Finite Group Databasebeta

← C3×#83353 #11871315

Subgroups — C3×#83353 #11871315

Order 1152.

999 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#533C211yesmaximal, characteristic, complemented
2C2×C3×C4×D12C211yesmaximal, characteristic
3C3×#5087C212yesmaximal, complemented
4C3×#5083C212yesmaximal, complemented
5C3×#5046C212yesmaximal
6C3×#5044C212yesmaximal, complemented
7C3×#5026C214yesmaximal, complemented
8C3×#4926C211yesmaximal, characteristic, complemented
9C3×#3110—31nomaximal
10#83353C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3—21no—
13C32×C4⋊Q8C2211yescharacteristic, complemented
14C32×C4⋊Q8C2211yescharacteristic, complemented
15C32×#125—21no—
16C32×C22⋊Q8—21no—
17C32×C22⋊Q8C2212yes—
18C32×C4×Q8—21no—
19C32×C4×Q8C2212yescomplemented
20C32×#117—21no—
21C2×C32×C42C2211yescharacteristic
22C2×C3×#215C2214yes—
23C2×C3×(C2×D3)⋊C4—21no—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C3×C4×D12—21no—
26C3×C4×D12—21no—
27C3×C4×D12C2212yes—
28C3×C4×D12C2212yes—
29C3×#835C2214yescomplemented
30C3×#830C2212yes—
31C3×#809C2214yes—
32C3×#807C2214yescomplemented
33C3×C4⋊(C3⋊C8)C2212yescomplemented
34C3×#803C2212yescomplemented
35C3×#533—21no—
36C3×#533D311yescharacteristic, complemented
37C3×#533C2×C311yescharacteristic, complemented
38C2×C3×C4×D4—31no—
39C2×C4×D12C2×C311yescharacteristic
40C3×#484—32no—
41C3×#474—32no—
42C3×#461—32no—
43C3×#450—32no—
44C3×#440—34no—
45C3×#432—31no—
46#5087C2×C312yescomplemented
47#5083C2×C312yescomplemented
48#5046C2×C312yes—
49#5044C2×C312yescomplemented
50#5026C2×C314yescomplemented
51#4926C2×C311yescharacteristic, complemented
52C23×C3×D3—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2312yes—
55C22×C32×C4D411yescharacteristic
56C22×C32×C4D411yescharacteristic
57C22×C32×C4C2311yescharacteristic
58C22×C3×Dic3—21no—
59C2×C3×D12—21no—
60C2×C3×D12—24no—
61C2×C3×D12C2312yes—
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3—44no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4—22no—
69C32×C4⋊C4—22no—
70C32×C4⋊C4C2312yes—
71C32×C22⋊C4—22no—
72C32×C42—22no—
73C32×C42D411yescharacteristic
74C32×C42D411yescharacteristic
75C32×C42C2311yescharacteristic
76C32×C42C2311yescharacteristic
77C3×#215—28no—
78C3×(C2×D3)⋊C4—22no—
79C3×(C2×D3)⋊C4—22no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3C2312yes—
82C2×C3×C3⋊C8C2314yes—
83#3110—31noSylow
84C22×C3×D4—31no—
85C23×C3×C4—61no—
86C22×D12C22×C311yescharacteristic
87C22×C4×D3—21no—
88C2×C3×SD16—34no—
89C3×C4⋊Q8—21no—
90C3×C4⋊Q8—21no—
91C3×C4⋊Q8C2×D311yescharacteristic, complemented
92C3×C4⋊Q8C2×D311yescharacteristic, complemented
93C3×C4⋊Q8C22×C311yescharacteristic, complemented
94C3×C4⋊Q8C22×C311yescharacteristic, complemented
95C3×#125—22no—
96C3×#125—21no—
97C3×#125—21no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—21no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8C2×D312yes—
103C3×C22⋊Q8C22×C312yes—
104C3×C4×Q8—22no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—22no—
108C3×C4×Q8C2×D312yescomplemented
109C3×C4×Q8C22×C312yescomplemented
110C3×C4×D4—32no—
111C3×C4×D4—32no—
112C3×C4×D4—61no—
113C3×C4×D4—61no—
114C3×#117—22no—
115C3×#117—21no—
116C3×#117—21no—
117C2×C3×C4⋊C4—31no—
118C2×C3×C22⋊C4—61no—
119C2×C3×C42—21no—
120C2×C3×C42C2×D311yescharacteristic
121C2×C3×C42C22×C311yescharacteristic
122C2×#215C22×C314yes—
123C2×(C2×D3)⋊C4—21no—
124C2×C4⋊Dic3C22×C311yescharacteristic
125C4×D12—21no—
126C4×D12—21no—
127C4×D12C22×C312yes—
128C4×D12C22×C312yes—
129C3×#107—34no—
130C3×C4⋊C8—32no—
131C3×#103—34no—
132C3×#102—34no—
133C3×C22⋊C8—32no—
134C3×#97—32no—
135#835C22×C314yescomplemented
136#830C22×C312yes—
137#809C22×C314yes—
138#807C22×C314yescomplemented
139C4⋊(C3⋊C8)C22×C312yescomplemented
140#803C22×C312yescomplemented
141C23×C32C2×D411yescharacteristic
142C22×C3×D3—24no—
143C22×C3×D3—22no—
144C22×C3×D3—42no—
145C22×C3×D3—42no—
146C32×Q8—24no—
147C32×Q8—42no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2411yescharacteristic
159C2×C3×Dic3—22no—
160C2×C3×Dic3—42no—
161C3×D12—24no—
162C3×D12—22no—
163C3×D12#4114yes—
164C3×C4×D3—44no—
165C3×C4×D3—44no—
166C3×C3⋊C8—24no—
167#533C3×D311yescharacteristic, complemented
168C2×C4×D4—31no—
169#484—32no—
170#474—32no—
171#461—32no—
172#450—32no—
173#440—34no—
174#432—31no—
175C24×C3—61no—
176C23×D3—21no—
177C2×C3×Q8—22no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8C22×D312yes—
182C2×C3×Q8C23×C312yes—
183C2×C3×D4—32no—
184C2×C3×D4—64no—
185C2×C3×D4—61no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—61no—
190C22×C3×C4—62no—
191C22×C3×C4—62no—
192C22×C3×C4—124no—
193C22×C3×C4#6711yescharacteristic
194C22×C3×C4#6711yescharacteristic
195C22×C3×C4C3×D411yescharacteristic
196C22×C3×C4C3×D411yescharacteristic
197C22×C3×C4C22×D311yescharacteristic
198C22×C3×C4C23×C311yescharacteristic
199C22×Dic3—21no—
200C2×D12—21no—
201C2×D12—24no—
202C2×D12C23×C312yes—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—44no—
206C3×SD16—68no—
207C2×C3×C8—34no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4C22×D312yes—
227C3×C4⋊C4C23×C312yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—62no—
232C3×C22⋊C4—62no—
233C3×C42—24no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42#6711yescharacteristic
241C3×C42#6711yescharacteristic
242C3×C42C3×D411yescharacteristic
243C3×C42C3×D411yescharacteristic
244C3×C42C22×D311yescharacteristic
245C3×C42C22×D311yescharacteristic
246C3×C42C23×C311yescharacteristic
247C3×C42C23×C311yescharacteristic
248#215—28no—
249(C2×D3)⋊C4—22no—
250(C2×D3)⋊C4—22no—
251C4⋊Dic3—21no—
252C4⋊Dic3C23×C312yes—
253C2×C3⋊C8C23×C314yes—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32#13712yes—
257C22×C32C22×D411yescharacteristic
258C2×C3×D3—24no—
259C2×C3×D3—48no—
260C2×C3×D3—42no—
261C2×C3×D3—42no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—42no—
267C32×C4—42no—
268C32×C4AGL(1,ℤ/8)12yes—
269C32×C4C2×#4112yes—
270C3×Dic3—42no—
271C22×D4—31no—
272C23×C4—61no—
273C2×SD16—34no—
274C4⋊Q8C2×C3×D311yescharacteristic, complemented
275C4⋊Q8C2×C3×D311yescharacteristic, complemented
276#125—21no—
277C22⋊Q8—21no—
278C22⋊Q8C2×C3×D312yes—
279C4×Q8—21no—
280C4×Q8C2×C3×D312yescomplemented
281C4×D4—32no—
282C4×D4—32no—
283C4×D4—61no—
284C4×D4—61no—
285#117—21no—
286C2×C4⋊C4—31no—
287C2×C22⋊C4—61no—
288C2×C42C2×C3×D311yescharacteristic
289#107—34no—
290C4⋊C8—32no—
291#103—34no—
292#102—34no—
293C22⋊C8—32no—
294#97—32no—
295C23×C3—21no—
296C23×C3—64no—
297C23×C3—62no—
298C23×C3—122no—
299C23×C3—122no—
300C23×C3C2×#6711yescharacteristic
301C23×C3C2×C3×D411yescharacteristic
302C22×D3—24no—
303C22×D3—22no—
304C22×D3—42no—
305C22×D3—42no—
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—44no—
309C3×Q8—44no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×D4—34no—
313C3×D4—64no—
314C3×D4—62no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—62no—
337C2×C3×C4—124no—
338C2×C3×C4—122no—
339C2×C3×C4—124no—
340C2×C3×C4C2×#6711yescharacteristic
341C2×C3×C4C2×#6711yescharacteristic
342C2×C3×C4C2×#6711yescharacteristic
343C2×C3×C4C2×#6711yescharacteristic
344C2×C3×C4C2×#6711yescharacteristic
345C2×C3×C4C2×C3×D411yescharacteristic
346C2×C3×C4C2×C3×D411yescharacteristic
347C2×C3×C4C2×C3×D411yescharacteristic
348C2×C3×C4C2×C3×D411yescharacteristic
349C2×C3×C4C2×C3×D411yescharacteristic
350C2×C3×C4C23×D311yescharacteristic
351C2×C3×C4C24×C311yescharacteristic
352C2×Dic3—22no—
353C2×Dic3—42no—
354D12—24no—
355D12—22no—
356D12C3×#4114yes—
357C4×D3—44no—
358C4×D3—44no—
359C3×C8—64no—
360C3⋊C8—24no—
361C2×C32—22no—
362C2×C32#52511yescharacteristic
363C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
364C2×C32C2×#13711yescharacteristic
365C3×D3—44no—
366C24—61no—
367C2×Q8—21no—
368C2×Q8C22×C3×D312yes—
369C2×D4—32no—
370C2×D4—64no—
371C2×D4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—124no—
376C22×C4C3×#6711yescharacteristic
377C22×C4C3×#6711yescharacteristic
378C22×C4C22×C3×D311yescharacteristic
379SD16—68no—
380C2×C8—34no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—32no—
387C4⋊C4—61no—
388C4⋊C4C22×C3×D312yes—
389C22⋊C4—22no—
390C22⋊C4—62no—
391C22⋊C4—62no—
392C42—22no—
393C42C3×#6711yescharacteristic
394C42C3×#6711yescharacteristic
395C42C22×C3×D311yescharacteristic
396C42C22×C3×D311yescharacteristic
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—64no—
406C22×C3—128no—
407C22×C3—122no—
408C22×C3—122no—
409C22×C3#94212yes—
410C22×C3C3×#13712yes—
411C22×C3C22×#6711yescharacteristic
412C22×C3C22×C3×D411yescharacteristic
413C2×D3—24no—
414C2×D3—48no—
415C2×D3—42no—
416C2×D3—42no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—44no—
432C3×C4—44no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—122no—
438C3×C4#94912yes—
439C3×C4C3×AGL(1,ℤ/8)12yes—
440C3×C4D3×#4112yes—
441C3×C4C2×C3×#4112yes—
442Dic3—42no—
443C32#311011yescharacteristic, complemented, Sylow
444C23—64no—
445C23—62no—
446C23—122no—
447C23—122no—
448C23C2×C3×#6711yescharacteristic
449Q8—24no—
450Q8—42no—
451D4—34no—
452D4—64no—
453D4—62no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—62no—
460C2×C4—124no—
461C2×C4—122no—
462C2×C4—124no—
463C2×C4C2×C3×#6711yescharacteristic
464C2×C4C2×C3×#6711yescharacteristic
465C2×C4C2×C3×#6711yescharacteristic
466C2×C4C2×C3×#6711yescharacteristic
467C2×C4C2×C3×#6711yescharacteristic
468C2×C4C23×C3×D311yescharacteristic
469C8—64no—
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3—124no—
477C2×C3C2×#94211yescharacteristic
478C2×C3C2×#94911yescharacteristic
479C2×C3#582811yescharacteristic
480C2×C3C3×#52511yescharacteristic
481C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
482C2×C3C2×C3×#13711yescharacteristic
483D3—44no—
484C22—21no—
485C22—21no—
486C22—64no—
487C22—128no—
488C22—122no—
489C22—122no—
490C22C3×#94212yes—
491C22C22×C3×#6711yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—42no—
498C4—122no—
499C4C3×#94912yes—
500C4C3×D3×#4112yes—
501C3—21no—
502C3#8335311yescharacteristic, direct product factor
503C3C3×#311011yescharacteristic, complemented
504C2—22no—
505C2—124no—
506C2C2×C3×#94211yescharacteristic
507C2C2×C3×#94911yescharacteristic
508C2C3×#582811yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#533C2
socleC22×C32C22×D4
prime radicalC22×C3×D12C22
hypercenterC3×#533D3
hypocenterC3C3×#3110
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C3×#533

Complements

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

NH ≅ G/Ntype
1. C32×#533C2semidirect
3. C3×#5087C2semidirect
4. C3×#5083C2semidirect
6. C3×#5044C2semidirect
7. C3×#5026C2semidirect
8. C3×#4926C2semidirect
10. #83353C3direct (N × H)
13. C32×C4⋊Q8C22semidirect
14. C32×C4⋊Q8C22semidirect
19. C32×C4×Q8C22semidirect
29. C3×#835C22semidirect
32. C3×#807C22semidirect
33. C3×C4⋊(C3⋊C8)C22semidirect
34. C3×#803C22semidirect
36. C3×#533D3semidirect
37. C3×#533C2×C3semidirect
46. #5087C2×C3semidirect
47. #5083C2×C3semidirect
49. #5044C2×C3semidirect
50. #5026C2×C3semidirect
51. #4926C2×C3semidirect
91. C3×C4⋊Q8C2×D3semidirect
92. C3×C4⋊Q8C2×D3semidirect
93. C3×C4⋊Q8C22×C3semidirect
94. C3×C4⋊Q8C22×C3semidirect
108. C3×C4×Q8C2×D3semidirect
109. C3×C4×Q8C22×C3semidirect
135. #835C22×C3semidirect
138. #807C22×C3semidirect
139. C4⋊(C3⋊C8)C22×C3semidirect
140. #803C22×C3semidirect
167. #533C3×D3semidirect
274. C4⋊Q8C2×C3×D3semidirect
275. C4⋊Q8C2×C3×D3semidirect
280. C4×Q8C2×C3×D3semidirect
443. C32#3110semidirect
502. C3#83353direct (N × H)
503. C3C3×#3110semidirect

Maximal subgroups

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

subgroupindex
C32×#5332
C2×C3×C4×D122
C3×#50872
C3×#50832
C3×#50462
C3×#50442
C3×#50262
C3×#49262
C3×#31103
#833533