Finite Group Databasebeta

← C2×C3×#5126 #11857352

Subgroups — C2×C3×#5126 #11857352

Order 1152.

2165 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×D3C211yesmaximal, characteristic
2C22×C32×M16C211yesmaximal, characteristic, complemented
3C22×C3×#208C211yesmaximal, characteristic, complemented
4C3×#5126C218yesmaximal, direct product factor
5C2×C3×(C2×D3)⋊C8C214yesmaximal, complemented
6C2×C3×#407—31nomaximal
7C2×#5126C311yesmaximal, characteristic, direct product factor
8C24×C3×D3C411yescharacteristic
9C23×C32×C4C2211yescharacteristic
10C23×C3×Dic3C411yescharacteristic
11C22×C3×C4×D3—24no—
12C22×C3×C4×D3—24no—
13C22×C3×C4×D3C414yes—
14C22×C3×C4×D3C412yes—
15C22×C3×C4×D3C2212yes—
16C22×C3×C4×D3C2214yes—
17C2×C32×M16—24no—
18C2×C32×M16C2214yescomplemented
19C22×C32×C8C2212yescomplemented
20C2×C3×#208—24no—
21C2×C3×#208C2214yescomplemented
22C22×C3×C3⋊C8C2212yescomplemented
23C3×(C2×D3)⋊C8C22116yescomplemented
24C24×C3×C4—31no—
25C23×C4×D3C2×C311yescharacteristic
26C22×C3×M16—21no—
27C22×C3×M16—31no—
28C22×C3×M16D311yescharacteristic, complemented
29C22×C3×M16C2×C311yescharacteristic, complemented
30C22×#208C2×C311yescharacteristic, complemented
31C3×#407—38no—
32C2×C3×C22⋊C8—34no—
33#5126C2×C318yesdirect product factor
34C2×(C2×D3)⋊C8C2×C314yescomplemented
35C24×C32C2×C411yescharacteristic
36C23×C3×D3—24no—
37C23×C3×D3—28no—
38C23×C3×D3C2×C414yes—
39C23×C3×D3C2×C412yes—
40C22×C32×C4C2×C412yes—
41C22×C32×C4C2×C411yescharacteristic
42C22×C32×C4D414yes—
43C22×C32×C4D414yes—
44C22×C32×C4C2312yes—
45C22×C32×C4C2311yescharacteristic
46C22×C3×Dic3—24no—
47C22×C3×Dic3C2×C414yes—
48C22×C3×Dic3C2×C412yes—
49C2×C3×C4×D3—216no—
50C2×C3×C4×D3—28no—
51C2×C3×C4×D3—24no—
52C2×C3×C4×D3—28no—
53C2×C3×C4×D3—24no—
54C2×C3×C4×D3—28no—
55C2×C3×C4×D3C2×C414yes—
56C2×C3×C4×D3C2×C418yes—
57C2×C3×C4×D3C2314yes—
58C32×M16—28no—
59C2×C32×C8—24no—
60C2×C32×C8C2314yescomplemented
61C3×#208—28no—
62C2×C3×C3⋊C8—24no—
63C2×C3×C3⋊C8C2314yescomplemented
64C2×#407—31noSylow
65C25×C3—31no—
66C24×D3C3×C411yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4—31no—
69C23×C3×C4—32no—
70C23×C3×C4—34no—
71C23×C3×C4—34no—
72C23×C3×C4—32no—
73C23×C3×C4—64no—
74C23×C3×C4—64no—
75C23×C3×C4C2×D311yescharacteristic
76C23×C3×C4C22×C311yescharacteristic
77C23×Dic3C3×C411yescharacteristic
78C22×C4×D3—24no—
79C22×C4×D3—24no—
80C22×C4×D3C3×C414yes—
81C22×C4×D3C3×C412yes—
82C22×C4×D3C22×C314yes—
83C22×C4×D3C22×C312yes—
84C2×C3×M16—28no—
85C2×C3×M16—24no—
86C2×C3×M16—24no—
87C2×C3×M16—24no—
88C2×C3×M16—34no—
89C2×C3×M16—64no—
90C2×C3×M16C2×D314yescomplemented
91C2×C3×M16C22×C314yescomplemented
92C22×C3×C8—22no—
93C22×C3×C8—32no—
94C22×C3×C8C2×D312yescomplemented
95C22×C3×C8C22×C312yescomplemented
96C2×#208—24no—
97C2×#208C22×C314yescomplemented
98C22×C3⋊C8C22×C312yescomplemented
99C3×C22⋊C8—316no—
100(C2×D3)⋊C8C22×C3116yescomplemented
101C23×C32—24no—
102C23×C32C22⋊C414yes—
103C23×C32C22×C412yes—
104C23×C32C22×C411yescharacteristic
105C22×C3×D3—232no—
106C22×C3×D3—28no—
107C22×C3×D3—24no—
108C22×C3×D3—24no—
109C22×C3×D3—216no—
110C22×C3×D3M1618yescomplemented
111C22×C3×D3C22×C414yes—
112C2×C32×C4C22⋊C418yes—
113C2×C32×C4C22⋊C414yes—
114C2×C32×C4C22×C412yes—
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C2×D412yes—
117C2×C32×C4C2×D414yes—
118C2×C32×C4C2×D412yes—
119C2×C32×C4C2×D414yes—
120C2×C32×C4C2411yescharacteristic
121C2×C3×Dic3—28no—
122C2×C3×Dic3—24no—
123C2×C3×Dic3C22×C414yes—
124C3×C4×D3—28no—
125C3×C4×D3—216no—
126C3×C4×D3—24no—
127C3×C4×D3—24no—
128C32×C8—24no—
129C3×C3⋊C8—24no—
130C24×C4—31no—
131C22×M16—31no—
132C22×M16C3×D311yescharacteristic, complemented
133#407—38no—
134C2×C22⋊C8—34no—
135C24×C3—21no—
136C24×C3—32no—
137C24×C3—34no—
138C24×C3—68no—
139C24×C3—64no—
140C24×C3C4×D311yescharacteristic
141C24×C3C2×C3×C411yescharacteristic
142C23×D3—24no—
143C23×D3—28no—
144C23×D3C2×C3×C414yes—
145C23×D3C2×C3×C412yes—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—34no—
153C22×C3×C4—32no—
154C22×C3×C4—38no—
155C22×C3×C4—34no—
156C22×C3×C4—34no—
157C22×C3×C4—64no—
158C22×C3×C4—616no—
159C22×C3×C4—68no—
160C22×C3×C4—64no—
161C22×C3×C4—68no—
162C22×C3×C4—64no—
163C22×C3×C4—68no—
164C22×C3×C4C4×D311yescharacteristic
165C22×C3×C4C4×D312yes—
166C22×C3×C4D1214yes—
167C22×C3×C4#6714yes—
168C22×C3×C4C2×C3×C412yes—
169C22×C3×C4C2×C3×C411yescharacteristic
170C22×C3×C4C3×D414yes—
171C22×C3×C4C3×D414yes—
172C22×C3×C4C22×D312yes—
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C23×C311yescharacteristic
175C22×C3×C4C23×C312yes—
176C22×Dic3—24no—
177C22×Dic3C2×C3×C414yes—
178C22×Dic3C2×C3×C412yes—
179C2×C4×D3—216no—
180C2×C4×D3—28no—
181C2×C4×D3—24no—
182C2×C4×D3—28no—
183C2×C4×D3—24no—
184C2×C4×D3—28no—
185C2×C4×D3C2×C3×C418yes—
186C2×C4×D3C2×C3×C414yes—
187C2×C4×D3C23×C314yes—
188C3×M16—216no—
189C3×M16—28no—
190C3×M16—28no—
191C3×M16—68no—
192C2×C3×C8—28no—
193C2×C3×C8—24no—
194C2×C3×C8—24no—
195C2×C3×C8—24no—
196C2×C3×C8—34no—
197C2×C3×C8—64no—
198C2×C3×C8C22×D314yescomplemented
199C2×C3×C8C23×C314yescomplemented
200#208—28no—
201C2×C3⋊C8—24no—
202C2×C3⋊C8C23×C314yescomplemented
203C22×C32—24no—
204C22×C32—28no—
205C22×C32C2×C22⋊C414yes—
206C22×C32C2×C22⋊C412yes—
207C22×C32C2×M1614yes—
208C22×C32C23×C411yescharacteristic
209C2×C3×D3—232no—
210C2×C3×D3—24no—
211C2×C3×D3—24no—
212C2×C3×D3—216no—
213C2×C3×D3C2×M1618yescomplemented
214C32×C4C2×C22⋊C412yes—
215C32×C4C2×C22⋊C414yes—
216C32×C4C22×D411yescharacteristic
217C32×C4C22×D411yescharacteristic
218C3×Dic3—24no—
219C25—31no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4—34no—
223C23×C4—32no—
224C23×C4—34no—
225C23×C4—64no—
226C23×C4—64no—
227C23×C4C2×C3×D311yescharacteristic
228C2×M16—24no—
229C2×M16—34no—
230C2×M16—64no—
231C2×M16C2×C3×D314yescomplemented
232C22×C8—32no—
233C22×C8C2×C3×D312yescomplemented
234C22⋊C8—316no—
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—21no—
239C23×C3—22no—
240C23×C3—38no—
241C23×C3—34no—
242C23×C3—44no—
243C23×C3—616no—
244C23×C3—632no—
245C23×C3—64no—
246C23×C3—68no—
247C23×C3—64no—
248C23×C3(C2×D3)⋊C414yes—
249C23×C3C3×C22⋊C414yes—
250C23×C3C2×C4×D312yes—
251C23×C3C2×C4×D311yescharacteristic
252C23×C3C22×C3×C412yes—
253C23×C3C22×C3×C411yescharacteristic
254C22×D3—232no—
255C22×D3—28no—
256C22×D3—24no—
257C22×D3—24no—
258C22×D3—216no—
259C22×D3C3×M1618yescomplemented
260C22×D3C22×C3×C414yes—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—28no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—34no—
271C2×C3×C4—68no—
272C2×C3×C4—64no—
273C2×C3×C4—616no—
274C2×C3×C4—68no—
275C2×C3×C4—64no—
276C2×C3×C4—64no—
277C2×C3×C4(C2×D3)⋊C418yes—
278C2×C3×C4(C2×D3)⋊C414yes—
279C2×C3×C4C3×C22⋊C418yes—
280C2×C3×C4C3×C22⋊C414yes—
281C2×C3×C4C2×C4×D312yes—
282C2×C3×C4C2×C4×D311yescharacteristic
283C2×C3×C4C2×D1212yes—
284C2×C3×C4C2×D1214yes—
285C2×C3×C4C2×#6712yes—
286C2×C3×C4C2×#6714yes—
287C2×C3×C4C22×C3×C411yescharacteristic
288C2×C3×C4C22×C3×C412yes—
289C2×C3×C4C2×C3×D412yes—
290C2×C3×C4C2×C3×D414yes—
291C2×C3×C4C2×C3×D412yes—
292C2×C3×C4C2×C3×D414yes—
293C2×C3×C4C23×D311yescharacteristic
294C2×C3×C4C24×C311yescharacteristic
295C2×Dic3—28no—
296C2×Dic3—24no—
297C2×Dic3C22×C3×C414yes—
298C4×D3—28no—
299C4×D3—216no—
300C4×D3—24no—
301C4×D3—24no—
302C3×C8—28no—
303C3×C8—24no—
304C3×C8—24no—
305C3×C8—64no—
306C3⋊C8—24no—
307C2×C32—24no—
308C2×C32#40714yescomplemented
309C2×C32C22×C22⋊C411yescharacteristic
310C2×C32C22×M1612yes—
311C3×D3—28no—
312C24—32no—
313C24—34no—
314C24—68no—
315C24—64no—
316C24C3×C4×D311yescharacteristic
317C22×C4—34no—
318C22×C4—32no—
319C22×C4—34no—
320C22×C4—34no—
321C22×C4—38no—
322C22×C4—64no—
323C22×C4—616no—
324C22×C4—68no—
325C22×C4—64no—
326C22×C4—68no—
327C22×C4—64no—
328C22×C4—68no—
329C22×C4C3×C4×D312yes—
330C22×C4C3×C4×D311yescharacteristic
331C22×C4C3×D1214yes—
332C22×C4C3×#6714yes—
333C22×C4C22×C3×D311yescharacteristic
334C22×C4C22×C3×D312yes—
335M16—28no—
336M16—68no—
337C2×C8—24no—
338C2×C8—34no—
339C2×C8—64no—
340C2×C8C22×C3×D314yescomplemented
341C22×C3—28no—
342C22×C3—24no—
343C22×C3—24no—
344C22×C3—28no—
345C22×C3—24no—
346C22×C3—22no—
347C22×C3—24no—
348C22×C3—21no—
349C22×C3—38no—
350C22×C3—44no—
351C22×C3—48no—
352C22×C3—632no—
353C22×C3—616no—
354C22×C3—64no—
355C22×C3—64no—
356C22×C3D3×M1614yes—
357C22×C3C2×(C2×D3)⋊C414yes—
358C22×C3C2×(C2×D3)⋊C412yes—
359C22×C3C2×C3×C22⋊C414yes—
360C22×C3C2×C3×C22⋊C412yes—
361C22×C3C2×C3×M1614yes—
362C22×C3C22×C4×D311yescharacteristic
363C22×C3C23×C3×C411yescharacteristic
364C2×D3—232no—
365C2×D3—24no—
366C2×D3—216no—
367C2×D3—24no—
368C2×D3C2×C3×M1618yescomplemented
369C3×C4—22no—
370C3×C4—24no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—64no—
374C3×C4C2×(C2×D3)⋊C414yes—
375C3×C4C2×(C2×D3)⋊C412yes—
376C3×C4C2×C3×C22⋊C412yes—
377C3×C4C2×C3×C22⋊C414yes—
378C3×C4C22×D1211yescharacteristic
379C3×C4C22×#6711yescharacteristic
380C3×C4C22×C3×D411yescharacteristic
381C3×C4C22×C3×D411yescharacteristic
382Dic3—24no—
383C32C2×#40711yescharacteristic, complemented, Sylow
384C23—24no—
385C23—38no—
386C23—34no—
387C23—616no—
388C23—632no—
389C23—64no—
390C23—68no—
391C23—64no—
392C23C3×(C2×D3)⋊C414yes—
393C23C2×C3×C4×D312yes—
394C23C2×C3×C4×D311yescharacteristic
395C2×C4—34no—
396C2×C4—68no—
397C2×C4—64no—
398C2×C4—68no—
399C2×C4—616no—
400C2×C4—64no—
401C2×C4—64no—
402C2×C4C3×(C2×D3)⋊C418yes—
403C2×C4C3×(C2×D3)⋊C414yes—
404C2×C4C2×C3×C4×D311yescharacteristic
405C2×C4C2×C3×C4×D312yes—
406C2×C4C2×C3×D1212yes—
407C2×C4C2×C3×D1214yes—
408C2×C4C2×C3×#6712yes—
409C2×C4C2×C3×#6714yes—
410C2×C4C23×C3×D311yescharacteristic
411C8—24no—
412C8—64no—
413C2×C3—24no—
414C2×C3—24no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—24no—
418C2×C3—44no—
419C2×C3—68no—
420C2×C3#512614yesdirect product factor
421C2×C3C3×#40714yescomplemented
422C2×C3C2×D3×M1612yes—
423C2×C3C22×(C2×D3)⋊C411yescharacteristic
424C2×C3C22×C3×C22⋊C411yescharacteristic
425C2×C3C22×C3×M1612yes—
426D3—28no—
427C22—24no—
428C22—28no—
429C22—38no—
430C22—632no—
431C22—64no—
432C22—64no—
433C22—616no—
434C22C3×D3×M1614yes—
435C22C2×C3×(C2×D3)⋊C414yes—
436C22C2×C3×(C2×D3)⋊C412yes—
437C22C22×C3×C4×D311yescharacteristic
438C4—64no—
439C4C2×C3×(C2×D3)⋊C412yes—
440C4C2×C3×(C2×D3)⋊C414yes—
441C4C22×C3×D1211yescharacteristic
442C4C22×C3×#6711yescharacteristic
443C3—21no—
444C3C2×#512611yescharacteristic, direct product factor
445C3C2×C3×#40711yescharacteristic, complemented
446C2—24no—
447C2—68no—
448C2C3×#512614yesdirect product factor
449C2C2×C3×D3×M1612yes—
450C2C22×C3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC22×C32×M16C2
socleC23×C32C22×C4
prime radicalC24×C3×D3C4
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×M16D3
hypocenterC3C2×C3×#407
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
2. C22×C32×M16C2semidirect
3. C22×C3×#208C2semidirect
4. C3×#5126C2direct (N × H)
5. C2×C3×(C2×D3)⋊C8C2semidirect
7. C2×#5126C3direct (N × H)
18. C2×C32×M16C22semidirect
19. C22×C32×C8C22semidirect
21. C2×C3×#208C22semidirect
22. C22×C3×C3⋊C8C22semidirect
23. C3×(C2×D3)⋊C8C22semidirect
28. C22×C3×M16D3semidirect
29. C22×C3×M16C2×C3semidirect
30. C22×#208C2×C3semidirect
33. #5126C2×C3direct (N × H)
34. C2×(C2×D3)⋊C8C2×C3semidirect
60. C2×C32×C8C23semidirect
63. C2×C3×C3⋊C8C23semidirect
90. C2×C3×M16C2×D3semidirect
91. C2×C3×M16C22×C3semidirect
94. C22×C3×C8C2×D3semidirect
95. C22×C3×C8C22×C3semidirect
97. C2×#208C22×C3semidirect
98. C22×C3⋊C8C22×C3semidirect
100. (C2×D3)⋊C8C22×C3semidirect
110. C22×C3×D3M16semidirect
132. C22×M16C3×D3semidirect
198. C2×C3×C8C22×D3semidirect
199. C2×C3×C8C23×C3semidirect
202. C2×C3⋊C8C23×C3semidirect
213. C2×C3×D3C2×M16semidirect
231. C2×M16C2×C3×D3semidirect
233. C22×C8C2×C3×D3semidirect
259. C22×D3C3×M16semidirect
308. C2×C32#407semidirect
340. C2×C8C22×C3×D3semidirect
368. C2×D3C2×C3×M16semidirect
383. C32C2×#407semidirect
420. C2×C3#5126direct (N × H)
421. C2×C3C3×#407semidirect
444. C3C2×#5126direct (N × H)
445. C3C2×C3×#407semidirect
448. C2C3×#5126direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×D32
C22×C32×M162
C22×C3×#2082
C3×#51262
C2×C3×(C2×D3)⋊C82
C2×C3×#4073
C2×#51263