Finite Group Databasebeta

← C2×C3×#5584 #11903370

Subgroups — C2×C3×#5584 #11903370

Order 1152.

2373 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#946C212yesmaximal, complemented
3C3×#5584C2116yesmaximal, direct product factor
4C2×C3×#898C214yesmaximal, complemented
5C2×C3×#892C212yesmaximal, complemented
6C2×C3×#875C212yesmaximal
7C2×C3×C4×D12C211yesmaximal, characteristic
8C2×C3×#869C211yesmaximal, characteristic, complemented
9C2×C3×C4×Dic6C212yesmaximal, complemented
10C2×C3×#556—31nomaximal
11C2×#5584C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2212yes—
15C22×C3×Dic6C2212yes—
16C32×C4×Q8C2218yescomplemented
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C42C2212yes—
20C2×C32×C42C2211yescharacteristic
21C3×#946C22116yescomplemented
22C2×C3×(C2×D3)⋊C4C2214yes—
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2214yes—
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×C4⋊Dic3C2212yes—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×Dic3⋊C4C2214yes—
29C2×C3×C4×Dic3C2214yes—
30C3×#898C22132yescomplemented
31C3×#892C22116yescomplemented
32C3×#875C22116yes—
33C3×C4×D12C2218yes—
34C3×#869C2218yescomplemented
35C3×C4×Dic6C22116yescomplemented
36C3×#556—316no—
37C2×C3×C4⋊Q8—31no—
38C2×C3×#126—34no—
39C2×C3×#124—32no—
40C2×C3×C22⋊Q8—32no—
41C2×C3×C4×Q8—21no—
42C2×C3×C4×Q8—32no—
43C2×C3×C4×Q8D311yescharacteristic, complemented
44C2×C3×C4×Q8C2×C311yescharacteristic, complemented
45C2×C3×C4×D4—31no—
46C2×C3×#117—32no—
47C2×#946C2×C312yescomplemented
48#5584C2×C3116yesdirect product factor
49C2×#898C2×C314yescomplemented
50C2×#892C2×C312yescomplemented
51C2×#875C2×C312yes—
52C2×C4×D12C2×C311yescharacteristic
53C2×#869C2×C311yescharacteristic, complemented
54C2×C4×Dic6C2×C312yescomplemented
55C23×C3×D3C2312yes—
56C2×C32×Q8—24no—
57C2×C32×Q8C2314yes—
58C22×C32×C4C2312yes—
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2314yes—
65C2×C3×D12—24no—
66C2×C3×D12C2314yes—
67C2×C3×C4×D3—28no—
68C2×C3×C4×D3C2318yes—
69C2×C3×Dic6—28no—
70C2×C3×Dic6C2318yes—
71C32×C4⋊C4C2314yes—
72C32×C4⋊C4C2318yes—
73C32×C42C2318yes—
74C32×C42C2314yes—
75C3×(C2×D3)⋊C4C23116yes—
76C3×(C2×D3)⋊C4C2318yes—
77C3×(C2×D3)⋊C4C23116yes—
78C3×C4⋊Dic3C2318yes—
79C3×C4⋊Dic3C2318yes—
80C3×C4⋊Dic3C2314yes—
81C3×Dic3⋊C4C23116yes—
82C3×C4×Dic3C23116yes—
83C2×#556—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—32no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C22×C3×D4—31no—
89C23×C3×C4—32no—
90C22×D12C22×C311yescharacteristic
91C22×C4×D3C22×C312yes—
92C22×Dic6C22×C312yes—
93C3×C4⋊Q8—38no—
94C3×#126—332no—
95C3×#124—316no—
96C3×C22⋊Q8—316no—
97C3×C4×Q8—28no—
98C3×C4×Q8—316no—
99C3×C4×Q8C2×D318yescomplemented
100C3×C4×Q8C22×C318yescomplemented
101C3×C4×D4—38no—
102C3×#117—316no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4—34no—
106C2×C3×C4⋊C4—32no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—32no—
109C2×C3×C4⋊C4C2×D311yescharacteristic
110C2×C3×C4⋊C4C2×D312yes—
111C2×C3×C4⋊C4C22×C311yescharacteristic
112C2×C3×C4⋊C4C22×C312yes—
113C2×C3×C22⋊C4—34no—
114C2×C3×C22⋊C4—32no—
115C2×C3×C22⋊C4—34no—
116C2×C3×C42—21no—
117C2×C3×C42—22no—
118C2×C3×C42—34no—
119C2×C3×C42C2×D311yescharacteristic
120C2×C3×C42C2×D312yes—
121C2×C3×C42C22×C312yes—
122C2×C3×C42C22×C311yescharacteristic
123#946C22×C3116yescomplemented
124C2×(C2×D3)⋊C4C22×C312yes—
125C2×(C2×D3)⋊C4C22×C314yes—
126C2×(C2×D3)⋊C4C22×C314yes—
127C2×C4⋊Dic3C22×C311yescharacteristic
128C2×C4⋊Dic3C22×C312yes—
129C2×C4⋊Dic3C22×C312yes—
130C2×Dic3⋊C4C22×C314yes—
131C2×C4×Dic3C22×C314yes—
132#898C22×C3132yescomplemented
133#892C22×C3116yescomplemented
134#875C22×C3116yes—
135C4×D12C22×C318yes—
136#869C22×C318yescomplemented
137C4×Dic6C22×C3116yescomplemented
138C23×C32C2411yescharacteristic
139C22×C3×D3—24no—
140C22×C3×D3—24no—
141C22×C3×D3—24no—
142C22×C3×D3C2414yes—
143C32×Q8—28no—
144C2×C32×C4—24no—
145C2×C32×C4—24no—
146C2×C32×C4—22no—
147C2×C32×C4#4114yes—
148C2×C32×C4#4114yes—
149C2×C32×C4C2414yes—
150C2×C32×C4C2414yes—
151C2×C32×C4C2412yes—
152C2×C32×C4C2412yes—
153C2×C32×C4C2412yes—
154C2×C3×Dic3—28no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3C2414yes—
157C2×C3×Dic3C2418yes—
158C3×D12—28no—
159C3×C4×D3—216no—
160C3×Dic6—216no—
161#556—316no—
162C2×C4⋊Q8—31no—
163C2×#126—34no—
164C2×#124—32no—
165C2×C22⋊Q8—32no—
166C2×C4×Q8—32no—
167C2×C4×Q8C3×D311yescharacteristic, complemented
168C2×C4×D4—31no—
169C2×#117—32no—
170C24×C3—32no—
171C23×D3C23×C312yes—
172C2×C3×Q8—28no—
173C2×C3×Q8—24no—
174C2×C3×Q8—24no—
175C2×C3×Q8—24no—
176C2×C3×Q8—38no—
177C2×C3×Q8—68no—
178C2×C3×Q8C22×D314yes—
179C2×C3×Q8C23×C314yes—
180C2×C3×D4—34no—
181C2×C3×D4—64no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—32no—
188C22×C3×C4—34no—
189C22×C3×C4—38no—
190C22×C3×C4—68no—
191C22×C3×C4C22×D312yes—
192C22×C3×C4C22×D311yescharacteristic
193C22×C3×C4C22×D312yes—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C23×C312yes—
197C22×C3×C4C23×C312yes—
198C22×C3×C4C23×C311yescharacteristic
199C22×C3×C4C23×C311yescharacteristic
200C22×C3×C4C23×C311yescharacteristic
201C22×Dic3C23×C312yes—
202C22×Dic3C23×C314yes—
203C2×D12—24no—
204C2×D12C23×C314yes—
205C2×C4×D3—28no—
206C2×C4×D3C23×C318yes—
207C2×Dic6—28no—
208C2×Dic6C23×C318yes—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—316no—
212C3×C4⋊C4—38no—
213C3×C4⋊C4—34no—
214C3×C4⋊C4—38no—
215C3×C4⋊C4C22×D314yes—
216C3×C4⋊C4C22×D318yes—
217C3×C4⋊C4C23×C314yes—
218C3×C4⋊C4C23×C318yes—
219C3×C22⋊C4—316no—
220C3×C22⋊C4—38no—
221C3×C22⋊C4—316no—
222C3×C42—24no—
223C3×C42—28no—
224C3×C42—316no—
225C3×C42C22×D314yes—
226C3×C42C22×D318yes—
227C3×C42C23×C318yes—
228C3×C42C23×C314yes—
229(C2×D3)⋊C4C23×C318yes—
230(C2×D3)⋊C4C23×C3116yes—
231(C2×D3)⋊C4C23×C3116yes—
232C4⋊Dic3C23×C314yes—
233C4⋊Dic3C23×C318yes—
234C4⋊Dic3C23×C318yes—
235Dic3⋊C4C23×C3116yes—
236C4×Dic3C23×C3116yes—
237C22×C32C2×#4112yes—
238C22×C32C2×#4112yes—
239C22×C322−1+412yes—
240C22×C32C2511yescharacteristic
241C2×C3×D3—24no—
242C2×C3×D3—24no—
243C2×C3×D3—24no—
244C2×C3×D3—48no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—22no—
248C32×C4C2×#4114yes—
249C32×C4C2×#4114yes—
250C3×Dic3—28no—
251C3×Dic3—24no—
252C22×Q8—32no—
253C22×Q8C2×C3×D311yescharacteristic
254C22×D4—31no—
255C23×C4—32no—
256C4⋊Q8—38no—
257#126—332no—
258#124—316no—
259C22⋊Q8—316no—
260C4×Q8—316no—
261C4×Q8C2×C3×D318yescomplemented
262C4×D4—38no—
263#117—316no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4C2×C3×D311yescharacteristic
269C2×C4⋊C4C2×C3×D312yes—
270C2×C22⋊C4—34no—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—34no—
273C2×C42—34no—
274C2×C42C2×C3×D311yescharacteristic
275C2×C42C2×C3×D312yes—
276C23×C3—21no—
277C23×C3—34no—
278C23×C3—64no—
279C23×C3—64no—
280C23×C3—64no—
281C23×C3C23×D311yescharacteristic
282C23×C3C24×C311yescharacteristic
283C22×D3—24no—
284C22×D3—24no—
285C22×D3—24no—
286C22×D3C24×C314yes—
287C3×Q8—216no—
288C3×Q8—28no—
289C3×Q8—28no—
290C3×Q8—616no—
291C3×D4—68no—
292C2×C3×C4—28no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—34no—
307C2×C3×C4—38no—
308C2×C3×C4—44no—
309C2×C3×C4—42no—
310C2×C3×C4—64no—
311C2×C3×C4—68no—
312C2×C3×C4—616no—
313C2×C3×C4#23514yes—
314C2×C3×C4#23914yes—
315C2×C3×C4C3×#4114yes—
316C2×C3×C4C3×#4114yes—
317C2×C3×C4C23×D314yes—
318C2×C3×C4C23×D312yes—
319C2×C3×C4C23×D314yes—
320C2×C3×C4C23×D312yes—
321C2×C3×C4C23×D312yes—
322C2×C3×C4C24×C314yes—
323C2×C3×C4C24×C314yes—
324C2×C3×C4C24×C312yes—
325C2×C3×C4C24×C312yes—
326C2×C3×C4C24×C312yes—
327C2×Dic3—28no—
328C2×Dic3—24no—
329C2×Dic3C24×C314yes—
330C2×Dic3C24×C318yes—
331D12—28no—
332C4×D3—216no—
333Dic6—216no—
334C2×C32#55614yescomplemented
335C2×C32C22×#4111yescharacteristic
336C2×C32C22×#4111yescharacteristic
337C2×C32C2×2−1+411yescharacteristic
338C3×D3—44no—
339C24—32no—
340C2×Q8—24no—
341C2×Q8—38no—
342C2×Q8—68no—
343C2×Q8C22×C3×D314yes—
344C2×D4—34no—
345C2×D4—64no—
346C22×C4—32no—
347C22×C4—34no—
348C22×C4—38no—
349C22×C4—68no—
350C22×C4C22×C3×D312yes—
351C22×C4C22×C3×D311yescharacteristic
352C22×C4C22×C3×D312yes—
353C22×C4C22×C3×D311yescharacteristic
354C22×C4C22×C3×D311yescharacteristic
355C4⋊C4—38no—
356C4⋊C4—34no—
357C4⋊C4—316no—
358C4⋊C4—38no—
359C4⋊C4C22×C3×D314yes—
360C4⋊C4C22×C3×D318yes—
361C22⋊C4—316no—
362C22⋊C4—38no—
363C22⋊C4—316no—
364C42—316no—
365C42C22×C3×D314yes—
366C42C22×C3×D318yes—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—21no—
371C22×C3—64no—
372C22×C3—64no—
373C22×C3—64no—
374C22×C3—128no—
375C22×C3C2×#23512yes—
376C22×C3C2×#23912yes—
377C22×C3#101012yes—
378C22×C3C2×C3×#4112yes—
379C22×C3C2×C3×#4112yes—
380C22×C3C3×2−1+412yes—
381C22×C3C24×D311yescharacteristic
382C22×C3C25×C311yescharacteristic
383C2×D3—24no—
384C2×D3—24no—
385C2×D3—24no—
386C2×D3—48no—
387C3×C4—28no—
388C3×C4—24no—
389C3×C4—24no—
390C3×C4—22no—
391C3×C4—24no—
392C3×C4—24no—
393C3×C4—22no—
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—44no—
397C3×C4—42no—
398C3×C4—64no—
399C3×C4—68no—
400C3×C4C2×#23514yes—
401C3×C4C2×#23914yes—
402C3×C4C2×C3×#4114yes—
403C3×C4C2×C3×#4114yes—
404Dic3—28no—
405Dic3—24no—
406C32C2×#55611yescharacteristic, complemented, Sylow
407C23—34no—
408C23—64no—
409C23—64no—
410C23—64no—
411C23C23×C3×D311yescharacteristic
412Q8—28no—
413Q8—616no—
414D4—68no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—34no—
419C2×C4—38no—
420C2×C4—64no—
421C2×C4—68no—
422C2×C4—616no—
423C2×C4C3×#23514yes—
424C2×C4C3×#23914yes—
425C2×C4C23×C3×D314yes—
426C2×C4C23×C3×D312yes—
427C2×C4C23×C3×D314yes—
428C2×C4C23×C3×D312yes—
429C2×C4C23×C3×D312yes—
430C2×C3—24no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—124no—
435C2×C3#558414yesdirect product factor
436C2×C3C3×#55614yescomplemented
437C2×C3C22×#23511yescharacteristic
438C2×C3C22×#23911yescharacteristic
439C2×C3C2×#101011yescharacteristic
440C2×C3C22×C3×#4111yescharacteristic
441C2×C3C22×C3×#4111yescharacteristic
442C2×C3C2×C3×2−1+411yescharacteristic
443D3—44no—
444C22—64no—
445C22—64no—
446C22—64no—
447C22—128no—
448C22C2×C3×#23512yes—
449C22C2×C3×#23912yes—
450C22C3×#101012yes—
451C22C24×C3×D311yescharacteristic
452C4—24no—
453C4—24no—
454C4—22no—
455C4—64no—
456C4—68no—
457C4C2×C3×#23514yes—
458C4C2×C3×#23914yes—
459C3—21no—
460C3C2×#558411yescharacteristic, direct product factor
461C3C2×C3×#55611yescharacteristic, complemented
462C2—124no—
463C2C3×#558414yesdirect product factor
464C2C22×C3×#23511yescharacteristic
465C2C22×C3×#23911yescharacteristic
466C2C2×C3×#101011yescharacteristic
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×#556
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. C2×C32×C4×Q8C2semidirect
2. C2×C3×#946C2semidirect
3. C3×#5584C2direct (N × H)
4. C2×C3×#898C2semidirect
5. C2×C3×#892C2semidirect
8. C2×C3×#869C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5584C3direct (N × H)
16. C32×C4×Q8C22semidirect
21. C3×#946C22semidirect
30. C3×#898C22semidirect
31. C3×#892C22semidirect
34. C3×#869C22semidirect
35. C3×C4×Dic6C22semidirect
43. C2×C3×C4×Q8D3semidirect
44. C2×C3×C4×Q8C2×C3semidirect
47. C2×#946C2×C3semidirect
48. #5584C2×C3direct (N × H)
49. C2×#898C2×C3semidirect
50. C2×#892C2×C3semidirect
53. C2×#869C2×C3semidirect
54. C2×C4×Dic6C2×C3semidirect
99. C3×C4×Q8C2×D3semidirect
100. C3×C4×Q8C22×C3semidirect
123. #946C22×C3semidirect
132. #898C22×C3semidirect
133. #892C22×C3semidirect
136. #869C22×C3semidirect
137. C4×Dic6C22×C3semidirect
167. C2×C4×Q8C3×D3semidirect
261. C4×Q8C2×C3×D3semidirect
334. C2×C32#556semidirect
406. C32C2×#556semidirect
435. C2×C3#5584direct (N × H)
436. C2×C3C3×#556semidirect
460. C3C2×#5584direct (N × H)
461. C3C2×C3×#556semidirect
463. C2C3×#5584direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C2×C3×#9462
C3×#55842
C2×C3×#8982
C2×C3×#8922
C2×C3×#8752
C2×C3×C4×D122
C2×C3×#8692
C2×C3×C4×Dic62
C2×C3×#5563
C2×#55843