Finite Group Databasebeta

← C2×C3×#5045 #11869476

Subgroups — C2×C3×#5045 #11869476

Order 1152.

993 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
1C2×C32×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#926C211yesmaximal, characteristic
3C2×C3×#835C211yesmaximal, characteristic, complemented
4C2×C3×#835C211yesmaximal, characteristic, complemented
5C2×C3×#830C211yesmaximal, characteristic
6C3×#5045C218yesmaximal, direct product factor
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C2×C3×#807C211yesmaximal, characteristic, complemented
9C2×C3×#485—31nomaximal
10C2×#5045C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C23×C32×C4C2211yescharacteristic
13C32×C22⋊Q8—22no—
14C32×C22⋊Q8C2214yes—
15C2×C32×C4⋊C4—21no—
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C22⋊C4—21no—
18C2×C3×#217—21no—
19C3×#926—22no—
20C3×#926C2214yes—
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3—21no—
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#835C2214yescomplemented
27C3×#835C2214yescomplemented
28C3×#830C2214yes—
29C3×#808C2214yescomplemented
30C3×#807C2214yescomplemented
31C2×C3×C22⋊Q8—21no—
32C2×C3×C22⋊Q8D311yescharacteristic
33C2×C3×C22⋊Q8C2×C311yescharacteristic
34C2×C3×#117—31no—
35C2×#926C2×C311yescharacteristic
36C3×#485—38no—
37C2×C3×#107—31no—
38C2×C3×#106—31no—
39C2×C3×#103—31no—
40C2×C3×#103—31no—
41C2×C3×C22⋊C8—31no—
42C2×#835C2×C311yescharacteristic, complemented
43C2×#835C2×C311yescharacteristic, complemented
44C2×#830C2×C311yescharacteristic
45#5045C2×C318yesdirect product factor
46C2×#808C2×C311yescharacteristic, complemented
47C2×#807C2×C311yescharacteristic, complemented
48C24×C32D411yescharacteristic
49C2×C32×Q8—21no—
50C2×C32×Q8—22no—
51C2×C32×Q8—22no—
52C2×C32×Q8C2312yes—
53C22×C32×C4—22no—
54C22×C32×C4—22no—
55C22×C32×C4—21no—
56C22×C32×C4—21no—
57C22×C32×C4D411yescharacteristic
58C22×C32×C4D412yes—
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—21no—
62C22×C3×Dic3—21no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2312yes—
66C32×C22⋊C4—24no—
67C3×#217—24no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3C2312yes—
71C3×C4⋊Dic3C2312yes—
72C3×C4×Dic3—24no—
73C2×C3×C3⋊C8—22no—
74C2×C3×C3⋊C8—22no—
75C2×C3×C3⋊C8C2312yes—
76C2×C3×C3⋊C8C2312yes—
77C2×#485—31noSylow
78C22×C3×Q8—21no—
79C22×C3×Q8C2×D311yescharacteristic
80C22×C3×Q8C22×C311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4C2×D311yescharacteristic
83C23×C3×C4C22×C311yescharacteristic
84C22×C3×C8—31no—
85C22×C3×C8—31no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—24no—
90C3×C22⋊Q8C2×D314yes—
91C3×C22⋊Q8C22×C314yes—
92C3×#117—34no—
93C3×#117—62no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4C2×D311yescharacteristic
101C2×C3×C4⋊C4C22×C311yescharacteristic
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—61no—
106C2×C3×C42—61no—
107C2×#217—21no—
108#926—22no—
109#926C22×C314yes—
110C2×C4⋊Dic3C22×C311yescharacteristic
111C2×C4⋊Dic3C22×C311yescharacteristic
112C2×C4×Dic3—21no—
113C22×C3⋊C8C22×C311yescharacteristic
114C22×C3⋊C8C22×C311yescharacteristic
115C3×#107—34no—
116C3×#106—34no—
117C3×#103—34no—
118C3×#103—34no—
119C3×C22⋊C8—34no—
120#835C22×C314yescomplemented
121#835C22×C314yescomplemented
122#830C22×C314yes—
123#808C22×C314yescomplemented
124#807C22×C314yescomplemented
125C23×C32—22no—
126C23×C32—22no—
127C23×C32—22no—
128C23×C32C2×D412yes—
129C23×C32C2×D411yescharacteristic
130C32×Q8—22no—
131C32×Q8—22no—
132C32×Q8—42no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—24no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—42no—
140C2×C32×C4—42no—
141C2×C32×C4C2×D412yes—
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4#4112yes—
144C2×C32×C4#4112yes—
145C2×C32×C4C2411yescharacteristic
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—42no—
149C2×C3×Dic3—42no—
150C3×C3⋊C8—22no—
151C3×C3⋊C8—22no—
152C2×C22⋊Q8C3×D311yescharacteristic
153C2×#117—31no—
154#485—38no—
155C2×#107—31no—
156C2×#106—31no—
157C2×#103—31no—
158C2×#103—31no—
159C2×C22⋊C8—31no—
160C24×C3—21no—
161C24×C3#6711yescharacteristic
162C24×C3C3×D411yescharacteristic
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—42no—
172C2×C3×Q8—42no—
173C2×C3×Q8C22×D312yes—
174C2×C3×Q8C23×C312yes—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4—61no—
192C22×C3×C4—61no—
193C22×C3×C4#6712yes—
194C22×C3×C4#6711yescharacteristic
195C22×C3×C4C3×D411yescharacteristic
196C22×C3×C4C3×D412yes—
197C22×C3×C4C22×D312yes—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C23×C312yes—
200C22×C3×C4C23×C311yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C2×C3×C8—32no—
204C2×C3×C8—32no—
205C2×C3×C8—62no—
206C2×C3×C8—62no—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—32no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4C22×D312yes—
219C3×C4⋊C4C23×C312yes—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—64no—
224C3×C42—64no—
225#217—24no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3C23×C312yes—
229C4⋊Dic3C23×C312yes—
230C4×Dic3—24no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8C23×C312yes—
234C2×C3⋊C8C23×C312yes—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—44no—
239C22×C32#12312yes—
240C22×C32#13512yes—
241C22×C32#13712yes—
242C22×C32C22×D411yescharacteristic
243C32×C4—22no—
244C32×C4—42no—
245C32×C4—42no—
246C32×C4#12312yes—
247C32×C4C2×#4111yescharacteristic
248C32×C4C2×#4111yescharacteristic
249C3×Dic3—42no—
250C3×Dic3—42no—
251C22×Q8C2×C3×D311yescharacteristic
252C23×C4C2×C3×D311yescharacteristic
253C22×C8—31no—
254C22×C8—31no—
255C22⋊Q8—22no—
256C22⋊Q8C2×C3×D314yes—
257#117—34no—
258#117—62no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4C2×C3×D311yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—61no—
265C2×C42—61no—
266#107—34no—
267#106—34no—
268#103—34no—
269#103—34no—
270C22⋊C8—34no—
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—21no—
279C23×C3—22no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3C2×#6712yes—
283C23×C3C2×#6711yescharacteristic
284C23×C3C2×C3×D412yes—
285C23×C3C2×C3×D411yescharacteristic
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8—42no—
291C3×Q8—42no—
292C3×Q8—44no—
293C3×Q8—42no—
294C3×Q8—42no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—44no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—122no—
327C2×C3×C4—122no—
328C2×C3×C4#23712yes—
329C2×C3×C4#23712yes—
330C2×C3×C4C2×#6712yes—
331C2×C3×C4C2×#6711yescharacteristic
332C2×C3×C4C2×C3×D412yes—
333C2×C3×C4C2×C3×D411yescharacteristic
334C2×C3×C4C3×#4112yes—
335C2×C3×C4C3×#4112yes—
336C2×C3×C4C23×D311yescharacteristic
337C2×C3×C4C24×C311yescharacteristic
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C3×C8—62no—
343C3×C8—62no—
344C3⋊C8—22no—
345C3⋊C8—22no—
346C2×C32—42no—
347C2×C32#48514yescomplemented
348C2×C32C2×#12311yescharacteristic
349C2×C32C2×#13511yescharacteristic
350C2×C32C2×#13711yescharacteristic
351C24C3×#6711yescharacteristic
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8C22×C3×D312yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4C3×#6711yescharacteristic
363C22×C4C3×#6712yes—
364C22×C4C22×C3×D312yes—
365C22×C4C22×C3×D311yescharacteristic
366C2×C8—32no—
367C2×C8—32no—
368C2×C8—62no—
369C2×C8—62no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—32no—
373C4⋊C4—32no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4C22×C3×D312yes—
377C22⋊C4—24no—
378C22⋊C4—64no—
379C42—64no—
380C22×C3—24no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—48no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3#93512yes—
397C22×C3#94212yes—
398C22×C3#95012yes—
399C22×C3C3×#12312yes—
400C22×C3C3×#13512yes—
401C22×C3C3×#13712yes—
402C22×C3C22×#6711yescharacteristic
403C22×C3C22×C3×D411yescharacteristic
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—42no—
410C3×C4—44no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—122no—
417C3×C4—122no—
418C3×C4#93512yes—
419C3×C4C3×#12312yes—
420C3×C4C2×#23711yescharacteristic
421C3×C4C2×#23711yescharacteristic
422C3×C4C2×C3×#4111yescharacteristic
423C3×C4C2×C3×#4111yescharacteristic
424Dic3—42no—
425Dic3—42no—
426C32C2×#48511yescharacteristic, complemented, Sylow
427C23—22no—
428C23—22no—
429C23—22no—
430C23C2×C3×#6712yes—
431C23C2×C3×#6711yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—42no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4C3×#23712yes—
448C2×C4C3×#23712yes—
449C2×C4C2×C3×#6712yes—
450C2×C4C2×C3×#6711yescharacteristic
451C2×C4C23×C3×D311yescharacteristic
452C8—62no—
453C8—62no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—24no—
458C2×C3—44no—
459C2×C3—42no—
460C2×C3—42no—
461C2×C3#504514yesdirect product factor
462C2×C3C3×#48514yescomplemented
463C2×C3C2×#93511yescharacteristic
464C2×C3C2×#94211yescharacteristic
465C2×C3C2×#95011yescharacteristic
466C2×C3C2×C3×#12311yescharacteristic
467C2×C3C2×C3×#13511yescharacteristic
468C2×C3C2×C3×#13711yescharacteristic
469C22—22no—
470C22—22no—
471C22—22no—
472C22—44no—
473C22C3×#93512yes—
474C22C3×#94212yes—
475C22C3×#95012yes—
476C22C22×C3×#6711yescharacteristic
477C4—22no—
478C4—42no—
479C4—42no—
480C4—122no—
481C4—122no—
482C4C3×#93512yes—
483C4C2×C3×#23711yescharacteristic
484C4C2×C3×#23711yescharacteristic
485C3—21no—
486C3C2×#504511yescharacteristic, direct product factor
487C3C2×C3×#48511yescharacteristic, complemented
488C2—42no—
489C2C3×#504514yesdirect product factor
490C2C2×C3×#93511yescharacteristic
491C2C2×C3×#94211yescharacteristic
492C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C22⋊Q8C2
socleC23×C32C2×D4
prime radicalC24×C32D4
hypercenterC2×C3×C22⋊Q8D3
hypocenterC3C2×C3×#485
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×C3×#835C2semidirect
4. C2×C3×#835C2semidirect
6. C3×#5045C2direct (N × H)
7. C2×C3×#808C2semidirect
8. C2×C3×#807C2semidirect
10. C2×#5045C3direct (N × H)
26. C3×#835C22semidirect
27. C3×#835C22semidirect
29. C3×#808C22semidirect
30. C3×#807C22semidirect
42. C2×#835C2×C3semidirect
43. C2×#835C2×C3semidirect
45. #5045C2×C3direct (N × H)
46. C2×#808C2×C3semidirect
47. C2×#807C2×C3semidirect
120. #835C22×C3semidirect
121. #835C22×C3semidirect
123. #808C22×C3semidirect
124. #807C22×C3semidirect
347. C2×C32#485semidirect
426. C32C2×#485semidirect
461. C2×C3#5045direct (N × H)
462. C2×C3C3×#485semidirect
486. C3C2×#5045direct (N × H)
487. C3C2×C3×#485semidirect
489. C2C3×#5045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊Q82
C2×C3×#9262
C2×C3×#8352
C2×C3×#8302
C3×#50452
C2×C3×#8082
C2×C3×#8072
C2×C3×#4853
C2×#50453