Finite Group Databasebeta

← C3×#82551 #11866321

Subgroups — C3×#82551 #11866321

Order 1152.

1113 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#519C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic, complemented
3C22×C3×#215C211yesmaximal, characteristic, complemented
4C3×#5525C211yesmaximal, characteristic
5C3×#5028C214yesmaximal, complemented
6C3×#5027C214yesmaximal, complemented
7C3×#5002C211yesmaximal, characteristic, complemented
8C3×#5001C211yesmaximal, characteristic, complemented
9C3×#4999C211yesmaximal, characteristic, complemented
10C3×#3043—31nomaximal
11#82551C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×D12C2211yescharacteristic
15C32×C4×Q8—22no—
16C32×C4×Q8C2214yescomplemented
17C32×#117—22no—
18C32×#117C2211yescharacteristic
19C32×#117C2211yescharacteristic
20C2×C3×#216C2212yescomplemented
21C3×#942—24no—
22C2×C3×#215—24no—
23C2×C3×#215C2214yes—
24C2×C3×#215C2212yescomplemented
25C2×C3×#208C2211yescharacteristic
26C22×C3×C3⋊C8C2211yescharacteristic
27C3×#895—22no—
28C3×#882—22no—
29C3×#875C2212yes—
30C3×#874C2212yes—
31C3×#810C2212yescomplemented
32C3×#809C2214yes—
33C3×#809C2212yescomplemented
34C3×C4⋊(C3⋊C8)C2214yescomplemented
35C2×C3×#137—31no—
36C22×C3×SD16—31no—
37C3×#535—31no—
38C3×#519—21no—
39C3×#519D311yescharacteristic, complemented
40C3×#519C2×C311yescharacteristic, complemented
41C2×#942C2×C311yescharacteristic, complemented
42C22×#215C2×C311yescharacteristic, complemented
43#5525C2×C311yescharacteristic
44C3×#464—34no—
45C3×#460—34no—
46C3×#424—31no—
47C3×#418—31no—
48C3×#416—31no—
49#5028C2×C314yescomplemented
50#5027C2×C314yescomplemented
51#5002C2×C311yescharacteristic, complemented
52#5001C2×C311yescharacteristic, complemented
53#4999C2×C311yescharacteristic, complemented
54C23×C3×D3—21no—
55C2×C32×Q8—21no—
56C2×C32×Q8—22no—
57C2×C32×Q8D414yescomplemented
58C2×C32×Q8C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C2×C3×#67—21no—
62C3×#235—22no—
63C3×#235—22no—
64C2×C3×D12—22no—
65C2×C3×D12—22no—
66C2×C3×D12—21no—
67C2×C3×D12—24no—
68C2×C3×D12C2312yes—
69C2×C3×D12C2311yescharacteristic
70C2×C3×C4×D3—21no—
71C2×C3×Dic6C2311yescharacteristic
72C32×C4⋊C4—24no—
73C32×C4⋊C4D412yes—
74C32×C4⋊C4C2312yes—
75C32×C22⋊C4—21no—
76C32×C22⋊C4D412yes—
77C32×C42—24no—
78C32×C42—21no—
79C32×C42C2312yes—
80C3×#216—24no—
81C3×#215—28no—
82C3×#215—24no—
83C3×(C2×D3)⋊C4—24no—
84C3×#208—22no—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2312yes—
87C2×C3×C3⋊C8C2312yes—
88#3043—31noSylow
89C2×C3×#41—31no—
90C22×C3×Q8—21no—
91C22×C3×Q8C2×D311yescharacteristic
92C22×C3×Q8C22×C311yescharacteristic
93C22×C3×D4—31no—
94C2×#235C22×C311yescharacteristic
95C22×D12C22×C311yescharacteristic
96C3×#137—64no—
97C2×C3×Q16—32no—
98C2×C3×SD16—34no—
99C2×C3×SD16—32no—
100C2×C3×SD16—64no—
101C2×C3×M16—31no—
102C22×C3×C8—31no—
103C3×#127—32no—
104C3×#124—32no—
105C3×#121—62no—
106C3×#120—62no—
107C3×C4×Q8—24no—
108C3×C4×Q8—22no—
109C3×C4×Q8—22no—
110C3×C4×Q8—24no—
111C3×C4×Q8C2×D314yescomplemented
112C3×C4×Q8C22×C314yescomplemented
113C3×#117—24no—
114C3×#117—22no—
115C3×#117—22no—
116C3×#117—21no—
117C3×#117—21no—
118C3×#117C2×D311yescharacteristic
119C3×#117C2×D311yescharacteristic
120C3×#117C22×C311yescharacteristic
121C3×#117C22×C311yescharacteristic
122C2×#216C22×C312yescomplemented
123#942—24no—
124C2×#215—24no—
125C2×#215C22×C314yes—
126C2×#215C22×C312yescomplemented
127C2×#208C22×C311yescharacteristic
128C22×C3⋊C8C22×C311yescharacteristic
129#895—22no—
130#882—22no—
131#875C22×C312yes—
132#874C22×C312yes—
133C3×C4⋊C8—34no—
134C3×#103—32no—
135C3×#102—34no—
136C3×#102—32no—
137#810C22×C312yescomplemented
138#809C22×C314yes—
139#809C22×C312yescomplemented
140C4⋊(C3⋊C8)C22×C314yescomplemented
141C23×C32C2×D411yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—21no—
144C22×C3×D3—42no—
145C22×C3×D3—82no—
146C32×Q8—24no—
147C32×Q8—22no—
148C32×Q8C2×D414yes—
149C2×C32×C4—24no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4C2×D412yes—
153C2×C32×C4C2×D414yes—
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4#4112yes—
156C2×C32×C4C2411yescharacteristic
157C3×#67—42no—
158C2×C3×Dic3—21no—
159C3×D12—22no—
160C3×D12—21no—
161C3×D12—22no—
162C3×D12—21no—
163C3×D12—42no—
164C3×D12—84no—
165C3×C4×D3—42no—
166C3×Dic6—21no—
167C3×Dic6—21no—
168C3×C3⋊C8—22no—
169C3×C3⋊C8—22no—
170C2×#137—31no—
171C22×SD16—31no—
172#535—31no—
173#519C3×D311yescharacteristic, complemented
174#464—34no—
175#460—34no—
176#424—31no—
177#418—31no—
178#416—31no—
179C24×C3—61no—
180C23×D3—21no—
181C3×#41—62no—
182C3×#41—62no—
183C2×C3×Q8—22no—
184C2×C3×Q8—24no—
185C2×C3×Q8—22no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—22no—
189C2×C3×Q8—24no—
190C2×C3×Q8—22no—
191C2×C3×Q8—31no—
192C2×C3×Q8D1214yescomplemented
193C2×C3×Q8C3×D414yescomplemented
194C2×C3×Q8C22×D312yes—
195C2×C3×Q8C23×C312yes—
196C2×C3×D4—32no—
197C2×C3×D4—31no—
198C2×C3×D4—61no—
199C2×C3×D4—62no—
200C2×C3×D4—62no—
201C2×C3×D4—61no—
202C2×C3×D4—64no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—61no—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C23×C311yescharacteristic
210C2×#67—21no—
211#235—22no—
212#235—22no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—21no—
216C2×D12—24no—
217C2×D12C23×C312yes—
218C2×D12C23×C311yescharacteristic
219C2×C4×D3—21no—
220C2×Dic6C23×C311yescharacteristic
221C3×Q16—64no—
222C3×SD16—68no—
223C3×SD16—64no—
224C3×M16—62no—
225C2×C3×C8—32no—
226C2×C3×C8—32no—
227C2×C3×C8—62no—
228C3×C4⋊C4—28no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4#6712yes—
234C3×C4⋊C4C3×D412yes—
235C3×C4⋊C4C22×D312yes—
236C3×C4⋊C4C23×C312yes—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—64no—
242C3×C22⋊C4#6712yes—
243C3×C22⋊C4C3×D412yes—
244C3×C42—28no—
245C3×C42—22no—
246C3×C42—24no—
247C3×C42—21no—
248C3×C42—24no—
249C3×C42—21no—
250C3×C42—22no—
251C3×C42C22×D312yes—
252C3×C42C23×C312yes—
253#216—24no—
254#215—28no—
255#215—24no—
256(C2×D3)⋊C4—24no—
257#208—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8C23×C312yes—
260C2×C3⋊C8C23×C312yes—
261C22×C32—22no—
262C22×C32#12112yes—
263C22×C32C22×D411yescharacteristic
264C2×C3×D3—41no—
265C2×C3×D3—42no—
266C2×C3×D3—42no—
267C2×C3×D3—44no—
268C2×C3×D3—84no—
269C32×C4—24no—
270C32×C4—24no—
271C32×C4—42no—
272C32×C4#12112yes—
273C32×C4C22×D411yescharacteristic
274C32×C4C2×#4111yescharacteristic
275C3×Dic3—41no—
276C2×#41—31no—
277C22×Q8C2×C3×D311yescharacteristic
278C22×D4—31no—
279#137—64no—
280C2×Q16—32no—
281C2×SD16—34no—
282C2×SD16—32no—
283C2×SD16—64no—
284C2×M16—31no—
285C22×C8—31no—
286#127—32no—
287#124—32no—
288#121—62no—
289#120—62no—
290C4×Q8—22no—
291C4×Q8C2×C3×D314yescomplemented
292#117—22no—
293#117C2×C3×D311yescharacteristic
294#117C2×C3×D311yescharacteristic
295C4⋊C8—34no—
296#103—32no—
297#102—34no—
298#102—32no—
299C23×C3—21no—
300C23×C3—62no—
301C23×C3—61no—
302C23×C3—122no—
303C23×C3—242no—
304C23×C3C2×#6711yescharacteristic
305C23×C3C2×C3×D411yescharacteristic
306C22×D3—22no—
307C22×D3—21no—
308C22×D3—42no—
309C22×D3—82no—
310C3×Q8—28no—
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—24no—
314C3×Q8—22no—
315C3×Q8—22no—
316C3×Q8—24no—
317C3×Q8—61no—
318C3×Q8—61no—
319C3×Q8C2×D1214yes—
320C3×Q8C2×C3×D414yes—
321C3×D4—62no—
322C3×D4—61no—
323C3×D4—62no—
324C3×D4—61no—
325C3×D4—122no—
326C3×D4—122no—
327C3×D4—244no—
328C2×C3×C4—24no—
329C2×C3×C4—28no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—24no—
342C2×C3×C4—61no—
343C2×C3×C4—122no—
344C2×C3×C4C2×D1212yes—
345C2×C3×C4#23512yes—
346C2×C3×C4C2×#6714yes—
347C2×C3×C4C2×#6711yescharacteristic
348C2×C3×C4C2×C3×D412yes—
349C2×C3×C4C2×C3×D414yes—
350C2×C3×C4C2×C3×D411yescharacteristic
351C2×C3×C4C3×#4112yes—
352C2×C3×C4C23×D311yescharacteristic
353C2×C3×C4C24×C311yescharacteristic
354#67—42no—
355C2×Dic3—21no—
356D12—22no—
357D12—21no—
358D12—22no—
359D12—21no—
360D12—42no—
361D12—84no—
362C4×D3—42no—
363Dic6—21no—
364Dic6—21no—
365C3×C8—62no—
366C3×C8—62no—
367C3⋊C8—22no—
368C3⋊C8—22no—
369C2×C32—22no—
370C2×C32C2×#12111yescharacteristic
371C2×C32#57712yes—
372C3×D3—81no—
373C3×D3—82no—
374C24—61no—
375#41—62no—
376#41—62no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—31no—
380C2×Q8C3×D1214yescomplemented
381C2×Q8C22×C3×D312yes—
382C2×D4—32no—
383C2×D4—31no—
384C2×D4—61no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—64no—
389C22×C4—21no—
390C22×C4—61no—
391C22×C4C22×C3×D311yescharacteristic
392Q16—64no—
393SD16—68no—
394SD16—64no—
395M16—62no—
396C2×C8—32no—
397C2×C8—32no—
398C2×C8—62no—
399C4⋊C4—24no—
400C4⋊C4C3×#6712yes—
401C4⋊C4C22×C3×D312yes—
402C22⋊C4—21no—
403C22⋊C4—64no—
404C22⋊C4C3×#6712yes—
405C42—24no—
406C42—21no—
407C42C22×C3×D312yes—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—42no—
413C22×C3—121no—
414C22×C3—122no—
415C22×C3—122no—
416C22×C3—124no—
417C22×C3—244no—
418C22×C3#93012yes—
419C22×C3C3×#12112yes—
420C22×C3C22×#6711yescharacteristic
421C22×C3C22×C3×D411yescharacteristic
422C2×D3—42no—
423C2×D3—44no—
424C2×D3—41no—
425C2×D3—42no—
426C2×D3—84no—
427C3×C4—28no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—44no—
436C3×C4—44no—
437C3×C4—42no—
438C3×C4—42no—
439C3×C4—121no—
440C3×C4#93012yes—
441C3×C4C3×#12112yes—
442C3×C4C22×D1211yescharacteristic
443C3×C4C2×#23511yescharacteristic
444C3×C4C22×C3×D411yescharacteristic
445C3×C4C2×C3×#4111yescharacteristic
446Dic3—41no—
447C32#304311yescharacteristic, complemented, Sylow
448C23—62no—
449C23—61no—
450C23—122no—
451C23—242no—
452C23C2×C3×#6711yescharacteristic
453Q8—24no—
454Q8—22no—
455Q8—61no—
456Q8—61no—
457Q8C2×C3×D1214yes—
458D4—62no—
459D4—61no—
460D4—62no—
461D4—61no—
462D4—122no—
463D4—122no—
464D4—244no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—61no—
469C2×C4—122no—
470C2×C4C2×C3×D1212yes—
471C2×C4C3×#23512yes—
472C2×C4C2×C3×#6714yes—
473C2×C4C2×C3×#6711yescharacteristic
474C2×C4C23×C3×D311yescharacteristic
475C8—62no—
476C8—62no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3—21no—
480C2×C3—22no—
481C2×C3—42no—
482C2×C3—242no—
483C2×C3—241no—
484C2×C3C2×#93011yescharacteristic
485C2×C3#583712yes—
486C2×C3C2×C3×#12111yescharacteristic
487C2×C3C3×#57712yes—
488D3—82no—
489D3—81no—
490C22—22no—
491C22—121no—
492C22—122no—
493C22—122no—
494C22—124no—
495C22—244no—
496C22C3×#93012yes—
497C22C22×C3×#6711yescharacteristic
498C4—24no—
499C4—24no—
500C4—42no—
501C4—121no—
502C4C3×#93012yes—
503C4C22×C3×D1211yescharacteristic
504C4C2×C3×#23511yescharacteristic
505C3—21no—
506C3#8255111yescharacteristic, direct product factor
507C3C3×#304311yescharacteristic, complemented
508C2—22no—
509C2—242no—
510C2—241no—
511C2C2×C3×#93011yescharacteristic
512C2C3×#583712yes—
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#519C2
socleC22×C32C22×D4
prime radicalC3×#5525C2
hypercenterC3×#519D3
hypocenterC3C3×#3043
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#519

Complements

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

NH ≅ G/Ntype
1. C32×#519C2semidirect
2. C2×C3×#942C2semidirect
3. C22×C3×#215C2semidirect
5. C3×#5028C2semidirect
6. C3×#5027C2semidirect
7. C3×#5002C2semidirect
8. C3×#5001C2semidirect
9. C3×#4999C2semidirect
11. #82551C3direct (N × H)
16. C32×C4×Q8C22semidirect
20. C2×C3×#216C22semidirect
24. C2×C3×#215C22semidirect
31. C3×#810C22semidirect
33. C3×#809C22semidirect
34. C3×C4⋊(C3⋊C8)C22semidirect
39. C3×#519D3semidirect
40. C3×#519C2×C3semidirect
41. C2×#942C2×C3semidirect
42. C22×#215C2×C3semidirect
49. #5028C2×C3semidirect
50. #5027C2×C3semidirect
51. #5002C2×C3semidirect
52. #5001C2×C3semidirect
53. #4999C2×C3semidirect
57. C2×C32×Q8D4semidirect
111. C3×C4×Q8C2×D3semidirect
112. C3×C4×Q8C22×C3semidirect
122. C2×#216C22×C3semidirect
126. C2×#215C22×C3semidirect
137. #810C22×C3semidirect
139. #809C22×C3semidirect
140. C4⋊(C3⋊C8)C22×C3semidirect
173. #519C3×D3semidirect
192. C2×C3×Q8D12semidirect
193. C2×C3×Q8C3×D4semidirect
291. C4×Q8C2×C3×D3semidirect
380. C2×Q8C3×D12semidirect
447. C32#3043semidirect
506. C3#82551direct (N × H)
507. C3C3×#3043semidirect

Maximal subgroups

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

subgroupindex
C32×#5192
C2×C3×#9422
C22×C3×#2152
C3×#55252
C3×#50282
C3×#50272
C3×#50022
C3×#50012
C3×#49992
C3×#30433
#825513