503D3—82no—
504C22—44no—
505C22—42no—
506C22—84no—
507C22—124no—
508C22—122no—
509C22—244no—
510C22C3×D3×SD1614yes—
511C22C3×#94012yes—
512C22C2×C3×D3×D411yescharacteristic
513C4—38no—
514C4—42no—
515C4—122no—
516C4C3×#94012yes—
517C4C2×C3×D3×D411yescharacteristic
518C4C22×C3×#6711yescharacteristic
519C3—21no—
520C3C2×#517511yescharacteristic, direct product factor
521C3C2×C3×#49211yescharacteristic, complemented
522C2—82no—
523C2—242no—
524C2C3×#517514yesdirect product factor
525C2C2×C3×D3×SD1612yes—
526C2C2×C3×#94011yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC2×C3×#940C2
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#492
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×#944C2semidirect
3. C22×C3×#215C2semidirect
5. C22×C3×#214C2semidirect
6. C22×C3×#204C2semidirect
7. C3×#5175C2direct (N × H)
8. C2×C3×C8×Dic3C2semidirect
10. C2×#5175C3direct (N × H)
17. C2×C32×SD16C22semidirect
20. C3×#944C22semidirect
22. C2×C3×#215C22semidirect
26. C2×C3×#214C22semidirect
31. C2×C3×#204C22semidirect
32. C3×C8×Dic3C22semidirect
37. C22×C3×SD16D3semidirect
38. C22×C3×SD16C2×C3semidirect
41. C2×#944C2×C3semidirect
42. C22×#215C2×C3semidirect
44. C22×#214C2×C3semidirect
45. C22×#204C2×C3semidirect
48. #5175C2×C3direct (N × H)
49. C2×C8×Dic3C2×C3semidirect
75. C2×C32×C8D4semidirect
82. C2×C3×C3⋊C8D4semidirect
108. C2×C3×SD16C2×D3semidirect
109. C2×C3×SD16C22×C3semidirect
121. #944C22×C3semidirect
123. C2×#215C22×C3semidirect
127. C2×#214C22×C3semidirect
132. C2×#204C22×C3semidirect
134. C8×Dic3C22×C3semidirect
157. C2×C3×Dic3SD16semidirect
165. C32×C8C2×D4semidirect
166. C3×C3⋊C8C2×D4semidirect
170. C22×SD16C3×D3semidirect
244. C2×C3×C8#67semidirect
245. C2×C3×C8C3×D4semidirect
256. C2×C3⋊C8C3×D4semidirect
273. C3×Dic3C2×SD16semidirect
286. C2×SD16C2×C3×D3semidirect
364. C2×Dic3C3×SD16semidirect
374. C3×C8C2×#67semidirect
375. C3×C8C2×C3×D4semidirect
376. C3⋊C8C2×C3×D4semidirect
378. C2×C32#492semidirect
412. C2×C8C3×#67semidirect
456. Dic3C2×C3×SD16semidirect
457. C32C2×#492semidirect
489. C8C2×C3×#67semidirect
497. C2×C3#5175direct (N × H)
498. C2×C3C3×#492semidirect
520. C3C2×#5175direct (N × H)
521. C3C2×C3×#492semidirect
524. C2C3×#5175direct (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×#9442
C22×C3×#2152
C2×C3×#9402
C22×C3×#2142
C22×C3×#2042
C3×#51752
C2×C3×C8×Dic32
C2×C3×#4923
C2×#51753