Finite Group Databasebeta

← C3×#78825 #11873467

Subgroups — C3×#78825 #11873467

Order 1152.

879 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5832C211yesmaximal, characteristic
2C3×#5517C211yesmaximal, characteristic, complemented
3C32×#416C211yesmaximal, characteristic, complemented
4C3×#5233C211yesmaximal, characteristic, complemented
5C3×#5113C211yesmaximal, characteristic, complemented
6C2×C3×C8×Dic3C211yesmaximal, characteristic, complemented
7C3×#5002C211yesmaximal, characteristic, complemented
8C3×#4800C212yesmaximal, complemented
9C3×#4792C212yesmaximal, complemented
10C3×#4768C212yesmaximal, complemented
11C3×#4760C212yesmaximal, complemented
12C3×#3131—31nomaximal
13#78825C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2211yescharacteristic
15C2×C3×#235C2211yescharacteristic
16C22×C32×C8C2211yescharacteristic
17C32×#117C2211yescharacteristic
18C3×#947C2211yescharacteristic
19C3×#944C2211yescharacteristic
20C3×#940C2211yescharacteristic
21C3×#939—22no—
22C3×#936C2211yescharacteristic
23C3×C4×#67—22no—
24C3×#926C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C22×C3×C3⋊C8C2211yescharacteristic
27C3×#890C2212yes—
28C3×#888C2212yes—
29C3×#878—22no—
30C3×C4×Dic6—22no—
31C32×#103C2212yes—
32C32×#102C2212yes—
33C3×#835C2212yes—
34C3×#832C2212yes—
35C3×#821C2212yes—
36C3×#816C2212yes—
37C3×C8×Dic3C2212yes—
38C3×C8×Dic3C2212yes—
39C3×#810C2212yes—
40C3×#809C2212yes—
41C3×#533—31no—
42C3×#532—31no—
43#5832C2×C311yescharacteristic
44#5517C2×C311yescharacteristic, complemented
45C3×#488—32no—
46C3×#488—32no—
47C3×#487—32no—
48C3×#486—32no—
49C3×#416—21no—
50C3×#416—31no—
51C3×#416—31no—
52C3×#416—31no—
53C3×#416D311yescharacteristic, complemented
54C3×#416C2×C311yescharacteristic, complemented
55C2×C3×C4×C8—31no—
56#5233C2×C311yescharacteristic, complemented
57#5113C2×C311yescharacteristic, complemented
58C2×C8×Dic3C2×C311yescharacteristic, complemented
59#5002C2×C311yescharacteristic, complemented
60#4800C2×C312yescomplemented
61#4792C2×C312yescomplemented
62#4768C2×C312yescomplemented
63#4760C2×C312yescomplemented
64C32×#41—22no—
65C32×#41—22no—
66C2×C32×Q8C2311yescharacteristic
67C2×C32×D4—21no—
68C2×C32×D4C2311yescharacteristic
69C22×C32×C4—21no—
70C22×C32×C4C2311yescharacteristic
71C2×C3×#67—21no—
72C2×C3×#67—22no—
73C22×C3×Dic3D412yes—
74C3×#235—22no—
75C3×#235—22no—
76C2×C3×D12C2311yescharacteristic
77C2×C3×C4×D3—21no—
78C2×C3×Dic6—22no—
79C2×C3×Dic6C2311yescharacteristic
80C2×C32×C8—22no—
81C2×C32×C8C2312yes—
82C32×C4⋊C4C2312yes—
83C32×C22⋊C4—21no—
84C32×C42—21no—
85C3×#217—21no—
86C3×#217—22no—
87C3×(C2×D3)⋊C4—22no—
88C3×C4⋊Dic3—22no—
89C3×C4⋊Dic3C2312yes—
90C3×Dic3⋊C4—22no—
91C3×Dic3⋊C4—22no—
92C3×Dic3⋊C4—22no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3D412yes—
95C3×C4×Dic3C2311yescharacteristic
96C3×C4×Dic3C2311yescharacteristic
97C2×C3×C3⋊C8—22no—
98C2×C3×C3⋊C8C2312yes—
99#3131—31noSylow
100C2×C3×#41—21no—
101C2×C3×#41—31no—
102C2×C3×#41C2×D311yescharacteristic
103C2×C3×#41C22×C311yescharacteristic
104C2×#235C22×C311yescharacteristic
105C22×C3×C8—21no—
106C22×C3×C8—31no—
107C22×C3×C8C2×D311yescharacteristic
108C22×C3×C8C22×C311yescharacteristic
109C3×C4⋊Q8—32no—
110C3×C4⋊Q8—31no—
111C3×#127—31no—
112C3×#125—32no—
113C3×#124—31no—
114C3×#124—31no—
115C3×C22⋊Q8—62no—
116C3×#121—62no—
117C3×C4×Q8—62no—
118C3×C4×D4—62no—
119C3×#117—21no—
120C3×#117—31no—
121C3×#117C2×D311yescharacteristic
122C3×#117C22×C311yescharacteristic
123C2×C3×C42—31no—
124#947C22×C311yescharacteristic
125#944C22×C311yescharacteristic
126#940C22×C311yescharacteristic
127#939—22no—
128#936C22×C311yescharacteristic
129C4×#67—22no—
130#926C22×C311yescharacteristic
131C2×C4×Dic3C22×C311yescharacteristic
132C22×C3⋊C8C22×C311yescharacteristic
133#890C22×C312yes—
134#888C22×C312yes—
135#878—22no—
136C4×Dic6—22no—
137C3×#103—22no—
138C3×#103—32no—
139C3×#103—32no—
140C3×#103—32no—
141C3×#103C2×D312yes—
142C3×#103C22×C312yes—
143C3×#102—22no—
144C3×#102—32no—
145C3×#102—32no—
146C3×#102—32no—
147C3×#102C2×D312yes—
148C3×#102C22×C312yes—
149C3×C4×C8—32no—
150C3×C4×C8—32no—
151#835C22×C312yes—
152#832C22×C312yes—
153#821C22×C312yes—
154#816C22×C312yes—
155C8×Dic3C22×C312yes—
156C8×Dic3C22×C312yes—
157#810C22×C312yes—
158#809C22×C312yes—
159C23×C32—21no—
160C23×C32C2×D411yescharacteristic
161C22×C3×D3—21no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×D4—21no—
165C32×D4—21no—
166C32×D4—42no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—42no—
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4#4112yes—
172C2×C32×C4#4112yes—
173C2×C32×C4C2411yescharacteristic
174C3×#67—42no—
175C3×#67—44no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3C2×D412yes—
180C2×C3×Dic3C2×D412yes—
181C3×D12—21no—
182C3×D12—21no—
183C3×C4×D3—42no—
184C3×Dic6—21no—
185C3×Dic6—21no—
186C3×Dic6—44no—
187C32×C8—22no—
188C3×C3⋊C8—22no—
189#533—31no—
190#532—31no—
191#488—32no—
192#488—32no—
193#487—32no—
194#486—32no—
195#416—31no—
196#416—31no—
197#416—31no—
198#416C3×D311yescharacteristic, complemented
199C2×C4×C8—31no—
200C3×#41—24no—
201C3×#41—22no—
202C3×#41—22no—
203C3×#41—22no—
204C3×#41—22no—
205C3×#41—42no—
206C3×#41—62no—
207C3×#41—62no—
208C2×C3×Q8—21no—
209C2×C3×Q8—31no—
210C2×C3×Q8—62no—
211C2×C3×Q8C22×D311yescharacteristic
212C2×C3×Q8C23×C311yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—31no—
217C2×C3×D4—41no—
218C2×C3×D4—62no—
219C2×C3×D4—61no—
220C2×C3×D4C22×D311yescharacteristic
221C2×C3×D4C23×C311yescharacteristic
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—32no—
226C22×C3×C4—41no—
227C22×C3×C4—61no—
228C22×C3×C4C22×D311yescharacteristic
229C22×C3×C4C23×C311yescharacteristic
230C2×#67—21no—
231C2×#67—22no—
232C22×Dic3C3×D412yes—
233#235—22no—
234#235—22no—
235C2×D12C23×C311yescharacteristic
236C2×C4×D3—21no—
237C2×Dic6—22no—
238C2×Dic6C23×C311yescharacteristic
239C2×C3×C8—22no—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—32no—
243C2×C3×C8—42no—
244C2×C3×C8—62no—
245C2×C3×C8C22×D312yes—
246C2×C3×C8C23×C312yes—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—32no—
249C3×C4⋊C4—62no—
250C3×C4⋊C4—62no—
251C3×C4⋊C4—62no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4C22×D312yes—
254C3×C4⋊C4C23×C312yes—
255C3×C22⋊C4—21no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—41no—
258C3×C22⋊C4—62no—
259C3×C22⋊C4—62no—
260C3×C22⋊C4—61no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—32no—
264C3×C42—31no—
265C3×C42—31no—
266C3×C42—41no—
267C3×C42—61no—
268#217—21no—
269#217—22no—
270(C2×D3)⋊C4—22no—
271C4⋊Dic3—22no—
272C4⋊Dic3C23×C312yes—
273Dic3⋊C4—22no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276C4×Dic3—21no—
277C4×Dic3C3×D412yes—
278C4×Dic3C23×C311yescharacteristic
279C4×Dic3C23×C311yescharacteristic
280C2×C3⋊C8—22no—
281C2×C3⋊C8C23×C312yes—
282C22×C32—22no—
283C22×C32—41no—
284C22×C32—42no—
285C22×C32#12412yes—
286C22×C32C22×D411yescharacteristic
287C2×C3×D3—41no—
288C2×C3×D3—42no—
289C32×C4—41no—
290C32×C4—42no—
291C32×C4#12412yes—
292C32×C4C2×#4111yescharacteristic
293C32×C4C2×#4111yescharacteristic
294C3×Dic3—22no—
295C3×Dic3—41no—
296C3×Dic3—42no—
297C3×Dic3#13514yes—
298C2×#41—31no—
299C2×#41C2×C3×D311yescharacteristic
300C22×C8—31no—
301C22×C8C2×C3×D311yescharacteristic
302C4⋊Q8—32no—
303C4⋊Q8—31no—
304#127—31no—
305#125—32no—
306#124—31no—
307#124—31no—
308C22⋊Q8—62no—
309#121—62no—
310C4×Q8—62no—
311C4×D4—62no—
312#117—31no—
313#117C2×C3×D311yescharacteristic
314C2×C42—31no—
315#103—32no—
316#103—32no—
317#103—32no—
318#103C2×C3×D312yes—
319#102—32no—
320#102—32no—
321#102—32no—
322#102C2×C3×D312yes—
323C4×C8—32no—
324C4×C8—32no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—41no—
329C23×C3—61no—
330C23×C3D3×D411yescharacteristic
331C23×C3C2×C3×D411yescharacteristic
332C22×D3—21no—
333C3×Q8—22no—
334C3×Q8—21no—
335C3×Q8—21no—
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—41no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—124no—
342C3×D4—22no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—21no—
346C3×D4—21no—
347C3×D4—41no—
348C3×D4—44no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—61no—
352C3×D4—61no—
353C3×D4—124no—
354C3×D4—122no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—32no—
364C2×C3×C4—32no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—44no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—64no—
371C2×C3×C4—62no—
372C2×C3×C4—61no—
373C2×C3×C4—122no—
374C2×C3×C4#23512yes—
375C2×C3×C4D3×D411yescharacteristic
376C2×C3×C4#23712yes—
377C2×C3×C4C2×C3×D411yescharacteristic
378C2×C3×C4C3×#4112yes—
379C2×C3×C4C3×#4112yes—
380C2×C3×C4C23×D311yescharacteristic
381C2×C3×C4C24×C311yescharacteristic
382#67—42no—
383#67—44no—
384C2×Dic3—24no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3C2×C3×D412yes—
388C2×Dic3C2×C3×D412yes—
389D12—21no—
390D12—21no—
391C4×D3—42no—
392Dic6—21no—
393Dic6—21no—
394Dic6—44no—
395C3×C8—22no—
396C3×C8—22no—
397C3×C8—42no—
398C3×C8—62no—
399C3⋊C8—22no—
400C2×C32—22no—
401C2×C32—81no—
402C2×C32C2×#12411yescharacteristic
403C2×C32C2×#13512yes—
404C3×D3—81no—
405#41—22no—
406#41—22no—
407#41—62no—
408#41—62no—
409C2×Q8—31no—
410C2×Q8—62no—
411C2×Q8C22×C3×D311yescharacteristic
412C2×D4—21no—
413C2×D4—31no—
414C2×D4—62no—
415C2×D4—61no—
416C2×D4C22×C3×D311yescharacteristic
417C22×C4—21no—
418C22×C4—32no—
419C22×C4—61no—
420C22×C4C22×C3×D311yescharacteristic
421C2×C8—22no—
422C2×C8—32no—
423C2×C8—62no—
424C2×C8C22×C3×D312yes—
425C4⋊C4—32no—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4C22×C3×D312yes—
431C22⋊C4—21no—
432C22⋊C4—62no—
433C22⋊C4—62no—
434C22⋊C4—61no—
435C42—21no—
436C42—31no—
437C42—31no—
438C42—32no—
439C42—61no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—21no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—41no—
448C22×C3—42no—
449C22×C3—41no—
450C22×C3—82no—
451C22×C3—121no—
452C22×C3—122no—
453C22×C3#88512yes—
454C22×C3C3×#12412yes—
455C22×C3C2×D3×D411yescharacteristic
456C22×C3C22×C3×D411yescharacteristic
457C2×D3—42no—
458C2×D3—41no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—34no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4—41no—
468C3×C4—62no—
469C3×C4—82no—
470C3×C4—121no—
471C3×C4—122no—
472C3×C4#88512yes—
473C3×C4C3×#12412yes—
474C3×C4C2×#23511yescharacteristic
475C3×C4C2×#23711yescharacteristic
476C3×C4C2×C3×#4111yescharacteristic
477C3×C4C2×C3×#4111yescharacteristic
478Dic3—22no—
479Dic3—42no—
480Dic3—41no—
481Dic3C3×#13514yes—
482C32#313111yescharacteristic, complemented, Sylow
483C23—21no—
484C23—61no—
485C23C3×D3×D411yescharacteristic
486Q8—21no—
487Q8—21no—
488Q8—61no—
489Q8—61no—
490Q8—124no—
491D4—21no—
492D4—21no—
493D4—42no—
494D4—61no—
495D4—61no—
496D4—124no—
497D4—122no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—42no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—61no—
506C2×C4—122no—
507C2×C4C3×#23512yes—
508C2×C4C3×D3×D411yescharacteristic
509C2×C4C3×#23712yes—
510C2×C4C23×C3×D311yescharacteristic
511C8—22no—
512C8—62no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—21no—
516C2×C3—22no—
517C2×C3—42no—
518C2×C3—82no—
519C2×C3—81no—
520C2×C3—81no—
521C2×C3—241no—
522C2×C3C2×#88511yescharacteristic
523C2×C3D3×#13512yes—
524C2×C3C2×C3×#12411yescharacteristic
525C2×C3C2×C3×#13512yes—
526D3—81no—
527C22—22no—
528C22—41no—
529C22—42no—
530C22—121no—
531C22—122no—
532C22C3×#88512yes—
533C22C2×C3×D3×D411yescharacteristic
534C4—34no—
535C4—41no—
536C4—42no—
537C4—62no—
538C4—121no—
539C4—122no—
540C4C3×#88512yes—
541C4C2×C3×#23511yescharacteristic
542C4C2×C3×#23711yescharacteristic
543C3—21no—
544C3#7882511yescharacteristic, direct product factor
545C3C3×#313111yescharacteristic, complemented
546C2—22no—
547C2—81no—
548C2—241no—
549C2C2×C3×#88511yescharacteristic
550C2C3×D3×#13512yes—
subgroupG/H
centerC2×C3×C4D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#416C2
socleC22×C32C22×D4
prime radicalC3×#5832C2
hypercenterC3×#416D3
hypocenterC3C3×#3131
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C3×#416

Complements

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

NH ≅ G/Ntype
2. C3×#5517C2semidirect
3. C32×#416C2semidirect
4. C3×#5233C2semidirect
5. C3×#5113C2semidirect
6. C2×C3×C8×Dic3C2semidirect
7. C3×#5002C2semidirect
8. C3×#4800C2semidirect
9. C3×#4792C2semidirect
10. C3×#4768C2semidirect
11. C3×#4760C2semidirect
13. #78825C3direct (N × H)
44. #5517C2×C3semidirect
53. C3×#416D3semidirect
54. C3×#416C2×C3semidirect
56. #5233C2×C3semidirect
57. #5113C2×C3semidirect
58. C2×C8×Dic3C2×C3semidirect
59. #5002C2×C3semidirect
60. #4800C2×C3semidirect
61. #4792C2×C3semidirect
62. #4768C2×C3semidirect
63. #4760C2×C3semidirect
198. #416C3×D3semidirect
482. C32#3131semidirect
544. C3#78825direct (N × H)
545. C3C3×#3131semidirect

Maximal subgroups

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

subgroupindex
C3×#58322
C3×#55172
C32×#4162
C3×#52332
C3×#51132
C2×C3×C8×Dic32
C3×#50022
C3×#48002
C3×#47922
C3×#47682
C3×#47602
C3×#31313
#788253