Finite Group Databasebeta

← C2×C3×#5133 #11857903

Subgroups — C2×C3×#5133 #11857903

Order 1152.

1781 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic
2C22×C32×M16C211yesmaximal, characteristic, complemented
3C22×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#5133C218yesmaximal, direct product factor
5C2×C3×#821C212yesmaximal, complemented
6C2×C3×#816C212yesmaximal, complemented
7C2×C3×#417—31nomaximal
8C2×#5133C311yesmaximal, characteristic, direct product factor
9C23×C32×C4C2211yescharacteristic
10C22×C3×#67—21no—
11C23×C3×Dic3—21no—
12C2×C3×#235—24no—
13C2×C3×#235—22no—
14C2×C3×#235C418yescomplemented
15C2×C3×#235C2214yes—
16C22×C3×D12C2211yescharacteristic
17C22×C3×C4×D3—21no—
18C22×C3×Dic6C2211yescharacteristic
19C2×C32×M16—24no—
20C2×C32×M16C2214yescomplemented
21C22×C32×C8C2212yes—
22C2×C3×C4⋊Dic3—22no—
23C2×C3×C4⋊Dic3—21no—
24C2×C3×C4⋊Dic3C2212yes—
25C2×C3×C4⋊Dic3C2214yescomplemented
26C3×#821C2218yescomplemented
27C3×#816C2218yescomplemented
28C22×C3×#41—31no—
29C22×#235C2×C311yescharacteristic
30C22×C3×M16—21no—
31C22×C3×M16D311yescharacteristic, complemented
32C22×C3×M16C2×C311yescharacteristic, complemented
33C22×C3×C4⋊C4—31no—
34C22×C4⋊Dic3C2×C311yescharacteristic, complemented
35C3×#417—38no—
36C2×C3×#103—32no—
37C2×C3×#102—32no—
38#5133C2×C318yesdirect product factor
39C2×#821C2×C312yescomplemented
40C2×#816C2×C312yescomplemented
41C24×C32D411yescharacteristic
42C23×C3×D3—21no—
43C22×C32×C4C2×C414yes—
44C22×C32×C4D411yescharacteristic
45C22×C32×C4D412yes—
46C22×C32×C4D414yes—
47C22×C32×C4C2311yescharacteristic
48C22×C32×C4C2312yes—
49C2×C3×#67—28no—
50C2×C3×#67—24no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—22no—
53C22×C3×Dic3—24no—
54C22×C3×Dic3—44no—
55C3×#235—216no—
56C3×#235—28no—
57C3×#235C2×C4116yescomplemented
58C2×C3×D12—21no—
59C2×C3×D12—22no—
60C2×C3×D12C2×C414yescomplemented
61C2×C3×D12C2312yes—
62C2×C3×C4×D3—28no—
63C2×C3×C4×D3—24no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—22no—
66C2×C3×Dic6C2×C414yescomplemented
67C2×C3×Dic6C2312yes—
68C32×M16—28no—
69C2×C32×C8—24no—
70C2×C32×C8C2314yes—
71C3×C4⋊Dic3—22no—
72C3×C4⋊Dic3—24no—
73C3×C4⋊Dic3C2314yes—
74C2×#417—31noSylow
75C2×C3×#41—38no—
76C2×C3×#41—34no—
77C2×C3×#41—62no—
78C2×C3×#41—64no—
79C22×C3×Q8—31no—
80C22×C3×D4—31no—
81C22×C3×D4—61no—
82C23×C3×C4—21no—
83C23×C3×C4—61no—
84C23×C3×C4—61no—
85C23×C3×C4C2×D311yescharacteristic
86C23×C3×C4C22×C311yescharacteristic
87C22×#67—21no—
88C23×Dic3—21no—
89C2×#235—22no—
90C2×#235—24no—
91C2×#235C3×C418yescomplemented
92C2×#235C22×C314yes—
93C22×D12C22×C311yescharacteristic
94C22×C4×D3—21no—
95C22×Dic6C22×C311yescharacteristic
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—22no—
103C22×C3×C8C2×D312yes—
104C22×C3×C8C22×C312yes—
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—34no—
107C2×C3×C4⋊C4—62no—
108C2×C3×C4⋊C4—61no—
109C2×C4⋊Dic3—22no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C22×C312yes—
112C2×C4⋊Dic3C22×C314yescomplemented
113C3×#103—38no—
114C3×#102—38no—
115#821C22×C318yescomplemented
116#816C22×C318yescomplemented
117C23×C32—22no—
118C23×C32—22no—
119C23×C32C22⋊C414yes—
120C23×C32C2×D412yes—
121C23×C32C2×D411yescharacteristic
122C22×C3×D3—24no—
123C22×C3×D3—22no—
124C22×C3×D3—42no—
125C22×C3×D3—42no—
126C2×C32×C4C22⋊C414yes—
127C2×C32×C4C22⋊C418yes—
128C2×C32×C4C22×C412yes—
129C2×C32×C4C22×C414yes—
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D412yes—
132C2×C32×C4C2×D412yes—
133C2×C32×C4C2×D414yes—
134C2×C32×C4C2411yescharacteristic
135C3×#67—216no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—48no—
140C2×C3×Dic3—44no—
141C3×D12—24no—
142C3×D12—22no—
143C3×D12C22×C414yescomplemented
144C3×C4×D3—216no—
145C3×Dic6—24no—
146C3×Dic6—22no—
147C3×Dic6C22×C414yescomplemented
148C32×C8—24no—
149C22×#41—31no—
150C22×M16C3×D311yescharacteristic, complemented
151C22×C4⋊C4—31no—
152#417—38no—
153C2×#103—32no—
154C2×#102—32no—
155C24×C3—21no—
156C24×C3—61no—
157C24×C3D1211yescharacteristic
158C24×C3C3×D411yescharacteristic
159C23×D3—21no—
160C3×#41—316no—
161C3×#41—616no—
162C3×#41—68no—
163C2×C3×Q8—34no—
164C2×C3×Q8—32no—
165C2×C3×Q8—61no—
166C2×C3×Q8—62no—
167C2×C3×D4—34no—
168C2×C3×D4—32no—
169C2×C3×D4—68no—
170C2×C3×D4—64no—
171C2×C3×D4—61no—
172C2×C3×D4—62no—
173C22×C3×C4—24no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—61no—
180C22×C3×C4—68no—
181C22×C3×C4—64no—
182C22×C3×C4—64no—
183C22×C3×C4—62no—
184C22×C3×C4—124no—
185C22×C3×C4C4×D314yes—
186C22×C3×C4D1212yes—
187C22×C3×C4D1211yescharacteristic
188C22×C3×C4#6714yes—
189C22×C3×C4C2×C3×C414yes—
190C22×C3×C4C3×D411yescharacteristic
191C22×C3×C4C3×D412yes—
192C22×C3×C4C3×D414yes—
193C22×C3×C4C22×D312yes—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C23×C311yescharacteristic
196C22×C3×C4C23×C312yes—
197C2×#67—28no—
198C2×#67—24no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—24no—
202C22×Dic3—44no—
203#235—216no—
204#235—28no—
205#235C2×C3×C4116yescomplemented
206C2×D12—21no—
207C2×D12—22no—
208C2×D12C2×C3×C414yescomplemented
209C2×D12C23×C312yes—
210C2×C4×D3—28no—
211C2×C4×D3—24no—
212C2×Dic6—21no—
213C2×Dic6—22no—
214C2×Dic6C2×C3×C414yescomplemented
215C2×Dic6C23×C312yes—
216C3×M16—216no—
217C3×M16—28no—
218C3×M16—28no—
219C2×C3×C8—28no—
220C2×C3×C8—24no—
221C2×C3×C8—24no—
222C2×C3×C8—24no—
223C2×C3×C8C22×D314yes—
224C2×C3×C8C23×C314yes—
225C3×C4⋊C4—34no—
226C3×C4⋊C4—64no—
227C3×C4⋊C4—62no—
228C4⋊Dic3—22no—
229C4⋊Dic3—24no—
230C4⋊Dic3C23×C314yes—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—28no—
234C22×C32C2×C22⋊C414yes—
235C22×C32C2×C22⋊C412yes—
236C22×C32AGL(1,ℤ/8)12yes—
237C22×C32#13712yes—
238C22×C32C22×D411yescharacteristic
239C2×C3×D3—24no—
240C2×C3×D3—42no—
241C2×C3×D3—42no—
242C2×C3×D3—48no—
243C32×C4C2×C22⋊C412yes—
244C32×C4C2×C22⋊C414yes—
245C32×C4C23×C411yescharacteristic
246C32×C4C22×D411yescharacteristic
247C3×Dic3—24no—
248C3×Dic3—44no—
249C2×#41—38no—
250C2×#41—34no—
251C2×#41—62no—
252C2×#41—64no—
253C22×Q8—31no—
254C22×D4—31no—
255C22×D4—61no—
256C23×C4—61no—
257C23×C4—61no—
258C23×C4C2×C3×D311yescharacteristic
259C2×M16—24no—
260C2×M16C2×C3×D314yescomplemented
261C22×C8C2×C3×D312yes—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266#103—38no—
267#102—38no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—24no—
273C23×C3—21no—
274C23×C3—22no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—64no—
278C23×C3—62no—
279C23×C3—122no—
280C23×C3—122no—
281C23×C3(C2×D3)⋊C414yes—
282C23×C3C3×C22⋊C414yes—
283C23×C3C2×D1212yes—
284C23×C3C2×D1211yescharacteristic
285C23×C3C2×C3×D412yes—
286C23×C3C2×C3×D411yescharacteristic
287C22×D3—24no—
288C22×D3—22no—
289C22×D3—42no—
290C22×D3—42no—
291C3×Q8—34no—
292C3×Q8—64no—
293C3×Q8—62no—
294C3×D4—34no—
295C3×D4—616no—
296C3×D4—64no—
297C3×D4—62no—
298C2×C3×C4—21no—
299C2×C3×C4—24no—
300C2×C3×C4—28no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—64no—
308C2×C3×C4—62no—
309C2×C3×C4—616no—
310C2×C3×C4—64no—
311C2×C3×C4—128no—
312C2×C3×C4—124no—
313C2×C3×C4(C2×D3)⋊C414yes—
314C2×C3×C4(C2×D3)⋊C418yes—
315C2×C3×C4C3×C22⋊C414yes—
316C2×C3×C4C3×C22⋊C418yes—
317C2×C3×C4C2×C4×D312yes—
318C2×C3×C4C2×C4×D314yes—
319C2×C3×C4C2×D1212yes—
320C2×C3×C4C2×D1211yescharacteristic
321C2×C3×C4C2×#6712yes—
322C2×C3×C4C2×#6714yes—
323C2×C3×C4C22×C3×C412yes—
324C2×C3×C4C22×C3×C414yes—
325C2×C3×C4C2×C3×D411yescharacteristic
326C2×C3×C4C2×C3×D412yes—
327C2×C3×C4C2×C3×D412yes—
328C2×C3×C4C2×C3×D414yes—
329C2×C3×C4C23×D311yescharacteristic
330C2×C3×C4C24×C311yescharacteristic
331#67—216no—
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—48no—
336C2×Dic3—44no—
337D12—24no—
338D12—22no—
339D12C22×C3×C414yescomplemented
340C4×D3—216no—
341Dic6—24no—
342Dic6—22no—
343Dic6C22×C3×C414yescomplemented
344C3×C8—28no—
345C3×C8—24no—
346C3×C8—24no—
347C2×C32—24no—
348C2×C32#41714yescomplemented
349C2×C32C22×C22⋊C411yescharacteristic
350C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
351C2×C32C2×#13711yescharacteristic
352C3×D3—44no—
353C24—61no—
354C24C3×D1211yescharacteristic
355#41—316no—
356#41—616no—
357#41—68no—
358C2×Q8—32no—
359C2×Q8—34no—
360C2×Q8—61no—
361C2×Q8—62no—
362C2×D4—32no—
363C2×D4—34no—
364C2×D4—68no—
365C2×D4—64no—
366C2×D4—61no—
367C2×D4—62no—
368C22×C4—61no—
369C22×C4—68no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—62no—
373C22×C4—124no—
374C22×C4C3×C4×D314yes—
375C22×C4C3×D1211yescharacteristic
376C22×C4C3×D1212yes—
377C22×C4C3×#6714yes—
378C22×C4C22×C3×D311yescharacteristic
379C22×C4C22×C3×D312yes—
380M16—28no—
381C2×C8—24no—
382C2×C8C22×C3×D314yes—
383C4⋊C4—34no—
384C4⋊C4—64no—
385C4⋊C4—62no—
386C22×C3—28no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—28no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—24no—
396C22×C3—21no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—48no—
400C22×C3—64no—
401C22×C3—128no—
402C22×C3—122no—
403C22×C3—122no—
404C22×C3#90812yes—
405C22×C3#90912yes—
406C22×C3C2×(C2×D3)⋊C414yes—
407C22×C3C2×(C2×D3)⋊C412yes—
408C22×C3C2×C3×C22⋊C414yes—
409C22×C3C2×C3×C22⋊C412yes—
410C22×C3C3×AGL(1,ℤ/8)12yes—
411C22×C3C3×#13712yes—
412C22×C3C22×D1211yescharacteristic
413C22×C3C22×C3×D411yescharacteristic
414C2×D3—24no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—48no—
418C3×C4—24no—
419C3×C4—22no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—64no—
423C3×C4—124no—
424C3×C4C2×(C2×D3)⋊C414yes—
425C3×C4C2×(C2×D3)⋊C412yes—
426C3×C4C2×C3×C22⋊C412yes—
427C3×C4C2×C3×C22⋊C414yes—
428C3×C4C22×C4×D311yescharacteristic
429C3×C4C22×#6711yescharacteristic
430C3×C4C23×C3×C411yescharacteristic
431C3×C4C22×C3×D411yescharacteristic
432Dic3—24no—
433Dic3—44no—
434C32C2×#41711yescharacteristic, complemented, Sylow
435C23—22no—
436C23—22no—
437C23—64no—
438C23—62no—
439C23—122no—
440C23—122no—
441C23C3×(C2×D3)⋊C414yes—
442C23C2×C3×D1212yes—
443C23C2×C3×D1211yescharacteristic
444Q8—34no—
445Q8—64no—
446Q8—62no—
447D4—34no—
448D4—616no—
449D4—64no—
450D4—62no—
451C2×C4—64no—
452C2×C4—62no—
453C2×C4—616no—
454C2×C4—64no—
455C2×C4—128no—
456C2×C4—124no—
457C2×C4C3×(C2×D3)⋊C414yes—
458C2×C4C3×(C2×D3)⋊C418yes—
459C2×C4C2×C3×C4×D312yes—
460C2×C4C2×C3×C4×D314yes—
461C2×C4C2×C3×D1212yes—
462C2×C4C2×C3×D1211yescharacteristic
463C2×C4C2×C3×#6712yes—
464C2×C4C2×C3×#6714yes—
465C2×C4C23×C3×D311yescharacteristic
466C8—24no—
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—24no—
473C2×C3—44no—
474C2×C3—124no—
475C2×C3#513314yesdirect product factor
476C2×C3C3×#41714yescomplemented
477C2×C3C2×#90811yescharacteristic
478C2×C3C2×#90911yescharacteristic
479C2×C3C22×(C2×D3)⋊C411yescharacteristic
480C2×C3C22×C3×C22⋊C411yescharacteristic
481C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
482C2×C3C2×C3×#13711yescharacteristic
483D3—44no—
484C22—22no—
485C22—22no—
486C22—28no—
487C22—64no—
488C22—122no—
489C22—122no—
490C22—128no—
491C22C3×#90812yes—
492C22C3×#90912yes—
493C22C2×C3×(C2×D3)⋊C414yes—
494C22C2×C3×(C2×D3)⋊C412yes—
495C22C22×C3×D1211yescharacteristic
496C4—64no—
497C4—124no—
498C4C2×C3×(C2×D3)⋊C414yes—
499C4C2×C3×(C2×D3)⋊C412yes—
500C4C22×C3×C4×D311yescharacteristic
501C4C22×C3×#6711yescharacteristic
502C3—21no—
503C3C2×#513311yescharacteristic, direct product factor
504C3C2×C3×#41711yescharacteristic, complemented
505C2—24no—
506C2—124no—
507C2C3×#513314yesdirect product factor
508C2C2×C3×#90811yescharacteristic
509C2C2×C3×#90911yescharacteristic
510C2C22×C3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC23×C3C2×D12
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC22×C32×M16C2
socleC23×C32C2×D4
prime radicalC22×C3×#235C2
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×M16D3
hypocenterC3C2×C3×#417
LCSG ▷ 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
2. C22×C32×M16C2semidirect
3. C22×C3×C4⋊Dic3C2semidirect
4. C3×#5133C2direct (N × H)
5. C2×C3×#821C2semidirect
6. C2×C3×#816C2semidirect
8. C2×#5133C3direct (N × H)
14. C2×C3×#235C4semidirect
20. C2×C32×M16C22semidirect
25. C2×C3×C4⋊Dic3C22semidirect
26. C3×#821C22semidirect
27. C3×#816C22semidirect
31. C22×C3×M16D3semidirect
32. C22×C3×M16C2×C3semidirect
34. C22×C4⋊Dic3C2×C3semidirect
38. #5133C2×C3direct (N × H)
39. C2×#821C2×C3semidirect
40. C2×#816C2×C3semidirect
57. C3×#235C2×C4semidirect
60. C2×C3×D12C2×C4semidirect
66. C2×C3×Dic6C2×C4semidirect
91. C2×#235C3×C4semidirect
100. C2×C3×M16C2×D3semidirect
101. C2×C3×M16C22×C3semidirect
112. C2×C4⋊Dic3C22×C3semidirect
115. #821C22×C3semidirect
116. #816C22×C3semidirect
143. C3×D12C22×C4semidirect
147. C3×Dic6C22×C4semidirect
150. C22×M16C3×D3semidirect
205. #235C2×C3×C4semidirect
208. C2×D12C2×C3×C4semidirect
214. C2×Dic6C2×C3×C4semidirect
260. C2×M16C2×C3×D3semidirect
339. D12C22×C3×C4semidirect
343. Dic6C22×C3×C4semidirect
348. C2×C32#417semidirect
434. C32C2×#417semidirect
475. C2×C3#5133direct (N × H)
476. C2×C3C3×#417semidirect
503. C3C2×#5133direct (N × H)
504. C3C2×C3×#417semidirect
507. C2C3×#5133direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C22×C32×M162
C22×C3×C4⋊Dic32
C3×#51332
C2×C3×#8212
C2×C3×#8162
C2×C3×#4173
C2×#51333