Finite Group Databasebeta

← C3×#72089 #11788473

Subgroups — C3×#72089 #11788473

Order 1152.

841 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C3×#5798C211yesmaximal, characteristic, complemented
3C3×#5240C211yesmaximal, characteristic, complemented
4C3×#5236C211yesmaximal, characteristic, complemented
5C3×#5215C211yesmaximal, characteristic, complemented
6C3×#4557C211yesmaximal, characteristic, complemented
7C3×#4556C211yesmaximal, characteristic, complemented
8C3×#1904—31nomaximal
9#72089C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C32×AGL(1,ℤ/8)—22no—
13C32×AGL(1,ℤ/8)—22no—
14C2×C32×SD16C412yescomplemented
15C2×C32×D8C412yescomplemented
16C2×C32×M16C2211yescharacteristic
17C2×C3×#217—22no—
18C3×D4×Dic3—22no—
19C3×D4×Dic3—22no—
20C3×#926C2211yescharacteristic
21C3×#926C2211yescharacteristic
22C3×#926C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C3×#837—22no—
25C3×#834—22no—
26C3×#832—22no—
27C2×C3×AGL(1,ℤ/8)—21no—
28C2×C3×AGL(1,ℤ/8)D311yescharacteristic
29C2×C3×AGL(1,ℤ/8)C2×C311yescharacteristic
30C3×#518—31no—
31#5798C2×C311yescharacteristic, complemented
32C3×#421—31no—
33C3×#418—31no—
34C3×#410—31no—
35#5240C2×C311yescharacteristic, complemented
36#5236C2×C311yescharacteristic, complemented
37#5215C2×C311yescharacteristic, complemented
38C3×#344—31no—
39C3×H(3,ℤ/4)—31no—
40#4557C2×C311yescharacteristic, complemented
41#4556C2×C311yescharacteristic, complemented
42C24×C32—21no—
43C32×#41—21no—
44C32×#41—21no—
45C32×#41—21no—
46C32×#41—21no—
47C2×C32×Q8C2×C411yescharacteristic, complemented
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4—21no—
51C2×C32×D4—21no—
52C2×C32×D4C2×C412yes—
53C2×C32×D4C2×C411yescharacteristic, complemented
54C2×C32×D4D412yes—
55C2×C32×D4D412yes—
56C22×C32×C4—21no—
57C22×C32×C4C2311yescharacteristic
58C22×C3×Dic3—24no—
59C32×SD16—24no—
60C32×D8—24no—
61C32×M16—21no—
62C32×M16—21no—
63C2×C32×C8C2×C412yescomplemented
64C3×#217—24no—
65C3×#217—21no—
66C3×#217—21no—
67C3×#217D412yes—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C3×C4×Dic3—21no—
72C3×C4×Dic3—21no—
73C3×C4×Dic3D412yes—
74C3×#208—21no—
75C3×#208—21no—
76C2×C3×C3⋊C8—21no—
77#1904—31noSylow
78C2×C3×#41—21no—
79C2×C3×#41C2×D311yescharacteristic
80C2×C3×#41C22×C311yescharacteristic
81C22×C3×D4—21no—
82C22×C3×D4C2×D311yescharacteristic
83C22×C3×D4C22×C311yescharacteristic
84C3×AGL(1,ℤ/8)—24no—
85C3×AGL(1,ℤ/8)—24no—
86C3×AGL(1,ℤ/8)—22no—
87C3×AGL(1,ℤ/8)—22no—
88C3×AGL(1,ℤ/8)—22no—
89C3×AGL(1,ℤ/8)—22no—
90C2×C3×SD16—22no—
91C2×C3×SD16Dic312yescomplemented
92C2×C3×SD16C3×C412yescomplemented
93C2×C3×D8—22no—
94C2×C3×D8Dic312yescomplemented
95C2×C3×D8C3×C412yescomplemented
96C2×C3×M16—21no—
97C2×C3×M16—31no—
98C2×C3×M16C2×D311yescharacteristic
99C2×C3×M16C22×C311yescharacteristic
100C3×C4×D4—62no—
101C3×C4×D4—62no—
102C3×#117—31no—
103C3×#117—31no—
104C3×#117—31no—
105C2×C3×C22⋊C4—62no—
106C2×#217—22no—
107D4×Dic3—22no—
108D4×Dic3—22no—
109#926C22×C311yescharacteristic
110#926C22×C311yescharacteristic
111#926C22×C311yescharacteristic
112C2×#208C22×C311yescharacteristic
113C3×#104—62no—
114C3×#102—62no—
115C3×#99—62no—
116#837—22no—
117#834—22no—
118#832—22no—
119C23×C32—22no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—21no—
123C23×C32—82no—
124C23×C32C2×D411yescharacteristic
125C32×Q8—21no—
126C32×Q8—21no—
127C32×D4—21no—
128C32×D4—22no—
129C32×D4—21no—
130C32×D4—22no—
131C32×D4—22no—
132C32×D4—41no—
133C32×D4—41no—
134C32×D4C22⋊C414yes—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4C22×C411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C2×C32×C4#4111yescharacteristic
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—84no—
148C32×C8—22no—
149C3×C3⋊C8—41no—
150C2×AGL(1,ℤ/8)C3×D311yescharacteristic
151#518—31no—
152#421—31no—
153#418—31no—
154#410—31no—
155#344—31no—
156H(3,ℤ/4)—31no—
157C24×C3—21no—
158C24×C3—21no—
159C24×C3—41no—
160C3×#41—22no—
161C3×#41—22no—
162C3×#41—21no—
163C3×#41—21no—
164C3×#41—21no—
165C3×#41—21no—
166C3×#41—21no—
167C3×#41—21no—
168C3×#41—21no—
169C3×#41—21no—
170C3×#41—41no—
171C3×#41—41no—
172C2×C3×Q8—21no—
173C2×C3×Q8C2×Dic311yescharacteristic, complemented
174C2×C3×Q8C2×C3×C411yescharacteristic, complemented
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—41no—
188C2×C3×D4—41no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4C2×Dic312yes—
192C2×C3×D4C2×Dic311yescharacteristic, complemented
193C2×C3×D4#6712yes—
194C2×C3×D4#6712yes—
195C2×C3×D4C2×C3×C412yes—
196C2×C3×D4C2×C3×C411yescharacteristic, complemented
197C2×C3×D4C3×D412yes—
198C2×C3×D4C3×D412yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—41no—
203C22×C3×C4—64no—
204C22×C3×C4C22×D311yescharacteristic
205C22×C3×C4C23×C311yescharacteristic
206C22×Dic3—24no—
207C3×SD16—28no—
208C3×SD16—24no—
209C3×SD16—24no—
210C3×D8—28no—
211C3×D8—24no—
212C3×D8—24no—
213C3×M16—22no—
214C3×M16—22no—
215C3×M16—21no—
216C3×M16—21no—
217C3×M16—21no—
218C3×M16—21no—
219C3×M16—61no—
220C3×M16—61no—
221C2×C3×C8—22no—
222C2×C3×C8—61no—
223C2×C3×C8C2×Dic312yescomplemented
224C2×C3×C8C2×C3×C412yescomplemented
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—61no—
228C3×C22⋊C4—32no—
229C3×C22⋊C4—64no—
230C3×C22⋊C4—61no—
231C3×C22⋊C4—61no—
232C3×C42—32no—
233C3×C42—61no—
234C3×C42—61no—
235#217—24no—
236#217—21no—
237#217—21no—
238#217C3×D412yes—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3C3×D412yes—
245#208—21no—
246#208—21no—
247C2×C3⋊C8—21no—
248C22×C32—24no—
249C22×C32—41no—
250C22×C32—41no—
251C22×C32—44no—
252C22×C32—82no—
253C22×C32—82no—
254C22×C32—81no—
255C22×C32C4×D411yescharacteristic
256C22×C32#12011yescharacteristic
257C22×C32#12111yescharacteristic
258C32×C4—41no—
259C32×C4C2×C22⋊C411yescharacteristic
260C32×C4C4×D411yescharacteristic
261C32×C4#12111yescharacteristic
262C32×C4#12311yescharacteristic
263C3×Dic3—42no—
264C3×Dic3—42no—
265C3×Dic3—81no—
266C2×#41C2×C3×D311yescharacteristic
267C22×D4C2×C3×D311yescharacteristic
268AGL(1,ℤ/8)—22no—
269AGL(1,ℤ/8)—22no—
270C2×SD16C3×Dic312yescomplemented
271C2×D8C3×Dic312yescomplemented
272C2×M16—31no—
273C2×M16C2×C3×D311yescharacteristic
274C4×D4—62no—
275C4×D4—62no—
276#117—31no—
277#117—31no—
278#117—31no—
279C2×C22⋊C4—62no—
280#104—62no—
281#102—62no—
282#99—62no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—82no—
297C23×C3—82no—
298C23×C3—84no—
299C23×C3D3×D411yescharacteristic
300C23×C3C2×C3×D411yescharacteristic
301C3×Q8—22no—
302C3×Q8—21no—
303C3×Q8—21no—
304C3×Q8—21no—
305C3×Q8—21no—
306C3×Q8—41no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—21no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—21no—
313C3×D4—21no—
314C3×D4—22no—
315C3×D4—21no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—24no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—41no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—41no—
329C3×D4#21714yes—
330C3×D4C3×C22⋊C414yes—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—121no—
349C2×C3×C4—244no—
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4#23711yescharacteristic
352C2×C3×C4#23711yescharacteristic
353C2×C3×C4C22×Dic311yescharacteristic
354C2×C3×C4C2×#6711yescharacteristic
355C2×C3×C4C2×#6711yescharacteristic
356C2×C3×C4C22×C3×C411yescharacteristic
357C2×C3×C4C2×C3×D411yescharacteristic
358C2×C3×C4C2×C3×D411yescharacteristic
359C2×C3×C4C2×C3×D411yescharacteristic
360C2×C3×C4C3×#4111yescharacteristic
361C2×C3×C4C3×#4111yescharacteristic
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—41no—
365C2×Dic3—84no—
366C3×C8—24no—
367C3×C8—22no—
368C3×C8—22no—
369C3×C8—121no—
370C3⋊C8—41no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—44no—
375C2×C32—81no—
376C2×C32#38611yescharacteristic
377C24—21no—
378#41—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382C2×Q8C2×C3×Dic311yescharacteristic, complemented
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4C2×C3×Dic312yes—
388C2×D4C2×C3×Dic311yescharacteristic, complemented
389C2×D4C3×#6712yes—
390C2×D4C3×#6712yes—
391C22×C4—21no—
392C22×C4—64no—
393C22×C4C22×C3×D311yescharacteristic
394SD16—24no—
395D8—24no—
396M16—21no—
397M16—21no—
398M16—61no—
399M16—61no—
400C2×C8—61no—
401C2×C8C2×C3×Dic312yescomplemented
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C22⋊C4—32no—
406C22⋊C4—64no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C42—32no—
410C42—61no—
411C42—61no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—44no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—84no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—84no—
434C22×C3—82no—
435C22×C3—84no—
436C22×C3D4×Dic311yescharacteristic
437C22×C3#93711yescharacteristic
438C22×C3#93911yescharacteristic
439C22×C3C3×C4×D411yescharacteristic
440C22×C3C3×#12011yescharacteristic
441C22×C3C3×#12111yescharacteristic
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—42no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4—241no—
452C3×C4D4×Dic311yescharacteristic
453C3×C4#93511yescharacteristic
454C3×C4#93911yescharacteristic
455C3×C4C2×#21711yescharacteristic
456C3×C4C2×C3×C22⋊C411yescharacteristic
457C3×C4C3×C4×D411yescharacteristic
458C3×C4C3×#12111yescharacteristic
459C3×C4C3×#12311yescharacteristic
460Dic3—42no—
461Dic3—42no—
462Dic3—81no—
463C32#190411yescharacteristic, complemented, Sylow
464C23—22no—
465C23—22no—
466C23—22no—
467C23—21no—
468C23—82no—
469C23C3×D3×D411yescharacteristic
470Q8—21no—
471Q8—21no—
472D4—22no—
473D4—21no—
474D4—22no—
475D4—22no—
476D4—21no—
477D4—41no—
478D4—41no—
479D4C3×#21714yes—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—121no—
486C2×C4—244no—
487C2×C4C3×D3×D411yescharacteristic
488C2×C4C3×#23711yescharacteristic
489C2×C4C3×#23711yescharacteristic
490C2×C4C22×C3×Dic311yescharacteristic
491C2×C4C2×C3×#6711yescharacteristic
492C2×C4C2×C3×#6711yescharacteristic
493C8—22no—
494C8—121no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—44no—
506C2×C3—44no—
507C2×C3—84no—
508C2×C3—82no—
509C2×C3—81no—
510C2×C3—81no—
511C2×C3#522011yescharacteristic
512C2×C3C3×#38611yescharacteristic
513C22—24no—
514C22—41no—
515C22—44no—
516C22—41no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22C3×D4×Dic311yescharacteristic
521C22C3×#93711yescharacteristic
522C22C3×#93911yescharacteristic
523C4—41no—
524C4—122no—
525C4—122no—
526C4—241no—
527C4C3×D4×Dic311yescharacteristic
528C4C3×#93511yescharacteristic
529C4C3×#93911yescharacteristic
530C4C2×C3×#21711yescharacteristic
531C3—21no—
532C3#7208911yescharacteristic, direct product factor
533C3C3×#190411yescharacteristic, complemented
534C2—21no—
535C2—21no—
536C2—21no—
537C2—44no—
538C2—81no—
539C2C3×#522011yescharacteristic
subgroupG/H
centerC2×C3#5220
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC2×C32#386
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×AGL(1,ℤ/8)D3
hypocenterC3C3×#1904
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. C3×#5798C2semidirect
3. C3×#5240C2semidirect
4. C3×#5236C2semidirect
5. C3×#5215C2semidirect
6. C3×#4557C2semidirect
7. C3×#4556C2semidirect
9. #72089C3direct (N × H)
14. C2×C32×SD16C4semidirect
15. C2×C32×D8C4semidirect
31. #5798C2×C3semidirect
35. #5240C2×C3semidirect
36. #5236C2×C3semidirect
37. #5215C2×C3semidirect
40. #4557C2×C3semidirect
41. #4556C2×C3semidirect
47. C2×C32×Q8C2×C4semidirect
53. C2×C32×D4C2×C4semidirect
63. C2×C32×C8C2×C4semidirect
91. C2×C3×SD16Dic3semidirect
92. C2×C3×SD16C3×C4semidirect
94. C2×C3×D8Dic3semidirect
95. C2×C3×D8C3×C4semidirect
173. C2×C3×Q8C2×Dic3semidirect
174. C2×C3×Q8C2×C3×C4semidirect
192. C2×C3×D4C2×Dic3semidirect
196. C2×C3×D4C2×C3×C4semidirect
223. C2×C3×C8C2×Dic3semidirect
224. C2×C3×C8C2×C3×C4semidirect
270. C2×SD16C3×Dic3semidirect
271. C2×D8C3×Dic3semidirect
382. C2×Q8C2×C3×Dic3semidirect
388. C2×D4C2×C3×Dic3semidirect
401. C2×C8C2×C3×Dic3semidirect
463. C32#1904semidirect
532. C3#72089direct (N × H)
533. C3C3×#1904semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
C3×#57982
C3×#52402
C3×#52362
C3×#52152
C3×#45572
C3×#45562
C3×#19043
#720893