Finite Group Databasebeta

← C2×C3×#5227 #11863602

Subgroups — C2×C3×#5227 #11863602

Order 1152.

2009 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×Q8C211yesmaximal, characteristic, complemented
2C22×C3×#215C212yesmaximal, complemented
3C2×C3×#930C211yesmaximal, characteristic
4C3×#5227C218yesmaximal, direct product factor
5C2×C3×#835C212yesmaximal, complemented
6C2×C3×#830C211yesmaximal, characteristic, complemented
7C2×C3×#448—31nomaximal
8C2×#5227C311yesmaximal, characteristic, direct product factor
9C22×C32×Q8—28no—
10C22×C32×Q8—21no—
11C22×C32×Q8—22no—
12C22×C32×Q8C2212yes—
13C22×C32×Q8C2214yescomplemented
14C23×C32×C4—21no—
15C23×C32×C4C2211yescharacteristic
16C22×C3×#67—21no—
17C22×C3×D12C2211yescharacteristic
18C2×C3×#215—28no—
19C2×C3×#215C2218yescomplemented
20C3×#930—22no—
21C3×#930C2214yes—
22C2×C3×(C2×D3)⋊C4—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C22×C3×C3⋊C8C2212yes—
25C3×#835C2218yescomplemented
26C3×#830C2214yescomplemented
27C23×C3×Q8—21no—
28C23×C3×Q8D311yescharacteristic, complemented
29C23×C3×Q8C2×C311yescharacteristic, complemented
30C22×C3×SD16—32no—
31C2×C3×#121—31no—
32C22×#215C2×C312yescomplemented
33C2×#930C2×C311yescharacteristic
34C3×#448—38no—
35C2×C3×#103—32no—
36C2×C3×C22⋊C8—31no—
37#5227C2×C318yesdirect product factor
38C2×#835C2×C312yescomplemented
39C2×#830C2×C311yescharacteristic, complemented
40C24×C32D411yescharacteristic
41C23×C3×D3—21no—
42C2×C32×Q8—216no—
43C2×C32×Q8—28no—
44C2×C32×Q8—216no—
45C2×C32×Q8—24no—
46C2×C32×Q8—22no—
47C2×C32×Q8—24no—
48C2×C32×Q8D418yescomplemented
49C2×C32×Q8C2314yes—
50C22×C32×C4—28no—
51C22×C32×C4—22no—
52C22×C32×C4—24no—
53C22×C32×C4—24no—
54C22×C32×C4D411yescharacteristic
55C22×C32×C4D412yes—
56C22×C32×C4C2312yes—
57C22×C32×C4C2311yescharacteristic
58C2×C3×#67—24no—
59C2×C3×#67—44no—
60C22×C3×Dic3—21no—
61C2×C3×D12—24no—
62C2×C3×D12—21no—
63C2×C3×D12C2312yes—
64C3×#215—216no—
65C3×(C2×D3)⋊C4—24no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3C2312yes—
68C2×C3×C3⋊C8—24no—
69C2×C3×C3⋊C8C2314yes—
70C2×#448—31noSylow
71C22×C3×Q8—216no—
72C22×C3×Q8—22no—
73C22×C3×Q8—24no—
74C22×C3×Q8—28no—
75C22×C3×Q8—28no—
76C22×C3×Q8—22no—
77C22×C3×Q8—21no—
78C22×C3×Q8—21no—
79C22×C3×Q8—22no—
80C22×C3×Q8—22no—
81C22×C3×Q8—24no—
82C22×C3×Q8C2×D312yes—
83C22×C3×Q8C2×D314yescomplemented
84C22×C3×Q8C22×C312yes—
85C22×C3×Q8C22×C314yescomplemented
86C22×C3×D4—31no—
87C22×C3×D4—61no—
88C23×C3×C4—22no—
89C23×C3×C4—21no—
90C23×C3×C4—21no—
91C23×C3×C4—21no—
92C23×C3×C4C2×D311yescharacteristic
93C23×C3×C4C22×C311yescharacteristic
94C22×#67—21no—
95C22×D12C22×C311yescharacteristic
96C2×C3×SD16—38no—
97C2×C3×SD16—68no—
98C22×C3×C8—32no—
99C3×#121—34no—
100C3×#121—62no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C22⋊C4—61no—
103C2×#215—28no—
104C2×#215C22×C318yescomplemented
105#930—22no—
106#930C22×C314yes—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C22×C311yescharacteristic
109C22×C3⋊C8C22×C312yes—
110C3×#103—38no—
111C3×C22⋊C8—34no—
112#835C22×C318yescomplemented
113#830C22×C314yescomplemented
114C23×C32—22no—
115C23×C32—22no—
116C23×C32SD1614yescomplemented
117C23×C32C2×D412yes—
118C23×C32C2×D411yescharacteristic
119C22×C3×D3—22no—
120C22×C3×D3—42no—
121C22×C3×D3—42no—
122C22×C3×D3—42no—
123C32×Q8—216no—
124C32×Q8—28no—
125C32×Q8—24no—
126C32×Q8C2×D418yescomplemented
127C2×C32×C4—216no—
128C2×C32×C4—28no—
129C2×C32×C4—22no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4C2×D412yes—
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D414yes—
136C2×C32×C4C2411yescharacteristic
137C3×#67—48no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—42no—
140C3×D12—24no—
141C3×D12—42no—
142C3×C3⋊C8—24no—
143C23×Q8C3×D311yescharacteristic, complemented
144C22×SD16—32no—
145C2×#121—31no—
146#448—38no—
147C2×#103—32no—
148C2×C22⋊C8—31no—
149C24×C3—21no—
150C24×C3—61no—
151C24×C3#6711yescharacteristic
152C24×C3C3×D411yescharacteristic
153C23×D3—21no—
154C2×C3×Q8—232no—
155C2×C3×Q8—216no—
156C2×C3×Q8—216no—
157C2×C3×Q8—216no—
158C2×C3×Q8—28no—
159C2×C3×Q8—28no—
160C2×C3×Q8—232no—
161C2×C3×Q8—28no—
162C2×C3×Q8—24no—
163C2×C3×Q8—28no—
164C2×C3×Q8—216no—
165C2×C3×Q8—216no—
166C2×C3×Q8—24no—
167C2×C3×Q8—22no—
168C2×C3×Q8—24no—
169C2×C3×Q8—24no—
170C2×C3×Q8—22no—
171C2×C3×Q8—24no—
172C2×C3×Q8—28no—
173C2×C3×Q8—24no—
174C2×C3×Q8#6718yescomplemented
175C2×C3×Q8C3×D418yescomplemented
176C2×C3×Q8C22×D314yes—
177C2×C3×Q8C23×C314yes—
178C2×C3×D4—32no—
179C2×C3×D4—64no—
180C2×C3×D4—64no—
181C2×C3×D4—61no—
182C2×C3×D4—124no—
183C22×C3×C4—216no—
184C22×C3×C4—28no—
185C22×C3×C4—28no—
186C22×C3×C4—24no—
187C22×C3×C4—28no—
188C22×C3×C4—28no—
189C22×C3×C4—24no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—61no—
200C22×C3×C4#6712yes—
201C22×C3×C4#6711yescharacteristic
202C22×C3×C4C3×D411yescharacteristic
203C22×C3×C4C3×D412yes—
204C22×C3×C4C22×D312yes—
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C23×C312yes—
207C22×C3×C4C23×C311yescharacteristic
208C2×#67—24no—
209C2×#67—44no—
210C22×Dic3—21no—
211C2×D12—24no—
212C2×D12—21no—
213C2×D12C23×C312yes—
214C3×SD16—616no—
215C2×C3×C8—34no—
216C2×C3×C8—64no—
217C3×C4⋊C4—32no—
218C3×C4⋊C4—61no—
219C3×C22⋊C4—64no—
220#215—216no—
221(C2×D3)⋊C4—24no—
222C4⋊Dic3—21no—
223C4⋊Dic3C23×C312yes—
224C2×C3⋊C8—24no—
225C2×C3⋊C8C23×C314yes—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—28no—
229C22×C32#12012yes—
230C22×C32C2×SD1612yes—
231C22×C32C2×SD1614yescomplemented
232C22×C32#13712yes—
233C22×C32C22×D411yescharacteristic
234C2×C3×D3—42no—
235C2×C3×D3—42no—
236C2×C3×D3—42no—
237C2×C3×D3—84no—
238C32×C4—28no—
239C32×C4—22no—
240C32×C4#12012yes—
241C32×C4C22×D412yes—
242C3×Dic3—42no—
243C22×Q8—28no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2×C3×D312yes—
247C22×Q8C2×C3×D314yescomplemented
248C22×D4—31no—
249C22×D4—61no—
250C23×C4—21no—
251C23×C4C2×C3×D311yescharacteristic
252C2×SD16—38no—
253C2×SD16—68no—
254C22×C8—32no—
255#121—34no—
256#121—62no—
257C2×C4⋊C4—31no—
258C2×C22⋊C4—61no—
259#103—38no—
260C22⋊C8—34no—
261C23×C3—24no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—21no—
270C23×C3—62no—
271C23×C3—122no—
272C23×C3—122no—
273C23×C3—122no—
274C23×C3#21514yescomplemented
275C23×C3C3×SD1614yescomplemented
276C23×C3C2×#6712yes—
277C23×C3C2×#6711yescharacteristic
278C23×C3C2×C3×D412yes—
279C23×C3C2×C3×D411yescharacteristic
280C22×D3—22no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C3×Q8—232no—
285C3×Q8—216no—
286C3×Q8—216no—
287C3×Q8—216no—
288C3×Q8—28no—
289C3×Q8—28no—
290C3×Q8—28no—
291C3×Q8—24no—
292C3×Q8—24no—
293C3×Q8—28no—
294C3×Q8C2×#6718yescomplemented
295C3×Q8C2×C3×D418yescomplemented
296C3×D4—64no—
297C3×D4—128no—
298C3×D4—122no—
299C2×C3×C4—232no—
300C2×C3×C4—216no—
301C2×C3×C4—216no—
302C2×C3×C4—216no—
303C2×C3×C4—24no—
304C2×C3×C4—28no—
305C2×C3×C4—28no—
306C2×C3×C4—28no—
307C2×C3×C4—28no—
308C2×C3×C4—22no—
309C2×C3×C4—28no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—24no—
321C2×C3×C4—62no—
322C2×C3×C4—122no—
323C2×C3×C4C2×#6712yes—
324C2×C3×C4C2×#6711yescharacteristic
325C2×C3×C4C2×#6714yes—
326C2×C3×C4C2×C3×D412yes—
327C2×C3×C4C2×C3×D411yescharacteristic
328C2×C3×C4C2×C3×D414yes—
329C2×C3×C4C23×D311yescharacteristic
330C2×C3×C4C24×C311yescharacteristic
331#67—48no—
332C2×Dic3—22no—
333C2×Dic3—42no—
334D12—24no—
335D12—42no—
336C3×C8—64no—
337C3⋊C8—24no—
338C2×C32—24no—
339C2×C32#44814yescomplemented
340C2×C32C2×#12011yescharacteristic
341C2×C32C22×SD1611yescharacteristic
342C2×C32C2×#13711yescharacteristic
343C3×D3—82no—
344C24—61no—
345C24C3×#6711yescharacteristic
346C2×Q8—216no—
347C2×Q8—28no—
348C2×Q8—216no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—24no—
352C2×Q8C3×#6718yescomplemented
353C2×Q8C22×C3×D314yes—
354C2×D4—32no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—61no—
358C2×D4—124no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—61no—
364C22×C4C3×#6711yescharacteristic
365C22×C4C3×#6712yes—
366C22×C4C22×C3×D312yes—
367C22×C4C22×C3×D311yescharacteristic
368SD16—616no—
369C2×C8—34no—
370C2×C8—64no—
371C4⋊C4—32no—
372C4⋊C4—61no—
373C22⋊C4—64no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—216no—
377C22×C3—22no—
378C22×C3—28no—
379C22×C3—22no—
380C22×C3—28no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—24no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—122no—
389C22×C3—122no—
390C22×C3—122no—
391C22×C3—244no—
392C22×C3C2×#21512yes—
393C22×C3C2×#21514yescomplemented
394C22×C3#94212yes—
395C22×C3#95312yes—
396C22×C3C3×#12012yes—
397C22×C3C2×C3×SD1612yes—
398C22×C3C2×C3×SD1614yescomplemented
399C22×C3C3×#13712yes—
400C22×C3C22×#6711yescharacteristic
401C22×C3C22×C3×D411yescharacteristic
402C2×D3—42no—
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—84no—
406C3×C4—216no—
407C3×C4—24no—
408C3×C4—28no—
409C3×C4—22no—
410C3×C4—28no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—122no—
415C3×C4#95312yes—
416C3×C4C3×#12012yes—
417C3×C4C22×#6712yes—
418C3×C4C22×C3×D412yes—
419Dic3—42no—
420C32C2×#44811yescharacteristic, complemented, Sylow
421C23—22no—
422C23—22no—
423C23—62no—
424C23—122no—
425C23—122no—
426C23—122no—
427C23C3×#21514yescomplemented
428C23C2×C3×#6712yes—
429C23C2×C3×#6711yescharacteristic
430Q8—216no—
431Q8—28no—
432Q8—24no—
433Q8C2×C3×#6718yescomplemented
434D4—64no—
435D4—128no—
436D4—122no—
437C2×C4—216no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—62no—
444C2×C4—122no—
445C2×C4C2×C3×#6712yes—
446C2×C4C2×C3×#6711yescharacteristic
447C2×C4C2×C3×#6714yes—
448C2×C4C23×C3×D311yescharacteristic
449C8—64no—
450C2×C3—28no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—24no—
456C2×C3—21no—
457C2×C3—242no—
458C2×C3#522714yesdirect product factor
459C2×C3C3×#44814yescomplemented
460C2×C3C22×#21511yescharacteristic
461C2×C3C2×#94211yescharacteristic
462C2×C3C2×#95311yescharacteristic
463C2×C3C2×C3×#12011yescharacteristic
464C2×C3C22×C3×SD1611yescharacteristic
465C2×C3C2×C3×#13711yescharacteristic
466D3—82no—
467C22—22no—
468C22—28no—
469C22—22no—
470C22—122no—
471C22—122no—
472C22—122no—
473C22—244no—
474C22C2×C3×#21512yes—
475C22C2×C3×#21514yescomplemented
476C22C3×#94212yes—
477C22C3×#95312yes—
478C22C22×C3×#6711yescharacteristic
479C4—28no—
480C4—22no—
481C4—122no—
482C4C3×#95312yes—
483C4C22×C3×#6712yes—
484C3—21no—
485C3C2×#522711yescharacteristic, direct product factor
486C3C2×C3×#44811yescharacteristic, complemented
487C2—24no—
488C2—242no—
489C2C3×#522714yesdirect product factor
490C2C22×C3×#21511yescharacteristic
491C2C2×C3×#94211yescharacteristic
492C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC23×C32×Q8C2
socleC23×C32C2×D4
prime radicalC2×C3×#930C2
hypercenterC23×C3×Q8D3
hypocenterC3C2×C3×#448
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C23×C32×Q8C2semidirect
2. C22×C3×#215C2semidirect
4. C3×#5227C2direct (N × H)
5. C2×C3×#835C2semidirect
6. C2×C3×#830C2semidirect
8. C2×#5227C3direct (N × H)
13. C22×C32×Q8C22semidirect
19. C2×C3×#215C22semidirect
25. C3×#835C22semidirect
26. C3×#830C22semidirect
28. C23×C3×Q8D3semidirect
29. C23×C3×Q8C2×C3semidirect
32. C22×#215C2×C3semidirect
37. #5227C2×C3direct (N × H)
38. C2×#835C2×C3semidirect
39. C2×#830C2×C3semidirect
48. C2×C32×Q8D4semidirect
83. C22×C3×Q8C2×D3semidirect
85. C22×C3×Q8C22×C3semidirect
104. C2×#215C22×C3semidirect
112. #835C22×C3semidirect
113. #830C22×C3semidirect
116. C23×C32SD16semidirect
126. C32×Q8C2×D4semidirect
143. C23×Q8C3×D3semidirect
174. C2×C3×Q8#67semidirect
175. C2×C3×Q8C3×D4semidirect
231. C22×C32C2×SD16semidirect
247. C22×Q8C2×C3×D3semidirect
274. C23×C3#215semidirect
275. C23×C3C3×SD16semidirect
294. C3×Q8C2×#67semidirect
295. C3×Q8C2×C3×D4semidirect
339. C2×C32#448semidirect
352. C2×Q8C3×#67semidirect
393. C22×C3C2×#215semidirect
398. C22×C3C2×C3×SD16semidirect
420. C32C2×#448semidirect
427. C23C3×#215semidirect
433. Q8C2×C3×#67semidirect
458. C2×C3#5227direct (N × H)
459. C2×C3C3×#448semidirect
475. C22C2×C3×#215semidirect
485. C3C2×#5227direct (N × H)
486. C3C2×C3×#448semidirect
489. C2C3×#5227direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×Q82
C22×C3×#2152
C2×C3×#9302
C3×#52272
C2×C3×#8352
C2×C3×#8302
C2×C3×#4483
C2×#52273