Finite Group Databasebeta

← C2×C3×#5542 #11903361

Subgroups — C2×C3×#5542 #11903361

Order 1152.

2437 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×C3×#936C212yesmaximal
3C2×C3×#926C212yesmaximal, complemented
4C2×C3×#924C212yesmaximal, complemented
5C3×#5542C2116yesmaximal, direct product factor
6C2×C3×#887C212yesmaximal, complemented
7C2×C3×#879C214yesmaximal, complemented
8C2×C3×#869C211yesmaximal, characteristic, complemented
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×#556—31nomaximal
11C2×#5542C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C23×C32×C4C2212yes—
14C22×C3×Dic6C2212yes—
15C22×C3×Dic6C2211yescharacteristic
16C32×C4×D4C2218yes—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4C2212yes—
19C2×C32×C42C2211yescharacteristic
20C2×C3×#217C2214yes—
21C2×C3×#217C2214yes—
22C3×#936C22116yes—
23C3×#926C22116yescomplemented
24C2×C3×C4⋊Dic3C2212yes—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×C4⋊Dic3C2212yes—
27C3×#924C22116yescomplemented
28C2×C3×Dic3⋊C4C2214yes—
29C2×C3×Dic3⋊C4C2212yes—
30C2×C3×C4×Dic3C2214yes—
31C2×C3×C4×Dic3C2212yes—
32C3×#887C22116yescomplemented
33C3×#879C22132yescomplemented
34C3×#869C2218yescomplemented
35C3×C4×Dic6C2218yescomplemented
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—32no—
42C2×C3×C4×Q8—31no—
43C2×C3×C4×D4—21no—
44C2×C3×C4×D4D311yescharacteristic
45C2×C3×C4×D4C2×C311yescharacteristic
46C2×C3×#117—32no—
47C2×#936C2×C312yes—
48C2×#926C2×C312yescomplemented
49C2×#924C2×C312yescomplemented
50#5542C2×C3116yesdirect product factor
51C2×#887C2×C312yescomplemented
52C2×#879C2×C314yescomplemented
53C2×#869C2×C311yescharacteristic, complemented
54C2×C4×Dic6C2×C311yescharacteristic, complemented
55C24×C32C2312yes—
56C2×C32×D4—24no—
57C2×C32×D4C2314yes—
58C22×C32×C4—28no—
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2318yes—
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2311yescharacteristic
64C22×C3×Dic3C2312yes—
65C22×C3×Dic3C2312yes—
66C22×C3×Dic3C2314yes—
67C2×C3×Dic6—28no—
68C2×C3×Dic6—24no—
69C2×C3×Dic6C2318yes—
70C2×C3×Dic6C2314yes—
71C32×C4⋊C4C2314yes—
72C32×C22⋊C4C2318yes—
73C32×C42C2314yes—
74C3×#217C23116yes—
75C3×#217C23116yes—
76C3×C4⋊Dic3C2318yes—
77C3×C4⋊Dic3C2318yes—
78C3×C4⋊Dic3C2314yes—
79C3×Dic3⋊C4C2318yes—
80C3×Dic3⋊C4C23116yes—
81C3×C4×Dic3C23116yes—
82C3×C4×Dic3C2318yes—
83C2×#556—31noSylow
84C22×C3×Q8—31no—
85C22×C3×Q8—32no—
86C22×C3×D4—21no—
87C22×C3×D4C2×D311yescharacteristic
88C22×C3×D4C22×C311yescharacteristic
89C23×C3×C4—22no—
90C23×C3×C4C2×D312yes—
91C23×C3×C4C22×C312yes—
92C22×Dic6C22×C311yescharacteristic
93C22×Dic6C22×C312yes—
94C3×C4⋊Q8—38no—
95C3×#126—332no—
96C3×#124—316no—
97C3×C22⋊Q8—316no—
98C3×C4×Q8—316no—
99C3×C4×Q8—38no—
100C3×C4×D4—28no—
101C3×C4×D4C2×D318yes—
102C3×C4×D4C22×C318yes—
103C3×#117—316no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—34no—
106C2×C3×C4⋊C4—32no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—32no—
109C2×C3×C4⋊C4—32no—
110C2×C3×C4⋊C4C2×D311yescharacteristic
111C2×C3×C4⋊C4C22×C311yescharacteristic
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—34no—
114C2×C3×C22⋊C4—34no—
115C2×C3×C22⋊C4C2×D312yes—
116C2×C3×C22⋊C4C22×C312yes—
117C2×C3×C42—21no—
118C2×C3×C42—34no—
119C2×C3×C42—32no—
120C2×C3×C42C2×D311yescharacteristic
121C2×C3×C42C22×C311yescharacteristic
122C2×#217C22×C314yes—
123C2×#217C22×C314yes—
124#936C22×C3116yes—
125#926C22×C3116yescomplemented
126C2×C4⋊Dic3C22×C311yescharacteristic
127C2×C4⋊Dic3C22×C312yes—
128C2×C4⋊Dic3C22×C312yes—
129#924C22×C3116yescomplemented
130C2×Dic3⋊C4C22×C312yes—
131C2×Dic3⋊C4C22×C314yes—
132C2×C4×Dic3C22×C312yes—
133C2×C4×Dic3C22×C314yes—
134#887C22×C3116yescomplemented
135#879C22×C3132yescomplemented
136#869C22×C318yescomplemented
137C4×Dic6C22×C318yescomplemented
138C23×C32—24no—
139C23×C32—24no—
140C23×C32—24no—
141C23×C32C2414yes—
142C23×C32C2411yescharacteristic
143C32×D4—28no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4—216no—
147C2×C32×C4#4114yes—
148C2×C32×C4#4114yes—
149C2×C32×C4C2414yes—
150C2×C32×C4C2412yes—
151C2×C32×C4C2412yes—
152C2×C32×C4C2412yes—
153C2×C3×Dic3—28no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3C2414yes—
157C2×C3×Dic3C2414yes—
158C2×C3×Dic3C2418yes—
159C3×Dic6—216no—
160C3×Dic6—28no—
161#556—316no—
162C2×C4⋊Q8—31no—
163C2×#126—34no—
164C2×#124—32no—
165C2×C22⋊Q8—32no—
166C2×C4×Q8—32no—
167C2×C4×Q8—31no—
168C2×C4×D4C3×D311yescharacteristic
169C2×#117—32no—
170C24×C3—22no—
171C24×C3C22×D312yes—
172C24×C3C23×C312yes—
173C2×C3×Q8—34no—
174C2×C3×Q8—38no—
175C2×C3×Q8—68no—
176C2×C3×Q8—64no—
177C2×C3×D4—28no—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4C22×D314yes—
182C2×C3×D4C23×C314yes—
183C22×C3×C4—28no—
184C22×C3×C4—28no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—28no—
190C22×C3×C4—32no—
191C22×C3×C4—34no—
192C22×C3×C4—32no—
193C22×C3×C4—48no—
194C22×C3×C4C22×D318yes—
195C22×C3×C4C22×D312yes—
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C22×D311yescharacteristic
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C23×C312yes—
200C22×C3×C4C23×C311yescharacteristic
201C22×C3×C4C23×C318yes—
202C22×C3×C4C23×C311yescharacteristic
203C22×C3×C4C23×C311yescharacteristic
204C22×Dic3C23×C312yes—
205C22×Dic3C23×C312yes—
206C22×Dic3C23×C314yes—
207C2×Dic6—28no—
208C2×Dic6—24no—
209C2×Dic6C23×C314yes—
210C2×Dic6C23×C318yes—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—316no—
213C3×C4⋊C4—38no—
214C3×C4⋊C4—34no—
215C3×C4⋊C4—38no—
216C3×C4⋊C4—38no—
217C3×C4⋊C4C22×D314yes—
218C3×C4⋊C4C23×C314yes—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—316no—
221C3×C22⋊C4—316no—
222C3×C22⋊C4C22×D318yes—
223C3×C22⋊C4C23×C318yes—
224C3×C42—24no—
225C3×C42—316no—
226C3×C42—38no—
227C3×C42C22×D314yes—
228C3×C42C23×C314yes—
229#217C23×C3116yes—
230#217C23×C3116yes—
231C4⋊Dic3C23×C314yes—
232C4⋊Dic3C23×C318yes—
233C4⋊Dic3C23×C318yes—
234Dic3⋊C4C23×C318yes—
235Dic3⋊C4C23×C3116yes—
236C4×Dic3C23×C318yes—
237C4×Dic3C23×C3116yes—
238C22×C32—24no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—48no—
242C22×C32C2×#4112yes—
243C22×C32C2×#4112yes—
244C22×C322−1+412yes—
245C22×C32C2511yescharacteristic
246C32×C4—24no—
247C32×C4—22no—
248C32×C4C2×#4114yes—
249C32×C4C2×#4114yes—
250C3×Dic3—28no—
251C3×Dic3—24no—
252C3×Dic3—24no—
253C22×Q8—31no—
254C22×Q8—32no—
255C22×D4C2×C3×D311yescharacteristic
256C23×C4C2×C3×D312yes—
257C4⋊Q8—38no—
258#126—332no—
259#124—316no—
260C22⋊Q8—316no—
261C4×Q8—316no—
262C4×Q8—38no—
263C4×D4C2×C3×D318yes—
264#117—316no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4C2×C3×D311yescharacteristic
271C2×C22⋊C4—34no—
272C2×C22⋊C4—34no—
273C2×C22⋊C4C2×C3×D312yes—
274C2×C42—34no—
275C2×C42—32no—
276C2×C42C2×C3×D311yescharacteristic
277C23×C3—28no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—21no—
285C23×C3—24no—
286C23×C3—44no—
287C23×C3—44no—
288C23×C3C23×D314yes—
289C23×C3C23×D311yescharacteristic
290C23×C3C24×C314yes—
291C23×C3C24×C311yescharacteristic
292C3×Q8—616no—
293C3×Q8—68no—
294C3×D4—216no—
295C3×D4—28no—
296C3×D4—28no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—216no—
302C2×C3×C4—216no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—34no—
310C2×C3×C4—38no—
311C2×C3×C4—34no—
312C2×C3×C4—416no—
313C2×C3×C4—42no—
314C2×C3×C4—44no—
315C2×C3×C4—68no—
316C2×C3×C4—64no—
317C2×C3×C4—64no—
318C2×C3×C4#23514yes—
319C2×C3×C4#23714yes—
320C2×C3×C4C3×#4114yes—
321C2×C3×C4C3×#4114yes—
322C2×C3×C4C23×D314yes—
323C2×C3×C4C23×D312yes—
324C2×C3×C4C23×D312yes—
325C2×C3×C4C23×D312yes—
326C2×C3×C4C24×C314yes—
327C2×C3×C4C24×C312yes—
328C2×C3×C4C24×C312yes—
329C2×C3×C4C24×C312yes—
330C2×Dic3—28no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3C24×C314yes—
334C2×Dic3C24×C314yes—
335C2×Dic3C24×C318yes—
336Dic6—216no—
337Dic6—28no—
338C2×C32—44no—
339C2×C32#55614yescomplemented
340C2×C32C22×#4111yescharacteristic
341C2×C32C22×#4111yescharacteristic
342C2×C32C2×2−1+411yescharacteristic
343C24C22×C3×D312yes—
344C2×Q8—34no—
345C2×Q8—38no—
346C2×Q8—68no—
347C2×Q8—64no—
348C2×D4—24no—
349C2×D4C22×C3×D314yes—
350C22×C4—28no—
351C22×C4—32no—
352C22×C4—34no—
353C22×C4—32no—
354C22×C4C22×C3×D312yes—
355C22×C4C22×C3×D311yescharacteristic
356C22×C4C22×C3×D318yes—
357C22×C4C22×C3×D311yescharacteristic
358C22×C4C22×C3×D311yescharacteristic
359C4⋊C4—34no—
360C4⋊C4—38no—
361C4⋊C4—316no—
362C4⋊C4—38no—
363C4⋊C4—38no—
364C4⋊C4C22×C3×D314yes—
365C22⋊C4—316no—
366C22⋊C4—316no—
367C22⋊C4C22×C3×D318yes—
368C42—316no—
369C42—38no—
370C42C22×C3×D314yes—
371C22×C3—28no—
372C22×C3—24no—
373C22×C3—24no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—21no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—416no—
385C22×C3—48no—
386C22×C3—48no—
387C22×C3C2×#23512yes—
388C22×C3C2×#23712yes—
389C22×C3#101012yes—
390C22×C3C2×C3×#4112yes—
391C22×C3C2×C3×#4112yes—
392C22×C3C3×2−1+412yes—
393C22×C3C24×D311yescharacteristic
394C22×C3C25×C311yescharacteristic
395C3×C4—24no—
396C3×C4—22no—
397C3×C4—24no—
398C3×C4—22no—
399C3×C4—24no—
400C3×C4—24no—
401C3×C4—44no—
402C3×C4—42no—
403C3×C4—68no—
404C3×C4—64no—
405C3×C4—64no—
406C3×C4C2×#23514yes—
407C3×C4C2×#23714yes—
408C3×C4C2×C3×#4114yes—
409C3×C4C2×C3×#4114yes—
410Dic3—28no—
411Dic3—24no—
412Dic3—24no—
413C32C2×#55611yescharacteristic, complemented, Sylow
414C23—24no—
415C23—24no—
416C23—24no—
417C23C23×C3×D314yes—
418C23C23×C3×D311yescharacteristic
419Q8—616no—
420Q8—68no—
421D4—28no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—216no—
425C2×C4—34no—
426C2×C4—38no—
427C2×C4—34no—
428C2×C4—68no—
429C2×C4—64no—
430C2×C4—64no—
431C2×C4C3×#23514yes—
432C2×C4C3×#23714yes—
433C2×C4C23×C3×D314yes—
434C2×C4C23×C3×D312yes—
435C2×C4C23×C3×D312yes—
436C2×C4C23×C3×D312yes—
437C2×C3—24no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—48no—
442C2×C3—44no—
443C2×C3—44no—
444C2×C3#554214yesdirect product factor
445C2×C3C3×#55614yescomplemented
446C2×C3C22×#23511yescharacteristic
447C2×C3C22×#23711yescharacteristic
448C2×C3C2×#101011yescharacteristic
449C2×C3C22×C3×#4111yescharacteristic
450C2×C3C22×C3×#4111yescharacteristic
451C2×C3C2×C3×2−1+411yescharacteristic
452C22—24no—
453C22—24no—
454C22—24no—
455C22—48no—
456C22C2×C3×#23512yes—
457C22C2×C3×#23712yes—
458C22C3×#101012yes—
459C22C24×C3×D311yescharacteristic
460C4—24no—
461C4—22no—
462C4—68no—
463C4—64no—
464C4—64no—
465C4C2×C3×#23514yes—
466C4C2×C3×#23714yes—
467C3—21no—
468C3C2×#554211yescharacteristic, direct product factor
469C3C2×C3×#55611yescharacteristic, complemented
470C2—44no—
471C2C3×#554214yesdirect product factor
472C2C22×C3×#23511yescharacteristic
473C2C22×C3×#23711yescharacteristic
474C2C2×C3×#101011yescharacteristic
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×#556
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. C2×C3×#926C2semidirect
4. C2×C3×#924C2semidirect
5. C3×#5542C2direct (N × H)
6. C2×C3×#887C2semidirect
7. C2×C3×#879C2semidirect
8. C2×C3×#869C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5542C3direct (N × H)
23. C3×#926C22semidirect
27. C3×#924C22semidirect
32. C3×#887C22semidirect
33. C3×#879C22semidirect
34. C3×#869C22semidirect
35. C3×C4×Dic6C22semidirect
48. C2×#926C2×C3semidirect
49. C2×#924C2×C3semidirect
50. #5542C2×C3direct (N × H)
51. C2×#887C2×C3semidirect
52. C2×#879C2×C3semidirect
53. C2×#869C2×C3semidirect
54. C2×C4×Dic6C2×C3semidirect
125. #926C22×C3semidirect
129. #924C22×C3semidirect
134. #887C22×C3semidirect
135. #879C22×C3semidirect
136. #869C22×C3semidirect
137. C4×Dic6C22×C3semidirect
339. C2×C32#556semidirect
413. C32C2×#556semidirect
444. C2×C3#5542direct (N × H)
445. C2×C3C3×#556semidirect
468. C3C2×#5542direct (N × H)
469. C3C2×C3×#556semidirect
471. C2C3×#5542direct (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×#9362
C2×C3×#9262
C2×C3×#9242
C3×#55422
C2×C3×#8872
C2×C3×#8792
C2×C3×#8692
C2×C3×C4×Dic62
C2×C3×#5563
C2×#55423