Finite Group Databasebeta

← C2×C3×#5727 #11902649

Subgroups — C2×C3×#5727 #11902649

Order 1152.

3281 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D3×Q8C212yesmaximal, complemented
2C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×#946C212yesmaximal, complemented
4C3×#5727C2116yesmaximal, direct product factor
5C2×C3×#896C214yesmaximal, complemented
6C2×C3×#893C212yesmaximal
7C2×C3×#888C212yesmaximal, complemented
8C2×C3×C4×D12C211yesmaximal, characteristic
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×D4×Q8—31nomaximal
11C2×#5727C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2212yes—
13C2×C3×D3×Q8—216no—
14C2×C3×D3×Q8C22116yescomplemented
15C22×C3×D12C2211yescharacteristic
16C22×C3×C4×D3C2214yes—
17C22×C3×C4×D3C2212yes—
18C22×C3×Dic6C2214yes—
19C22×C3×Dic6C2211yescharacteristic
20C32×C4⋊Q8C2218yescomplemented
21C2×C32×C4⋊C4C2212yes—
22C2×C32×C4⋊C4C2212yes—
23C2×C32×C42C2211yescharacteristic
24C3×#946C22116yescomplemented
25C2×C3×(C2×D3)⋊C4C2214yes—
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×C4⋊Dic3C2211yescharacteristic
29C2×C3×Dic3⋊C4C2214yes—
30C2×C3×Dic3⋊C4C2212yes—
31C2×C3×C4×Dic3C2212yes—
32C3×#896C22132yescomplemented
33C3×#893C22116yes—
34C3×#888C22116yescomplemented
35C3×C4×D12C2218yes—
36C3×C4×Dic6C2218yescomplemented
37C23×C3×Q8—32no—
38C22×D3×Q8C2×C312yescomplemented
39C3×D4×Q8—316no—
40C2×C3×C4⋊Q8—21no—
41C2×C3×C4⋊Q8—32no—
42C2×C3×C4⋊Q8D311yescharacteristic, complemented
43C2×C3×C4⋊Q8C2×C311yescharacteristic, complemented
44C2×C3×C22⋊Q8—32no—
45C2×C3×C22⋊Q8—34no—
46C2×C3×C4×Q8—31no—
47C2×C3×C4×D4—32no—
48C2×C3×C4×D4—31no—
49C2×#946C2×C312yescomplemented
50#5727C2×C3116yesdirect product factor
51C2×#896C2×C314yescomplemented
52C2×#893C2×C312yes—
53C2×#888C2×C312yescomplemented
54C2×C4×D12C2×C311yescharacteristic
55C2×C4×Dic6C2×C311yescharacteristic, complemented
56C23×C3×D3C2312yes—
57C2×C32×Q8—28no—
58C2×C32×Q8C2318yes—
59C22×C32×C4C2311yescharacteristic
60C22×C32×C4C2314yes—
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2312yes—
65C22×C3×Dic3C2312yes—
66C3×D3×Q8—264no—
67C2×C3×D12Q818yescomplemented
68C2×C3×D12C2314yes—
69C2×C3×C4×D3—28no—
70C2×C3×C4×D3—216no—
71C2×C3×C4×D3C23116yes—
72C2×C3×C4×D3C2318yes—
73C2×C3×Dic6—216no—
74C2×C3×Dic6D418yescomplemented
75C2×C3×Dic6C23116yes—
76C2×C3×Dic6C2314yes—
77C32×C4⋊C4C2318yes—
78C32×C4⋊C4C2318yes—
79C32×C42C2314yes—
80C3×(C2×D3)⋊C4C23116yes—
81C3×(C2×D3)⋊C4C2318yes—
82C3×C4⋊Dic3C2314yes—
83C3×C4⋊Dic3C2314yes—
84C3×Dic3⋊C4C23116yes—
85C3×Dic3⋊C4C2318yes—
86C3×C4×Dic3C2318yes—
87C2×D4×Q8—31noSylow
88C22×C3×Q8—22no—
89C22×C3×Q8—316no—
90C22×C3×Q8—31no—
91C22×C3×Q8—34no—
92C22×C3×Q8—616no—
93C22×C3×Q8C2×D312yes—
94C22×C3×Q8C22×C312yes—
95C22×C3×D4—31no—
96C23×C3×C4—34no—
97C23×C3×C4—32no—
98C2×D3×Q8—216no—
99C2×D3×Q8C22×C3116yescomplemented
100C22×D12C22×C311yescharacteristic
101C22×C4×D3C22×C312yes—
102C22×C4×D3C22×C314yes—
103C22×Dic6C22×C314yes—
104C22×Dic6C22×C311yescharacteristic
105C3×C4⋊Q8—28no—
106C3×C4⋊Q8—316no—
107C3×C4⋊Q8C2×D318yescomplemented
108C3×C4⋊Q8C22×C318yescomplemented
109C3×C22⋊Q8—316no—
110C3×C22⋊Q8—332no—
111C3×C4×Q8—38no—
112C3×C4×D4—316no—
113C3×C4×D4—38no—
114C2×C3×C4⋊C4—22no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—32no—
117C2×C3×C4⋊C4—34no—
118C2×C3×C4⋊C4—31no—
119C2×C3×C4⋊C4—31no—
120C2×C3×C4⋊C4C2×D312yes—
121C2×C3×C4⋊C4C2×D312yes—
122C2×C3×C4⋊C4C22×C312yes—
123C2×C3×C4⋊C4C22×C312yes—
124C2×C3×C22⋊C4—34no—
125C2×C3×C22⋊C4—32no—
126C2×C3×C42—21no—
127C2×C3×C42—32no—
128C2×C3×C42C2×D311yescharacteristic
129C2×C3×C42C22×C311yescharacteristic
130#946C22×C3116yescomplemented
131C2×(C2×D3)⋊C4C22×C312yes—
132C2×(C2×D3)⋊C4C22×C314yes—
133C2×C4⋊Dic3C22×C311yescharacteristic
134C2×C4⋊Dic3C22×C311yescharacteristic
135C2×Dic3⋊C4C22×C312yes—
136C2×Dic3⋊C4C22×C314yes—
137C2×C4×Dic3C22×C312yes—
138#896C22×C3132yescomplemented
139#893C22×C3116yes—
140#888C22×C3116yescomplemented
141C4×D12C22×C318yes—
142C4×Dic6C22×C318yescomplemented
143C23×C32C2411yescharacteristic
144C22×C3×D3—24no—
145C22×C3×D3—24no—
146C22×C3×D3C2×Q818yescomplemented
147C22×C3×D3C2414yes—
148C32×Q8—216no—
149C2×C32×C4—22no—
150C2×C32×C4—28no—
151C2×C32×C4C2×D414yes—
152C2×C32×C4C2×Q814yescomplemented
153C2×C32×C4C2412yes—
154C2×C32×C4C2418yes—
155C2×C32×C4C2412yes—
156C2×C32×C4C2412yes—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3C2×D418yes—
160C2×C3×Dic3C2414yes—
161C2×C3×Dic3C2414yes—
162C2×C3×Dic3C2414yes—
163C3×D12C2×Q8116yescomplemented
164C3×C4×D3—216no—
165C3×C4×D3—232no—
166C3×Dic6—232no—
167C3×Dic6C2×D4116yescomplemented
168C23×Q8—32no—
169D4×Q8—316no—
170C2×C4⋊Q8—32no—
171C2×C4⋊Q8C3×D311yescharacteristic, complemented
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—34no—
174C2×C4×Q8—31no—
175C2×C4×D4—32no—
176C2×C4×D4—31no—
177C24×C3—32no—
178C23×D3C23×C312yes—
179C2×C3×Q8—28no—
180C2×C3×Q8—28no—
181C2×C3×Q8—28no—
182C2×C3×Q8—38no—
183C2×C3×Q8—34no—
184C2×C3×Q8—316no—
185C2×C3×Q8—48no—
186C2×C3×Q8—616no—
187C2×C3×Q8—664no—
188C2×C3×Q8C22×D318yes—
189C2×C3×Q8C23×C318yes—
190C2×C3×D4—38no—
191C2×C3×D4—34no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—32no—
197C22×C3×C4—32no—
198C22×C3×C4—32no—
199C22×C3×C4—316no—
200C22×C3×C4—38no—
201C22×C3×C4—616no—
202C22×C3×C4—68no—
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C22×D314yes—
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C22×D311yescharacteristic
207C22×C3×C4C23×C311yescharacteristic
208C22×C3×C4C23×C314yes—
209C22×C3×C4C23×C311yescharacteristic
210C22×C3×C4C23×C311yescharacteristic
211C22×Dic3C23×C312yes—
212C22×Dic3C23×C312yes—
213C22×Dic3C23×C312yes—
214D3×Q8—264no—
215C2×D12C3×Q818yescomplemented
216C2×D12C23×C314yes—
217C2×C4×D3—28no—
218C2×C4×D3—216no—
219C2×C4×D3C23×C318yes—
220C2×C4×D3C23×C3116yes—
221C2×Dic6—216no—
222C2×Dic6C3×D418yescomplemented
223C2×Dic6C23×C3116yes—
224C2×Dic6C23×C314yes—
225C3×C4⋊C4—28no—
226C3×C4⋊C4—28no—
227C3×C4⋊C4—38no—
228C3×C4⋊C4—316no—
229C3×C4⋊C4—34no—
230C3×C4⋊C4—34no—
231C3×C4⋊C4C22×D318yes—
232C3×C4⋊C4C22×D318yes—
233C3×C4⋊C4C23×C318yes—
234C3×C4⋊C4C23×C318yes—
235C3×C22⋊C4—316no—
236C3×C22⋊C4—38no—
237C3×C42—24no—
238C3×C42—38no—
239C3×C42C22×D314yes—
240C3×C42C23×C314yes—
241(C2×D3)⋊C4C23×C318yes—
242(C2×D3)⋊C4C23×C3116yes—
243C4⋊Dic3C23×C314yes—
244C4⋊Dic3C23×C314yes—
245Dic3⋊C4C23×C318yes—
246Dic3⋊C4C23×C3116yes—
247C4×Dic3C23×C318yes—
248C22×C32C22×D412yes—
249C22×C32C22×Q812yes—
250C22×C322−1+412yes—
251C22×C32C2511yescharacteristic
252C2×C3×D3—216no—
253C2×C3×D3—24no—
254C2×C3×D3—24no—
255C2×C3×D3C22×Q818yescomplemented
256C32×C4—22no—
257C32×C4—28no—
258C32×C4C22×D414yes—
259C32×C4C22×Q814yescomplemented
260C3×Dic3—24no—
261C3×Dic3—24no—
262C3×Dic3C22×D418yes—
263C22×Q8—316no—
264C22×Q8—31no—
265C22×Q8—34no—
266C22×Q8—616no—
267C22×Q8C2×C3×D312yes—
268C22×D4—31no—
269C23×C4—34no—
270C23×C4—32no—
271C4⋊Q8—316no—
272C4⋊Q8C2×C3×D318yescomplemented
273C22⋊Q8—316no—
274C22⋊Q8—332no—
275C4×Q8—38no—
276C4×D4—316no—
277C4×D4—38no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—34no—
282C2×C4⋊C4C2×C3×D312yes—
283C2×C4⋊C4C2×C3×D312yes—
284C2×C22⋊C4—34no—
285C2×C22⋊C4—32no—
286C2×C42—32no—
287C2×C42C2×C3×D311yescharacteristic
288C23×C3—21no—
289C23×C3—38no—
290C23×C3—34no—
291C23×C3—64no—
292C23×C3—64no—
293C23×C3C23×D311yescharacteristic
294C23×C3C24×C311yescharacteristic
295C22×D3—24no—
296C22×D3—24no—
297C22×D3C2×C3×Q818yescomplemented
298C22×D3C24×C314yes—
299C3×Q8—216no—
300C3×Q8—216no—
301C3×Q8—316no—
302C3×Q8—416no—
303C3×Q8—632no—
304C3×D4—316no—
305C2×C3×C4—24no—
306C2×C3×C4—28no—
307C2×C3×C4—22no—
308C2×C3×C4—28no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—28no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—38no—
317C2×C3×C4—34no—
318C2×C3×C4—34no—
319C2×C3×C4—34no—
320C2×C3×C4—48no—
321C2×C3×C4—632no—
322C2×C3×C4—64no—
323C2×C3×C4—64no—
324C2×C3×C4—616no—
325C2×C3×C4D3×D414yes—
326C2×C3×C4D3×Q814yescomplemented
327C2×C3×C4C2×C3×D414yes—
328C2×C3×C4C2×C3×Q814yescomplemented
329C2×C3×C4C23×D312yes—
330C2×C3×C4C23×D318yes—
331C2×C3×C4C23×D312yes—
332C2×C3×C4C23×D312yes—
333C2×C3×C4C24×C312yes—
334C2×C3×C4C24×C318yes—
335C2×C3×C4C24×C312yes—
336C2×C3×C4C24×C312yes—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3C2×C3×D418yes—
340C2×Dic3C24×C314yes—
341C2×Dic3C24×C314yes—
342C2×Dic3C24×C314yes—
343D12C2×C3×Q8116yescomplemented
344C4×D3—216no—
345C4×D3—232no—
346Dic6—232no—
347Dic6C2×C3×D4116yescomplemented
348C2×C32D4×Q814yescomplemented
349C2×C32C23×D411yescharacteristic
350C2×C32C23×Q811yescharacteristic
351C2×C32C2×2−1+411yescharacteristic
352C3×D3—28no—
353C24—32no—
354C2×Q8—28no—
355C2×Q8—34no—
356C2×Q8—316no—
357C2×Q8—38no—
358C2×Q8—616no—
359C2×Q8—664no—
360C2×Q8C22×C3×D318yes—
361C2×D4—34no—
362C2×D4—38no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—32no—
366C22×C4—316no—
367C22×C4—38no—
368C22×C4—616no—
369C22×C4—68no—
370C22×C4C22×C3×D311yescharacteristic
371C22×C4C22×C3×D314yes—
372C22×C4C22×C3×D311yescharacteristic
373C22×C4C22×C3×D311yescharacteristic
374C4⋊C4—34no—
375C4⋊C4—38no—
376C4⋊C4—34no—
377C4⋊C4—316no—
378C4⋊C4C22×C3×D318yes—
379C4⋊C4C22×C3×D318yes—
380C22⋊C4—316no—
381C22⋊C4—38no—
382C42—38no—
383C42C22×C3×D314yes—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—38no—
389C22×C3—64no—
390C22×C3—64no—
391C22×C3—616no—
392C22×C3C2×D3×D412yes—
393C22×C3C2×D3×Q812yes—
394C22×C3#100712yes—
395C22×C3C22×C3×D412yes—
396C22×C3C22×C3×Q812yes—
397C22×C3C3×2−1+412yes—
398C22×C3C24×D311yescharacteristic
399C22×C3C25×C311yescharacteristic
400C2×D3—24no—
401C2×D3—24no—
402C2×D3—216no—
403C2×D3C22×C3×Q818yescomplemented
404C3×C4—24no—
405C3×C4—28no—
406C3×C4—22no—
407C3×C4—28no—
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—24no—
411C3×C4—38no—
412C3×C4—48no—
413C3×C4—64no—
414C3×C4—64no—
415C3×C4C2×D3×D414yes—
416C3×C4C2×D3×Q814yescomplemented
417C3×C4C22×C3×D414yes—
418C3×C4C22×C3×Q814yescomplemented
419Dic3—24no—
420Dic3—24no—
421Dic3C22×C3×D418yes—
422C32C2×D4×Q811yescharacteristic, complemented, Sylow
423C23—38no—
424C23—34no—
425C23—64no—
426C23—64no—
427C23C23×C3×D311yescharacteristic
428Q8—216no—
429Q8—316no—
430Q8—632no—
431D4—316no—
432C2×C4—28no—
433C2×C4—22no—
434C2×C4—38no—
435C2×C4—34no—
436C2×C4—34no—
437C2×C4—34no—
438C2×C4—632no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4—616no—
442C2×C4C3×D3×D414yes—
443C2×C4C3×D3×Q814yescomplemented
444C2×C4C23×C3×D312yes—
445C2×C4C23×C3×D318yes—
446C2×C4C23×C3×D312yes—
447C2×C4C23×C3×D312yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—24no—
452C2×C3—68no—
453C2×C3#572714yesdirect product factor
454C2×C3C3×D4×Q814yescomplemented
455C2×C3C22×D3×D411yescharacteristic
456C2×C3C22×D3×Q811yescharacteristic
457C2×C3C2×#100711yescharacteristic
458C2×C3C23×C3×D411yescharacteristic
459C2×C3C23×C3×Q811yescharacteristic
460C2×C3C2×C3×2−1+411yescharacteristic
461D3—28no—
462C22—38no—
463C22—64no—
464C22—64no—
465C22—616no—
466C22C2×C3×D3×D412yes—
467C22C2×C3×D3×Q812yes—
468C22C3×#100712yes—
469C22C24×C3×D311yescharacteristic
470C4—28no—
471C4—22no—
472C4—38no—
473C4—64no—
474C4—64no—
475C4C2×C3×D3×D414yes—
476C4C2×C3×D3×Q814yescomplemented
477C3—21no—
478C3C2×#572711yescharacteristic, direct product factor
479C3C2×C3×D4×Q811yescharacteristic, complemented
480C2—68no—
481C2C3×#572714yesdirect product factor
482C2C22×C3×D3×D411yescharacteristic
483C2C22×C3×D3×Q811yescharacteristic
484C2C2×C3×#100711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4⋊Q8C2
socleC23×C32C24
prime radicalC22×C3×D12C22
hypercenterC2×C3×C4⋊Q8D3
hypocenterC3C2×C3×D4×Q8
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×D3×Q8C2semidirect
2. C2×C32×C4⋊Q8C2semidirect
3. C2×C3×#946C2semidirect
4. C3×#5727C2direct (N × H)
5. C2×C3×#896C2semidirect
7. C2×C3×#888C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5727C3direct (N × H)
14. C2×C3×D3×Q8C22semidirect
20. C32×C4⋊Q8C22semidirect
24. C3×#946C22semidirect
32. C3×#896C22semidirect
34. C3×#888C22semidirect
36. C3×C4×Dic6C22semidirect
38. C22×D3×Q8C2×C3semidirect
42. C2×C3×C4⋊Q8D3semidirect
43. C2×C3×C4⋊Q8C2×C3semidirect
49. C2×#946C2×C3semidirect
50. #5727C2×C3direct (N × H)
51. C2×#896C2×C3semidirect
53. C2×#888C2×C3semidirect
55. C2×C4×Dic6C2×C3semidirect
67. C2×C3×D12Q8semidirect
74. C2×C3×Dic6D4semidirect
99. C2×D3×Q8C22×C3semidirect
107. C3×C4⋊Q8C2×D3semidirect
108. C3×C4⋊Q8C22×C3semidirect
130. #946C22×C3semidirect
138. #896C22×C3semidirect
140. #888C22×C3semidirect
142. C4×Dic6C22×C3semidirect
146. C22×C3×D3C2×Q8semidirect
152. C2×C32×C4C2×Q8semidirect
163. C3×D12C2×Q8semidirect
167. C3×Dic6C2×D4semidirect
171. C2×C4⋊Q8C3×D3semidirect
215. C2×D12C3×Q8semidirect
222. C2×Dic6C3×D4semidirect
255. C2×C3×D3C22×Q8semidirect
259. C32×C4C22×Q8semidirect
272. C4⋊Q8C2×C3×D3semidirect
297. C22×D3C2×C3×Q8semidirect
326. C2×C3×C4D3×Q8semidirect
328. C2×C3×C4C2×C3×Q8semidirect
343. D12C2×C3×Q8semidirect
347. Dic6C2×C3×D4semidirect
348. C2×C32D4×Q8semidirect
403. C2×D3C22×C3×Q8semidirect
416. C3×C4C2×D3×Q8semidirect
418. C3×C4C22×C3×Q8semidirect
422. C32C2×D4×Q8semidirect
443. C2×C4C3×D3×Q8semidirect
453. C2×C3#5727direct (N × H)
454. C2×C3C3×D4×Q8semidirect
476. C4C2×C3×D3×Q8semidirect
478. C3C2×#5727direct (N × H)
479. C3C2×C3×D4×Q8semidirect
481. C2C3×#5727direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D3×Q82
C2×C32×C4⋊Q82
C2×C3×#9462
C3×#57272
C2×C3×#8962
C2×C3×#8932
C2×C3×#8882
C2×C3×C4×D122
C2×C3×C4×Dic62
C2×C3×D4×Q83
C2×#57273