Finite Group Databasebeta

← C3×#68760 #11787864

Subgroups — C3×#68760 #11787864

Order 1152.

943 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5521C211yesmaximal, characteristic, complemented
2C2×C3×#874C211yesmaximal, characteristic
3C3×#5131C211yesmaximal, characteristic, complemented
4C2×C3×#823C211yesmaximal, characteristic, complemented
5C3×#5001C211yesmaximal, characteristic, complemented
6C32×#339C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#1887—31nomaximal
9#68760C311yesmaximal, characteristic, direct product factor
10C22×C3×D12—22no—
11C22×C3×D12C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C32×#117C2211yescharacteristic
15C2×C32×C42C2211yescharacteristic
16C3×#926C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C3×#893—22no—
19C3×D3×C22⋊C4—22no—
20C3×#874—22no—
21C3×#874C414yescomplemented
22C3×C4×D12—22no—
23C3×#823—22no—
24C3×#821—22no—
25C3×#809—22no—
26C2×C3×#127—31no—
27C3×#518—31no—
28#5521C2×C311yescharacteristic, complemented
29C2×#874C2×C311yescharacteristic
30C3×#418—31no—
31C3×#418—31no—
32C2×C3×#100—31no—
33#5131C2×C311yescharacteristic, complemented
34C2×#823C2×C311yescharacteristic, complemented
35#5001C2×C311yescharacteristic, complemented
36C3×#339—21no—
37C3×#339—31no—
38C3×#339D311yescharacteristic, complemented
39C3×#339C2×C311yescharacteristic, complemented
40#4523C2×C311yescharacteristic, complemented
41C23×C3×D3—21no—
42C23×C3×D3D412yes—
43C22×C32×C4—21no—
44C22×C32×C4C2311yescharacteristic
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C2×C3×D12—21no—
49C2×C3×D12—21no—
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12—28no—
54C2×C3×D12—21no—
55C2×C3×D12—48no—
56C2×C3×D12C2×C412yes—
57C2×C3×D12C2×C412yescomplemented
58C2×C3×D12D412yes—
59C2×C3×C4×D3—24no—
60C32×M16—21no—
61C32×M16—21no—
62C2×C32×C8—21no—
63C32×C4⋊C4D412yescomplemented
64C32×C22⋊C4—21no—
65C32×C42—21no—
66C32×C42—21no—
67C32×C42C2×C412yescomplemented
68C3×#217—21no—
69C3×(C2×D3)⋊C4—24no—
70C3×C4⋊Dic3D412yescomplemented
71C3×C4×Dic3—21no—
72C3×#208—21no—
73C3×#208—21no—
74C2×C3×C3⋊C8—21no—
75#1887—31noSylow
76C22×C3×D4—31no—
77C22×C3×D4—31no—
78C22×C3×D4—62no—
79C22×D12—22no—
80C22×D12C22×C311yescharacteristic
81C22×D12C22×C311yescharacteristic
82C2×C3×M16—21no—
83C2×C3×M16—31no—
84C2×C3×M16C2×D311yescharacteristic
85C2×C3×M16C22×C311yescharacteristic
86C3×#127—34no—
87C3×#127—62no—
88C3×C4×D4—62no—
89C3×C4×D4—62no—
90C3×#117—21no—
91C3×#117—31no—
92C3×#117C2×D311yescharacteristic
93C3×#117C22×C311yescharacteristic
94C2×C3×C22⋊C4—62no—
95C2×C3×C42—21no—
96C2×C3×C42C2×D311yescharacteristic
97C2×C3×C42C22×C311yescharacteristic
98#926C22×C311yescharacteristic
99C2×#208C22×C311yescharacteristic
100#893—22no—
101D3×C22⋊C4—22no—
102#874—22no—
103#874C3×C414yescomplemented
104C4×D12—22no—
105C3×#102—62no—
106C3×#102—62no—
107C3×#100—62no—
108#823—22no—
109#821—22no—
110#809—22no—
111C23×C32C2×D411yescharacteristic
112C22×C3×D3—22no—
113C22×C3×D3—22no—
114C22×C3×D3—22no—
115C22×C3×D3—44no—
116C22×C3×D3—44no—
117C22×C3×D3—41no—
118C22×C3×D3—41no—
119C22×C3×D3C22⋊C414yes—
120C2×C32×C4—24no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4C22×C411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4#4111yescharacteristic
128C2×C32×C4#4111yescharacteristic
129C2×C3×Dic3—22no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12—22no—
134C3×D12—22no—
135C3×D12—22no—
136C3×D12—416no—
137C3×D12—41no—
138C3×D12—41no—
139C3×C4×D3—84no—
140C32×C8—41no—
141C3×C3⋊C8—41no—
142C2×#127—31no—
143#518—31no—
144#418—31no—
145#418—31no—
146C2×#100—31no—
147#339—31no—
148#339C3×D311yescharacteristic, complemented
149C24×C3—32no—
150C24×C3—61no—
151C23×D3—21no—
152C23×D3C3×D412yes—
153C2×C3×D4—32no—
154C2×C3×D4—32no—
155C2×C3×D4—32no—
156C2×C3×D4—68no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C2×C3×D4—61no—
160C2×C3×D4—61no—
161C2×C3×D4—61no—
162C2×C3×D4—61no—
163C2×C3×D4—61no—
164C2×C3×D4—61no—
165C2×C3×D4—61no—
166C2×C3×D4—128no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—41no—
171C22×C3×C4—64no—
172C22×C3×C4C22×D311yescharacteristic
173C22×C3×C4C23×C311yescharacteristic
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—28no—
183C2×D12—21no—
184C2×D12—48no—
185C2×D12C2×C3×C412yes—
186C2×D12C2×C3×C412yescomplemented
187C2×D12C3×D412yes—
188C2×C4×D3—24no—
189C3×M16—22no—
190C3×M16—21no—
191C3×M16—21no—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—41no—
195C3×M16—61no—
196C3×M16—61no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—41no—
200C2×C3×C8—61no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—32no—
203C3×C4⋊C4D1212yescomplemented
204C3×C4⋊C4C3×D412yescomplemented
205C3×C22⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—64no—
209C3×C22⋊C4—61no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—22no—
215C3×C42—41no—
216C3×C42—41no—
217C3×C42—61no—
218C3×C42C4×D312yescomplemented
219C3×C42C2×C3×C412yescomplemented
220#217—21no—
221(C2×D3)⋊C4—24no—
222C4⋊Dic3C3×D412yescomplemented
223C4×Dic3—21no—
224#208—21no—
225#208—21no—
226C2×C3⋊C8—21no—
227C22×C32—22no—
228C22×C32C2×C22⋊C411yescharacteristic
229C22×C32#12011yescharacteristic
230C22×C32#12311yescharacteristic
231C2×C3×D3—24no—
232C2×C3×D3—44no—
233C2×C3×D3—44no—
234C2×C3×D3—44no—
235C2×C3×D3—42no—
236C2×C3×D3—82no—
237C2×C3×D3—82no—
238C2×C3×D3—82no—
239C32×C4—24no—
240C32×C4—42no—
241C32×C4C4×D411yescharacteristic
242C32×C4C4×D411yescharacteristic
243C32×C4#12111yescharacteristic
244C32×C4#12111yescharacteristic
245C3×Dic3—42no—
246C22×D4—31no—
247C22×D4—31no—
248C22×D4—62no—
249C2×M16—31no—
250C2×M16C2×C3×D311yescharacteristic
251#127—34no—
252#127—62no—
253C4×D4—62no—
254C4×D4—62no—
255#117—31no—
256#117C2×C3×D311yescharacteristic
257C2×C22⋊C4—62no—
258C2×C42C2×C3×D311yescharacteristic
259#102—62no—
260#102—62no—
261#100—62no—
262C23×C3—21no—
263C23×C3—34no—
264C23×C3—62no—
265C23×C3—62no—
266C23×C3—62no—
267C23×C3—124no—
268C23×C3—124no—
269C23×C3—121no—
270C23×C3—121no—
271C23×C3D3×D411yescharacteristic
272C23×C3C2×C3×D411yescharacteristic
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—44no—
277C22×D3—44no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3C3×C22⋊C414yes—
281C3×D4—62no—
282C3×D4—62no—
283C3×D4—62no—
284C3×D4—62no—
285C3×D4—62no—
286C3×D4—62no—
287C3×D4—1216no—
288C3×D4—121no—
289C3×D4—121no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—42no—
303C2×C3×C4—44no—
304C2×C3×C4—42no—
305C2×C3×C4—62no—
306C2×C3×C4—244no—
307C2×C3×C4C2×C4×D311yescharacteristic
308C2×C3×C4C2×D1211yescharacteristic
309C2×C3×C4#23511yescharacteristic
310C2×C3×C4D3×D411yescharacteristic
311C2×C3×C4D3×D411yescharacteristic
312C2×C3×C4#23911yescharacteristic
313C2×C3×C4C22×C3×C411yescharacteristic
314C2×C3×C4C2×C3×D411yescharacteristic
315C2×C3×C4C2×C3×D411yescharacteristic
316C2×C3×C4C2×C3×D411yescharacteristic
317C2×C3×C4C3×#4111yescharacteristic
318C2×C3×C4C3×#4111yescharacteristic
319C2×Dic3—22no—
320D12—22no—
321D12—22no—
322D12—22no—
323D12—22no—
324D12—22no—
325D12—22no—
326D12—416no—
327D12—41no—
328D12—41no—
329C4×D3—84no—
330C3×C8—42no—
331C3×C8—41no—
332C3×C8—41no—
333C3×C8—121no—
334C3⋊C8—41no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32#38611yescharacteristic
338C2×C32#45312yes—
339C3×D3—44no—
340C3×D3—82no—
341C24—32no—
342C24—61no—
343C2×D4—32no—
344C2×D4—32no—
345C2×D4—32no—
346C2×D4—68no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—128no—
357C22×C4—21no—
358C22×C4—64no—
359C22×C4C22×C3×D311yescharacteristic
360M16—21no—
361M16—21no—
362M16—61no—
363M16—61no—
364C2×C8—21no—
365C2×C8—61no—
366C4⋊C4—32no—
367C4⋊C4C3×D1212yescomplemented
368C22⋊C4—21no—
369C22⋊C4—64no—
370C22⋊C4—61no—
371C42—21no—
372C42—21no—
373C42—61no—
374C42C3×C4×D312yescomplemented
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—42no—
381C22×C3—64no—
382C22×C3—124no—
383C22×C3—124no—
384C22×C3—124no—
385C22×C3—122no—
386C22×C3—242no—
387C22×C3—242no—
388C22×C3—242no—
389C22×C3D3×C22⋊C411yescharacteristic
390C22×C3#88211yescharacteristic
391C22×C3#89411yescharacteristic
392C22×C3C2×C3×C22⋊C411yescharacteristic
393C22×C3C3×#12011yescharacteristic
394C22×C3C3×#12311yescharacteristic
395C2×D3—24no—
396C2×D3—44no—
397C2×D3—44no—
398C2×D3—44no—
399C2×D3—42no—
400C2×D3—82no—
401C2×D3—82no—
402C2×D3—82no—
403C3×C4—24no—
404C3×C4—24no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—44no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—82no—
413C3×C4—122no—
414C3×C4C4×D1211yescharacteristic
415C3×C4#88411yescharacteristic
416C3×C4#89311yescharacteristic
417C3×C4#89511yescharacteristic
418C3×C4C3×C4×D411yescharacteristic
419C3×C4C3×C4×D411yescharacteristic
420C3×C4C3×#12111yescharacteristic
421C3×C4C3×#12111yescharacteristic
422Dic3—42no—
423C32#188711yescharacteristic, complemented, Sylow
424C23—34no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—124no—
429C23—124no—
430C23—121no—
431C23—121no—
432C23C3×D3×D411yescharacteristic
433D4—62no—
434D4—62no—
435D4—62no—
436D4—62no—
437D4—62no—
438D4—62no—
439D4—1216no—
440D4—121no—
441D4—121no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—62no—
446C2×C4—244no—
447C2×C4C2×C3×C4×D311yescharacteristic
448C2×C4C2×C3×D1211yescharacteristic
449C2×C4C3×#23511yescharacteristic
450C2×C4C3×D3×D411yescharacteristic
451C2×C4C3×D3×D411yescharacteristic
452C2×C4C3×#23911yescharacteristic
453C8—41no—
454C8—121no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—124no—
464C2×C3—242no—
465C2×C3#466511yescharacteristic
466C2×C3#482212yes—
467C2×C3C3×#38611yescharacteristic
468C2×C3C3×#45312yes—
469D3—44no—
470D3—82no—
471C22—22no—
472C22—64no—
473C22—124no—
474C22—124no—
475C22—124no—
476C22—122no—
477C22—242no—
478C22—242no—
479C22—242no—
480C22C3×D3×C22⋊C411yescharacteristic
481C22C3×#88211yescharacteristic
482C22C3×#89411yescharacteristic
483C4—24no—
484C4—42no—
485C4—122no—
486C4C3×C4×D1211yescharacteristic
487C4C3×#88411yescharacteristic
488C4C3×#89311yescharacteristic
489C4C3×#89511yescharacteristic
490C3—21no—
491C3#6876011yescharacteristic, direct product factor
492C3C3×#188711yescharacteristic, complemented
493C2—21no—
494C2—21no—
495C2—124no—
496C2—242no—
497C2C3×#466511yescharacteristic
498C2C3×#482212yes—
subgroupG/H
centerC22×C3#882
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#339C2
socleC22×C32#120
prime radicalC2×C3×#874C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#339D3
hypocenterC3C3×#1887
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#339

Complements

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

NH ≅ G/Ntype
1. C3×#5521C2semidirect
3. C3×#5131C2semidirect
4. C2×C3×#823C2semidirect
5. C3×#5001C2semidirect
6. C32×#339C2semidirect
7. C3×#4523C2semidirect
9. #68760C3direct (N × H)
21. C3×#874C4semidirect
28. #5521C2×C3semidirect
33. #5131C2×C3semidirect
34. C2×#823C2×C3semidirect
35. #5001C2×C3semidirect
38. C3×#339D3semidirect
39. C3×#339C2×C3semidirect
40. #4523C2×C3semidirect
57. C2×C3×D12C2×C4semidirect
63. C32×C4⋊C4D4semidirect
67. C32×C42C2×C4semidirect
70. C3×C4⋊Dic3D4semidirect
103. #874C3×C4semidirect
148. #339C3×D3semidirect
186. C2×D12C2×C3×C4semidirect
203. C3×C4⋊C4D12semidirect
204. C3×C4⋊C4C3×D4semidirect
218. C3×C42C4×D3semidirect
219. C3×C42C2×C3×C4semidirect
222. C4⋊Dic3C3×D4semidirect
367. C4⋊C4C3×D12semidirect
374. C42C3×C4×D3semidirect
423. C32#1887semidirect
491. C3#68760direct (N × H)
492. C3C3×#1887semidirect

Maximal subgroups

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

subgroupindex
C3×#55212
C2×C3×#8742
C3×#51312
C2×C3×#8232
C3×#50012
C32×#3392
C3×#45232
C3×#18873
#687603