Finite Group Databasebeta

← C3×#68873 #11785807

Subgroups — C3×#68873 #11785807

Order 1152.

657 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#892C211yesmaximal, characteristic
2C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic
3C3×#4987C211yesmaximal, characteristic
4C32×#340C211yesmaximal, characteristic, complemented
5C3×#4551C211yesmaximal, characteristic, complemented
6C3×#4529C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8C3×#1825—31nomaximal
9#68873C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×D3C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C3×(C2×D3)⋊C4—21no—
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C4×Dic3—21no—
19C22×C3×C3⋊C8C2211yescharacteristic
20C3×#892—22no—
21C3×#892C414yescomplemented
22C3×C3⋊#95—21no—
23C3×(C2×D3)⋊C8—22no—
24C2×C3×#117—31no—
25C2×#892C2×C311yescharacteristic
26C2×C3×C22⋊C8—31no—
27C3×#380—31no—
28C2×(C2×D3)⋊C8C2×C311yescharacteristic
29#4987C2×C311yescharacteristic
30C3×#340—21no—
31C3×#340—31no—
32C3×#340—31no—
33C3×#340—31no—
34C3×#340D311yescharacteristic, complemented
35C3×#340C2×C311yescharacteristic, complemented
36#4551C2×C311yescharacteristic, complemented
37#4529C2×C311yescharacteristic, complemented
38#4529C2×C311yescharacteristic, complemented
39C23×C3×D3D411yescharacteristic
40C22×C32×C4—21no—
41C22×C32×C4—21no—
42C22×C32×C4C2311yescharacteristic
43C22×C3×Dic3—21no—
44C22×C3×Dic3—21no—
45C22×C3×Dic3D411yescharacteristic
46C2×C3×C4×D3—21no—
47C2×C3×C4×D3—21no—
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3—21no—
50C2×C3×C4×D3C2×C412yes—
51C2×C3×C4×D3D412yes—
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×C412yes—
59C3×(C2×D3)⋊C4—24no—
60C3×(C2×D3)⋊C4—42no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3C2×C412yes—
65C3×C4×Dic3—24no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8—21no—
69#1825—31noSylow
70C23×C3×C4—31no—
71C22×C4×D3C22×C311yescharacteristic
72C22×C3×C8—21no—
73C22×C3×C8—31no—
74C22×C3×C8C2×D311yescharacteristic
75C22×C3×C8C22×C311yescharacteristic
76C3×#117—34no—
77C3×#117—62no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—31no—
81C2×C3×C4⋊C4—31no—
82C2×C3×C4⋊C4C2×D311yescharacteristic
83C2×C3×C4⋊C4C2×D311yescharacteristic
84C2×C3×C4⋊C4C22×C311yescharacteristic
85C2×C3×C4⋊C4C22×C311yescharacteristic
86C2×C3×C22⋊C4—61no—
87C2×C3×C22⋊C4—61no—
88C2×C3×C42—61no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3C22×C311yescharacteristic
92C2×C4⋊Dic3C22×C311yescharacteristic
93C2×C4×Dic3—21no—
94C22×C3⋊C8C22×C311yescharacteristic
95#892—22no—
96#892C3×C414yescomplemented
97C3×C22⋊C8—62no—
98C3×#95—61no—
99C3⋊#95—21no—
100(C2×D3)⋊C8—22no—
101C23×C32C2×D411yescharacteristic
102C22×C3×D3—21no—
103C22×C3×D3—21no—
104C22×C3×D3—21no—
105C22×C3×D3—21no—
106C22×C3×D3—21no—
107C22×C3×D3—21no—
108C22×C3×D3C22⋊C412yes—
109C2×C32×C4—22no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4—41no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4#4111yescharacteristic
118C2×C32×C4#4111yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—41no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—41no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3C22⋊C412yes—
130C3×C4×D3—22no—
131C3×C4×D3—22no—
132C3×C4×D3—22no—
133C3×C4×D3—22no—
134C32×C8—41no—
135C3×C3⋊C8—41no—
136C2×#117—31no—
137C2×C22⋊C8—31no—
138#380—31no—
139#340—31no—
140#340—31no—
141#340—31no—
142#340C3×D311yescharacteristic, complemented
143C24×C3—31no—
144C23×D3C3×D411yescharacteristic
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—31no—
151C22×C3×C4—32no—
152C22×C3×C4—32no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—61no—
156C22×C3×C4—61no—
157C22×C3×C4—61no—
158C22×C3×C4—61no—
159C22×C3×C4—61no—
160C22×C3×C4—61no—
161C22×C3×C4C22×D311yescharacteristic
162C22×C3×C4C23×C311yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3C3×D411yescharacteristic
166C2×C4×D3—21no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3C2×C3×C412yes—
171C2×C4×D3C3×D412yes—
172C2×C3×C8—22no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—41no—
180C2×C3×C8—41no—
181C2×C3×C8—61no—
182C2×C3×C8—61no—
183C2×C3×C8—61no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—32no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—41no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4C4×D312yes—
199C3×C4⋊C4C2×C3×C412yes—
200C3×C22⋊C4—64no—
201C3×C22⋊C4—122no—
202C3×C42—64no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—42no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3C2×C3×C412yes—
209C4×Dic3—24no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C22×C32C2×C22⋊C411yescharacteristic
214C22×C32#12311yescharacteristic
215C22×C32#12311yescharacteristic
216C22×C32#13511yescharacteristic
217C22×C32#13511yescharacteristic
218C22×C32AGL(1,ℤ/8)11yescharacteristic
219C22×C32#13711yescharacteristic
220C2×C3×D3—22no—
221C2×C3×D3—22no—
222C2×C3×D3—22no—
223C2×C3×D3—41no—
224C2×C3×D3—41no—
225C2×C3×D3—41no—
226C2×C3×D3—41no—
227C32×C4—42no—
228C32×C4—81no—
229C32×C4#11711yescharacteristic
230C32×C4#11711yescharacteristic
231C32×C4#12311yescharacteristic
232C32×C4#12311yescharacteristic
233C3×Dic3—22no—
234C3×Dic3—42no—
235C3×Dic3—81no—
236C23×C4—31no—
237C22×C8—31no—
238C22×C8C2×C3×D311yescharacteristic
239#117—34no—
240#117—62no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4C2×C3×D311yescharacteristic
244C2×C4⋊C4C2×C3×D311yescharacteristic
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C42—61no—
248C22⋊C8—62no—
249#95—61no—
250C23×C3—21no—
251C23×C3—32no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—61no—
256C23×C3—61no—
257C23×C3—61no—
258C23×C3D3×D411yescharacteristic
259C23×C3C2×C3×D411yescharacteristic
260C22×D3—21no—
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3—21no—
266C22×D3C3×C22⋊C412yes—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—32no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—81no—
296C2×C3×C4—81no—
297C2×C3×C4—81no—
298C2×C3×C4—81no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4—121no—
303C2×C3×C4—121no—
304C2×C3×C4C2×C4×D311yescharacteristic
305C2×C3×C4#23511yescharacteristic
306C2×C3×C4#23511yescharacteristic
307C2×C3×C4D3×D411yescharacteristic
308C2×C3×C4#23711yescharacteristic
309C2×C3×C4#23911yescharacteristic
310C2×C3×C4C22×C3×C411yescharacteristic
311C2×C3×C4C2×C3×D411yescharacteristic
312C2×C3×C4C3×#4111yescharacteristic
313C2×C3×C4C3×#4111yescharacteristic
314C2×C3×C4C3×#4111yescharacteristic
315C2×C3×C4C3×#4111yescharacteristic
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3C3×C22⋊C412yes—
325C4×D3—22no—
326C4×D3—22no—
327C4×D3—22no—
328C4×D3—22no—
329C3×C8—42no—
330C3×C8—41no—
331C3×C8—41no—
332C3×C8—121no—
333C3⋊C8—41no—
334C2×C32#38111yescharacteristic
335C2×C32#41611yescharacteristic
336C2×C32#41711yescharacteristic
337C2×C32#48211yescharacteristic
338C2×C32#48211yescharacteristic
339C2×C32#48511yescharacteristic
340C2×C32#48511yescharacteristic
341C3×D3—42no—
342C24—31no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—31no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4C22×C3×D311yescharacteristic
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—61no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—32no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4C3×C4×D312yes—
369C22⋊C4—64no—
370C22⋊C4—122no—
371C42—64no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—121no—
383C22×C3—121no—
384C22×C3—121no—
385C22×C3—121no—
386C22×C3D3×C22⋊C411yescharacteristic
387C22×C3#88311yescharacteristic
388C22×C3#89411yescharacteristic
389C22×C3#91211yescharacteristic
390C22×C3#91411yescharacteristic
391C22×C3#91611yescharacteristic
392C22×C3#91811yescharacteristic
393C22×C3C2×C3×C22⋊C411yescharacteristic
394C22×C3C3×#12311yescharacteristic
395C22×C3C3×#12311yescharacteristic
396C22×C3C3×#13511yescharacteristic
397C22×C3C3×#13511yescharacteristic
398C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
399C22×C3C3×#13711yescharacteristic
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—41no—
406C2×D3—41no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—62no—
414C3×C4—82no—
415C3×C4—82no—
416C3×C4—81no—
417C3×C4—81no—
418C3×C4—122no—
419C3×C4—241no—
420C3×C4#87211yescharacteristic
421C3×C4#88311yescharacteristic
422C3×C4#89211yescharacteristic
423C3×C4#89411yescharacteristic
424C3×C4C3×#11711yescharacteristic
425C3×C4C3×#11711yescharacteristic
426C3×C4C3×#12311yescharacteristic
427C3×C4C3×#12311yescharacteristic
428Dic3—22no—
429Dic3—42no—
430Dic3—81no—
431C32#182511yescharacteristic, complemented, Sylow
432C23—32no—
433C23—61no—
434C23—61no—
435C23—61no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23C3×D3×D411yescharacteristic
440C2×C4—22no—
441C2×C4—32no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4C2×C3×C4×D311yescharacteristic
460C2×C4C3×#23511yescharacteristic
461C2×C4C3×#23511yescharacteristic
462C2×C4C3×D3×D411yescharacteristic
463C2×C4C3×#23711yescharacteristic
464C2×C4C3×#23911yescharacteristic
465C8—41no—
466C8—121no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—122no—
475C2×C3#466411yescharacteristic
476C2×C3#477211yescharacteristic
477C2×C3#478411yescharacteristic
478C2×C3#480311yescharacteristic
479C2×C3#481511yescharacteristic
480C2×C3#486611yescharacteristic
481C2×C3#488811yescharacteristic
482C2×C3C3×#38111yescharacteristic
483C2×C3C3×#41611yescharacteristic
484C2×C3C3×#41711yescharacteristic
485C2×C3C3×#48211yescharacteristic
486C2×C3C3×#48211yescharacteristic
487C2×C3C3×#48511yescharacteristic
488C2×C3C3×#48511yescharacteristic
489D3—42no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22C3×D3×C22⋊C411yescharacteristic
498C22C3×#88311yescharacteristic
499C22C3×#89411yescharacteristic
500C22C3×#91211yescharacteristic
501C22C3×#91411yescharacteristic
502C22C3×#91611yescharacteristic
503C22C3×#91811yescharacteristic
504C4—42no—
505C4—62no—
506C4—81no—
507C4—122no—
508C4—241no—
509C4C3×#87211yescharacteristic
510C4C3×#88311yescharacteristic
511C4C3×#89211yescharacteristic
512C4C3×#89411yescharacteristic
513C3—21no—
514C3#6887311yescharacteristic, direct product factor
515C3C3×#182511yescharacteristic, complemented
516C2—122no—
517C2C3×#466411yescharacteristic
518C2C3×#477211yescharacteristic
519C2C3×#478411yescharacteristic
520C2C3×#480311yescharacteristic
521C2C3×#481511yescharacteristic
522C2C3×#486611yescharacteristic
523C2C3×#488811yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#340C2
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
cosocleC22×C3×C4C23×C3
hypercenterC3×#340D3
hypocenterC3C3×#1825
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#340

Complements

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

NH ≅ G/Ntype
4. C32×#340C2semidirect
5. C3×#4551C2semidirect
6. C3×#4529C2semidirect
7. C3×#4529C2semidirect
9. #68873C3direct (N × H)
21. C3×#892C4semidirect
34. C3×#340D3semidirect
35. C3×#340C2×C3semidirect
36. #4551C2×C3semidirect
37. #4529C2×C3semidirect
38. #4529C2×C3semidirect
96. #892C3×C4semidirect
142. #340C3×D3semidirect
431. C32#1825semidirect
514. C3#68873direct (N × H)
515. C3C3×#1825semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8922
C2×C3×(C2×D3)⋊C82
C3×#49872
C32×#3402
C3×#45512
C3×#45292
C3×#18253
#688733