Finite Group Databasebeta

← C2×C3×#5028 #11865676

Subgroups — C2×C3×#5028 #11865676

Order 1152.

1149 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C22×C3×#216C211yesmaximal, characteristic, complemented
3C22×C3×#215C211yesmaximal, characteristic
4C2×C3×#875C211yesmaximal, characteristic
5C3×#5028C218yesmaximal, direct product factor
6C2×C3×#810C211yesmaximal, characteristic, complemented
7C2×C3×#809C211yesmaximal, characteristic
8C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C2×C3×#464—31nomaximal
10C2×#5028C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C32×C4×Q8—22no—
15C32×C4×Q8C2214yescomplemented
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C42—21no—
19C2×C32×C42C2211yescharacteristic
20C2×C3×#216—24no—
21C2×C3×#216C2214yescomplemented
22C2×C3×#215—24no—
23C2×C3×#215C2214yes—
24C2×C3×(C2×D3)⋊C4—21no—
25C2×C3×(C2×D3)⋊C4—21no—
26C22×C3×C3⋊C8C2211yescharacteristic
27C22×C3×C3⋊C8C2211yescharacteristic
28C3×#875—22no—
29C3×#875C2214yes—
30C3×#810C2214yescomplemented
31C3×#809C2214yes—
32C3×C4⋊(C3⋊C8)C2214yescomplemented
33C22×C3×Q16—31no—
34C22×C3×SD16—31no—
35C2×C3×#124—31no—
36C2×C3×C4×Q8—21no—
37C2×C3×C4×Q8D311yescharacteristic, complemented
38C2×C3×C4×Q8C2×C311yescharacteristic, complemented
39C22×#216C2×C311yescharacteristic, complemented
40C22×#215C2×C311yescharacteristic
41C2×#875C2×C311yescharacteristic
42C3×#464—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46#5028C2×C318yesdirect product factor
47C2×#810C2×C311yescharacteristic, complemented
48C2×#809C2×C311yescharacteristic
49C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
50C23×C3×D3—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8D414yes—
54C2×C32×Q8C2312yes—
55C22×C32×C4—21no—
56C22×C32×C4—21no—
57C22×C32×C4D411yescharacteristic
58C22×C32×C4D411yescharacteristic
59C22×C32×C4C2311yescharacteristic
60C22×C3×Dic3—21no—
61C2×C3×D12—22no—
62C2×C3×D12—22no—
63C2×C3×D12—21no—
64C2×C3×D12C2312yes—
65C2×C3×Dic6—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2312yes—
69C32×C4⋊C4—24no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4C2312yes—
72C32×C42—24no—
73C32×C42D412yes—
74C32×C42C2312yes—
75C3×#216—28no—
76C3×#215—28no—
77C3×(C2×D3)⋊C4—24no—
78C3×(C2×D3)⋊C4—24no—
79C2×C3×C3⋊C8—22no—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8C2312yes—
82C2×C3×C3⋊C8C2312yes—
83C2×#464—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—31no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C22×C3×D4—31no—
89C22×D12C22×C311yescharacteristic
90C22×Dic6C22×C311yescharacteristic
91C2×C3×Q16—34no—
92C2×C3×Q16—64no—
93C2×C3×SD16—34no—
94C2×C3×SD16—64no—
95C22×C3×C8—31no—
96C22×C3×C8—31no—
97C3×#124—34no—
98C3×#124—62no—
99C3×C4×Q8—24no—
100C3×C4×Q8—22no—
101C3×C4×Q8—22no—
102C3×C4×Q8—24no—
103C3×C4×Q8C2×D314yescomplemented
104C3×C4×Q8C22×C314yescomplemented
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4C2×D311yescharacteristic
110C2×C3×C4⋊C4C22×C311yescharacteristic
111C2×C3×C22⋊C4—61no—
112C2×C3×C22⋊C4—61no—
113C2×C3×C42—22no—
114C2×C3×C42—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2×D311yescharacteristic
118C2×C3×C42C22×C311yescharacteristic
119C2×#216—24no—
120C2×#216C22×C314yescomplemented
121C2×#215—24no—
122C2×#215C22×C314yes—
123C2×(C2×D3)⋊C4—21no—
124C2×(C2×D3)⋊C4—21no—
125C22×C3⋊C8C22×C311yescharacteristic
126C22×C3⋊C8C22×C311yescharacteristic
127#875—22no—
128#875C22×C314yes—
129C3×C4⋊C8—34no—
130C3×#103—34no—
131C3×#102—34no—
132#810C22×C314yescomplemented
133#809C22×C314yes—
134C4⋊(C3⋊C8)C22×C314yescomplemented
135C23×C32C2×D411yescharacteristic
136C22×C3×D3—22no—
137C22×C3×D3—42no—
138C22×C3×D3—42no—
139C22×C3×D3—42no—
140C32×Q8—24no—
141C32×Q8—22no—
142C32×Q8C2×D414yes—
143C2×C32×C4—24no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—42no—
149C2×C32×C4C2×D412yes—
150C2×C32×C4C2×D412yes—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4#4112yes—
154C2×C32×C4C2411yescharacteristic
155C2×C3×Dic3—22no—
156C2×C3×Dic3—42no—
157C3×D12—22no—
158C3×D12—22no—
159C3×D12—42no—
160C3×Dic6—22no—
161C3×Dic6—22no—
162C3×Dic6—42no—
163C3×C3⋊C8—22no—
164C3×C3⋊C8—22no—
165C22×Q16—31no—
166C22×SD16—31no—
167C2×#124—31no—
168C2×C4×Q8C3×D311yescharacteristic, complemented
169#464—38no—
170C2×C4⋊C8—31no—
171C2×#103—31no—
172C2×#102—31no—
173C24×C3—61no—
174C23×D3—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—24no—
177C2×C3×Q8—22no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8—24no—
182C2×C3×Q8—22no—
183C2×C3×Q8—32no—
184C2×C3×Q8—61no—
185C2×C3×Q8—62no—
186C2×C3×Q8—62no—
187C2×C3×Q8D1214yes—
188C2×C3×Q8C3×D414yes—
189C2×C3×Q8C22×D312yes—
190C2×C3×Q8C23×C312yes—
191C2×C3×D4—32no—
192C2×C3×D4—62no—
193C2×C3×D4—62no—
194C2×C3×D4—61no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—61no—
205C22×C3×C4#6711yescharacteristic
206C22×C3×C4#6711yescharacteristic
207C22×C3×C4C3×D411yescharacteristic
208C22×C3×C4C3×D411yescharacteristic
209C22×C3×C4C22×D311yescharacteristic
210C22×C3×C4C23×C311yescharacteristic
211C22×Dic3—21no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—21no—
215C2×D12C23×C312yes—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—21no—
219C2×Dic6C23×C312yes—
220C3×Q16—68no—
221C3×SD16—68no—
222C2×C3×C8—32no—
223C2×C3×C8—32no—
224C2×C3×C8—62no—
225C2×C3×C8—62no—
226C3×C4⋊C4—28no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4C22×D312yes—
234C3×C4⋊C4C23×C312yes—
235C3×C22⋊C4—64no—
236C3×C22⋊C4—64no—
237C3×C42—28no—
238C3×C42—24no—
239C3×C42—24no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42#6712yes—
243C3×C42C3×D412yes—
244C3×C42C22×D312yes—
245C3×C42C23×C312yes—
246#216—28no—
247#215—28no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—24no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8C23×C312yes—
253C2×C3⋊C8C23×C312yes—
254C22×C32#12112yes—
255C22×C32#13512yes—
256C22×C32#13712yes—
257C22×C32C22×D411yescharacteristic
258C2×C3×D3—42no—
259C2×C3×D3—42no—
260C2×C3×D3—42no—
261C2×C3×D3—84no—
262C32×C4—24no—
263C32×C4—22no—
264C32×C4—22no—
265C32×C4—42no—
266C32×C4#12112yes—
267C32×C4C22×D411yescharacteristic
268C32×C4C2×#4111yescharacteristic
269C3×Dic3—42no—
270C22×Q8—31no—
271C22×Q8C2×C3×D311yescharacteristic
272C22×D4—31no—
273C2×Q16—34no—
274C2×Q16—64no—
275C2×SD16—34no—
276C2×SD16—64no—
277C22×C8—31no—
278C22×C8—31no—
279#124—34no—
280#124—62no—
281C4×Q8—22no—
282C4×Q8C2×C3×D314yescomplemented
283C2×C4⋊C4—21no—
284C2×C4⋊C4C2×C3×D311yescharacteristic
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—21no—
288C2×C42C2×C3×D311yescharacteristic
289C4⋊C8—34no—
290#103—34no—
291#102—34no—
292C23×C3—21no—
293C23×C3—62no—
294C23×C3—122no—
295C23×C3—122no—
296C23×C3—122no—
297C23×C3C2×#6711yescharacteristic
298C23×C3C2×C3×D411yescharacteristic
299C22×D3—22no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C3×Q8—28no—
304C3×Q8—24no—
305C3×Q8—24no—
306C3×Q8—24no—
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—24no—
310C3×Q8—62no—
311C3×Q8—62no—
312C3×Q8—122no—
313C3×Q8C2×D1214yes—
314C3×Q8C2×C3×D414yes—
315C3×D4—62no—
316C3×D4—62no—
317C3×D4—122no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—28no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—62no—
343C2×C3×C4—122no—
344C2×C3×C4C2×D1212yes—
345C2×C3×C4#23512yes—
346C2×C3×C4C2×#6712yes—
347C2×C3×C4C2×#6712yes—
348C2×C3×C4C2×#6711yescharacteristic
349C2×C3×C4C2×C3×D412yes—
350C2×C3×C4C2×C3×D412yes—
351C2×C3×C4C2×C3×D412yes—
352C2×C3×C4C2×C3×D411yescharacteristic
353C2×C3×C4C3×#4112yes—
354C2×C3×C4C23×D311yescharacteristic
355C2×C3×C4C24×C311yescharacteristic
356C2×Dic3—22no—
357C2×Dic3—42no—
358D12—22no—
359D12—22no—
360D12—42no—
361Dic6—22no—
362Dic6—22no—
363Dic6—42no—
364C3×C8—62no—
365C3×C8—62no—
366C3⋊C8—22no—
367C3⋊C8—22no—
368C2×C32#46414yescomplemented
369C2×C32C2×#12111yescharacteristic
370C2×C32C2×#13511yescharacteristic
371C2×C32C2×#13711yescharacteristic
372C3×D3—82no—
373C24—61no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—32no—
377C2×Q8—61no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8C3×D1214yes—
381C2×Q8C22×C3×D312yes—
382C2×D4—32no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—61no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—61no—
389C22×C4C3×#6711yescharacteristic
390C22×C4C3×#6711yescharacteristic
391C22×C4C22×C3×D311yescharacteristic
392Q16—68no—
393SD16—68no—
394C2×C8—32no—
395C2×C8—32no—
396C2×C8—62no—
397C2×C8—62no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4C22×C3×D312yes—
401C22⋊C4—64no—
402C22⋊C4—64no—
403C42—24no—
404C42C3×#6712yes—
405C42C22×C3×D312yes—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—122no—
411C22×C3—122no—
412C22×C3—122no—
413C22×C3—244no—
414C22×C3#93012yes—
415C22×C3#94212yes—
416C22×C3#95012yes—
417C22×C3C3×#12112yes—
418C22×C3C3×#13512yes—
419C22×C3C3×#13712yes—
420C22×C3C22×#6711yescharacteristic
421C22×C3C22×C3×D411yescharacteristic
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—84no—
426C3×C4—28no—
427C3×C4—24no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—24no—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—24no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—42no—
438C3×C4—44no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—122no—
442C3×C4#93012yes—
443C3×C4C3×#12112yes—
444C3×C4C22×D1211yescharacteristic
445C3×C4C2×#23511yescharacteristic
446C3×C4C22×C3×D411yescharacteristic
447C3×C4C2×C3×#4111yescharacteristic
448Dic3—42no—
449C32C2×#46411yescharacteristic, complemented, Sylow
450C23—62no—
451C23—122no—
452C23—122no—
453C23—122no—
454C23C2×C3×#6711yescharacteristic
455Q8—24no—
456Q8—22no—
457Q8—62no—
458Q8—62no—
459Q8—122no—
460Q8C2×C3×D1214yes—
461D4—62no—
462D4—62no—
463D4—122no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—62no—
471C2×C4—122no—
472C2×C4C2×C3×D1212yes—
473C2×C4C3×#23512yes—
474C2×C4C2×C3×#6712yes—
475C2×C4C2×C3×#6712yes—
476C2×C4C2×C3×#6711yescharacteristic
477C2×C4C23×C3×D311yescharacteristic
478C8—62no—
479C8—62no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—24no—
484C2×C3—242no—
485C2×C3#502814yesdirect product factor
486C2×C3C3×#46414yescomplemented
487C2×C3C2×#93011yescharacteristic
488C2×C3C2×#94211yescharacteristic
489C2×C3C2×#95011yescharacteristic
490C2×C3C2×C3×#12111yescharacteristic
491C2×C3C2×C3×#13511yescharacteristic
492C2×C3C2×C3×#13711yescharacteristic
493D3—82no—
494C22—122no—
495C22—122no—
496C22—122no—
497C22—244no—
498C22C3×#93012yes—
499C22C3×#94212yes—
500C22C3×#95012yes—
501C22C22×C3×#6711yescharacteristic
502C4—24no—
503C4—22no—
504C4—22no—
505C4—42no—
506C4—122no—
507C4C3×#93012yes—
508C4C22×C3×D1211yescharacteristic
509C4C2×C3×#23511yescharacteristic
510C3—21no—
511C3C2×#502811yescharacteristic, direct product factor
512C3C2×C3×#46411yescharacteristic, complemented
513C2—242no—
514C2C3×#502814yesdirect product factor
515C2C2×C3×#93011yescharacteristic
516C2C2×C3×#94211yescharacteristic
517C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×Q8C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×#464
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
2. C22×C3×#216C2semidirect
5. C3×#5028C2direct (N × H)
6. C2×C3×#810C2semidirect
8. C2×C3×C4⋊(C3⋊C8)C2semidirect
10. C2×#5028C3direct (N × H)
15. C32×C4×Q8C22semidirect
21. C2×C3×#216C22semidirect
30. C3×#810C22semidirect
32. C3×C4⋊(C3⋊C8)C22semidirect
37. C2×C3×C4×Q8D3semidirect
38. C2×C3×C4×Q8C2×C3semidirect
39. C22×#216C2×C3semidirect
46. #5028C2×C3direct (N × H)
47. C2×#810C2×C3semidirect
49. C2×C4⋊(C3⋊C8)C2×C3semidirect
103. C3×C4×Q8C2×D3semidirect
104. C3×C4×Q8C22×C3semidirect
120. C2×#216C22×C3semidirect
132. #810C22×C3semidirect
134. C4⋊(C3⋊C8)C22×C3semidirect
168. C2×C4×Q8C3×D3semidirect
282. C4×Q8C2×C3×D3semidirect
368. C2×C32#464semidirect
449. C32C2×#464semidirect
485. C2×C3#5028direct (N × H)
486. C2×C3C3×#464semidirect
511. C3C2×#5028direct (N × H)
512. C3C2×C3×#464semidirect
514. C2C3×#5028direct (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
C22×C3×#2162
C22×C3×#2152
C2×C3×#8752
C3×#50282
C2×C3×#8102
C2×C3×#8092
C2×C3×C4⋊(C3⋊C8)2
C2×C3×#4643
C2×#50283