Finite Group Databasebeta

← C2×C3×#5127 #11859196

Subgroups — C2×C3×#5127 #11859196

Order 1152.

1277 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×M16C211yesmaximal, characteristic, complemented
2C2×C3×C4×#67C211yesmaximal, characteristic
3C22×C3×D3×C8C211yesmaximal, characteristic, complemented
4C2×C3×#830C211yesmaximal, characteristic, complemented
5C3×#5127C218yesmaximal, direct product factor
6C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7C2×C3×#815C211yesmaximal, characteristic, complemented
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×#435—31nomaximal
10C2×#5127C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67C411yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C2×C32×M16—24no—
15C2×C32×M16C2214yescomplemented
16C22×C32×C8C2211yescharacteristic
17C22×C32×C8C2211yescharacteristic
18C2×C3×#217C411yescharacteristic
19C3×C4×#67C414yes—
20C3×C4×#67C2214yes—
21C2×C3×(C2×D3)⋊C4C411yescharacteristic
22C2×C3×Dic3⋊C4C411yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C2×C3×D3×C8—24no—
27C2×C3×D3×C8C2214yescomplemented
28C3×#830C2214yescomplemented
29C3×(C2×D3)⋊C8C2214yescomplemented
30C3×#815C2214yescomplemented
31C3×Dic3⋊C8C2214yescomplemented
32C22×C3×M16—21no—
33C22×C3×M16D311yescharacteristic, complemented
34C22×C3×M16C2×C311yescharacteristic, complemented
35C23×C3×C8—31no—
36C2×C3×C4×D4—31no—
37C2×C4×#67C2×C311yescharacteristic
38C22×D3×C8C2×C311yescharacteristic, complemented
39C3×#435—38no—
40C2×C3×C4⋊C8—31no—
41C2×C3×C22⋊C8—31no—
42C2×C3×C22⋊C8—31no—
43C2×C3×#97—31no—
44C2×#830C2×C311yescharacteristic, complemented
45#5127C2×C318yesdirect product factor
46C2×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
47C2×#815C2×C311yescharacteristic, complemented
48C2×Dic3⋊C8C2×C311yescharacteristic, complemented
49C24×C32C2×C411yescharacteristic
50C23×C3×D3C2×C411yescharacteristic
51C22×C32×C4—22no—
52C22×C32×C4—22no—
53C22×C32×C4C2×C411yescharacteristic
54C22×C32×C4C2×C412yes—
55C22×C32×C4C2312yes—
56C22×C32×C4C2311yescharacteristic
57C2×C3×#67—24no—
58C2×C3×#67C2×C414yes—
59C22×C3×Dic3C2×C411yescharacteristic
60C22×C3×Dic3C2×C411yescharacteristic
61C22×C3×Dic3C2×C411yescharacteristic
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3C2×C412yes—
65C2×C3×C4×D3C2312yes—
66C32×M16—28no—
67C2×C32×C8—22no—
68C2×C32×C8D414yescomplemented
69C2×C32×C8C2312yes—
70C2×C32×C8C2312yes—
71C3×#217C2×C414yes—
72C3×(C2×D3)⋊C4C2×C414yes—
73C3×Dic3⋊C4C2×C414yes—
74C3×C4×Dic3C2×C412yes—
75C3×C4×Dic3C2312yes—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8—22no—
78C2×C3×C3⋊C8C2312yes—
79C2×C3×C3⋊C8C2312yes—
80C3×D3×C8—28no—
81C2×#435—31noSylow
82C22×C3×D4—31no—
83C23×C3×C4—21no—
84C23×C3×C4—31no—
85C23×C3×C4C2×D311yescharacteristic
86C23×C3×C4C22×C311yescharacteristic
87C22×#67C3×C411yescharacteristic
88C22×C4×D3C22×C311yescharacteristic
89C2×C3×M16—28no—
90C2×C3×M16—24no—
91C2×C3×M16—24no—
92C2×C3×M16—24no—
93C2×C3×M16C2×D314yescomplemented
94C2×C3×M16C22×C314yescomplemented
95C22×C3×C8—21no—
96C22×C3×C8—21no—
97C22×C3×C8—31no—
98C22×C3×C8—31no—
99C22×C3×C8—34no—
100C22×C3×C8—64no—
101C22×C3×C8C2×D311yescharacteristic
102C22×C3×C8C2×D311yescharacteristic
103C22×C3×C8C22×C311yescharacteristic
104C22×C3×C8C22×C311yescharacteristic
105C3×C4×D4—34no—
106C3×C4×D4—34no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C22⋊C4—31no—
109C2×C3×C22⋊C4—31no—
110C2×C3×C42—31no—
111C2×#217C3×C411yescharacteristic
112C4×#67C3×C414yes—
113C4×#67C22×C314yes—
114C2×(C2×D3)⋊C4C3×C411yescharacteristic
115C2×Dic3⋊C4C3×C411yescharacteristic
116C2×C4×Dic3C22×C311yescharacteristic
117C22×C3⋊C8C22×C311yescharacteristic
118C22×C3⋊C8C22×C311yescharacteristic
119C2×D3×C8—24no—
120C2×D3×C8C22×C314yescomplemented
121C3×C4⋊C8—34no—
122C3×C22⋊C8—34no—
123C3×C22⋊C8—34no—
124C3×#97—34no—
125#830C22×C314yescomplemented
126(C2×D3)⋊C8C22×C314yescomplemented
127#815C22×C314yescomplemented
128Dic3⋊C8C22×C314yescomplemented
129C23×C32—22no—
130C23×C32—22no—
131C23×C32—22no—
132C23×C32C22×C412yes—
133C23×C32C22×C411yescharacteristic
134C22×C3×D3—22no—
135C22×C3×D3—22no—
136C22×C3×D3M1614yescomplemented
137C22×C3×D3C22×C412yes—
138C2×C32×C4—22no—
139C2×C32×C4—24no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4C22×C412yes—
143C2×C32×C4C22×C411yescharacteristic
144C2×C32×C4C2×D412yes—
145C2×C32×C4#4112yes—
146C2×C32×C4C2411yescharacteristic
147C3×#67—28no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3C22×C412yes—
152C2×C3×Dic3C22×C412yes—
153C2×C3×Dic3C22×C412yes—
154C3×C4×D3—24no—
155C3×C4×D3—22no—
156C3×C4×D3—22no—
157C32×C8—22no—
158C32×C8C2×D414yescomplemented
159C3×C3⋊C8—22no—
160C3×C3⋊C8—22no—
161C22×M16C3×D311yescharacteristic, complemented
162C23×C8—31no—
163C2×C4×D4—31no—
164#435—38no—
165C2×C4⋊C8—31no—
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×#97—31no—
169C24×C3—21no—
170C24×C3—31no—
171C24×C3C4×D311yescharacteristic
172C24×C3C2×C3×C411yescharacteristic
173C23×D3C2×C3×C411yescharacteristic
174C2×C3×D4—34no—
175C2×C3×D4—64no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—62no—
192C22×C3×C4—62no—
193C22×C3×C4C4×D311yescharacteristic
194C22×C3×C4C4×D312yes—
195C22×C3×C4C2×C3×C411yescharacteristic
196C22×C3×C4C2×C3×C412yes—
197C22×C3×C4C22×D312yes—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C23×C312yes—
200C22×C3×C4C23×C311yescharacteristic
201C2×#67—24no—
202C2×#67C2×C3×C414yes—
203C22×Dic3C2×C3×C411yescharacteristic
204C22×Dic3C2×C3×C411yescharacteristic
205C22×Dic3C2×C3×C411yescharacteristic
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3C2×C3×C412yes—
209C2×C4×D3C23×C312yes—
210C3×M16—216no—
211C3×M16—28no—
212C3×M16—28no—
213C2×C3×C8—24no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—24no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—32no—
220C2×C3×C8—32no—
221C2×C3×C8—62no—
222C2×C3×C8—62no—
223C2×C3×C8—68no—
224C2×C3×C8#6714yescomplemented
225C2×C3×C8C3×D414yescomplemented
226C2×C3×C8C22×D312yes—
227C2×C3×C8C22×D312yes—
228C2×C3×C8C23×C312yes—
229C2×C3×C8C23×C312yes—
230C3×C4⋊C4—34no—
231C3×C22⋊C4—34no—
232C3×C22⋊C4—34no—
233C3×C42—32no—
234C3×C42—32no—
235#217C2×C3×C414yes—
236(C2×D3)⋊C4C2×C3×C414yes—
237Dic3⋊C4C2×C3×C414yes—
238C4×Dic3C2×C3×C412yes—
239C4×Dic3C23×C312yes—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C23×C312yes—
243C2×C3⋊C8C23×C312yes—
244D3×C8—28no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—44no—
249C22×C32C4×D412yes—
250C22×C32C2×M1612yes—
251C22×C32#13112yes—
252C22×C32C23×C411yescharacteristic
253C2×C3×D3—22no—
254C2×C3×D3—22no—
255C2×C3×D3—28no—
256C2×C3×D3C2×M1614yescomplemented
257C32×C4—22no—
258C32×C4C4×D412yes—
259C32×C4C22×D411yescharacteristic
260C32×C4C2×#4111yescharacteristic
261C3×Dic3—22no—
262C3×Dic3—22no—
263C3×Dic3—22no—
264C22×D4—31no—
265C23×C4—31no—
266C23×C4C2×C3×D311yescharacteristic
267C2×M16—24no—
268C2×M16C2×C3×D314yescomplemented
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—34no—
272C22×C8—64no—
273C22×C8C2×C3×D311yescharacteristic
274C22×C8C2×C3×D311yescharacteristic
275C4×D4—34no—
276C4×D4—34no—
277C2×C4⋊C4—31no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—31no—
280C2×C42—31no—
281C4⋊C8—34no—
282C22⋊C8—34no—
283C22⋊C8—34no—
284#97—34no—
285C23×C3—24no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—34no—
295C23×C3—32no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—62no—
299C23×C3—62no—
300C23×C3C2×C4×D312yes—
301C23×C3C2×C4×D311yescharacteristic
302C23×C3C22×C3×C412yes—
303C23×C3C22×C3×C411yescharacteristic
304C22×D3—22no—
305C22×D3—22no—
306C22×D3C3×M1614yescomplemented
307C22×D3C22×C3×C412yes—
308C3×D4—68no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—28no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—64no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4C2×C4×D312yes—
336C2×C3×C4C2×C4×D311yescharacteristic
337C2×C3×C4#23512yes—
338C2×C3×C4C2×#6712yes—
339C2×C3×C4C22×C3×C412yes—
340C2×C3×C4C22×C3×C411yescharacteristic
341C2×C3×C4C2×C3×D412yes—
342C2×C3×C4C3×#4112yes—
343C2×C3×C4C23×D311yescharacteristic
344C2×C3×C4C24×C311yescharacteristic
345#67—28no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3C22×C3×C412yes—
350C2×Dic3C22×C3×C412yes—
351C2×Dic3C22×C3×C412yes—
352C4×D3—24no—
353C4×D3—22no—
354C4×D3—22no—
355C3×C8—24no—
356C3×C8—22no—
357C3×C8—22no—
358C3×C8—24no—
359C3×C8—62no—
360C3×C8—62no—
361C3×C8C2×#6714yescomplemented
362C3×C8C2×C3×D414yescomplemented
363C3⋊C8—22no—
364C3⋊C8—22no—
365C2×C32—42no—
366C2×C32#43514yescomplemented
367C2×C32C2×C4×D411yescharacteristic
368C2×C32C22×M1611yescharacteristic
369C2×C32C2×#13111yescharacteristic
370C3×D3—24no—
371C24—31no—
372C24C3×C4×D311yescharacteristic
373C2×D4—34no—
374C2×D4—64no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4C3×C4×D311yescharacteristic
385C22×C4C3×C4×D312yes—
386C22×C4C22×C3×D312yes—
387C22×C4C22×C3×D311yescharacteristic
388M16—28no—
389C2×C8—22no—
390C2×C8—32no—
391C2×C8—32no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—68no—
395C2×C8C3×#6714yescomplemented
396C2×C8C22×C3×D312yes—
397C2×C8C22×C3×D312yes—
398C4⋊C4—34no—
399C22⋊C4—34no—
400C22⋊C4—34no—
401C42—32no—
402C42—32no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—34no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—48no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—68no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3D3×M1612yes—
424C22×C3#90712yes—
425C22×C3C4×#6712yes—
426C22×C3C3×C4×D412yes—
427C22×C3C2×C3×M1612yes—
428C22×C3C3×#13112yes—
429C22×C3C22×C4×D311yescharacteristic
430C22×C3C23×C3×C411yescharacteristic
431C2×D3—22no—
432C2×D3—28no—
433C2×D3—22no—
434C2×D3C2×C3×M1614yescomplemented
435C3×C4—24no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—62no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4C4×#6712yes—
445C3×C4C3×C4×D412yes—
446C3×C4C2×#23511yescharacteristic
447C3×C4C22×#6711yescharacteristic
448C3×C4C22×C3×D411yescharacteristic
449C3×C4C2×C3×#4111yescharacteristic
450Dic3—22no—
451Dic3—22no—
452Dic3—22no—
453C32C2×#43511yescharacteristic, complemented, Sylow
454C23—22no—
455C23—22no—
456C23—22no—
457C23—34no—
458C23—32no—
459C23—62no—
460C23—62no—
461C23C2×C3×C4×D312yes—
462C23C2×C3×C4×D311yescharacteristic
463D4—68no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—32no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4C2×C3×C4×D312yes—
478C2×C4C2×C3×C4×D311yescharacteristic
479C2×C4C3×#23512yes—
480C2×C4C2×C3×#6712yes—
481C2×C4C23×C3×D311yescharacteristic
482C8—22no—
483C8—62no—
484C8—62no—
485C8C2×C3×#6714yescomplemented
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—24no—
490C2×C3—44no—
491C2×C3—42no—
492C2×C3—42no—
493C2×C3—64no—
494C2×C3#512714yesdirect product factor
495C2×C3C3×#43514yescomplemented
496C2×C3C2×D3×M1611yescharacteristic
497C2×C3C2×#90711yescharacteristic
498C2×C3C2×C4×#6711yescharacteristic
499C2×C3C2×C3×C4×D411yescharacteristic
500C2×C3C22×C3×M1611yescharacteristic
501C2×C3C2×C3×#13111yescharacteristic
502D3—24no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—34no—
507C22—44no—
508C22—62no—
509C22—62no—
510C22—68no—
511C22C3×D3×M1612yes—
512C22C3×#90712yes—
513C22C3×C4×#6712yes—
514C22C22×C3×C4×D311yescharacteristic
515C4—22no—
516C4—62no—
517C4—62no—
518C4—62no—
519C4C3×C4×#6712yes—
520C4C2×C3×#23511yescharacteristic
521C4C22×C3×#6711yescharacteristic
522C3—21no—
523C3C2×#512711yescharacteristic, direct product factor
524C3C2×C3×#43511yescharacteristic, complemented
525C2—42no—
526C2—64no—
527C2C3×#512714yesdirect product factor
528C2C2×C3×D3×M1611yescharacteristic
529C2C2×C3×#90711yescharacteristic
530C2C2×C3×C4×#6711yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC22×C32×M16C2
socleC23×C32C22×C4
prime radicalC22×C3×#67C4
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×M16D3
hypocenterC3C2×C3×#435
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C22×C3×M16

Complements

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

NH ≅ G/Ntype
1. C22×C32×M16C2semidirect
3. C22×C3×D3×C8C2semidirect
4. C2×C3×#830C2semidirect
5. C3×#5127C2direct (N × H)
6. C2×C3×(C2×D3)⋊C8C2semidirect
7. C2×C3×#815C2semidirect
8. C2×C3×Dic3⋊C8C2semidirect
10. C2×#5127C3direct (N × H)
15. C2×C32×M16C22semidirect
27. C2×C3×D3×C8C22semidirect
28. C3×#830C22semidirect
29. C3×(C2×D3)⋊C8C22semidirect
30. C3×#815C22semidirect
31. C3×Dic3⋊C8C22semidirect
33. C22×C3×M16D3semidirect
34. C22×C3×M16C2×C3semidirect
38. C22×D3×C8C2×C3semidirect
44. C2×#830C2×C3semidirect
45. #5127C2×C3direct (N × H)
46. C2×(C2×D3)⋊C8C2×C3semidirect
47. C2×#815C2×C3semidirect
48. C2×Dic3⋊C8C2×C3semidirect
68. C2×C32×C8D4semidirect
93. C2×C3×M16C2×D3semidirect
94. C2×C3×M16C22×C3semidirect
120. C2×D3×C8C22×C3semidirect
125. #830C22×C3semidirect
126. (C2×D3)⋊C8C22×C3semidirect
127. #815C22×C3semidirect
128. Dic3⋊C8C22×C3semidirect
136. C22×C3×D3M16semidirect
158. C32×C8C2×D4semidirect
161. C22×M16C3×D3semidirect
224. C2×C3×C8#67semidirect
225. C2×C3×C8C3×D4semidirect
256. C2×C3×D3C2×M16semidirect
268. C2×M16C2×C3×D3semidirect
306. C22×D3C3×M16semidirect
361. C3×C8C2×#67semidirect
362. C3×C8C2×C3×D4semidirect
366. C2×C32#435semidirect
395. C2×C8C3×#67semidirect
434. C2×D3C2×C3×M16semidirect
453. C32C2×#435semidirect
485. C8C2×C3×#67semidirect
494. C2×C3#5127direct (N × H)
495. C2×C3C3×#435semidirect
523. C3C2×#5127direct (N × H)
524. C3C2×C3×#435semidirect
527. C2C3×#5127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×M162
C2×C3×C4×#672
C22×C3×D3×C82
C2×C3×#8302
C3×#51272
C2×C3×(C2×D3)⋊C82
C2×C3×#8152
C2×C3×Dic3⋊C82
C2×C3×#4353
C2×#51273