Finite Group Databasebeta

← C3×#83935 #11866252

Subgroups — C3×#83935 #11866252

Order 1152.

1137 conjugacy classes of subgroups.

The subgroup lattice has 529 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×#5825C211yesmaximal, characteristic, complemented
3C3×#5798C211yesmaximal, characteristic, complemented
4C22×C3×#214C211yesmaximal, characteristic, complemented
5C3×#5233C211yesmaximal, characteristic, complemented
6C3×#5162C214yesmaximal, complemented
7C3×#5153C214yesmaximal, complemented
8C3×#5120C211yesmaximal, characteristic, complemented
9C3×#5119C211yesmaximal, characteristic, complemented
10C3×#3042—31nomaximal
11#83935C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C22×C32×D4C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C32×AGL(1,ℤ/8)—24no—
16C2×C32×SD16C2212yes—
17C2×C32×D8C2212yes—
18C2×C32×M16C2211yescharacteristic
19C2×C3×#217—22no—
20C3×#944C2212yescomplemented
21C3×#936C2212yescomplemented
22C3×#935—22no—
23C3×D4×Dic3—22no—
24C3×D4×Dic3C2214yes—
25C2×C3×#214—24no—
26C2×C3×#214C2214yes—
27C3×#926C2211yescharacteristic
28C3×#926C2211yescharacteristic
29C3×#924—22no—
30C22×C3×C3⋊C8C2211yescharacteristic
31C3×#835C2212yescomplemented
32C3×#832C2212yescomplemented
33C3×#816C2214yescomplemented
34C3×Dic3⋊C8C2214yescomplemented
35C2×C3×AGL(1,ℤ/8)—21no—
36C2×C3×AGL(1,ℤ/8)D311yescharacteristic
37C2×C3×AGL(1,ℤ/8)C2×C311yescharacteristic
38C22×C3×SD16—31no—
39C3×#536—31no—
40C3×#518—31no—
41#5825C2×C311yescharacteristic, complemented
42#5798C2×C311yescharacteristic, complemented
43C22×#214C2×C311yescharacteristic, complemented
44C3×#463—34no—
45C3×#461—34no—
46C3×#424—31no—
47C3×#419—31no—
48C3×#416—31no—
49#5233C2×C311yescharacteristic, complemented
50#5162C2×C314yescomplemented
51#5153C2×C314yescomplemented
52#5120C2×C311yescharacteristic, complemented
53#5119C2×C311yescharacteristic, complemented
54C24×C32—21no—
55C32×#41—22no—
56C32×#41—22no—
57C2×C32×Q8C2311yescharacteristic
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4—22no—
61C2×C32×D4D414yes—
62C2×C32×D4C2311yescharacteristic
63C2×C32×D4C2312yes—
64C22×C32×C4—21no—
65C22×C32×C4C2311yescharacteristic
66C22×C3×Dic3—21no—
67C22×C3×Dic3—24no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2312yes—
72C32×SD16—24no—
73C32×D8—24no—
74C32×M16—22no—
75C2×C32×C8C2312yes—
76C3×#217—21no—
77C3×#217—24no—
78C3×#217—24no—
79C3×#217D412yes—
80C3×#214—28no—
81C3×C4⋊Dic3D412yes—
82C3×C4⋊Dic3C2312yes—
83C3×Dic3⋊C4—24no—
84C3×C4×Dic3—21no—
85C3×C4×Dic3C2312yes—
86C2×C3×C3⋊C8—22no—
87C2×C3×C3⋊C8C2312yes—
88#3042—31noSylow
89C2×C3×#41—21no—
90C2×C3×#41C2×D311yescharacteristic
91C2×C3×#41C22×C311yescharacteristic
92C22×C3×Q8—31no—
93C22×C3×D4—21no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C22×Dic6C22×C311yescharacteristic
97C3×AGL(1,ℤ/8)—28no—
98C3×AGL(1,ℤ/8)—24no—
99C3×AGL(1,ℤ/8)—24no—
100C2×C3×SD16—22no—
101C2×C3×SD16—34no—
102C2×C3×SD16—64no—
103C2×C3×SD16C2×D312yes—
104C2×C3×SD16C22×C312yes—
105C2×C3×D8—22no—
106C2×C3×D8C2×D312yes—
107C2×C3×D8C22×C312yes—
108C2×C3×M16—21no—
109C2×C3×M16C2×D311yescharacteristic
110C2×C3×M16C22×C311yescharacteristic
111C22×C3×C8—31no—
112C3×C4⋊Q8—32no—
113C3×#124—32no—
114C3×#123—62no—
115C3×C22⋊Q8—62no—
116C3×C4×D4—34no—
117C3×C4×D4—62no—
118C3×#117—31no—
119C3×#117—31no—
120C2×C3×C22⋊C4—62no—
121C2×#217—22no—
122#944C22×C312yescomplemented
123#936C22×C312yescomplemented
124#935—22no—
125D4×Dic3—22no—
126D4×Dic3C22×C314yes—
127C2×#214—24no—
128C2×#214C22×C314yes—
129#926C22×C311yescharacteristic
130#926C22×C311yescharacteristic
131#924—22no—
132C22×C3⋊C8C22×C311yescharacteristic
133C3×C4⋊C8—34no—
134C3×#103—32no—
135C3×#103—34no—
136C3×#102—32no—
137#835C22×C312yescomplemented
138#832C22×C312yescomplemented
139#816C22×C314yescomplemented
140Dic3⋊C8C22×C314yescomplemented
141C23×C32—22no—
142C23×C32—24no—
143C23×C32—21no—
144C23×C32—82no—
145C23×C32C2×D411yescharacteristic
146C32×Q8—21no—
147C32×Q8—21no—
148C32×D4—21no—
149C32×D4—24no—
150C32×D4—22no—
151C32×D4—21no—
152C32×D4—42no—
153C32×D4C2×D414yes—
154C2×C32×C4—21no—
155C2×C32×C4—42no—
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D412yes—
158C2×C32×C4#4112yes—
159C2×C32×C4C2411yescharacteristic
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—42no—
163C2×C3×Dic3—84no—
164C2×C3×Dic3C2×D414yes—
165C3×Dic6—22no—
166C3×Dic6—22no—
167C3×Dic6—42no—
168C32×C8—22no—
169C3×C3⋊C8—22no—
170C2×AGL(1,ℤ/8)C3×D311yescharacteristic
171C22×SD16—31no—
172#536—31no—
173#518—31no—
174#463—34no—
175#461—34no—
176#424—31no—
177#419—31no—
178#416—31no—
179C24×C3—21no—
180C24×C3—21no—
181C24×C3—41no—
182C3×#41—24no—
183C3×#41—22no—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—42no—
188C2×C3×Q8—21no—
189C2×C3×Q8—32no—
190C2×C3×Q8—61no—
191C2×C3×Q8—62no—
192C2×C3×Q8—62no—
193C2×C3×Q8C22×D311yescharacteristic
194C2×C3×Q8C23×C311yescharacteristic
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—22no—
201C2×C3×D4—24no—
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4—41no—
205C2×C3×D4—42no—
206C2×C3×D4—41no—
207C2×C3×D4#6714yes—
208C2×C3×D4C3×D414yes—
209C2×C3×D4C22×D311yescharacteristic
210C2×C3×D4C22×D312yes—
211C2×C3×D4C23×C311yescharacteristic
212C2×C3×D4C23×C312yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—41no—
217C22×C3×C4—64no—
218C22×C3×C4—61no—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C23×C311yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—24no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—21no—
226C2×Dic6C23×C312yes—
227C3×SD16—28no—
228C3×SD16—24no—
229C3×SD16—24no—
230C3×SD16—68no—
231C3×D8—28no—
232C3×D8—24no—
233C3×D8—24no—
234C3×M16—24no—
235C3×M16—22no—
236C3×M16—22no—
237C2×C3×C8—22no—
238C2×C3×C8—32no—
239C2×C3×C8—62no—
240C2×C3×C8C22×D312yes—
241C2×C3×C8C23×C312yes—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—32no—
244C3×C4⋊C4—64no—
245C3×C22⋊C4—32no—
246C3×C22⋊C4—64no—
247C3×C22⋊C4—64no—
248C3×C22⋊C4—61no—
249C3×C42—32no—
250C3×C42—61no—
251#217—21no—
252#217—24no—
253#217—24no—
254#217C3×D412yes—
255#214—28no—
256C4⋊Dic3C3×D412yes—
257C4⋊Dic3C23×C312yes—
258Dic3⋊C4—24no—
259C4×Dic3—21no—
260C4×Dic3C23×C312yes—
261C2×C3⋊C8—22no—
262C2×C3⋊C8C23×C312yes—
263C22×C32—22no—
264C22×C32—24no—
265C22×C32—41no—
266C22×C32—44no—
267C22×C32—42no—
268C22×C32—84no—
269C22×C32#12112yes—
270C22×C32C22×D411yescharacteristic
271C32×C4—41no—
272C32×C4#12112yes—
273C32×C4C22×D411yescharacteristic
274C32×C4C2×#4111yescharacteristic
275C3×Dic3—24no—
276C3×Dic3—42no—
277C3×Dic3—42no—
278C2×#41C2×C3×D311yescharacteristic
279C22×Q8—31no—
280C22×D4C2×C3×D311yescharacteristic
281AGL(1,ℤ/8)—24no—
282C2×SD16—34no—
283C2×SD16—64no—
284C2×SD16C2×C3×D312yes—
285C2×D8C2×C3×D312yes—
286C2×M16C2×C3×D311yescharacteristic
287C22×C8—31no—
288C4⋊Q8—32no—
289#124—32no—
290#123—62no—
291C22⋊Q8—62no—
292C4×D4—34no—
293C4×D4—62no—
294#117—31no—
295#117—31no—
296C2×C22⋊C4—62no—
297C4⋊C8—34no—
298#103—32no—
299#103—34no—
300#102—32no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—44no—
311C23×C3—82no—
312C23×C3—82no—
313C23×C3—84no—
314C23×C3D3×D411yescharacteristic
315C23×C3C2×C3×D411yescharacteristic
316C3×Q8—22no—
317C3×Q8—21no—
318C3×Q8—21no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—41no—
322C3×Q8—62no—
323C3×Q8—62no—
324C3×Q8—122no—
325C3×D4—22no—
326C3×D4—24no—
327C3×D4—24no—
328C3×D4—22no—
329C3×D4—21no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—22no—
333C3×D4—21no—
334C3×D4—24no—
335C3×D4—44no—
336C3×D4—42no—
337C3×D4—41no—
338C3×D4—44no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4C2×#6714yes—
342C3×D4C2×C3×D414yes—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—34no—
350C2×C3×C4—41no—
351C2×C3×C4—44no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—122no—
357C2×C3×C4—244no—
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4#23712yes—
360C2×C3×C4C2×#6712yes—
361C2×C3×C4C2×C3×D411yescharacteristic
362C2×C3×C4C2×C3×D412yes—
363C2×C3×C4C3×#4112yes—
364C2×C3×C4C23×D311yescharacteristic
365C2×C3×C4C24×C311yescharacteristic
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—42no—
369C2×Dic3—84no—
370C2×Dic3C2×C3×D414yes—
371Dic6—22no—
372Dic6—22no—
373Dic6—42no—
374C3×C8—24no—
375C3×C8—22no—
376C3×C8—22no—
377C3×C8—62no—
378C3⋊C8—22no—
379C2×C32—22no—
380C2×C32—44no—
381C2×C32—81no—
382C2×C32C2×#12111yescharacteristic
383C2×C32#57712yes—
384C24—21no—
385#41—22no—
386#41—22no—
387C2×Q8—32no—
388C2×Q8—61no—
389C2×Q8—62no—
390C2×Q8—62no—
391C2×Q8C22×C3×D311yescharacteristic
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4C3×#6714yes—
396C2×D4C22×C3×D311yescharacteristic
397C2×D4C22×C3×D312yes—
398C22×C4—21no—
399C22×C4—64no—
400C22×C4—61no—
401C22×C4C22×C3×D311yescharacteristic
402SD16—24no—
403SD16—68no—
404D8—24no—
405M16—22no—
406C2×C8—32no—
407C2×C8—62no—
408C2×C8C22×C3×D312yes—
409C4⋊C4—32no—
410C4⋊C4—32no—
411C4⋊C4—64no—
412C22⋊C4—32no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22⋊C4—61no—
416C42—32no—
417C42—61no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—22no—
423C22×C3—21no—
424C22×C3—42no—
425C22×C3—44no—
426C22×C3—42no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—84no—
434C22×C3—82no—
435C22×C3—84no—
436C22×C3—84no—
437C22×C3—88no—
438C22×C3#93912yes—
439C22×C3C3×#12112yes—
440C22×C3C2×D3×D411yescharacteristic
441C22×C3C22×C3×D411yescharacteristic
442C3×C4—22no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—64no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4#93912yes—
452C3×C4C3×#12112yes—
453C3×C4C2×#23711yescharacteristic
454C3×C4C22×#6711yescharacteristic
455C3×C4C22×C3×D411yescharacteristic
456C3×C4C2×C3×#4111yescharacteristic
457Dic3—24no—
458Dic3—42no—
459Dic3—42no—
460C32#304211yescharacteristic, complemented, Sylow
461C23—22no—
462C23—24no—
463C23—21no—
464C23—82no—
465C23C3×D3×D411yescharacteristic
466Q8—21no—
467Q8—21no—
468Q8—62no—
469Q8—62no—
470Q8—122no—
471D4—24no—
472D4—22no—
473D4—21no—
474D4—21no—
475D4—42no—
476D4C2×C3×#6714yes—
477C2×C4—21no—
478C2×C4—34no—
479C2×C4—42no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—122no—
483C2×C4—244no—
484C2×C4C3×D3×D411yescharacteristic
485C2×C4C3×#23712yes—
486C2×C4C2×C3×#6712yes—
487C2×C4C23×C3×D311yescharacteristic
488C8—22no—
489C8—62no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—42no—
495C2×C3—44no—
496C2×C3—44no—
497C2×C3—84no—
498C2×C3—82no—
499C2×C3—81no—
500C2×C3—81no—
501C2×C3#577512yes—
502C2×C3C2×#93911yescharacteristic
503C2×C3C2×C3×#12111yescharacteristic
504C2×C3C3×#57712yes—
505C22—22no—
506C22—24no—
507C22—44no—
508C22—42no—
509C22—41no—
510C22—84no—
511C22C3×#93912yes—
512C22C2×C3×D3×D411yescharacteristic
513C4—41no—
514C4—64no—
515C4—122no—
516C4—122no—
517C4C3×#93912yes—
518C4C2×C3×#23711yescharacteristic
519C4C22×C3×#6711yescharacteristic
520C3—21no—
521C3#8393511yescharacteristic, direct product factor
522C3C3×#304211yescharacteristic, complemented
523C2—22no—
524C2—44no—
525C2—81no—
526C2C3×#577512yes—
527C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC22×C32C22×D4
hypercenterC2×C3×AGL(1,ℤ/8)D3
hypocenterC3C3×#3042
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×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×#5825C2semidirect
3. C3×#5798C2semidirect
4. C22×C3×#214C2semidirect
5. C3×#5233C2semidirect
6. C3×#5162C2semidirect
7. C3×#5153C2semidirect
8. C3×#5120C2semidirect
9. C3×#5119C2semidirect
11. #83935C3direct (N × H)
20. C3×#944C22semidirect
21. C3×#936C22semidirect
31. C3×#835C22semidirect
32. C3×#832C22semidirect
33. C3×#816C22semidirect
34. C3×Dic3⋊C8C22semidirect
41. #5825C2×C3semidirect
42. #5798C2×C3semidirect
43. C22×#214C2×C3semidirect
49. #5233C2×C3semidirect
50. #5162C2×C3semidirect
51. #5153C2×C3semidirect
52. #5120C2×C3semidirect
53. #5119C2×C3semidirect
122. #944C22×C3semidirect
123. #936C22×C3semidirect
137. #835C22×C3semidirect
138. #832C22×C3semidirect
139. #816C22×C3semidirect
140. Dic3⋊C8C22×C3semidirect
460. C32#3042semidirect
521. C3#83935direct (N × H)
522. C3C3×#3042semidirect

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×#58252
C3×#57982
C22×C3×#2142
C3×#52332
C3×#51622
C3×#51532
C3×#51202
C3×#51192
C3×#30423
#839353