Finite Group Databasebeta

← C2×C3×#4729 #11859258

Subgroups — C2×C3×#4729 #11859258

Order 1152.

1205 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#67C211yesmaximal, characteristic
2C22×C3×#208C211yesmaximal, characteristic, complemented
3C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C2×C32×C22⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C2×C3×C8×Dic3C211yesmaximal, characteristic, complemented
8C3×#4729C218yesmaximal, direct product factor
9C2×C3×#436—31nomaximal
10C2×#4729C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67C411yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×C3×#217C411yescharacteristic
17C3×C4×#67C414yes—
18C3×C4×#67C2214yes—
19C2×C3×(C2×D3)⋊C4C411yescharacteristic
20C2×C3×Dic3⋊C4C411yescharacteristic
21C2×C3×C4×Dic3C2211yescharacteristic
22C2×C3×#208—24no—
23C2×C3×#208C2214yescomplemented
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C2×C3×ΓL(1,𝔽25)—24no—
27C2×C3×ΓL(1,𝔽25)C2214yescomplemented
28C32×C22⋊C8C2214yescomplemented
29C3×(C2×D3)⋊C8C2214yescomplemented
30C3×Dic3⋊C8C2214yescomplemented
31C3×C8×Dic3C2214yescomplemented
32C22×C3×M16—31no—
33C22×C3×M16—31no—
34C2×C3×C4×D4—31no—
35C2×C4×#67C2×C311yescharacteristic
36C22×#208C2×C311yescharacteristic, complemented
37C22×ΓL(1,𝔽25)C2×C311yescharacteristic, complemented
38C3×#436—38no—
39C2×C3×C4⋊C8—31no—
40C2×C3×C22⋊C8—21no—
41C2×C3×C22⋊C8—31no—
42C2×C3×C22⋊C8D311yescharacteristic, complemented
43C2×C3×C22⋊C8C2×C311yescharacteristic, complemented
44C2×C3×C4×C8—31no—
45C2×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
46C2×Dic3⋊C8C2×C311yescharacteristic, complemented
47C2×C8×Dic3C2×C311yescharacteristic, complemented
48#4729C2×C318yesdirect product factor
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—
66C2×C32×C8—22no—
67C2×C32×C8—22no—
68C2×C32×C8C2312yes—
69C2×C32×C8C2312yes—
70C3×#217C2×C414yes—
71C3×(C2×D3)⋊C4C2×C414yes—
72C3×Dic3⋊C4C2×C414yes—
73C3×C4×Dic3C2×C412yes—
74C3×C4×Dic3C2312yes—
75C3×#208—28no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8D414yescomplemented
78C2×C3×C3⋊C8C2312yes—
79C2×C3×C3⋊C8C2312yes—
80C3×ΓL(1,𝔽25)—28no—
81C2×#436—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—34no—
90C2×C3×M16—34no—
91C2×C3×M16—64no—
92C2×C3×M16—64no—
93C22×C3×C8—21no—
94C22×C3×C8—21no—
95C22×C3×C8—31no—
96C22×C3×C8—31no—
97C22×C3×C8C2×D311yescharacteristic
98C22×C3×C8C2×D311yescharacteristic
99C22×C3×C8C22×C311yescharacteristic
100C22×C3×C8C22×C311yescharacteristic
101C3×C4×D4—34no—
102C3×C4×D4—34no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C22⋊C4—31no—
105C2×C3×C22⋊C4—31no—
106C2×C3×C42—31no—
107C2×#217C3×C411yescharacteristic
108C4×#67C3×C414yes—
109C4×#67C22×C314yes—
110C2×(C2×D3)⋊C4C3×C411yescharacteristic
111C2×Dic3⋊C4C3×C411yescharacteristic
112C2×C4×Dic3C22×C311yescharacteristic
113C2×#208—24no—
114C2×#208C22×C314yescomplemented
115C22×C3⋊C8C22×C311yescharacteristic
116C22×C3⋊C8C22×C311yescharacteristic
117C2×ΓL(1,𝔽25)—24no—
118C2×ΓL(1,𝔽25)C22×C314yescomplemented
119C3×C4⋊C8—34no—
120C3×C22⋊C8—24no—
121C3×C22⋊C8—34no—
122C3×C22⋊C8C2×D314yescomplemented
123C3×C22⋊C8C22×C314yescomplemented
124C3×C4×C8—34no—
125(C2×D3)⋊C8C22×C314yescomplemented
126Dic3⋊C8C22×C314yescomplemented
127C8×Dic3C22×C314yescomplemented
128C23×C32—22no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32C22×C412yes—
132C23×C32C22×C411yescharacteristic
133C22×C3×D3—22no—
134C22×C3×D3—22no—
135C22×C3×D3—22no—
136C22×C3×D3C22×C412yes—
137C2×C32×C4—22no—
138C2×C32×C4—24no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4C22×C412yes—
142C2×C32×C4C22×C411yescharacteristic
143C2×C32×C4C2×D412yes—
144C2×C32×C4#4112yes—
145C2×C32×C4C2411yescharacteristic
146C3×#67—28no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3M1614yescomplemented
150C2×C3×Dic3C22×C412yes—
151C2×C3×Dic3C22×C412yes—
152C2×C3×Dic3C22×C412yes—
153C3×C4×D3—24no—
154C3×C4×D3—22no—
155C3×C4×D3—22no—
156C32×C8—22no—
157C32×C8—22no—
158C3×C3⋊C8—22no—
159C3×C3⋊C8C2×D414yescomplemented
160C22×M16—31no—
161C22×M16—31no—
162C2×C4×D4—31no—
163#436—38no—
164C2×C4⋊C8—31no—
165C2×C22⋊C8—31no—
166C2×C22⋊C8C3×D311yescharacteristic, complemented
167C2×C4×C8—31no—
168C24×C3—21no—
169C24×C3—31no—
170C24×C3C4×D311yescharacteristic
171C24×C3C2×C3×C411yescharacteristic
172C23×D3C2×C3×C411yescharacteristic
173C2×C3×D4—34no—
174C2×C3×D4—64no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4—62no—
191C22×C3×C4—62no—
192C22×C3×C4C4×D311yescharacteristic
193C22×C3×C4C4×D312yes—
194C22×C3×C4C2×C3×C411yescharacteristic
195C22×C3×C4C2×C3×C412yes—
196C22×C3×C4C22×D312yes—
197C22×C3×C4C22×D311yescharacteristic
198C22×C3×C4C23×C312yes—
199C22×C3×C4C23×C311yescharacteristic
200C2×#67—24no—
201C2×#67C2×C3×C414yes—
202C22×Dic3C2×C3×C411yescharacteristic
203C22×Dic3C2×C3×C411yescharacteristic
204C22×Dic3C2×C3×C411yescharacteristic
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3C2×C3×C412yes—
208C2×C4×D3C23×C312yes—
209C3×M16—68no—
210C3×M16—68no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—34no—
218C2×C3×C8—32no—
219C2×C3×C8—32no—
220C2×C3×C8—42no—
221C2×C3×C8—42no—
222C2×C3×C8—62no—
223C2×C3×C8C22×D312yes—
224C2×C3×C8C22×D312yes—
225C2×C3×C8C23×C312yes—
226C2×C3×C8C23×C312yes—
227C3×C4⋊C4—34no—
228C3×C22⋊C4—34no—
229C3×C22⋊C4—34no—
230C3×C42—32no—
231C3×C42—32no—
232#217C2×C3×C414yes—
233(C2×D3)⋊C4C2×C3×C414yes—
234Dic3⋊C4C2×C3×C414yes—
235C4×Dic3C2×C3×C412yes—
236C4×Dic3C23×C312yes—
237#208—28no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8C3×D414yescomplemented
240C2×C3⋊C8C23×C312yes—
241C2×C3⋊C8C23×C312yes—
242ΓL(1,𝔽25)—28no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—44no—
247C22×C32C4×D412yes—
248C22×C32C2×M1612yes—
249C22×C32#13112yes—
250C22×C32C23×C411yescharacteristic
251C2×C3×D3—22no—
252C2×C3×D3—22no—
253C2×C3×D3—22no—
254C2×C3×D3—44no—
255C32×C4—22no—
256C32×C4C4×D412yes—
257C32×C4C22×D411yescharacteristic
258C32×C4C2×#4111yescharacteristic
259C3×Dic3—22no—
260C3×Dic3—22no—
261C3×Dic3C2×M1614yescomplemented
262C22×D4—31no—
263C23×C4—31no—
264C23×C4C2×C3×D311yescharacteristic
265C2×M16—34no—
266C2×M16—34no—
267C2×M16—64no—
268C2×M16—64no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C2×C3×D311yescharacteristic
272C22×C8C2×C3×D311yescharacteristic
273C4×D4—34no—
274C4×D4—34no—
275C2×C4⋊C4—31no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—31no—
278C2×C42—31no—
279C4⋊C8—34no—
280C22⋊C8—34no—
281C22⋊C8C2×C3×D314yescomplemented
282C4×C8—34no—
283C23×C3—24no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—21no—
291C23×C3—22no—
292C23×C3—32no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—62no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3C2×C4×D312yes—
299C23×C3C2×C4×D311yescharacteristic
300C23×C3C22×C3×C412yes—
301C23×C3C22×C3×C411yescharacteristic
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3C22×C3×C412yes—
306C3×D4—68no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—34no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—64no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4C2×C4×D312yes—
334C2×C3×C4C2×C4×D311yescharacteristic
335C2×C3×C4D3×D412yes—
336C2×C3×C4#23712yes—
337C2×C3×C4C22×C3×C412yes—
338C2×C3×C4C22×C3×C411yescharacteristic
339C2×C3×C4C2×C3×D412yes—
340C2×C3×C4C3×#4112yes—
341C2×C3×C4C23×D311yescharacteristic
342C2×C3×C4C24×C311yescharacteristic
343#67—28no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3C3×M1614yescomplemented
347C2×Dic3C22×C3×C412yes—
348C2×Dic3C22×C3×C412yes—
349C2×Dic3C22×C3×C412yes—
350C4×D3—24no—
351C4×D3—22no—
352C4×D3—22no—
353C3×C8—22no—
354C3×C8—22no—
355C3×C8—22no—
356C3×C8—22no—
357C3×C8—34no—
358C3×C8—42no—
359C3×C8—42no—
360C3×C8—62no—
361C3⋊C8—22no—
362C3⋊C8C2×C3×D414yescomplemented
363C2×C32—42no—
364C2×C32#43614yescomplemented
365C2×C32C2×C4×D411yescharacteristic
366C2×C32C22×M1611yescharacteristic
367C2×C32C2×#13111yescharacteristic
368C3×D3—42no—
369C24—31no—
370C24C3×C4×D311yescharacteristic
371C2×D4—34no—
372C2×D4—64no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4C3×C4×D311yescharacteristic
383C22×C4C3×C4×D312yes—
384C22×C4C22×C3×D312yes—
385C22×C4C22×C3×D311yescharacteristic
386M16—68no—
387M16—68no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—32no—
391C2×C8—34no—
392C2×C8—32no—
393C2×C8—62no—
394C2×C8C22×C3×D312yes—
395C2×C8C22×C3×D312yes—
396C4⋊C4—34no—
397C22⋊C4—34no—
398C22⋊C4—34no—
399C42—32no—
400C42—32no—
401C22×C3—24no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—48no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—62no—
420C22×C3—124no—
421C22×C3#67⋊C412yes—
422C22×C3#90112yes—
423C22×C3D3×M1612yes—
424C22×C3C3×C4×D412yes—
425C22×C3C2×C3×M1612yes—
426C22×C3C3×#13112yes—
427C22×C3C22×C4×D311yescharacteristic
428C22×C3C23×C3×C411yescharacteristic
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—44no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—34no—
439C3×C4—42no—
440C3×C4—62no—
441C3×C4—62no—
442C3×C4#67⋊C412yes—
443C3×C4C3×C4×D412yes—
444C3×C4C2×D3×D411yescharacteristic
445C3×C4C2×#23711yescharacteristic
446C3×C4C22×C3×D411yescharacteristic
447C3×C4C2×C3×#4111yescharacteristic
448Dic3—22no—
449Dic3—22no—
450Dic3C2×C3×M1614yescomplemented
451C32C2×#43611yescharacteristic, complemented, Sylow
452C23—22no—
453C23—22no—
454C23—22no—
455C23—32no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23C2×C3×C4×D312yes—
460C23C2×C3×C4×D311yescharacteristic
461D4—68no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—34no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4C2×C3×C4×D312yes—
476C2×C4C2×C3×C4×D311yescharacteristic
477C2×C4C3×D3×D412yes—
478C2×C4C3×#23712yes—
479C2×C4C23×C3×D311yescharacteristic
480C8—22no—
481C8—22no—
482C8—34no—
483C8—62no—
484C2×C3—21no—
485C2×C3—24no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—44no—
489C2×C3—42no—
490C2×C3—42no—
491C2×C3—122no—
492C2×C3#472914yesdirect product factor
493C2×C3C3×#43614yescomplemented
494C2×C3C2×#67⋊C411yescharacteristic
495C2×C3C2×#90111yescharacteristic
496C2×C3C2×D3×M1611yescharacteristic
497C2×C3C2×C3×C4×D411yescharacteristic
498C2×C3C22×C3×M1611yescharacteristic
499C2×C3C2×C3×#13111yescharacteristic
500D3—42no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—44no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—124no—
509C22C3×#67⋊C412yes—
510C22C3×#90112yes—
511C22C3×D3×M1612yes—
512C22C22×C3×C4×D311yescharacteristic
513C4—22no—
514C4—34no—
515C4—62no—
516C4—62no—
517C4C3×#67⋊C412yes—
518C4C2×C3×D3×D411yescharacteristic
519C4C2×C3×#23711yescharacteristic
520C3—21no—
521C3C2×#472911yescharacteristic, direct product factor
522C3C2×C3×#43611yescharacteristic, complemented
523C2—42no—
524C2—122no—
525C2C3×#472914yesdirect product factor
526C2C2×C3×#67⋊C411yescharacteristic
527C2C2×C3×#90111yescharacteristic
528C2C2×C3×D3×M1611yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C22⋊C8C2
socleC23×C32C22×C4
prime radicalC22×C3×#67C4
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×C22⋊C8D3
hypocenterC3C2×C3×#436
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×C3×#208C2semidirect
3. C22×C3×ΓL(1,𝔽25)C2semidirect
4. C2×C32×C22⋊C8C2semidirect
5. C2×C3×(C2×D3)⋊C8C2semidirect
6. C2×C3×Dic3⋊C8C2semidirect
7. C2×C3×C8×Dic3C2semidirect
8. C3×#4729C2direct (N × H)
10. C2×#4729C3direct (N × H)
23. C2×C3×#208C22semidirect
27. C2×C3×ΓL(1,𝔽25)C22semidirect
28. C32×C22⋊C8C22semidirect
29. C3×(C2×D3)⋊C8C22semidirect
30. C3×Dic3⋊C8C22semidirect
31. C3×C8×Dic3C22semidirect
36. C22×#208C2×C3semidirect
37. C22×ΓL(1,𝔽25)C2×C3semidirect
42. C2×C3×C22⋊C8D3semidirect
43. C2×C3×C22⋊C8C2×C3semidirect
45. C2×(C2×D3)⋊C8C2×C3semidirect
46. C2×Dic3⋊C8C2×C3semidirect
47. C2×C8×Dic3C2×C3semidirect
48. #4729C2×C3direct (N × H)
77. C2×C3×C3⋊C8D4semidirect
114. C2×#208C22×C3semidirect
118. C2×ΓL(1,𝔽25)C22×C3semidirect
122. C3×C22⋊C8C2×D3semidirect
123. C3×C22⋊C8C22×C3semidirect
125. (C2×D3)⋊C8C22×C3semidirect
126. Dic3⋊C8C22×C3semidirect
127. C8×Dic3C22×C3semidirect
149. C2×C3×Dic3M16semidirect
159. C3×C3⋊C8C2×D4semidirect
166. C2×C22⋊C8C3×D3semidirect
239. C2×C3⋊C8C3×D4semidirect
261. C3×Dic3C2×M16semidirect
281. C22⋊C8C2×C3×D3semidirect
346. C2×Dic3C3×M16semidirect
362. C3⋊C8C2×C3×D4semidirect
364. C2×C32#436semidirect
450. Dic3C2×C3×M16semidirect
451. C32C2×#436semidirect
492. C2×C3#4729direct (N × H)
493. C2×C3C3×#436semidirect
521. C3C2×#4729direct (N × H)
522. C3C2×C3×#436semidirect
525. C2C3×#4729direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#672
C22×C3×#2082
C22×C3×ΓL(1,𝔽25)2
C2×C32×C22⋊C82
C2×C3×(C2×D3)⋊C82
C2×C3×Dic3⋊C82
C2×C3×C8×Dic32
C3×#47292
C2×C3×#4363
C2×#47293