Finite Group Databasebeta

← C3×#82544 #11866246

Subgroups — C3×#82544 #11866246

Order 1152.

1157 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
1C32×#518C211yesmaximal, characteristic, complemented
2C22×C3×#214C211yesmaximal, characteristic, complemented
3C2×C3×#932C211yesmaximal, characteristic
4C3×#5526C211yesmaximal, characteristic, complemented
5C3×#5019C214yesmaximal, complemented
6C3×#5016C214yesmaximal, complemented
7C3×#5004C211yesmaximal, characteristic, complemented
8C3×#5002C211yesmaximal, characteristic, complemented
9C3×#4999C211yesmaximal, characteristic, complemented
10C3×#3042—31nomaximal
11#82544C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C32×C4×D4—22no—
16C32×C4×D4C2214yes—
17C32×#117C2211yescharacteristic
18C32×#117C2211yescharacteristic
19C2×C32×C22⋊C4—22no—
20C2×C3×#214—24no—
21C2×C3×#214C2212yes—
22C2×C3×#214C2214yes—
23C3×#932—24no—
24C2×C3×#213C2212yes—
25C2×C3×#208C2211yescharacteristic
26C22×C3×C3⋊C8C2211yescharacteristic
27C3×#897—22no—
28C3×#886—22no—
29C3×#875C2212yescomplemented
30C3×#869C2212yescomplemented
31C3×#810C2212yescomplemented
32C3×#810C2214yescomplemented
33C3×#809C2212yescomplemented
34C3×C4⋊(C3⋊C8)C2214yescomplemented
35C2×C3×AGL(1,ℤ/8)—31no—
36C22×C3×SD16—31no—
37C3×#536—31no—
38C3×#518—21no—
39C3×#518D311yescharacteristic, complemented
40C3×#518C2×C311yescharacteristic, complemented
41C22×#214C2×C311yescharacteristic, complemented
42C2×#932C2×C311yescharacteristic
43#5526C2×C311yescharacteristic, complemented
44C3×#463—34no—
45C3×#461—34no—
46C3×#424—31no—
47C3×#419—31no—
48C3×#416—31no—
49#5019C2×C314yescomplemented
50#5016C2×C314yescomplemented
51#5004C2×C311yescharacteristic, complemented
52#5002C2×C311yescharacteristic, complemented
53#4999C2×C311yescharacteristic, complemented
54C24×C32—21no—
55C2×C32×D4—21no—
56C2×C32×D4—22no—
57C2×C32×D4D414yes—
58C2×C32×D4C2312yes—
59C22×C32×C4—24no—
60C22×C32×C4C2311yescharacteristic
61C2×C3×#67—21no—
62C22×C3×Dic3—21no—
63C3×#235—22no—
64C3×#235—22no—
65C2×C3×D12C2311yescharacteristic
66C2×C3×C4×D3—21no—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C2×C3×Dic6C2311yescharacteristic
72C32×C4⋊C4D412yes—
73C32×C4⋊C4C2312yes—
74C32×C22⋊C4—21no—
75C32×C22⋊C4—24no—
76C32×C22⋊C4D412yes—
77C32×C42—21no—
78C32×C42C2312yes—
79C3×#214—24no—
80C3×#214—28no—
81C3×#213—24no—
82C3×(C2×D3)⋊C4—24no—
83C3×C4⋊Dic3—24no—
84C3×#208—22no—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2312yes—
87C2×C3×C3⋊C8C2312yes—
88#3042—31noSylow
89C2×C3×#41—31no—
90C22×C3×Q8—31no—
91C22×C3×D4—21no—
92C22×C3×D4C2×D311yescharacteristic
93C22×C3×D4C22×C311yescharacteristic
94C2×#235C22×C311yescharacteristic
95C22×Dic6C22×C311yescharacteristic
96C3×AGL(1,ℤ/8)—64no—
97C2×C3×SD16—34no—
98C2×C3×SD16—32no—
99C2×C3×SD16—64no—
100C2×C3×D8—32no—
101C2×C3×M16—31no—
102C22×C3×C8—31no—
103C3×C4⋊Q8—32no—
104C3×#124—32no—
105C3×#123—62no—
106C3×C22⋊Q8—62no—
107C3×C4×D4—24no—
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4—24no—
111C3×C4×D4C2×D314yes—
112C3×C4×D4C22×C314yes—
113C3×#117—21no—
114C3×#117—21no—
115C3×#117C2×D311yescharacteristic
116C3×#117C2×D311yescharacteristic
117C3×#117C22×C311yescharacteristic
118C3×#117C22×C311yescharacteristic
119C2×C3×C22⋊C4—24no—
120C2×C3×C22⋊C4—22no—
121C2×C3×C22⋊C4—22no—
122C2×#214—24no—
123C2×#214C22×C314yes—
124C2×#214C22×C312yes—
125#932—24no—
126C2×#213C22×C312yes—
127C2×#208C22×C311yescharacteristic
128C22×C3⋊C8C22×C311yescharacteristic
129#897—22no—
130#886—22no—
131#875C22×C312yescomplemented
132#869C22×C312yescomplemented
133C3×C4⋊C8—34no—
134C3×#103—32no—
135C3×#103—34no—
136C3×#102—32no—
137#810C22×C312yescomplemented
138#810C22×C314yescomplemented
139#809C22×C312yescomplemented
140C4⋊(C3⋊C8)C22×C314yescomplemented
141C23×C32—22no—
142C23×C32—24no—
143C23×C32—82no—
144C23×C32C2×D411yescharacteristic
145C22×C3×D3—21no—
146C32×D4—24no—
147C32×D4—22no—
148C32×D4C2×D414yes—
149C2×C32×C4—22no—
150C2×C32×C4—84no—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D414yes—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4#4112yes—
155C2×C32×C4C2411yescharacteristic
156C3×#67—42no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—42no—
160C3×D12—21no—
161C3×D12—21no—
162C3×C4×D3—42no—
163C3×Dic6—22no—
164C3×Dic6—21no—
165C3×Dic6—22no—
166C3×Dic6—21no—
167C3×Dic6—42no—
168C3×C3⋊C8—22no—
169C3×C3⋊C8—22no—
170C2×AGL(1,ℤ/8)—31no—
171C22×SD16—31no—
172#536—31no—
173#518C3×D311yescharacteristic, complemented
174#463—34no—
175#461—34no—
176#424—31no—
177#419—31no—
178#416—31no—
179C24×C3—22no—
180C24×C3—21no—
181C24×C3—21no—
182C3×#41—62no—
183C3×#41—62no—
184C2×C3×Q8—32no—
185C2×C3×Q8—31no—
186C2×C3×Q8—61no—
187C2×C3×Q8—62no—
188C2×C3×Q8—62no—
189C2×C3×D4—22no—
190C2×C3×D4—24no—
191C2×C3×D4—22no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4—31no—
198C2×C3×D4—61no—
199C2×C3×D4D1214yes—
200C2×C3×D4C3×D414yes—
201C2×C3×D4C22×D312yes—
202C2×C3×D4C23×C312yes—
203C22×C3×C4—28no—
204C22×C3×C4—24no—
205C22×C3×C4—24no—
206C22×C3×C4—21no—
207C22×C3×C4—61no—
208C22×C3×C4—61no—
209C22×C3×C4C22×D311yescharacteristic
210C22×C3×C4C23×C311yescharacteristic
211C2×#67—21no—
212C22×Dic3—21no—
213#235—22no—
214#235—22no—
215C2×D12C23×C311yescharacteristic
216C2×C4×D3—21no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—21no—
220C2×Dic6C23×C311yescharacteristic
221C2×Dic6C23×C312yes—
222C3×SD16—68no—
223C3×SD16—64no—
224C3×D8—64no—
225C3×M16—62no—
226C2×C3×C8—32no—
227C2×C3×C8—32no—
228C2×C3×C8—62no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—64no—
232C3×C4⋊C4#6712yes—
233C3×C4⋊C4C3×D412yes—
234C3×C4⋊C4C22×D312yes—
235C3×C4⋊C4C23×C312yes—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—64no—
244C3×C22⋊C4#6712yes—
245C3×C22⋊C4C3×D412yes—
246C3×C42—22no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—22no—
250C3×C42C22×D312yes—
251C3×C42C23×C312yes—
252#214—24no—
253#214—28no—
254#213—24no—
255(C2×D3)⋊C4—24no—
256C4⋊Dic3—24no—
257#208—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8C23×C312yes—
260C2×C3⋊C8C23×C312yes—
261C22×C32—24no—
262C22×C32—22no—
263C22×C32—44no—
264C22×C32—84no—
265C22×C32#12112yes—
266C22×C32C22×D411yescharacteristic
267C2×C3×D3—41no—
268C2×C3×D3—42no—
269C32×C4—24no—
270C32×C4—42no—
271C32×C4#12112yes—
272C32×C4C22×D411yescharacteristic
273C32×C4C2×#4111yescharacteristic
274C3×Dic3—42no—
275C3×Dic3—41no—
276C2×#41—31no—
277C22×Q8—31no—
278C22×D4C2×C3×D311yescharacteristic
279AGL(1,ℤ/8)—64no—
280C2×SD16—34no—
281C2×SD16—32no—
282C2×SD16—64no—
283C2×D8—32no—
284C2×M16—31no—
285C22×C8—31no—
286C4⋊Q8—32no—
287#124—32no—
288#123—62no—
289C22⋊Q8—62no—
290C4×D4—22no—
291C4×D4C2×C3×D314yes—
292#117C2×C3×D311yescharacteristic
293#117C2×C3×D311yescharacteristic
294C2×C22⋊C4—22no—
295C4⋊C8—34no—
296#103—32no—
297#103—34no—
298#102—32no—
299C23×C3—24no—
300C23×C3—28no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—21no—
306C23×C3—61no—
307C23×C3—82no—
308C23×C3—82no—
309C23×C3—84no—
310C23×C3C2×#6711yescharacteristic
311C23×C3C2×C3×D411yescharacteristic
312C22×D3—21no—
313C3×Q8—62no—
314C3×Q8—61no—
315C3×Q8—62no—
316C3×Q8—61no—
317C3×Q8—122no—
318C3×D4—28no—
319C3×D4—24no—
320C3×D4—24no—
321C3×D4—24no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—24no—
325C3×D4—61no—
326C3×D4—61no—
327C3×D4—122no—
328C3×D4C2×D1214yes—
329C3×D4C2×C3×D414yes—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—24no—
338C2×C3×C4—62no—
339C2×C3×C4—61no—
340C2×C3×C4—84no—
341C2×C3×C4—84no—
342C2×C3×C4—88no—
343C2×C3×C4—122no—
344C2×C3×C4—122no—
345C2×C3×C4C2×D1212yes—
346C2×C3×C4#23512yes—
347C2×C3×C4C2×#6714yes—
348C2×C3×C4C2×#6711yescharacteristic
349C2×C3×C4C2×C3×D412yes—
350C2×C3×C4C2×C3×D414yes—
351C2×C3×C4C2×C3×D411yescharacteristic
352C2×C3×C4C3×#4112yes—
353C2×C3×C4C23×D311yescharacteristic
354C2×C3×C4C24×C311yescharacteristic
355#67—42no—
356C2×Dic3—21no—
357C2×Dic3—22no—
358C2×Dic3—42no—
359D12—21no—
360D12—21no—
361C4×D3—42no—
362Dic6—22no—
363Dic6—21no—
364Dic6—22no—
365Dic6—21no—
366Dic6—42no—
367C3×C8—62no—
368C3×C8—62no—
369C3⋊C8—22no—
370C3⋊C8—22no—
371C2×C32—22no—
372C2×C32—44no—
373C2×C32C2×#12111yescharacteristic
374C2×C32#57712yes—
375C3×D3—81no—
376C24—21no—
377#41—62no—
378#41—62no—
379C2×Q8—32no—
380C2×Q8—31no—
381C2×Q8—61no—
382C2×Q8—62no—
383C2×Q8—62no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—31no—
387C2×D4—61no—
388C2×D4C3×D1214yes—
389C2×D4C22×C3×D312yes—
390C22×C4—24no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4C22×C3×D311yescharacteristic
394SD16—68no—
395SD16—64no—
396D8—64no—
397M16—62no—
398C2×C8—32no—
399C2×C8—32no—
400C2×C8—62no—
401C4⋊C4—64no—
402C4⋊C4C3×#6712yes—
403C4⋊C4C22×C3×D312yes—
404C22⋊C4—21no—
405C22⋊C4—24no—
406C22⋊C4—64no—
407C22⋊C4C3×#6712yes—
408C42—21no—
409C42C22×C3×D312yes—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—28no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—42no—
418C22×C3—48no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—84no—
422C22×C3—84no—
423C22×C3—88no—
424C22×C3—121no—
425C22×C3—122no—
426C22×C3#93012yes—
427C22×C3C3×#12112yes—
428C22×C3C22×#6711yescharacteristic
429C22×C3C22×C3×D411yescharacteristic
430C2×D3—42no—
431C2×D3—41no—
432C3×C4—24no—
433C3×C4—24no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—44no—
438C3×C4—44no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—121no—
442C3×C4—122no—
443C3×C4#93012yes—
444C3×C4C3×#12112yes—
445C3×C4C22×D1211yescharacteristic
446C3×C4C2×#23511yescharacteristic
447C3×C4C22×C3×D411yescharacteristic
448C3×C4C2×C3×#4111yescharacteristic
449Dic3—42no—
450Dic3—41no—
451C32#304211yescharacteristic, complemented, Sylow
452C23—22no—
453C23—24no—
454C23—61no—
455C23—82no—
456C23C2×C3×#6711yescharacteristic
457Q8—62no—
458Q8—61no—
459Q8—62no—
460Q8—61no—
461Q8—122no—
462D4—24no—
463D4—22no—
464D4—61no—
465D4—61no—
466D4—122no—
467D4C2×C3×D1214yes—
468C2×C4—22no—
469C2×C4—62no—
470C2×C4—61no—
471C2×C4—84no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4C2×C3×D1212yes—
475C2×C4C3×#23512yes—
476C2×C4C2×C3×#6714yes—
477C2×C4C2×C3×#6711yescharacteristic
478C2×C4C23×C3×D311yescharacteristic
479C8—62no—
480C8—62no—
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—21no—
484C2×C3—22no—
485C2×C3—42no—
486C2×C3—48no—
487C2×C3—44no—
488C2×C3—44no—
489C2×C3—241no—
490C2×C3C2×#93011yescharacteristic
491C2×C3#583612yes—
492C2×C3C2×C3×#12111yescharacteristic
493C2×C3C3×#57712yes—
494D3—81no—
495C22—22no—
496C22—24no—
497C22—44no—
498C22—84no—
499C22—121no—
500C22—122no—
501C22C3×#93012yes—
502C22C22×C3×#6711yescharacteristic
503C4—24no—
504C4—42no—
505C4—121no—
506C4—122no—
507C4C3×#93012yes—
508C4C22×C3×D1211yescharacteristic
509C4C2×C3×#23511yescharacteristic
510C3—21no—
511C3#8254411yescharacteristic, direct product factor
512C3C3×#304211yescharacteristic, complemented
513C2—22no—
514C2—44no—
515C2—241no—
516C2C2×C3×#93011yescharacteristic
517C2C3×#583612yes—
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#518C2
socleC22×C32C22×D4
prime radicalC2×C3×#932C2
hypercenterC3×#518D3
hypocenterC3C3×#3042
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#518

Complements

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

NH ≅ G/Ntype
1. C32×#518C2semidirect
2. C22×C3×#214C2semidirect
4. C3×#5526C2semidirect
5. C3×#5019C2semidirect
6. C3×#5016C2semidirect
7. C3×#5004C2semidirect
8. C3×#5002C2semidirect
9. C3×#4999C2semidirect
11. #82544C3direct (N × H)
29. C3×#875C22semidirect
30. C3×#869C22semidirect
31. C3×#810C22semidirect
32. C3×#810C22semidirect
33. C3×#809C22semidirect
34. C3×C4⋊(C3⋊C8)C22semidirect
39. C3×#518D3semidirect
40. C3×#518C2×C3semidirect
41. C22×#214C2×C3semidirect
43. #5526C2×C3semidirect
49. #5019C2×C3semidirect
50. #5016C2×C3semidirect
51. #5004C2×C3semidirect
52. #5002C2×C3semidirect
53. #4999C2×C3semidirect
131. #875C22×C3semidirect
132. #869C22×C3semidirect
137. #810C22×C3semidirect
138. #810C22×C3semidirect
139. #809C22×C3semidirect
140. C4⋊(C3⋊C8)C22×C3semidirect
173. #518C3×D3semidirect
451. C32#3042semidirect
511. C3#82544direct (N × H)
512. C3C3×#3042semidirect

Maximal subgroups

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

subgroupindex
C32×#5182
C22×C3×#2142
C2×C3×#9322
C3×#55262
C3×#50192
C3×#50162
C3×#50042
C3×#50022
C3×#49992
C3×#30423
#825443