Finite Group Databasebeta

← C2×C3×#4706 #11859208

Subgroups — C2×C3×#4706 #11859208

Order 1152.

1201 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
1C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
2C22×C3×D3×C8C211yesmaximal, characteristic, complemented
3C2×C3×C4×D12C211yesmaximal, characteristic
4C2×C32×#97C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#4706C218yesmaximal, direct product factor
9C2×C3×#435—31nomaximal
10C2×#4706C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C411yescharacteristic
12C22×C3×C4×D3C2211yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×C32×C42C2211yescharacteristic
17C2×C3×(C2×D3)⋊C4C411yescharacteristic
18C2×C3×(C2×D3)⋊C4C411yescharacteristic
19C2×C3×C4⋊Dic3C411yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C2×C3×ΓL(1,𝔽25)—24no—
23C2×C3×ΓL(1,𝔽25)C2214yescomplemented
24C2×C3×D3×C8—24no—
25C2×C3×D3×C8C2214yescomplemented
26C3×C4×D12C414yes—
27C3×C4×D12C2214yes—
28C32×#97C2214yescomplemented
29C3×(C2×D3)⋊C8C2214yescomplemented
30C3×(C2×D3)⋊C8C2214yescomplemented
31C3×C4⋊(C3⋊C8)C2214yescomplemented
32C22×C3×M16—31no—
33C23×C3×C8—31no—
34C2×C3×C4×D4—31no—
35C22×ΓL(1,𝔽25)C2×C311yescharacteristic, complemented
36C22×D3×C8C2×C311yescharacteristic, complemented
37C2×C4×D12C2×C311yescharacteristic
38C3×#435—38no—
39C2×C3×C4⋊C8—31no—
40C2×C3×C22⋊C8—31no—
41C2×C3×C22⋊C8—31no—
42C2×C3×#97—21no—
43C2×C3×#97D311yescharacteristic, complemented
44C2×C3×#97C2×C311yescharacteristic, complemented
45C2×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
46C2×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
47C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48#4706C2×C318yesdirect product factor
49C23×C3×D3C2×C411yescharacteristic
50C23×C3×D3C2×C411yescharacteristic
51C22×C32×C4C2×C411yescharacteristic
52C22×C32×C4C2×C411yescharacteristic
53C22×C32×C4C2311yescharacteristic
54C22×C3×Dic3C2×C411yescharacteristic
55C22×C3×Dic3C2×C411yescharacteristic
56C2×C3×D12—24no—
57C2×C3×D12C2×C414yes—
58C2×C3×C4×D3—22no—
59C2×C3×C4×D3—22no—
60C2×C3×C4×D3—22no—
61C2×C3×C4×D3—22no—
62C2×C3×C4×D3C2×C412yes—
63C2×C3×C4×D3C2×C412yes—
64C2×C3×C4×D3C2312yes—
65C2×C3×C4×D3C2312yes—
66C2×C32×C8—22no—
67C2×C32×C8D414yescomplemented
68C2×C32×C8C2312yes—
69C2×C32×C8C2312yes—
70C32×C42C2×C412yes—
71C32×C42C2312yes—
72C3×(C2×D3)⋊C4C2×C414yes—
73C3×(C2×D3)⋊C4C2×C414yes—
74C3×C4⋊Dic3C2×C414yes—
75C2×C3×C3⋊C8—22no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8C2312yes—
78C2×C3×C3⋊C8C2312yes—
79C3×ΓL(1,𝔽25)—28no—
80C3×D3×C8—28no—
81C2×#435—31noSylow
82C22×C3×D4—31no—
83C23×C3×C4—31no—
84C23×C3×C4—31no—
85C22×D12C3×C411yescharacteristic
86C22×C4×D3C22×C311yescharacteristic
87C22×C4×D3C22×C311yescharacteristic
88C2×C3×M16—34no—
89C2×C3×M16—64no—
90C22×C3×C8—21no—
91C22×C3×C8—21no—
92C22×C3×C8—31no—
93C22×C3×C8—31no—
94C22×C3×C8—34no—
95C22×C3×C8—64no—
96C22×C3×C8C2×D311yescharacteristic
97C22×C3×C8C2×D311yescharacteristic
98C22×C3×C8C22×C311yescharacteristic
99C22×C3×C8C22×C311yescharacteristic
100C3×C4×D4—34no—
101C3×C4×D4—34no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C22⋊C4—31no—
104C2×C3×C22⋊C4—31no—
105C2×C3×C42—21no—
106C2×C3×C42C2×D311yescharacteristic
107C2×C3×C42C22×C311yescharacteristic
108C2×(C2×D3)⋊C4C3×C411yescharacteristic
109C2×(C2×D3)⋊C4C3×C411yescharacteristic
110C2×C4⋊Dic3C3×C411yescharacteristic
111C22×C3⋊C8C22×C311yescharacteristic
112C22×C3⋊C8C22×C311yescharacteristic
113C2×ΓL(1,𝔽25)—24no—
114C2×ΓL(1,𝔽25)C22×C314yescomplemented
115C2×D3×C8—24no—
116C2×D3×C8C22×C314yescomplemented
117C4×D12C3×C414yes—
118C4×D12C22×C314yes—
119C3×C4⋊C8—34no—
120C3×C22⋊C8—34no—
121C3×C22⋊C8—34no—
122C3×#97—24no—
123C3×#97C2×D314yescomplemented
124C3×#97C22×C314yescomplemented
125(C2×D3)⋊C8C22×C314yescomplemented
126(C2×D3)⋊C8C22×C314yescomplemented
127C4⋊(C3⋊C8)C22×C314yescomplemented
128C23×C32C22×C411yescharacteristic
129C22×C3×D3—22no—
130C22×C3×D3—22no—
131C22×C3×D3—22no—
132C22×C3×D3—22no—
133C22×C3×D3—22no—
134C22×C3×D3M1614yescomplemented
135C22×C3×D3C22×C412yes—
136C22×C3×D3C22×C412yes—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4C22×C412yes—
140C2×C32×C4C22×C412yes—
141C2×C32×C4C22×C411yescharacteristic
142C2×C32×C4C2×D412yes—
143C2×C32×C4#4112yes—
144C2×C32×C4C2411yescharacteristic
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3C22×C412yes—
148C2×C3×Dic3C22×C412yes—
149C3×D12—28no—
150C3×C4×D3—24no—
151C3×C4×D3—24no—
152C3×C4×D3—22no—
153C3×C4×D3—22no—
154C3×C4×D3—22no—
155C3×C4×D3—22no—
156C32×C8—22no—
157C32×C8C2×D414yescomplemented
158C3×C3⋊C8—22no—
159C3×C3⋊C8—22no—
160C22×M16—31no—
161C23×C8—31no—
162C2×C4×D4—31no—
163#435—38no—
164C2×C4⋊C8—31no—
165C2×C22⋊C8—31no—
166C2×C22⋊C8—31no—
167C2×#97C3×D311yescharacteristic, complemented
168C24×C3—31no—
169C24×C3—31no—
170C23×D3C2×C3×C411yescharacteristic
171C23×D3C2×C3×C411yescharacteristic
172C2×C3×D4—34no—
173C2×C3×D4—64no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—32no—
180C22×C3×C4—32no—
181C22×C3×C4—32no—
182C22×C3×C4—32no—
183C22×C3×C4—62no—
184C22×C3×C4—62no—
185C22×C3×C4—62no—
186C22×C3×C4—62no—
187C22×C3×C4C4×D311yescharacteristic
188C22×C3×C4C4×D311yescharacteristic
189C22×C3×C4C2×C3×C411yescharacteristic
190C22×C3×C4C2×C3×C411yescharacteristic
191C22×C3×C4C22×D311yescharacteristic
192C22×C3×C4C23×C311yescharacteristic
193C22×Dic3C2×C3×C411yescharacteristic
194C22×Dic3C2×C3×C411yescharacteristic
195C2×D12—24no—
196C2×D12C2×C3×C414yes—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3C2×C3×C412yes—
202C2×C4×D3C2×C3×C412yes—
203C2×C4×D3C23×C312yes—
204C2×C4×D3C23×C312yes—
205C3×M16—68no—
206C2×C3×C8—24no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—24no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—32no—
213C2×C3×C8—32no—
214C2×C3×C8—62no—
215C2×C3×C8—62no—
216C2×C3×C8—68no—
217C2×C3×C8D1214yescomplemented
218C2×C3×C8C3×D414yescomplemented
219C2×C3×C8C22×D312yes—
220C2×C3×C8C22×D312yes—
221C2×C3×C8C23×C312yes—
222C2×C3×C8C23×C312yes—
223C3×C4⋊C4—34no—
224C3×C22⋊C4—34no—
225C3×C22⋊C4—34no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42C4×D312yes—
229C3×C42C2×C3×C412yes—
230C3×C42C22×D312yes—
231C3×C42C23×C312yes—
232(C2×D3)⋊C4C2×C3×C414yes—
233(C2×D3)⋊C4C2×C3×C414yes—
234C4⋊Dic3C2×C3×C414yes—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8C23×C312yes—
238C2×C3⋊C8C23×C312yes—
239ΓL(1,𝔽25)—28no—
240D3×C8—28no—
241C22×C32C4×D412yes—
242C22×C32C2×M1612yes—
243C22×C32#13112yes—
244C22×C32C23×C411yescharacteristic
245C2×C3×D3—22no—
246C2×C3×D3—22no—
247C2×C3×D3—28no—
248C2×C3×D3—22no—
249C2×C3×D3—22no—
250C2×C3×D3—22no—
251C2×C3×D3—44no—
252C2×C3×D3C2×M1614yescomplemented
253C32×C4—22no—
254C32×C4—22no—
255C32×C4C4×D412yes—
256C32×C4C22×D411yescharacteristic
257C32×C4C2×#4111yescharacteristic
258C3×Dic3—22no—
259C3×Dic3—22no—
260C22×D4—31no—
261C23×C4—31no—
262C23×C4—31no—
263C2×M16—34no—
264C2×M16—64no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—34no—
268C22×C8—64no—
269C22×C8C2×C3×D311yescharacteristic
270C22×C8C2×C3×D311yescharacteristic
271C4×D4—34no—
272C4×D4—34no—
273C2×C4⋊C4—31no—
274C2×C22⋊C4—31no—
275C2×C22⋊C4—31no—
276C2×C42C2×C3×D311yescharacteristic
277C4⋊C8—34no—
278C22⋊C8—34no—
279C22⋊C8—34no—
280#97C2×C3×D314yescomplemented
281C23×C3—21no—
282C23×C3—34no—
283C23×C3—32no—
284C23×C3—32no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—62no—
288C23×C3—62no—
289C23×C3—62no—
290C23×C3C2×C4×D311yescharacteristic
291C23×C3C22×C3×C411yescharacteristic
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3C3×M1614yescomplemented
298C22×D3C22×C3×C412yes—
299C22×D3C22×C3×C412yes—
300C3×D4—68no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—32no—
312C2×C3×C4—32no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—64no—
318C2×C3×C4—64no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4C2×C4×D312yes—
324C2×C3×C4C2×C4×D312yes—
325C2×C3×C4C2×C4×D311yescharacteristic
326C2×C3×C4C2×D1212yes—
327C2×C3×C4#23512yes—
328C2×C3×C4C22×C3×C412yes—
329C2×C3×C4C22×C3×C412yes—
330C2×C3×C4C22×C3×C411yescharacteristic
331C2×C3×C4C2×C3×D412yes—
332C2×C3×C4C3×#4112yes—
333C2×C3×C4C23×D311yescharacteristic
334C2×C3×C4C24×C311yescharacteristic
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3C22×C3×C412yes—
338C2×Dic3C22×C3×C412yes—
339D12—28no—
340C4×D3—24no—
341C4×D3—24no—
342C4×D3—22no—
343C4×D3—22no—
344C4×D3—22no—
345C4×D3—22no—
346C3×C8—24no—
347C3×C8—22no—
348C3×C8—22no—
349C3×C8—24no—
350C3×C8—62no—
351C3×C8—62no—
352C3×C8C2×D1214yescomplemented
353C3×C8C2×C3×D414yescomplemented
354C3⋊C8—22no—
355C3⋊C8—22no—
356C2×C32#43514yescomplemented
357C2×C32C2×C4×D411yescharacteristic
358C2×C32C22×M1611yescharacteristic
359C2×C32C2×#13111yescharacteristic
360C3×D3—24no—
361C3×D3—42no—
362C24—31no—
363C24—31no—
364C2×D4—34no—
365C2×D4—64no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4C3×C4×D311yescharacteristic
377C22×C4C3×C4×D311yescharacteristic
378C22×C4C22×C3×D311yescharacteristic
379M16—68no—
380C2×C8—22no—
381C2×C8—32no—
382C2×C8—32no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—68no—
386C2×C8C3×D1214yescomplemented
387C2×C8C22×C3×D312yes—
388C2×C8C22×C3×D312yes—
389C4⋊C4—34no—
390C22⋊C4—34no—
391C22⋊C4—34no—
392C42C3×C4×D312yes—
393C42C22×C3×D312yes—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—34no—
399C22×C3—62no—
400C22×C3—68no—
401C22×C3—62no—
402C22×C3—62no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—124no—
406C22×C3C4×D1212yes—
407C22×C3D3×M1612yes—
408C22×C3#90712yes—
409C22×C3C3×C4×D412yes—
410C22×C3C2×C3×M1612yes—
411C22×C3C3×#13112yes—
412C22×C3C22×C4×D311yescharacteristic
413C22×C3C23×C3×C411yescharacteristic
414C2×D3—22no—
415C2×D3—28no—
416C2×D3—22no—
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—22no—
420C2×D3—44no—
421C2×D3C2×C3×M1614yescomplemented
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—62no—
432C3×C4—62no—
433C3×C4C4×D1212yes—
434C3×C4C3×C4×D412yes—
435C3×C4C22×D1211yescharacteristic
436C3×C4C2×#23511yescharacteristic
437C3×C4C22×C3×D411yescharacteristic
438C3×C4C2×C3×#4111yescharacteristic
439Dic3—22no—
440Dic3—22no—
441C32C2×#43511yescharacteristic, complemented, Sylow
442C23—34no—
443C23—32no—
444C23—32no—
445C23—62no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23C2×C3×C4×D311yescharacteristic
451D4—68no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—64no—
463C2×C4—64no—
464C2×C4C2×C3×C4×D312yes—
465C2×C4C2×C3×C4×D312yes—
466C2×C4C2×C3×C4×D311yescharacteristic
467C2×C4C2×C3×D1212yes—
468C2×C4C3×#23512yes—
469C2×C4C23×C3×D311yescharacteristic
470C8—22no—
471C8—62no—
472C8—62no—
473C8C2×C3×D1214yescomplemented
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—24no—
477C2×C3—21no—
478C2×C3—64no—
479C2×C3—122no—
480C2×C3#470614yesdirect product factor
481C2×C3C3×#43514yescomplemented
482C2×C3C2×C4×D1211yescharacteristic
483C2×C3C2×D3×M1611yescharacteristic
484C2×C3C2×#90711yescharacteristic
485C2×C3C2×C3×C4×D411yescharacteristic
486C2×C3C22×C3×M1611yescharacteristic
487C2×C3C2×C3×#13111yescharacteristic
488D3—24no—
489D3—42no—
490C22—34no—
491C22—62no—
492C22—68no—
493C22—62no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—124no—
498C22C3×C4×D1212yes—
499C22C3×D3×M1612yes—
500C22C3×#90712yes—
501C22C22×C3×C4×D311yescharacteristic
502C4—22no—
503C4—22no—
504C4—62no—
505C4—62no—
506C4C3×C4×D1212yes—
507C4C22×C3×D1211yescharacteristic
508C4C2×C3×#23511yescharacteristic
509C3—21no—
510C3C2×#470611yescharacteristic, direct product factor
511C3C2×C3×#43511yescharacteristic, complemented
512C2—64no—
513C2—122no—
514C2C3×#470614yesdirect product factor
515C2C2×C3×C4×D1211yescharacteristic
516C2C2×C3×D3×M1611yescharacteristic
517C2C2×C3×#90711yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#97C2
socleC23×C32C22×C4
prime radicalC22×C3×D12C4
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#97D3
hypocenterC3C2×C3×#435
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×#97

Complements

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

NH ≅ G/Ntype
1. C22×C3×ΓL(1,𝔽25)C2semidirect
2. C22×C3×D3×C8C2semidirect
4. C2×C32×#97C2semidirect
5. C2×C3×(C2×D3)⋊C8C2semidirect
6. C2×C3×(C2×D3)⋊C8C2semidirect
7. C2×C3×C4⋊(C3⋊C8)C2semidirect
8. C3×#4706C2direct (N × H)
10. C2×#4706C3direct (N × H)
23. C2×C3×ΓL(1,𝔽25)C22semidirect
25. C2×C3×D3×C8C22semidirect
28. C32×#97C22semidirect
29. C3×(C2×D3)⋊C8C22semidirect
30. C3×(C2×D3)⋊C8C22semidirect
31. C3×C4⋊(C3⋊C8)C22semidirect
35. C22×ΓL(1,𝔽25)C2×C3semidirect
36. C22×D3×C8C2×C3semidirect
43. C2×C3×#97D3semidirect
44. C2×C3×#97C2×C3semidirect
45. C2×(C2×D3)⋊C8C2×C3semidirect
46. C2×(C2×D3)⋊C8C2×C3semidirect
47. C2×C4⋊(C3⋊C8)C2×C3semidirect
48. #4706C2×C3direct (N × H)
67. C2×C32×C8D4semidirect
114. C2×ΓL(1,𝔽25)C22×C3semidirect
116. C2×D3×C8C22×C3semidirect
123. C3×#97C2×D3semidirect
124. C3×#97C22×C3semidirect
125. (C2×D3)⋊C8C22×C3semidirect
126. (C2×D3)⋊C8C22×C3semidirect
127. C4⋊(C3⋊C8)C22×C3semidirect
134. C22×C3×D3M16semidirect
157. C32×C8C2×D4semidirect
167. C2×#97C3×D3semidirect
217. C2×C3×C8D12semidirect
218. C2×C3×C8C3×D4semidirect
252. C2×C3×D3C2×M16semidirect
280. #97C2×C3×D3semidirect
297. C22×D3C3×M16semidirect
352. C3×C8C2×D12semidirect
353. C3×C8C2×C3×D4semidirect
356. C2×C32#435semidirect
386. C2×C8C3×D12semidirect
421. C2×D3C2×C3×M16semidirect
441. C32C2×#435semidirect
473. C8C2×C3×D12semidirect
480. C2×C3#4706direct (N × H)
481. C2×C3C3×#435semidirect
510. C3C2×#4706direct (N × H)
511. C3C2×C3×#435semidirect
514. C2C3×#4706direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽25)2
C22×C3×D3×C82
C2×C3×C4×D122
C2×C32×#972
C2×C3×(C2×D3)⋊C82
C2×C3×C4⋊(C3⋊C8)2
C3×#47062
C2×C3×#4353
C2×#47063