Finite Group Databasebeta

← C2×C3×#5543 #11903225

Subgroups — C2×C3×#5543 #11903225

Order 1152.

2613 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×C3×D4×Dic3C212yesmaximal
3C22×C3×C4⋊Dic3C212yesmaximal, complemented
4C2×C3×#924C212yesmaximal, complemented
5C3×#5543C2116yesmaximal, direct product factor
6C2×C3×#890C212yesmaximal, complemented
7C2×C3×#878C214yesmaximal, complemented
8C2×C3×#869C211yesmaximal, characteristic, complemented
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×#554—31nomaximal
11C2×#5543C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C23×C32×C4C2212yes—
14C23×C3×Dic3C2214yes—
15C22×C3×Dic6C2211yescharacteristic
16C22×C3×Dic6C2212yes—
17C32×C4×D4C2218yes—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C22⋊C4C2212yes—
20C2×C32×C42C2211yescharacteristic
21C2×C3×#217C2214yes—
22C3×D4×Dic3C22116yes—
23C2×C3×C4⋊Dic3C2214yes—
24C2×C3×C4⋊Dic3C2212yes—
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×C4⋊Dic3C22116yescomplemented
28C3×#924C22116yescomplemented
29C2×C3×Dic3⋊C4C2214yes—
30C2×C3×Dic3⋊C4C2212yes—
31C2×C3×C4×Dic3C2212yes—
32C3×#890C22116yescomplemented
33C3×#878C22132yescomplemented
34C3×#869C2218yescomplemented
35C3×C4×Dic6C2218yescomplemented
36C3×#554—316no—
37C2×C3×C4⋊Q8—31no—
38C2×C3×#125—32no—
39C2×C3×C22⋊Q8—32no—
40C2×C3×C22⋊Q8—34no—
41C2×C3×C4×Q8—31no—
42C2×C3×C4×D4—21no—
43C2×C3×C4×D4—32no—
44C2×C3×C4×D4D311yescharacteristic
45C2×C3×C4×D4C2×C311yescharacteristic
46C22×C3×C4⋊C4—32no—
47C2×D4×Dic3C2×C312yes—
48C22×C4⋊Dic3C2×C312yescomplemented
49C2×#924C2×C312yescomplemented
50#5543C2×C3116yesdirect product factor
51C2×#890C2×C312yescomplemented
52C2×#878C2×C314yescomplemented
53C2×#869C2×C311yescharacteristic, complemented
54C2×C4×Dic6C2×C311yescharacteristic, complemented
55C24×C32C2312yes—
56C2×C32×D4Q818yescomplemented
57C2×C32×D4C2314yes—
58C22×C32×C4—28no—
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2318yes—
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2311yescharacteristic
64C22×C3×Dic3—48no—
65C22×C3×Dic3C2312yes—
66C22×C3×Dic3C2312yes—
67C22×C3×Dic3C2314yes—
68C22×C3×Dic3C23116yes—
69C2×C3×Dic6—24no—
70C2×C3×Dic6—44no—
71C2×C3×Dic6C2314yes—
72C2×C3×Dic6C2318yes—
73C32×C4⋊C4C2314yes—
74C32×C22⋊C4C2318yes—
75C32×C42C2314yes—
76C3×#217C23116yes—
77C3×C4⋊Dic3C23116yes—
78C3×C4⋊Dic3C2318yes—
79C3×C4⋊Dic3C2318yes—
80C3×C4⋊Dic3C2314yes—
81C3×Dic3⋊C4C2318yes—
82C3×Dic3⋊C4C23116yes—
83C3×C4×Dic3C2318yes—
84C2×#554—31noSylow
85C22×C3×Q8—31no—
86C22×C3×Q8—32no—
87C22×C3×D4—21no—
88C22×C3×D4C2×D311yescharacteristic
89C22×C3×D4C22×C311yescharacteristic
90C23×C3×C4—22no—
91C23×C3×C4—34no—
92C23×C3×C4C2×D312yes—
93C23×C3×C4C22×C312yes—
94C23×Dic3C22×C314yes—
95C22×Dic6C22×C311yescharacteristic
96C22×Dic6C22×C312yes—
97C3×C4⋊Q8—38no—
98C3×#125—316no—
99C3×C22⋊Q8—316no—
100C3×C22⋊Q8—332no—
101C3×C4×Q8—38no—
102C3×C4×D4—28no—
103C3×C4×D4—316no—
104C3×C4×D4C2×D318yes—
105C3×C4×D4C22×C318yes—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—32no—
108C2×C3×C4⋊C4—34no—
109C2×C3×C4⋊C4—31no—
110C2×C3×C4⋊C4—34no—
111C2×C3×C4⋊C4—32no—
112C2×C3×C4⋊C4—32no—
113C2×C3×C4⋊C4—316no—
114C2×C3×C4⋊C4C2×D311yescharacteristic
115C2×C3×C4⋊C4C22×C311yescharacteristic
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4—34no—
118C2×C3×C22⋊C4C2×D312yes—
119C2×C3×C22⋊C4C22×C312yes—
120C2×C3×C42—21no—
121C2×C3×C42—32no—
122C2×C3×C42C2×D311yescharacteristic
123C2×C3×C42C22×C311yescharacteristic
124C2×#217C22×C314yes—
125D4×Dic3C22×C3116yes—
126C2×C4⋊Dic3C22×C311yescharacteristic
127C2×C4⋊Dic3C22×C312yes—
128C2×C4⋊Dic3C22×C312yes—
129C2×C4⋊Dic3C22×C314yes—
130C2×C4⋊Dic3C22×C3116yescomplemented
131#924C22×C3116yescomplemented
132C2×Dic3⋊C4C22×C312yes—
133C2×Dic3⋊C4C22×C314yes—
134C2×C4×Dic3C22×C312yes—
135#890C22×C3116yescomplemented
136#878C22×C3132yescomplemented
137#869C22×C318yescomplemented
138C4×Dic6C22×C318yescomplemented
139C23×C32—24no—
140C23×C32—24no—
141C23×C32C2×Q818yes—
142C23×C32C2414yes—
143C23×C32C2411yescharacteristic
144C32×D4C2×Q8116yescomplemented
145C2×C32×C4—24no—
146C2×C32×C4—22no—
147C2×C32×C4—216no—
148C2×C32×C4C2×Q814yes—
149C2×C32×C4#4114yes—
150C2×C32×C4C2414yes—
151C2×C32×C4C2412yes—
152C2×C32×C4C2412yes—
153C2×C32×C4C2412yes—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—28no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—416no—
158C2×C3×Dic3C2414yes—
159C2×C3×Dic3C2414yes—
160C2×C3×Dic3C2418yes—
161C3×Dic6—28no—
162C3×Dic6—48no—
163#554—316no—
164C2×C4⋊Q8—31no—
165C2×#125—32no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—34no—
168C2×C4×Q8—31no—
169C2×C4×D4—32no—
170C2×C4×D4C3×D311yescharacteristic
171C22×C4⋊C4—32no—
172C24×C3—22no—
173C24×C3C22×D312yes—
174C24×C3C23×C312yes—
175C2×C3×Q8—34no—
176C2×C3×Q8—38no—
177C2×C3×Q8—64no—
178C2×C3×Q8—124no—
179C2×C3×D4—28no—
180C2×C3×D4—24no—
181C2×C3×D4Dic618yescomplemented
182C2×C3×D4C3×Q818yescomplemented
183C2×C3×D4C22×D314yes—
184C2×C3×D4C23×C314yes—
185C22×C3×C4—28no—
186C22×C3×C4—28no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—28no—
192C22×C3×C4—32no—
193C22×C3×C4—32no—
194C22×C3×C4—34no—
195C22×C3×C4—316no—
196C22×C3×C4—48no—
197C22×C3×C4—128no—
198C22×C3×C4C22×D318yes—
199C22×C3×C4C22×D311yescharacteristic
200C22×C3×C4C22×D312yes—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C23×C312yes—
204C22×C3×C4C23×C311yescharacteristic
205C22×C3×C4C23×C318yes—
206C22×C3×C4C23×C311yescharacteristic
207C22×C3×C4C23×C311yescharacteristic
208C22×Dic3—48no—
209C22×Dic3C23×C312yes—
210C22×Dic3C23×C312yes—
211C22×Dic3C23×C314yes—
212C22×Dic3C23×C3116yes—
213C2×Dic6—24no—
214C2×Dic6—44no—
215C2×Dic6C23×C314yes—
216C2×Dic6C23×C318yes—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—38no—
219C3×C4⋊C4—316no—
220C3×C4⋊C4—34no—
221C3×C4⋊C4—316no—
222C3×C4⋊C4—38no—
223C3×C4⋊C4—38no—
224C3×C4⋊C4C22×D314yes—
225C3×C4⋊C4C23×C314yes—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—316no—
228C3×C22⋊C4C22×D318yes—
229C3×C22⋊C4C23×C318yes—
230C3×C42—24no—
231C3×C42—38no—
232C3×C42C22×D314yes—
233C3×C42C23×C314yes—
234#217C23×C3116yes—
235C4⋊Dic3C23×C314yes—
236C4⋊Dic3C23×C318yes—
237C4⋊Dic3C23×C318yes—
238C4⋊Dic3C23×C3116yes—
239Dic3⋊C4C23×C318yes—
240Dic3⋊C4C23×C3116yes—
241C4×Dic3C23×C318yes—
242C22×C32—216no—
243C22×C32—24no—
244C22×C32—24no—
245C22×C32C22×Q818yes—
246C22×C32C22×Q812yes—
247C22×C32C2×#4112yes—
248C22×C32H(5,𝔽2)12yes—
249C22×C32C2511yescharacteristic
250C32×C4—24no—
251C32×C4—22no—
252C32×C4C22×Q814yes—
253C32×C4C2×#4114yes—
254C3×Dic3—24no—
255C3×Dic3—28no—
256C3×Dic3—24no—
257C22×Q8—31no—
258C22×Q8—32no—
259C22×D4C2×C3×D311yescharacteristic
260C23×C4—34no—
261C23×C4C2×C3×D312yes—
262C4⋊Q8—38no—
263#125—316no—
264C22⋊Q8—316no—
265C22⋊Q8—332no—
266C4×Q8—38no—
267C4×D4—316no—
268C4×D4C2×C3×D318yes—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—34no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—316no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4C2×C3×D311yescharacteristic
277C2×C22⋊C4—34no—
278C2×C22⋊C4C2×C3×D312yes—
279C2×C42—32no—
280C2×C42C2×C3×D311yescharacteristic
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—21no—
286C23×C3—24no—
287C23×C3—28no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3C2×Dic618yes—
291C23×C3C2×C3×Q818yes—
292C23×C3C23×D314yes—
293C23×C3C23×D311yescharacteristic
294C23×C3C24×C314yes—
295C23×C3C24×C311yescharacteristic
296C3×Q8—68no—
297C3×Q8—128no—
298C3×D4—216no—
299C3×D4C2×Dic6116yescomplemented
300C3×D4C2×C3×Q8116yescomplemented
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—216no—
306C2×C3×C4—216no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—34no—
314C2×C3×C4—34no—
315C2×C3×C4—38no—
316C2×C3×C4—416no—
317C2×C3×C4—42no—
318C2×C3×C4—44no—
319C2×C3×C4—64no—
320C2×C3×C4—64no—
321C2×C3×C4—68no—
322C2×C3×C4—1216no—
323C2×C3×C4C2×Dic614yes—
324C2×C3×C4#23714yes—
325C2×C3×C4C2×C3×Q814yes—
326C2×C3×C4C3×#4114yes—
327C2×C3×C4C23×D312yes—
328C2×C3×C4C23×D314yes—
329C2×C3×C4C23×D312yes—
330C2×C3×C4C23×D312yes—
331C2×C3×C4C24×C314yes—
332C2×C3×C4C24×C312yes—
333C2×C3×C4C24×C312yes—
334C2×C3×C4C24×C312yes—
335C2×Dic3—24no—
336C2×Dic3—28no—
337C2×Dic3—24no—
338C2×Dic3—416no—
339C2×Dic3C24×C314yes—
340C2×Dic3C24×C314yes—
341C2×Dic3C24×C318yes—
342Dic6—28no—
343Dic6—48no—
344C2×C32—28no—
345C2×C32#55414yescomplemented
346C2×C32C23×Q811yescharacteristic
347C2×C32C22×#4111yescharacteristic
348C2×C32C2×H(5,𝔽2)11yescharacteristic
349C24C22×C3×D312yes—
350C2×Q8—34no—
351C2×Q8—38no—
352C2×Q8—64no—
353C2×Q8—124no—
354C2×D4C3×Dic618yescomplemented
355C2×D4C22×C3×D314yes—
356C22×C4—28no—
357C22×C4—32no—
358C22×C4—32no—
359C22×C4—34no—
360C22×C4—316no—
361C22×C4—128no—
362C22×C4C22×C3×D311yescharacteristic
363C22×C4C22×C3×D312yes—
364C22×C4C22×C3×D318yes—
365C22×C4C22×C3×D311yescharacteristic
366C22×C4C22×C3×D311yescharacteristic
367C4⋊C4—34no—
368C4⋊C4—316no—
369C4⋊C4—38no—
370C4⋊C4—38no—
371C4⋊C4—38no—
372C4⋊C4—316no—
373C4⋊C4C22×C3×D314yes—
374C22⋊C4—316no—
375C22⋊C4C22×C3×D318yes—
376C42—38no—
377C42C22×C3×D314yes—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—216no—
383C22×C3—216no—
384C22×C3—22no—
385C22×C3—28no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—21no—
389C22×C3—44no—
390C22×C3—416no—
391C22×C3—44no—
392C22×C3C22×Dic618yes—
393C22×C3C22×Dic612yes—
394C22×C3C2×#23712yes—
395C22×C3#100912yes—
396C22×C3C22×C3×Q818yes—
397C22×C3C22×C3×Q812yes—
398C22×C3C2×C3×#4112yes—
399C22×C3C3×H(5,𝔽2)12yes—
400C22×C3C24×D311yescharacteristic
401C22×C3C25×C311yescharacteristic
402C3×C4—24no—
403C3×C4—22no—
404C3×C4—24no—
405C3×C4—22no—
406C3×C4—24no—
407C3×C4—24no—
408C3×C4—44no—
409C3×C4—42no—
410C3×C4—64no—
411C3×C4—64no—
412C3×C4—68no—
413C3×C4C22×Dic614yes—
414C3×C4C2×#23714yes—
415C3×C4C22×C3×Q814yes—
416C3×C4C2×C3×#4114yes—
417Dic3—24no—
418Dic3—28no—
419Dic3—24no—
420C32C2×#55411yescharacteristic, complemented, Sylow
421C23—24no—
422C23—24no—
423C23C2×C3×Dic618yes—
424C23C23×C3×D314yes—
425C23C23×C3×D311yescharacteristic
426Q8—68no—
427Q8—128no—
428D4C2×C3×Dic6116yescomplemented
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—216no—
432C2×C4—34no—
433C2×C4—34no—
434C2×C4—38no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—68no—
438C2×C4—1216no—
439C2×C4C2×C3×Dic614yes—
440C2×C4C3×#23714yes—
441C2×C4C23×C3×D312yes—
442C2×C4C23×C3×D314yes—
443C2×C4C23×C3×D312yes—
444C2×C4C23×C3×D312yes—
445C2×C3—28no—
446C2×C3—28no—
447C2×C3—24no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—48no—
452C2×C3#554314yesdirect product factor
453C2×C3C3×#55414yescomplemented
454C2×C3C23×Dic611yescharacteristic
455C2×C3C22×#23711yescharacteristic
456C2×C3C2×#100911yescharacteristic
457C2×C3C23×C3×Q811yescharacteristic
458C2×C3C22×C3×#4111yescharacteristic
459C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
460C22—216no—
461C22—24no—
462C22—24no—
463C22C22×C3×Dic618yes—
464C22C22×C3×Dic612yes—
465C22C2×C3×#23712yes—
466C22C3×#100912yes—
467C22C24×C3×D311yescharacteristic
468C4—24no—
469C4—22no—
470C4—64no—
471C4—64no—
472C4—68no—
473C4C22×C3×Dic614yes—
474C4C2×C3×#23714yes—
475C3—21no—
476C3C2×#554311yescharacteristic, direct product factor
477C3C2×C3×#55411yescharacteristic, complemented
478C2—28no—
479C2C3×#554314yesdirect product factor
480C2C23×C3×Dic611yescharacteristic
481C2C22×C3×#23711yescharacteristic
482C2C2×C3×#100911yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C24
prime radicalC22×C32×D4C22
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×#554
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C22×C3×C4⋊Dic3C2semidirect
4. C2×C3×#924C2semidirect
5. C3×#5543C2direct (N × H)
6. C2×C3×#890C2semidirect
7. C2×C3×#878C2semidirect
8. C2×C3×#869C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5543C3direct (N × H)
27. C2×C3×C4⋊Dic3C22semidirect
28. C3×#924C22semidirect
32. C3×#890C22semidirect
33. C3×#878C22semidirect
34. C3×#869C22semidirect
35. C3×C4×Dic6C22semidirect
48. C22×C4⋊Dic3C2×C3semidirect
49. C2×#924C2×C3semidirect
50. #5543C2×C3direct (N × H)
51. C2×#890C2×C3semidirect
52. C2×#878C2×C3semidirect
53. C2×#869C2×C3semidirect
54. C2×C4×Dic6C2×C3semidirect
56. C2×C32×D4Q8semidirect
130. C2×C4⋊Dic3C22×C3semidirect
131. #924C22×C3semidirect
135. #890C22×C3semidirect
136. #878C22×C3semidirect
137. #869C22×C3semidirect
138. C4×Dic6C22×C3semidirect
144. C32×D4C2×Q8semidirect
181. C2×C3×D4Dic6semidirect
182. C2×C3×D4C3×Q8semidirect
299. C3×D4C2×Dic6semidirect
300. C3×D4C2×C3×Q8semidirect
345. C2×C32#554semidirect
354. C2×D4C3×Dic6semidirect
420. C32C2×#554semidirect
428. D4C2×C3×Dic6semidirect
452. C2×C3#5543direct (N × H)
453. C2×C3C3×#554semidirect
476. C3C2×#5543direct (N × H)
477. C3C2×C3×#554semidirect
479. C2C3×#5543direct (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×D42
C2×C3×D4×Dic32
C22×C3×C4⋊Dic32
C2×C3×#9242
C3×#55432
C2×C3×#8902
C2×C3×#8782
C2×C3×#8692
C2×C3×C4×Dic62
C2×C3×#5543
C2×#55433