Finite Group Databasebeta

← C2×C3×#5218 #11863556

Subgroups — C2×C3×#5218 #11863556

Order 1152.

2673 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×D4C211yesmaximal, characteristic
2C22×C3×#214C212yesmaximal, complemented
3C2×C3×#924C211yesmaximal, characteristic, complemented
4C3×#5218C218yesmaximal, direct product factor
5C2×C3×#832C212yesmaximal, complemented
6C2×C3×#830C211yesmaximal, characteristic, complemented
7C2×C3×#450—31nomaximal
8C2×#5218C311yesmaximal, characteristic, direct product factor
9C25×C32—21no—
10C22×C32×D4—28no—
11C22×C32×D4—21no—
12C22×C32×D4—22no—
13C22×C32×D4C2214yes—
14C22×C32×D4C2212yes—
15C23×C32×C4C2211yescharacteristic
16C22×C3×Dic6C2211yescharacteristic, complemented
17C2×C3×#217—21no—
18C2×C3×#214—28no—
19C2×C3×#214C2218yescomplemented
20C2×C3×C4⋊Dic3C2211yescharacteristic, complemented
21C3×#924—22no—
22C3×#924C2214yescomplemented
23C2×C3×Dic3⋊C4—21no—
24C22×C3×C3⋊C8C2212yescomplemented
25C3×#832C2218yescomplemented
26C3×#830C2214yescomplemented
27C23×C3×D4—21no—
28C23×C3×D4D311yescharacteristic
29C23×C3×D4C2×C311yescharacteristic
30C22×C3×SD16—32no—
31C2×C3×C22⋊Q8—31no—
32C22×#214C2×C312yescomplemented
33C2×#924C2×C311yescharacteristic, complemented
34C3×#450—38no—
35C2×C3×#102—32no—
36C2×C3×C22⋊C8—31no—
37#5218C2×C318yesdirect product factor
38C2×#832C2×C312yescomplemented
39C2×#830C2×C311yescharacteristic, complemented
40C24×C32—28no—
41C24×C32—24no—
42C24×C32—22no—
43C24×C32—44no—
44C24×C32—44no—
45C24×C32D411yescharacteristic
46C2×C32×D4—28no—
47C2×C32×D4—216no—
48C2×C32×D4—216no—
49C2×C32×D4—24no—
50C2×C32×D4—22no—
51C2×C32×D4—24no—
52C2×C32×D4D418yes—
53C2×C32×D4C2314yes—
54C22×C32×C4—24no—
55C22×C32×C4D411yescharacteristic
56C22×C32×C4D412yes—
57C22×C32×C4C2312yes—
58C22×C32×C4C2311yescharacteristic
59C22×C3×Dic3—21no—
60C22×C3×Dic3—21no—
61C2×C3×Dic6—24no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6C2312yescomplemented
64C3×#217—24no—
65C3×#214—216no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3C2312yescomplemented
68C3×Dic3⋊C4—24no—
69C2×C3×C3⋊C8—24no—
70C2×C3×C3⋊C8C2314yescomplemented
71C2×#450—31noSylow
72C25×C3—22no—
73C25×C3—21no—
74C25×C3—21no—
75C22×C3×Q8—31no—
76C22×C3×D4—216no—
77C22×C3×D4—22no—
78C22×C3×D4—24no—
79C22×C3×D4—28no—
80C22×C3×D4—28no—
81C22×C3×D4—22no—
82C22×C3×D4—21no—
83C22×C3×D4—21no—
84C22×C3×D4—22no—
85C22×C3×D4—24no—
86C22×C3×D4—22no—
87C22×C3×D4C2×D314yes—
88C22×C3×D4C2×D312yes—
89C22×C3×D4C22×C314yes—
90C22×C3×D4C22×C312yes—
91C23×C3×C4—21no—
92C23×C3×C4C2×D311yescharacteristic
93C23×C3×C4C22×C311yescharacteristic
94C22×Dic6C22×C311yescharacteristic, complemented
95C2×C3×SD16—38no—
96C2×C3×SD16—68no—
97C22×C3×C8—32no—
98C3×C22⋊Q8—34no—
99C3×C22⋊Q8—62no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—61no—
102C2×C3×C22⋊C4—61no—
103C2×#217—21no—
104C2×#214—28no—
105C2×#214C22×C318yescomplemented
106C2×C4⋊Dic3C22×C311yescharacteristic, complemented
107#924—22no—
108#924C22×C314yescomplemented
109C2×Dic3⋊C4—21no—
110C22×C3⋊C8C22×C312yescomplemented
111C3×#102—38no—
112C3×C22⋊C8—34no—
113#832C22×C318yescomplemented
114#830C22×C314yescomplemented
115C23×C32—22no—
116C23×C32—22no—
117C23×C32—216no—
118C23×C32—28no—
119C23×C32—24no—
120C23×C32—48no—
121C23×C32—432no—
122C23×C32—44no—
123C23×C32—48no—
124C23×C32—44no—
125C23×C32SD1614yescomplemented
126C23×C32C2×D412yes—
127C23×C32C2×D411yescharacteristic
128C32×D4—216no—
129C32×D4—28no—
130C32×D4—24no—
131C32×D4C2×D418yes—
132C2×C32×C4—22no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D414yes—
138C2×C32×C4C2411yescharacteristic
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3—42no—
143C3×Dic6—24no—
144C3×Dic6—42no—
145C3×C3⋊C8—24no—
146C23×D4C3×D311yescharacteristic
147C22×SD16—32no—
148C2×C22⋊Q8—31no—
149#450—38no—
150C2×#102—32no—
151C2×C22⋊C8—31no—
152C24×C3—216no—
153C24×C3—28no—
154C24×C3—28no—
155C24×C3—28no—
156C24×C3—24no—
157C24×C3—24no—
158C24×C3—22no—
159C24×C3—24no—
160C24×C3—22no—
161C24×C3—21no—
162C24×C3—48no—
163C24×C3—44no—
164C24×C3—44no—
165C24×C3—48no—
166C24×C3—44no—
167C24×C3—44no—
168C24×C3#6711yescharacteristic
169C24×C3C3×D411yescharacteristic
170C2×C3×Q8—32no—
171C2×C3×Q8—61no—
172C2×C3×Q8—64no—
173C2×C3×D4—216no—
174C2×C3×D4—232no—
175C2×C3×D4—28no—
176C2×C3×D4—28no—
177C2×C3×D4—216no—
178C2×C3×D4—216no—
179C2×C3×D4—232no—
180C2×C3×D4—28no—
181C2×C3×D4—24no—
182C2×C3×D4—28no—
183C2×C3×D4—216no—
184C2×C3×D4—216no—
185C2×C3×D4—24no—
186C2×C3×D4—22no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—22no—
190C2×C3×D4—24no—
191C2×C3×D4—28no—
192C2×C3×D4—24no—
193C2×C3×D4#6718yes—
194C2×C3×D4C3×D418yes—
195C2×C3×D4C22×D314yes—
196C2×C3×D4C23×C314yes—
197C22×C3×C4—28no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—21no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C22×C3×C4#6712yes—
207C22×C3×C4#6711yescharacteristic
208C22×C3×C4C3×D411yescharacteristic
209C22×C3×C4C3×D412yes—
210C22×C3×C4C22×D312yes—
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C23×C312yes—
213C22×C3×C4C23×C311yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C2×Dic6—24no—
217C2×Dic6—21no—
218C2×Dic6C23×C312yescomplemented
219C3×SD16—616no—
220C2×C3×C8—34no—
221C2×C3×C8—64no—
222C3×C4⋊C4—32no—
223C3×C4⋊C4—64no—
224C3×C4⋊C4—61no—
225C3×C22⋊C4—64no—
226#217—24no—
227#214—216no—
228C4⋊Dic3—21no—
229C4⋊Dic3C23×C312yescomplemented
230Dic3⋊C4—24no—
231C2×C3⋊C8—24no—
232C2×C3⋊C8C23×C314yescomplemented
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—28no—
236C22×C32—28no—
237C22×C32—432no—
238C22×C32—44no—
239C22×C32—44no—
240C22×C32—416no—
241C22×C32#12012yes—
242C22×C32C2×SD1612yes—
243C22×C32C2×SD1614yescomplemented
244C22×C32AGL(1,ℤ/8)12yes—
245C22×C32C22×D411yescharacteristic
246C32×C4—22no—
247C32×C4#12012yes—
248C32×C4C22×D412yes—
249C3×Dic3—42no—
250C3×Dic3—42no—
251C25—21no—
252C22×Q8—31no—
253C22×D4—28no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4C2×C3×D314yes—
257C22×D4C2×C3×D312yes—
258C23×C4C2×C3×D311yescharacteristic
259C2×SD16—38no—
260C2×SD16—68no—
261C22×C8—32no—
262C22⋊Q8—34no—
263C22⋊Q8—62no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—61no—
266C2×C22⋊C4—61no—
267#102—38no—
268C22⋊C8—34no—
269C23×C3—24no—
270C23×C3—24no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—232no—
276C23×C3—216no—
277C23×C3—216no—
278C23×C3—216no—
279C23×C3—28no—
280C23×C3—28no—
281C23×C3—28no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—21no—
287C23×C3—416no—
288C23×C3—464no—
289C23×C3—48no—
290C23×C3—432no—
291C23×C3—48no—
292C23×C3—432no—
293C23×C3—48no—
294C23×C3—48no—
295C23×C3—416no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—48no—
299C23×C3—44no—
300C23×C3—48no—
301C23×C3—44no—
302C23×C3#21414yescomplemented
303C23×C3C3×SD1614yescomplemented
304C23×C3C2×#6712yes—
305C23×C3C2×#6711yescharacteristic
306C23×C3C2×C3×D412yes—
307C23×C3C2×C3×D411yescharacteristic
308C3×Q8—64no—
309C3×Q8—122no—
310C3×D4—232no—
311C3×D4—216no—
312C3×D4—216no—
313C3×D4—216no—
314C3×D4—28no—
315C3×D4—28no—
316C3×D4—28no—
317C3×D4—24no—
318C3×D4—24no—
319C3×D4—28no—
320C3×D4C2×#6718yes—
321C3×D4C2×C3×D418yes—
322C2×C3×C4—24no—
323C2×C3×C4—28no—
324C2×C3×C4—28no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—24no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—122no—
338C2×C3×C4—122no—
339C2×C3×C4C2×#6712yes—
340C2×C3×C4C2×#6711yescharacteristic
341C2×C3×C4C2×#6714yes—
342C2×C3×C4C2×C3×D412yes—
343C2×C3×C4C2×C3×D411yescharacteristic
344C2×C3×C4C2×C3×D414yes—
345C2×C3×C4C23×D311yescharacteristic
346C2×C3×C4C24×C311yescharacteristic
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351Dic6—24no—
352Dic6—42no—
353C3×C8—64no—
354C3⋊C8—24no—
355C2×C32—24no—
356C2×C32—48no—
357C2×C32#45014yescomplemented
358C2×C32C2×#12011yescharacteristic
359C2×C32C22×SD1611yescharacteristic
360C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
361C24—28no—
362C24—24no—
363C24—22no—
364C24—44no—
365C24—44no—
366C24C3×#6711yescharacteristic
367C2×Q8—32no—
368C2×Q8—61no—
369C2×Q8—64no—
370C2×D4—28no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4C3×#6718yes—
377C2×D4C22×C3×D314yes—
378C22×C4—24no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4C3×#6711yescharacteristic
382C22×C4C3×#6712yes—
383C22×C4C22×C3×D312yes—
384C22×C4C22×C3×D311yescharacteristic
385SD16—616no—
386C2×C8—34no—
387C2×C8—64no—
388C4⋊C4—32no—
389C4⋊C4—61no—
390C4⋊C4—64no—
391C22⋊C4—64no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—216no—
395C22×C3—22no—
396C22×C3—28no—
397C22×C3—22no—
398C22×C3—28no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—216no—
402C22×C3—28no—
403C22×C3—28no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—24no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—464no—
410C22×C3—432no—
411C22×C3—432no—
412C22×C3—48no—
413C22×C3—48no—
414C22×C3—432no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—416no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—416no—
421C22×C3#93212yes—
422C22×C3C2×#21412yes—
423C22×C3C2×#21414yescomplemented
424C22×C3#95312yes—
425C22×C3C3×#12012yes—
426C22×C3C2×C3×SD1612yes—
427C22×C3C2×C3×SD1614yescomplemented
428C22×C3C3×AGL(1,ℤ/8)12yes—
429C22×C3C22×#6711yescharacteristic
430C22×C3C22×C3×D411yescharacteristic
431C3×C4—24no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—122no—
437C3×C4—122no—
438C3×C4#95312yes—
439C3×C4C3×#12012yes—
440C3×C4C22×#6712yes—
441C3×C4C22×C3×D412yes—
442Dic3—42no—
443Dic3—42no—
444C32C2×#45011yescharacteristic, complemented, Sylow
445C23—22no—
446C23—22no—
447C23—216no—
448C23—28no—
449C23—24no—
450C23—48no—
451C23—432no—
452C23—44no—
453C23—44no—
454C23—48no—
455C23C3×#21414yescomplemented
456C23C2×C3×#6712yes—
457C23C2×C3×#6711yescharacteristic
458Q8—64no—
459Q8—122no—
460D4—216no—
461D4—28no—
462D4—24no—
463D4C2×C3×#6718yes—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4C2×C3×#6712yes—
472C2×C4C2×C3×#6711yescharacteristic
473C2×C4C2×C3×#6714yes—
474C2×C4C23×C3×D311yescharacteristic
475C8—64no—
476C2×C3—28no—
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—24no—
482C2×C3—21no—
483C2×C3—416no—
484C2×C3—48no—
485C2×C3—48no—
486C2×C3#521814yesdirect product factor
487C2×C3C3×#45014yescomplemented
488C2×C3C2×#93211yescharacteristic
489C2×C3C22×#21411yescharacteristic
490C2×C3C2×#95311yescharacteristic
491C2×C3C2×C3×#12011yescharacteristic
492C2×C3C22×C3×SD1611yescharacteristic
493C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
494C22—22no—
495C22—28no—
496C22—22no—
497C22—28no—
498C22—432no—
499C22—44no—
500C22—44no—
501C22—416no—
502C22C3×#93212yes—
503C22C2×C3×#21412yes—
504C22C2×C3×#21414yescomplemented
505C22C3×#95312yes—
506C22C22×C3×#6711yescharacteristic
507C4—22no—
508C4—122no—
509C4—122no—
510C4C3×#95312yes—
511C4C22×C3×#6712yes—
512C3—21no—
513C3C2×#521811yescharacteristic, direct product factor
514C3C2×C3×#45011yescharacteristic, complemented
515C2—24no—
516C2—48no—
517C2C3×#521814yesdirect product factor
518C2C2×C3×#93211yescharacteristic
519C2C22×C3×#21411yescharacteristic
520C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC23×C32×D4C2
socleC23×C32C2×D4
hypercenterC23×C3×D4D3
hypocenterC3C2×C3×#450
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#214C2semidirect
3. C2×C3×#924C2semidirect
4. C3×#5218C2direct (N × H)
5. C2×C3×#832C2semidirect
6. C2×C3×#830C2semidirect
8. C2×#5218C3direct (N × H)
16. C22×C3×Dic6C22semidirect
19. C2×C3×#214C22semidirect
20. C2×C3×C4⋊Dic3C22semidirect
22. C3×#924C22semidirect
24. C22×C3×C3⋊C8C22semidirect
25. C3×#832C22semidirect
26. C3×#830C22semidirect
32. C22×#214C2×C3semidirect
33. C2×#924C2×C3semidirect
37. #5218C2×C3direct (N × H)
38. C2×#832C2×C3semidirect
39. C2×#830C2×C3semidirect
63. C2×C3×Dic6C23semidirect
67. C3×C4⋊Dic3C23semidirect
70. C2×C3×C3⋊C8C23semidirect
94. C22×Dic6C22×C3semidirect
105. C2×#214C22×C3semidirect
106. C2×C4⋊Dic3C22×C3semidirect
108. #924C22×C3semidirect
110. C22×C3⋊C8C22×C3semidirect
113. #832C22×C3semidirect
114. #830C22×C3semidirect
125. C23×C32SD16semidirect
218. C2×Dic6C23×C3semidirect
229. C4⋊Dic3C23×C3semidirect
232. C2×C3⋊C8C23×C3semidirect
243. C22×C32C2×SD16semidirect
302. C23×C3#214semidirect
303. C23×C3C3×SD16semidirect
357. C2×C32#450semidirect
423. C22×C3C2×#214semidirect
427. C22×C3C2×C3×SD16semidirect
444. C32C2×#450semidirect
455. C23C3×#214semidirect
486. C2×C3#5218direct (N × H)
487. C2×C3C3×#450semidirect
504. C22C2×C3×#214semidirect
513. C3C2×#5218direct (N × H)
514. C3C2×C3×#450semidirect
517. C2C3×#5218direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×D42
C22×C3×#2142
C2×C3×#9242
C3×#52182
C2×C3×#8322
C2×C3×#8302
C2×C3×#4503
C2×#52183