Finite Group Databasebeta

← C3×#82267 #11801037

Subgroups — C3×#82267 #11801037

Order 1152.

2341 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×D3C211yesmaximal, characteristic
2C2×C32×#117C211yesmaximal, characteristic, complemented
3C2×C3×#926C211yesmaximal, characteristic, complemented
4C3×#4987C214yesmaximal, complemented
5C3×#4955C212yesmaximal
6C3×#4943C212yesmaximal, complemented
7C3×C4×(C2×D3)⋊C4C214yesmaximal, complemented
8C3×#2280—31nomaximal
9#82267C311yesmaximal, characteristic, direct product factor
10C24×C3×D3C2211yescharacteristic
11C23×C32×C4C2211yescharacteristic
12C23×C3×Dic3C2211yescharacteristic
13C22×C3×C4×D3—24no—
14C22×C3×C4×D3—24no—
15C22×C3×C4×D3C418yescomplemented
16C22×C3×C4×D3C2212yes—
17C22×C3×C4×D3C2212yes—
18C32×#117—24no—
19C2×C32×C4⋊C4C2212yescomplemented
20C2×C32×C22⋊C4C2212yes—
21C2×C32×C42C2212yescomplemented
22C2×C3×#217C2212yes—
23C2×C3×(C2×D3)⋊C4C2218yes—
24C3×#926—24no—
25C2×C3×C4⋊Dic3C2212yescomplemented
26C2×C3×C4×Dic3C2212yescomplemented
27C3×C3⋊#95C2218yescomplemented
28C24×C3×C4—31no—
29C23×C4×D3C2×C311yescharacteristic
30C2×C3×#117—21no—
31C2×C3×#117—31no—
32C2×C3×#117D311yescharacteristic, complemented
33C2×C3×#117C2×C311yescharacteristic, complemented
34C2×#926C2×C311yescharacteristic, complemented
35C3×#381—32no—
36C3×#380—34no—
37C3×#379—32no—
38C3×C4×C22⋊C4—34no—
39#4987C2×C314yescomplemented
40#4955C2×C312yes—
41#4943C2×C312yescomplemented
42C4×(C2×D3)⋊C4C2×C314yescomplemented
43C24×C32C2311yescharacteristic
44C23×C3×D3—22no—
45C23×C3×D3—22no—
46C23×C3×D3—28no—
47C23×C3×D3C2×C414yes—
48C23×C3×D3C2312yes—
49C22×C32×C4C2×C414yes—
50C22×C32×C4D414yes—
51C22×C32×C4D414yes—
52C22×C32×C4C2314yes—
53C22×C32×C4C2311yescharacteristic
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3—22no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3C2×C414yes—
58C22×C3×Dic3C2314yes—
59C22×C3×Dic3C2312yes—
60C2×C3×C4×D3—232no—
61C2×C3×C4×D3—24no—
62C2×C3×C4×D3—24no—
63C2×C3×C4×D3—24no—
64C2×C3×C4×D3—24no—
65C2×C3×C4×D3C2×C418yes—
66C2×C3×C4×D3C2×C418yes—
67C32×C4⋊C4—24no—
68C32×C22⋊C4—24no—
69C32×C42—24no—
70C3×#217—24no—
71C3×(C2×D3)⋊C4—216no—
72C3×C4⋊Dic3—24no—
73C3×C4×Dic3—24no—
74#2280—31noSylow
75C25×C3—31no—
76C24×D3C22×C311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—31no—
79C23×C3×C4—38no—
80C23×C3×C4—32no—
81C23×C3×C4—32no—
82C23×C3×C4—64no—
83C23×C3×C4—64no—
84C23×C3×C4C2×D311yescharacteristic
85C23×C3×C4C22×C311yescharacteristic
86C23×Dic3C22×C311yescharacteristic
87C22×C4×D3—24no—
88C22×C4×D3—24no—
89C22×C4×D3C3×C418yescomplemented
90C22×C4×D3C22×C312yes—
91C22×C4×D3C22×C312yes—
92C3×#117—28no—
93C3×#117—24no—
94C3×#117—24no—
95C3×#117—64no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—32no—
98C2×C3×C4⋊C4C2×D312yescomplemented
99C2×C3×C4⋊C4C22×C312yescomplemented
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—32no—
102C2×C3×C22⋊C4—38no—
103C2×C3×C22⋊C4C2×D312yes—
104C2×C3×C22⋊C4C22×C312yes—
105C2×C3×C42—22no—
106C2×C3×C42—32no—
107C2×C3×C42C2×D312yescomplemented
108C2×C3×C42C22×C312yescomplemented
109C2×#217C22×C312yes—
110C2×(C2×D3)⋊C4C22×C318yes—
111#926—24no—
112C2×C4⋊Dic3C22×C312yescomplemented
113C2×C4×Dic3C22×C312yescomplemented
114C3×#95—38no—
115C3⋊#95C22×C318yescomplemented
116C23×C32—24no—
117C23×C32C22×C412yes—
118C23×C32C2×D412yes—
119C23×C32C2×D412yes—
120C23×C32C2411yescharacteristic
121C22×C3×D3—216no—
122C22×C3×D3—216no—
123C22×C3×D3—28no—
124C22×C3×D3—22no—
125C22×C3×D3—22no—
126C22×C3×D3—24no—
127C22×C3×D3—216no—
128C22×C3×D3C22×C414yes—
129C22×C3×D3#4118yes—
130C2×C32×C4—24no—
131C2×C32×C4—28no—
132C2×C32×C4C22⋊C4116yescomplemented
133C2×C32×C4C22×C412yes—
134C2×C32×C4C22×C412yes—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×D412yes—
137C2×C32×C4C2×D412yes—
138C2×C32×C4C2×D412yes—
139C2×C3×Dic3—28no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—24no—
143C2×C3×Dic3—28no—
144C2×C3×Dic3C22×C414yes—
145C3×C4×D3—216no—
146C3×C4×D3—216no—
147C24×C4—31no—
148C2×#117—31no—
149C2×#117C3×D311yescharacteristic, complemented
150#381—32no—
151#380—34no—
152#379—32no—
153C4×C22⋊C4—34no—
154C24×C3—21no—
155C24×C3—32no—
156C24×C3—34no—
157C24×C3—68no—
158C24×C3—62no—
159C24×C3—62no—
160C24×C3C22×D311yescharacteristic
161C24×C3C23×C311yescharacteristic
162C23×D3—22no—
163C23×D3—22no—
164C23×D3—28no—
165C23×D3C2×C3×C414yes—
166C23×D3C23×C312yes—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—24no—
173C22×C3×C4—34no—
174C22×C3×C4—32no—
175C22×C3×C4—38no—
176C22×C3×C4—38no—
177C22×C3×C4—34no—
178C22×C3×C4—62no—
179C22×C3×C4—62no—
180C22×C3×C4—632no—
181C22×C3×C4—64no—
182C22×C3×C4—64no—
183C22×C3×C4—64no—
184C22×C3×C4—64no—
185C22×C3×C4C4×D314yes—
186C22×C3×C4D1214yes—
187C22×C3×C4#6714yes—
188C22×C3×C4C2×C3×C414yes—
189C22×C3×C4C3×D414yes—
190C22×C3×C4C3×D414yes—
191C22×C3×C4C22×D314yes—
192C22×C3×C4C22×D311yescharacteristic
193C22×C3×C4C22×D311yescharacteristic
194C22×C3×C4C23×C314yes—
195C22×C3×C4C23×C311yescharacteristic
196C22×C3×C4C23×C311yescharacteristic
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3C2×C3×C414yes—
200C22×Dic3C23×C314yes—
201C22×Dic3C23×C312yes—
202C2×C4×D3—232no—
203C2×C4×D3—24no—
204C2×C4×D3—24no—
205C2×C4×D3—24no—
206C2×C4×D3—24no—
207C2×C4×D3C2×C3×C418yes—
208C2×C4×D3C2×C3×C418yes—
209C3×C4⋊C4—28no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—64no—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—616no—
217C3×C22⋊C4—64no—
218C3×C42—28no—
219C3×C42—24no—
220C3×C42—24no—
221C3×C42—64no—
222#217—24no—
223(C2×D3)⋊C4—216no—
224C4⋊Dic3—24no—
225C4×Dic3—24no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32C2×C22⋊C414yes—
230C22×C32C23×C411yescharacteristic
231C22×C32C22×D411yescharacteristic
232C22×C32C22×D411yescharacteristic
233C22×C32C2×#4114yes—
234C2×C3×D3—232no—
235C2×C3×D3—24no—
236C2×C3×D3—24no—
237C2×C3×D3—28no—
238C2×C3×D3—28no—
239C2×C3×D3#11718yescomplemented
240C32×C4—44no—
241C32×C4C2×C22⋊C414yes—
242C32×C4C2×C22⋊C414yes—
243C3×Dic3—24no—
244C3×Dic3—44no—
245C25—31no—
246C23×C4—31no—
247C23×C4—38no—
248C23×C4—32no—
249C23×C4—32no—
250C23×C4—64no—
251C23×C4—64no—
252C23×C4C2×C3×D311yescharacteristic
253#117—24no—
254#117—64no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4C2×C3×D312yescomplemented
257C2×C22⋊C4—32no—
258C2×C22⋊C4—38no—
259C2×C22⋊C4C2×C3×D312yes—
260C2×C42—32no—
261C2×C42C2×C3×D312yescomplemented
262#95—38no—
263C23×C3—24no—
264C23×C3—24no—
265C23×C3—21no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—34no—
270C23×C3—38no—
271C23×C3—44no—
272C23×C3—616no—
273C23×C3—616no—
274C23×C3—616no—
275C23×C3—68no—
276C23×C3—62no—
277C23×C3—64no—
278C23×C3—62no—
279C23×C3C2×C4×D312yes—
280C23×C3C2×D1212yes—
281C23×C3C2×#6712yes—
282C23×C3C22×C3×C412yes—
283C23×C3C2×C3×D412yes—
284C23×C3C2×C3×D412yes—
285C23×C3C23×D311yescharacteristic
286C23×C3C24×C311yescharacteristic
287C22×D3—216no—
288C22×D3—216no—
289C22×D3—28no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—24no—
293C22×D3—216no—
294C22×D3C22×C3×C414yes—
295C22×D3C3×#4118yes—
296C2×C3×C4—28no—
297C2×C3×C4—28no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—28no—
301C2×C3×C4—216no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—34no—
309C2×C3×C4—48no—
310C2×C3×C4—68no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—616no—
314C2×C3×C4—616no—
315C2×C3×C4—64no—
316C2×C3×C4—68no—
317C2×C3×C4(C2×D3)⋊C4116yescomplemented
318C2×C3×C4C3×C22⋊C4116yescomplemented
319C2×C3×C4C2×C4×D312yes—
320C2×C3×C4C2×C4×D312yes—
321C2×C3×C4C2×D1212yes—
322C2×C3×C4C2×D1212yes—
323C2×C3×C4C2×#6712yes—
324C2×C3×C4C2×#6712yes—
325C2×C3×C4C22×C3×C412yes—
326C2×C3×C4C22×C3×C412yes—
327C2×C3×C4C2×C3×D412yes—
328C2×C3×C4C2×C3×D412yes—
329C2×C3×C4C2×C3×D412yes—
330C2×C3×C4C2×C3×D412yes—
331C2×Dic3—28no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—28no—
336C2×Dic3C22×C3×C414yes—
337C4×D3—216no—
338C4×D3—216no—
339C2×C32—24no—
340C2×C32C22×C22⋊C411yescharacteristic
341C2×C32C2×#11712yes—
342C2×C32#52512yes—
343C2×C32#52512yes—
344C3×D3—28no—
345C24—32no—
346C24—34no—
347C24—68no—
348C24—62no—
349C24—62no—
350C24C22×C3×D311yescharacteristic
351C22×C4—32no—
352C22×C4—34no—
353C22×C4—38no—
354C22×C4—38no—
355C22×C4—34no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—632no—
359C22×C4—64no—
360C22×C4—64no—
361C22×C4—64no—
362C22×C4—64no—
363C22×C4C3×C4×D314yes—
364C22×C4C3×D1214yes—
365C22×C4C3×#6714yes—
366C22×C4C22×C3×D314yes—
367C22×C4C22×C3×D311yescharacteristic
368C22×C4C22×C3×D311yescharacteristic
369C4⋊C4—24no—
370C4⋊C4—64no—
371C22⋊C4—24no—
372C22⋊C4—616no—
373C22⋊C4—64no—
374C42—24no—
375C42—64no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—24no—
386C22×C3—21no—
387C22×C3—38no—
388C22×C3—44no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—632no—
392C22×C3—64no—
393C22×C3—68no—
394C22×C3—64no—
395C22×C3—68no—
396C22×C3C2×(C2×D3)⋊C414yes—
397C22×C3C2×C3×C22⋊C414yes—
398C22×C3C22×C4×D311yescharacteristic
399C22×C3C22×D1211yescharacteristic
400C22×C3D3×#4114yes—
401C22×C3C22×#6711yescharacteristic
402C22×C3C23×C3×C411yescharacteristic
403C22×C3C22×C3×D411yescharacteristic
404C22×C3C22×C3×D411yescharacteristic
405C22×C3C2×C3×#4114yes—
406C2×D3—232no—
407C2×D3—24no—
408C2×D3—24no—
409C2×D3—28no—
410C2×D3—28no—
411C2×D3C3×#11718yescomplemented
412C3×C4—24no—
413C3×C4—24no—
414C3×C4—48no—
415C3×C4—44no—
416C3×C4—44no—
417C3×C4—64no—
418C3×C4—124no—
419C3×C4C2×(C2×D3)⋊C414yes—
420C3×C4C2×(C2×D3)⋊C414yes—
421C3×C4C2×C3×C22⋊C414yes—
422C3×C4C2×C3×C22⋊C414yes—
423Dic3—24no—
424Dic3—44no—
425C32#228011yescharacteristic, complemented, Sylow
426C23—24no—
427C23—34no—
428C23—38no—
429C23—616no—
430C23—616no—
431C23—616no—
432C23—68no—
433C23—62no—
434C23—64no—
435C23—62no—
436C23C2×C3×C4×D312yes—
437C23C2×C3×D1212yes—
438C23C2×C3×#6712yes—
439C23C23×C3×D311yescharacteristic
440C2×C4—28no—
441C2×C4—24no—
442C2×C4—34no—
443C2×C4—68no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—616no—
447C2×C4—616no—
448C2×C4—64no—
449C2×C4—68no—
450C2×C4C3×(C2×D3)⋊C4116yescomplemented
451C2×C4C2×C3×C4×D312yes—
452C2×C4C2×C3×C4×D312yes—
453C2×C4C2×C3×D1212yes—
454C2×C4C2×C3×D1212yes—
455C2×C4C2×C3×#6712yes—
456C2×C4C2×C3×#6712yes—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—22no—
463C2×C3—44no—
464C2×C3—68no—
465C2×C3D3×#11712yes—
466C2×C3#552412yes—
467C2×C3C22×(C2×D3)⋊C411yescharacteristic
468C2×C3#582812yes—
469C2×C3C22×C3×C22⋊C411yescharacteristic
470C2×C3C2×C3×#11712yes—
471C2×C3C3×#52512yes—
472C2×C3C3×#52512yes—
473D3—28no—
474C22—24no—
475C22—24no—
476C22—24no—
477C22—38no—
478C22—632no—
479C22—64no—
480C22—68no—
481C22—64no—
482C22—68no—
483C22C2×C3×(C2×D3)⋊C414yes—
484C22C22×C3×C4×D311yescharacteristic
485C22C22×C3×D1211yescharacteristic
486C22C3×D3×#4114yes—
487C22C22×C3×#6711yescharacteristic
488C4—44no—
489C4—64no—
490C4—124no—
491C4C2×C3×(C2×D3)⋊C414yes—
492C4C2×C3×(C2×D3)⋊C414yes—
493C3—21no—
494C3#8226711yescharacteristic, direct product factor
495C3C3×#228011yescharacteristic, complemented
496C2—24no—
497C2—68no—
498C2C3×D3×#11712yes—
499C2C3×#552412yes—
500C2C22×C3×(C2×D3)⋊C411yescharacteristic
501C2C3×#582812yes—
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC2×C32×#117C2
socleC23×C32C24
prime radicalC24×C3×D3C22
cosocleC23×C3C24×C3
hypercenterC2×C3×#117D3
hypocenterC3C3×#2280
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
2. C2×C32×#117C2semidirect
3. C2×C3×#926C2semidirect
4. C3×#4987C2semidirect
6. C3×#4943C2semidirect
7. C3×C4×(C2×D3)⋊C4C2semidirect
9. #82267C3direct (N × H)
15. C22×C3×C4×D3C4semidirect
19. C2×C32×C4⋊C4C22semidirect
21. C2×C32×C42C22semidirect
25. C2×C3×C4⋊Dic3C22semidirect
26. C2×C3×C4×Dic3C22semidirect
27. C3×C3⋊#95C22semidirect
32. C2×C3×#117D3semidirect
33. C2×C3×#117C2×C3semidirect
34. C2×#926C2×C3semidirect
39. #4987C2×C3semidirect
41. #4943C2×C3semidirect
42. C4×(C2×D3)⋊C4C2×C3semidirect
89. C22×C4×D3C3×C4semidirect
98. C2×C3×C4⋊C4C2×D3semidirect
99. C2×C3×C4⋊C4C22×C3semidirect
107. C2×C3×C42C2×D3semidirect
108. C2×C3×C42C22×C3semidirect
112. C2×C4⋊Dic3C22×C3semidirect
113. C2×C4×Dic3C22×C3semidirect
115. C3⋊#95C22×C3semidirect
132. C2×C32×C4C22⋊C4semidirect
149. C2×#117C3×D3semidirect
239. C2×C3×D3#117semidirect
256. C2×C4⋊C4C2×C3×D3semidirect
261. C2×C42C2×C3×D3semidirect
317. C2×C3×C4(C2×D3)⋊C4semidirect
318. C2×C3×C4C3×C22⋊C4semidirect
411. C2×D3C3×#117semidirect
425. C32#2280semidirect
450. C2×C4C3×(C2×D3)⋊C4semidirect
494. C3#82267direct (N × H)
495. C3C3×#2280semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C4×D32
C2×C32×#1172
C2×C3×#9262
C3×#49872
C3×#49552
C3×#49432
C3×C4×(C2×D3)⋊C42
C3×#22803
#822673