Finite Group Databasebeta

← C3×#70314 #11788461

Subgroups — C3×#70314 #11788461

Order 1152.

787 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
1C3×#5798C211yesmaximal, characteristic, complemented
2C2×C3×#932C211yesmaximal, characteristic
3C32×#418C211yesmaximal, characteristic, complemented
4C3×#5139C211yesmaximal, characteristic, complemented
5C3×#5006C211yesmaximal, characteristic, complemented
6C3×#4559C211yesmaximal, characteristic, complemented
7C3×#4531C211yesmaximal, characteristic, complemented
8C3×#1904—31nomaximal
9#70314C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C2×C3×#217—22no—
15C3×D4×Dic3—22no—
16C3×D4×Dic3—22no—
17C2×C3×#214C412yescomplemented
18C3×#932—22no—
19C3×#932—22no—
20C2×C3×#213C412yescomplemented
21C3×#926C2211yescharacteristic
22C3×#926C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C32×#102—22no—
25C3×#825—22no—
26C3×#806—22no—
27C2×C3×AGL(1,ℤ/8)—31no—
28C3×#518—31no—
29#5798C2×C311yescharacteristic, complemented
30C2×#932C2×C311yescharacteristic
31C3×#421—31no—
32C3×#418—21no—
33C3×#418D311yescharacteristic, complemented
34C3×#418C2×C311yescharacteristic, complemented
35C3×#410—31no—
36#5139C2×C311yescharacteristic, complemented
37#5006C2×C311yescharacteristic, complemented
38C3×#344—31no—
39C3×H(3,ℤ/4)—31no—
40#4559C2×C311yescharacteristic, complemented
41#4531C2×C311yescharacteristic, complemented
42C24×C32—21no—
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×C32×D4—21no—
46C2×C32×D4C2×C412yes—
47C2×C32×D4D412yes—
48C2×C32×D4D412yes—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—21no—
51C22×C3×Dic3—24no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—21no—
56C2×C3×D12C2×C411yescharacteristic, complemented
57C2×C3×C4×D3—21no—
58C2×C3×Dic6C2×C411yescharacteristic, complemented
59C32×M16—21no—
60C32×M16—21no—
61C2×C32×C8—21no—
62C32×C4⋊C4—21no—
63C32×C22⋊C4—21no—
64C32×C42—21no—
65C3×#217—24no—
66C3×#217—21no—
67C3×#217D412yes—
68C3×#214—24no—
69C3×#213—24no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3—21no—
72C3×C4×Dic3—21no—
73C3×C4×Dic3D412yes—
74C3×#208—21no—
75C3×#208—21no—
76C2×C3×C3⋊C8C2×C412yescomplemented
77#1904—31noSylow
78C2×C3×#41—31no—
79C22×C3×D4—21no—
80C22×C3×D4C2×D311yescharacteristic
81C22×C3×D4C22×C311yescharacteristic
82C2×#235C22×C311yescharacteristic
83C3×AGL(1,ℤ/8)—62no—
84C3×AGL(1,ℤ/8)—62no—
85C2×C3×SD16—32no—
86C2×C3×D8—32no—
87C2×C3×M16—21no—
88C2×C3×M16—31no—
89C2×C3×M16C2×D311yescharacteristic
90C2×C3×M16C22×C311yescharacteristic
91C3×C4×D4—62no—
92C3×C4×D4—62no—
93C3×#117—21no—
94C3×#117—31no—
95C3×#117—31no—
96C3×#117C2×D311yescharacteristic
97C3×#117C22×C311yescharacteristic
98C2×C3×C22⋊C4—62no—
99C2×#217—22no—
100D4×Dic3—22no—
101D4×Dic3—22no—
102C2×#214C3×C412yescomplemented
103#932—22no—
104#932—22no—
105C2×#213C3×C412yescomplemented
106#926C22×C311yescharacteristic
107#926C22×C311yescharacteristic
108C2×#208C22×C311yescharacteristic
109C3×#104—62no—
110C3×#102—24no—
111C3×#102—22no—
112C3×#102—22no—
113C3×#99—62no—
114#825—22no—
115#806—22no—
116C23×C32—22no—
117C23×C32—22no—
118C23×C32—22no—
119C23×C32—82no—
120C23×C32C2×D411yescharacteristic
121C22×C3×D3—21no—
122C32×D4—22no—
123C32×D4—22no—
124C32×D4—22no—
125C32×D4C22⋊C414yes—
126C2×C32×C4—41no—
127C2×C32×C4C22×C411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4#4111yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C3×#67—41no—
134C3×#67—41no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—84no—
139C3×D12—21no—
140C3×D12—21no—
141C3×C4×D3—41no—
142C3×C4×D3—41no—
143C3×Dic6—21no—
144C3×Dic6—21no—
145C32×C8—41no—
146C3×C3⋊C8—22no—
147C2×AGL(1,ℤ/8)—31no—
148#518—31no—
149#421—31no—
150#418C3×D311yescharacteristic, complemented
151#410—31no—
152#344—31no—
153H(3,ℤ/4)—31no—
154C24×C3—21no—
155C24×C3—21no—
156C24×C3—41no—
157C3×#41—61no—
158C3×#41—61no—
159C3×#41—61no—
160C3×#41—61no—
161C2×C3×Q8—31no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—31no—
172C2×C3×D4—41no—
173C2×C3×D4—41no—
174C2×C3×D4—41no—
175C2×C3×D4—61no—
176C2×C3×D4C4×D312yes—
177C2×C3×D4D1212yes—
178C2×C3×D4#6712yes—
179C2×C3×D4C2×C3×C412yes—
180C2×C3×D4C3×D412yes—
181C2×C3×D4C3×D412yes—
182C22×C3×C4—21no—
183C22×C3×C4—64no—
184C22×C3×C4—61no—
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C23×C311yescharacteristic
187C2×#67—21no—
188C22×Dic3—24no—
189#235—21no—
190#235—21no—
191#235—21no—
192#235—21no—
193C2×D12C2×C3×C411yescharacteristic, complemented
194C2×C4×D3—21no—
195C2×Dic6C2×C3×C411yescharacteristic, complemented
196C3×SD16—64no—
197C3×D8—64no—
198C3×M16—21no—
199C3×M16—21no—
200C3×M16—21no—
201C3×M16—21no—
202C3×M16—41no—
203C3×M16—41no—
204C3×M16—61no—
205C3×M16—61no—
206C2×C3×C8—22no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—32no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—61no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4—32no—
218C3×C22⋊C4—41no—
219C3×C22⋊C4—64no—
220C3×C22⋊C4—61no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—32no—
224C3×C42—41no—
225C3×C42—61no—
226#217—24no—
227#217—21no—
228#217C3×D412yes—
229#214—24no—
230#213—24no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4×Dic3—21no—
234C4×Dic3C3×D412yes—
235#208—21no—
236#208—21no—
237C2×C3⋊C8C2×C3×C412yescomplemented
238C22×C32—24no—
239C22×C32—41no—
240C22×C32—44no—
241C22×C32—82no—
242C22×C32—82no—
243C22×C32C4×D411yescharacteristic
244C22×C32#12011yescharacteristic
245C22×C32#12111yescharacteristic
246C2×C3×D3—41no—
247C2×C3×D3—81no—
248C32×C4—81no—
249C32×C4C2×C22⋊C411yescharacteristic
250C32×C4C4×D411yescharacteristic
251C32×C4#12111yescharacteristic
252C32×C4#12311yescharacteristic
253C3×Dic3—42no—
254C3×Dic3—42no—
255C3×Dic3—41no—
256C2×#41—31no—
257C22×D4C2×C3×D311yescharacteristic
258AGL(1,ℤ/8)—62no—
259AGL(1,ℤ/8)—62no—
260C2×SD16—32no—
261C2×D8—32no—
262C2×M16—31no—
263C2×M16C2×C3×D311yescharacteristic
264C4×D4—62no—
265C4×D4—62no—
266#117—31no—
267#117—31no—
268#117C2×C3×D311yescharacteristic
269C2×C22⋊C4—62no—
270#104—62no—
271#102—22no—
272#99—62no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—61no—
284C23×C3—82no—
285C23×C3—82no—
286C23×C3—84no—
287C23×C3D3×D411yescharacteristic
288C23×C3C2×C3×D411yescharacteristic
289C22×D3—21no—
290C3×Q8—61no—
291C3×Q8—61no—
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—24no—
299C3×D4—42no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—61no—
303C3×D4—61no—
304C3×D4—121no—
305C3×D4—121no—
306C3×D4(C2×D3)⋊C414yes—
307C3×D4C3×C22⋊C414yes—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—61no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—121no—
321C2×C3×C4—121no—
322C2×C3×C4—244no—
323C2×C3×C4C2×C4×D311yescharacteristic
324C2×C3×C4C2×D1211yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4#23711yescharacteristic
327C2×C3×C4#23711yescharacteristic
328C2×C3×C4C2×#6711yescharacteristic
329C2×C3×C4C22×C3×C411yescharacteristic
330C2×C3×C4C2×C3×D411yescharacteristic
331C2×C3×C4C2×C3×D411yescharacteristic
332C2×C3×C4C2×C3×D411yescharacteristic
333C2×C3×C4C3×#4111yescharacteristic
334C2×C3×C4C3×#4111yescharacteristic
335#67—41no—
336#67—41no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—84no—
341D12—21no—
342D12—21no—
343C4×D3—41no—
344C4×D3—41no—
345Dic6—21no—
346Dic6—21no—
347C3×C8—42no—
348C3×C8—41no—
349C3×C8—41no—
350C3×C8—62no—
351C3⋊C8—22no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—44no—
356C2×C32#38611yescharacteristic
357C3×D3—81no—
358C24—21no—
359#41—61no—
360#41—61no—
361#41—61no—
362#41—61no—
363C2×Q8—31no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—31no—
368C2×D4—61no—
369C2×D4C3×C4×D312yes—
370C2×D4C3×D1212yes—
371C2×D4C3×#6712yes—
372C22×C4—64no—
373C22×C4—61no—
374C22×C4C22×C3×D311yescharacteristic
375SD16—64no—
376D8—64no—
377M16—21no—
378M16—21no—
379M16—61no—
380M16—61no—
381C2×C8—21no—
382C2×C8—32no—
383C4⋊C4—21no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C22⋊C4—21no—
387C22⋊C4—32no—
388C22⋊C4—64no—
389C22⋊C4—61no—
390C42—21no—
391C42—32no—
392C42—61no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—44no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3—42no—
404C22×C3—84no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—84no—
410C22×C3—84no—
411C22×C3—121no—
412C22×C3—241no—
413C22×C3#67⋊C411yescharacteristic
414C22×C3#88211yescharacteristic
415C22×C3#93911yescharacteristic
416C22×C3C3×C4×D411yescharacteristic
417C22×C3C3×#12011yescharacteristic
418C22×C3C3×#12111yescharacteristic
419C2×D3—41no—
420C2×D3—81no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—82no—
426C3×C4—81no—
427C3×C4—81no—
428C3×C4—122no—
429C3×C4—122no—
430C3×C4—121no—
431C3×C4#67⋊C411yescharacteristic
432C3×C4#88611yescharacteristic
433C3×C4C2×(C2×D3)⋊C411yescharacteristic
434C3×C4#93911yescharacteristic
435C3×C4C2×C3×C22⋊C411yescharacteristic
436C3×C4C3×C4×D411yescharacteristic
437C3×C4C3×#12111yescharacteristic
438C3×C4C3×#12311yescharacteristic
439Dic3—42no—
440Dic3—42no—
441Dic3—41no—
442C32#190411yescharacteristic, complemented, Sylow
443C23—22no—
444C23—22no—
445C23—22no—
446C23—61no—
447C23—82no—
448C23C3×D3×D411yescharacteristic
449Q8—61no—
450Q8—61no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—61no—
455D4—61no—
456D4—121no—
457D4—121no—
458D4C3×(C2×D3)⋊C414yes—
459C2×C4—41no—
460C2×C4—61no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—244no—
466C2×C4C2×C3×C4×D311yescharacteristic
467C2×C4C2×C3×D1211yescharacteristic
468C2×C4C3×D3×D411yescharacteristic
469C2×C4C3×#23711yescharacteristic
470C2×C4C3×#23711yescharacteristic
471C2×C4C2×C3×#6711yescharacteristic
472C8—41no—
473C8—62no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—44no—
485C2×C3—44no—
486C2×C3—84no—
487C2×C3—241no—
488C2×C3#495811yescharacteristic
489C2×C3C3×#38611yescharacteristic
490D3—81no—
491C22—24no—
492C22—41no—
493C22—44no—
494C22—82no—
495C22—82no—
496C22—121no—
497C22—241no—
498C22C3×#67⋊C411yescharacteristic
499C22C3×#88211yescharacteristic
500C22C3×#93911yescharacteristic
501C4—81no—
502C4—122no—
503C4—122no—
504C4—121no—
505C4C3×#67⋊C411yescharacteristic
506C4C3×#88611yescharacteristic
507C4C2×C3×(C2×D3)⋊C411yescharacteristic
508C4C3×#93911yescharacteristic
509C3—21no—
510C3#7031411yescharacteristic, direct product factor
511C3C3×#190411yescharacteristic, complemented
512C2—21no—
513C2—21no—
514C2—21no—
515C2—44no—
516C2—241no—
517C2C3×#495811yescharacteristic
subgroupG/H
centerC2×C3#4958
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#418C2
socleC2×C32#386
prime radicalC2×C3×#932C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#418D3
hypocenterC3C3×#1904
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#418

Complements

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

NH ≅ G/Ntype
1. C3×#5798C2semidirect
3. C32×#418C2semidirect
4. C3×#5139C2semidirect
5. C3×#5006C2semidirect
6. C3×#4559C2semidirect
7. C3×#4531C2semidirect
9. #70314C3direct (N × H)
17. C2×C3×#214C4semidirect
20. C2×C3×#213C4semidirect
29. #5798C2×C3semidirect
33. C3×#418D3semidirect
34. C3×#418C2×C3semidirect
36. #5139C2×C3semidirect
37. #5006C2×C3semidirect
40. #4559C2×C3semidirect
41. #4531C2×C3semidirect
56. C2×C3×D12C2×C4semidirect
58. C2×C3×Dic6C2×C4semidirect
76. C2×C3×C3⋊C8C2×C4semidirect
102. C2×#214C3×C4semidirect
105. C2×#213C3×C4semidirect
150. #418C3×D3semidirect
193. C2×D12C2×C3×C4semidirect
195. C2×Dic6C2×C3×C4semidirect
237. C2×C3⋊C8C2×C3×C4semidirect
442. C32#1904semidirect
510. C3#70314direct (N × H)
511. C3C3×#1904semidirect

Maximal subgroups

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

subgroupindex
C3×#57982
C2×C3×#9322
C32×#4182
C3×#51392
C3×#50062
C3×#45592
C3×#45312
C3×#19043
#703143