Finite Group Databasebeta

← C3×#78717 #11865147

Subgroups — C3×#78717 #11865147

Order 1152.

1547 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic, complemented
2C3×#5830C211yesmaximal, characteristic, complemented
3C2×C3×#908C212yesmaximal, complemented
4C32×#413C211yesmaximal, characteristic, complemented
5C3×#5139C212yesmaximal, complemented
6C3×#4753C214yesmaximal, complemented
7C3×#4747C214yesmaximal, complemented
8C3×#3021—31nomaximal
9#78717C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1010—22no—
12C3×#1009—22no—
13C3×D3×#41—24no—
14C3×#1007C2212yescomplemented
15C3×#1004—22no—
16C3×#1004C2212yescomplemented
17C2×C3×#235—22no—
18C2×C3×#235C2212yes—
19C22×C3×D12C2211yescharacteristic, complemented
20C2×C32×M16C2212yescomplemented
21C3×#947C2212yescomplemented
22C3×#940C2212yescomplemented
23C3×#937—22no—
24C3×#930—22no—
25C3×#926C2211yescharacteristic, complemented
26C3×#908—24no—
27C3×#908C2218yescomplemented
28C2×C3×D24—22no—
29C2×C3×#204—22no—
30C32×#101C2212yescomplemented
31C32×#100C2212yescomplemented
32C3×#825C2218yescomplemented
33C3×#585—31no—
34#5964C2×C311yescharacteristic, complemented
35C2×C3×AGL(1,ℤ/8)—32no—
36C3×#535—31no—
37#5830C2×C311yescharacteristic, complemented
38C2×#908C2×C312yescomplemented
39C3×#454—34no—
40C3×#453—34no—
41C3×#421—32no—
42C3×#413—21no—
43C3×#413D311yescharacteristic, complemented
44C3×#413C2×C311yescharacteristic, complemented
45#5139C2×C312yescomplemented
46#4753C2×C314yescomplemented
47#4747C2×C314yescomplemented
48C23×C3×D3—21no—
49C32×#41—22no—
50C32×#41—22no—
51C2×C32×Q8C2311yescharacteristic
52C2×C32×D4D412yes—
53C2×C32×D4C2311yescharacteristic
54C22×C32×C4D412yes—
55C22×C32×C4C2311yescharacteristic
56C2×C3×#67—24no—
57C2×C3×#67—22no—
58C2×C3×#67—24no—
59C3×#239—24no—
60C3×D3×Q8—24no—
61C3×#237—28no—
62C3×#237—24no—
63C3×D3×D4—28no—
64C3×D3×D4—24no—
65C3×#235—24no—
66C3×#235—24no—
67C3×#235—22no—
68C3×#235—24no—
69C3×#235—22no—
70C3×#235D414yescomplemented
71C3×#235C2314yes—
72C2×C3×D12—22no—
73C2×C3×D12—22no—
74C2×C3×D12—22no—
75C2×C3×D12—21no—
76C2×C3×D12D412yescomplemented
77C2×C3×D12C2312yes—
78C2×C3×C4×D3—24no—
79C2×C3×C4×D3—22no—
80C2×C3×Dic6—22no—
81C2×C3×Dic6D412yescomplemented
82C32×M16—22no—
83C32×M16C2314yes—
84C2×C32×C8—22no—
85C3×#217—21no—
86C3×(C2×D3)⋊C4—24no—
87C3×C4⋊Dic3—21no—
88C3×C4×Dic3C2312yes—
89C3×D24—28no—
90C3×#204—28no—
91#3021—31noSylow
92C3×2−1+4—32no—
93C3×2−1+4—62no—
94C3×H(5,𝔽2)—32no—
95C3×H(5,𝔽2)—62no—
96C3×H(5,𝔽2)—62no—
97C2×C3×#41—21no—
98C2×C3×#41—32no—
99C2×C3×#41—64no—
100C2×C3×#41—62no—
101C2×C3×#41C2×D311yescharacteristic
102C2×C3×#41C22×C311yescharacteristic
103C22×C3×D4—31no—
104#1010—22no—
105#1009—22no—
106D3×#41—24no—
107#1007C22×C312yescomplemented
108#1004—22no—
109#1004C22×C312yescomplemented
110C2×#235—22no—
111C2×#235C22×C312yes—
112C22×D12C22×C311yescharacteristic, complemented
113C3×AGL(1,ℤ/8)—38no—
114C3×AGL(1,ℤ/8)—64no—
115C2×C3×SD16—62no—
116C2×C3×D8—62no—
117C2×C3×M16—22no—
118C2×C3×M16C2×D312yescomplemented
119C2×C3×M16C22×C312yescomplemented
120C3×#127—32no—
121C3×#124—32no—
122C3×#121—62no—
123C3×#120—62no—
124C3×#117—31no—
125#947C22×C312yescomplemented
126#940C22×C312yescomplemented
127#937—22no—
128#930—22no—
129#926C22×C311yescharacteristic, complemented
130#908—24no—
131#908C22×C318yescomplemented
132C2×D24—22no—
133C2×#204—22no—
134C3×#104—38no—
135C3×#101—22no—
136C3×#101C2×D312yescomplemented
137C3×#101C22×C312yescomplemented
138C3×#100—22no—
139C3×#100C2×D312yescomplemented
140C3×#100C22×C312yescomplemented
141#825C22×C318yescomplemented
142C23×C32C2×D412yes—
143C23×C32C2×D411yescharacteristic
144C22×C3×D3—24no—
145C22×C3×D3—22no—
146C22×C3×D3—22no—
147C22×C3×D3—42no—
148C22×C3×D3—82no—
149C32×Q8—22no—
150C32×D4—24no—
151C32×D4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D412yes—
157C2×C32×C4C2×D412yes—
158C2×C32×C4C2411yescharacteristic
159C3×#67—28no—
160C3×#67—24no—
161C3×#67—24no—
162C3×#67—84no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C3×D12—24no—
167C3×D12—22no—
168C3×D12—24no—
169C3×D12—41no—
170C3×D12—41no—
171C3×D12C2×D414yes—
172C3×C4×D3—28no—
173C3×C4×D3—24no—
174C3×C4×D3—24no—
175C3×Dic6—24no—
176C3×Dic6—22no—
177C3×Dic6C2×D414yes—
178C32×C8—24no—
179#585—31no—
180C2×AGL(1,ℤ/8)—32no—
181#535—31no—
182#454—34no—
183#453—34no—
184#421—32no—
185#413C3×D311yescharacteristic, complemented
186C24×C3—61no—
187C23×D3—21no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41—34no—
193C3×#41—34no—
194C3×#41—42no—
195C3×#41—42no—
196C3×#41—64no—
197C3×#41—64no—
198C3×#41—64no—
199C3×#41—68no—
200C3×#41—64no—
201C3×#41—62no—
202C3×#41—64no—
203C3×#41—62no—
204C2×C3×Q8—21no—
205C2×C3×Q8—32no—
206C2×C3×Q8—62no—
207C2×C3×Q8—64no—
208C2×C3×Q8C22×D311yescharacteristic
209C2×C3×Q8C23×C311yescharacteristic
210C2×C3×D4—22no—
211C2×C3×D4—21no—
212C2×C3×D4—32no—
213C2×C3×D4—32no—
214C2×C3×D4—62no—
215C2×C3×D4—64no—
216C2×C3×D4—68no—
217C2×C3×D4—62no—
218C2×C3×D4—64no—
219C2×C3×D4—64no—
220C2×C3×D4—61no—
221C2×C3×D4—62no—
222C2×C3×D4—62no—
223C2×C3×D4D1212yes—
224C2×C3×D4C3×D412yes—
225C2×C3×D4C22×D311yescharacteristic
226C2×C3×D4C23×C311yescharacteristic
227C22×C3×C4—22no—
228C22×C3×C4—21no—
229C22×C3×C4—64no—
230C22×C3×C4—62no—
231C22×C3×C4D1212yes—
232C22×C3×C4C3×D412yes—
233C22×C3×C4C22×D311yescharacteristic
234C22×C3×C4C23×C311yescharacteristic
235C2×#67—24no—
236C2×#67—22no—
237C2×#67—24no—
238#239—24no—
239D3×Q8—24no—
240#237—28no—
241#237—24no—
242D3×D4—28no—
243D3×D4—24no—
244#235—24no—
245#235—24no—
246#235—22no—
247#235—24no—
248#235—22no—
249#235C3×D414yescomplemented
250#235C23×C314yes—
251C2×D12—22no—
252C2×D12—22no—
253C2×D12—22no—
254C2×D12—21no—
255C2×D12C3×D412yescomplemented
256C2×D12C23×C312yes—
257C2×C4×D3—24no—
258C2×C4×D3—22no—
259C2×Dic6—22no—
260C2×Dic6C3×D412yescomplemented
261C3×SD16—68no—
262C3×D8—68no—
263C3×M16—22no—
264C3×M16—22no—
265C3×M16—24no—
266C3×M16—42no—
267C3×M16C22×D314yes—
268C3×M16C23×C314yes—
269C2×C3×C8—22no—
270C2×C3×C8—22no—
271C2×C3×C8—42no—
272C3×C4⋊C4—61no—
273C3×C22⋊C4—61no—
274C3×C22⋊C4—64no—
275C3×C42—32no—
276#217—21no—
277(C2×D3)⋊C4—24no—
278C4⋊Dic3—21no—
279C4×Dic3C23×C312yes—
280D24—28no—
281#204—28no—
282C22×C32—22no—
283C22×C32—41no—
284C22×C32—42no—
285C22×C32#12012yes—
286C22×C32C22×D411yescharacteristic
287C2×C3×D3—24no—
288C2×C3×D3—44no—
289C2×C3×D3—42no—
290C2×C3×D3—82no—
291C2×C3×D3—84no—
292C2×C3×D3—84no—
293C32×C4—22no—
294C32×C4#12011yescharacteristic
295C32×C4#12011yescharacteristic
296C32×C4C22×D412yes—
297C3×Dic3—24no—
298C3×Dic3—42no—
2992−1+4—32no—
3002−1+4—62no—
301H(5,𝔽2)—32no—
302H(5,𝔽2)—62no—
303H(5,𝔽2)—62no—
304C2×#41—32no—
305C2×#41—64no—
306C2×#41—62no—
307C2×#41C2×C3×D311yescharacteristic
308C22×D4—31no—
309AGL(1,ℤ/8)—38no—
310AGL(1,ℤ/8)—64no—
311C2×SD16—62no—
312C2×D8—62no—
313C2×M16C2×C3×D312yescomplemented
314#127—32no—
315#124—32no—
316#121—62no—
317#120—62no—
318#117—31no—
319#104—38no—
320#101C2×C3×D312yescomplemented
321#100C2×C3×D312yescomplemented
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—64no—
325C23×C3—62no—
326C23×C3—62no—
327C23×C3—122no—
328C23×C3—242no—
329C23×C3C2×D1212yes—
330C23×C3C2×D1211yescharacteristic
331C23×C3C2×C3×D412yes—
332C23×C3C2×C3×D411yescharacteristic
333C22×D3—24no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—42no—
337C22×D3—82no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—34no—
341C3×Q8—42no—
342C3×Q8—64no—
343C3×Q8—62no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—34no—
349C3×D4—42no—
350C3×D4—44no—
351C3×D4—64no—
352C3×D4—68no—
353C3×D4—64no—
354C3×D4—64no—
355C3×D4—62no—
356C3×D4—64no—
357C3×D4—121no—
358C3×D4—121no—
359C3×D4—244no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—68no—
372C2×C3×C4—64no—
373C2×C3×C4—62no—
374C2×C3×C4—64no—
375C2×C3×C4—64no—
376C2×C3×C4—62no—
377C2×C3×C4C2×D1212yes—
378C2×C3×C4C2×D1211yescharacteristic
379C2×C3×C4D3×D412yes—
380C2×C3×C4D3×D412yes—
381C2×C3×C4C2×C3×D412yes—
382C2×C3×C4C2×C3×D411yescharacteristic
383C2×C3×C4C2×C3×D412yes—
384C2×C3×C4C2×C3×D412yes—
385C2×C3×C4C23×D311yescharacteristic
386C2×C3×C4C24×C311yescharacteristic
387#67—28no—
388#67—24no—
389#67—24no—
390#67—84no—
391C2×Dic3—24no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394D12—24no—
395D12—22no—
396D12—24no—
397D12—41no—
398D12—41no—
399D12C2×C3×D414yes—
400C4×D3—28no—
401C4×D3—24no—
402C4×D3—24no—
403Dic6—24no—
404Dic6—22no—
405Dic6C2×C3×D414yes—
406C3×C8—24no—
407C3×C8—24no—
408C3×C8—44no—
409C2×C32—21no—
410C2×C32—22no—
411C2×C32—42no—
412C2×C32C2×#12011yescharacteristic
413C3×D3—44no—
414C3×D3—82no—
415C24—61no—
416#41—22no—
417#41—22no—
418#41—34no—
419#41—34no—
420#41—64no—
421#41—64no—
422#41—64no—
423#41—68no—
424#41—64no—
425#41—62no—
426#41—64no—
427#41—62no—
428C2×Q8—32no—
429C2×Q8—62no—
430C2×Q8—64no—
431C2×Q8C22×C3×D311yescharacteristic
432C2×D4—32no—
433C2×D4—32no—
434C2×D4—62no—
435C2×D4—64no—
436C2×D4—68no—
437C2×D4—62no—
438C2×D4—64no—
439C2×D4—64no—
440C2×D4—61no—
441C2×D4—62no—
442C2×D4—62no—
443C2×D4C3×D1212yes—
444C2×D4C22×C3×D311yescharacteristic
445C22×C4—64no—
446C22×C4—62no—
447C22×C4C3×D1212yes—
448C22×C4C22×C3×D311yescharacteristic
449SD16—68no—
450D8—68no—
451M16—22no—
452M16C22×C3×D314yes—
453C2×C8—22no—
454C4⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—64no—
457C42—32no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—21no—
462C22×C3—42no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—64no—
468C22×C3—81no—
469C22×C3—82no—
470C22×C3—124no—
471C22×C3—122no—
472C22×C3—242no—
473C22×C3—244no—
474C22×C3—244no—
475C22×C3#88212yes—
476C22×C3C3×#12012yes—
477C22×C3C22×D1211yescharacteristic
478C22×C3C22×C3×D411yescharacteristic
479C2×D3—24no—
480C2×D3—44no—
481C2×D3—42no—
482C2×D3—82no—
483C2×D3—84no—
484C2×D3—84no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—22no—
490C3×C4—42no—
491C3×C4—64no—
492C3×C4—122no—
493C3×C4#88211yescharacteristic
494C3×C4#88211yescharacteristic
495C3×C4C3×#12011yescharacteristic
496C3×C4C3×#12011yescharacteristic
497C3×C4C2×D3×D412yes—
498C3×C4C22×C3×D412yes—
499Dic3—24no—
500Dic3—42no—
501C32#302111yescharacteristic, complemented, Sylow
502C23—64no—
503C23—62no—
504C23—62no—
505C23—122no—
506C23—242no—
507C23C2×C3×D1212yes—
508C23C2×C3×D1211yescharacteristic
509Q8—22no—
510Q8—34no—
511Q8—64no—
512Q8—62no—
513D4—24no—
514D4—22no—
515D4—34no—
516D4—64no—
517D4—68no—
518D4—64no—
519D4—64no—
520D4—62no—
521D4—64no—
522D4—121no—
523D4—121no—
524D4—244no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—68no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4C2×C3×D1212yes—
534C2×C4C2×C3×D1211yescharacteristic
535C2×C4C3×D3×D412yes—
536C2×C4C3×D3×D412yes—
537C2×C4C23×C3×D311yescharacteristic
538C8—24no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—22no—
543C2×C3—21no—
544C2×C3—41no—
545C2×C3—42no—
546C2×C3—42no—
547C2×C3—42no—
548C2×C3—82no—
549C2×C3—124no—
550C2×C3—242no—
551C2×C3C2×#88211yescharacteristic
552C2×C3C2×C3×#12011yescharacteristic
553D3—44no—
554D3—82no—
555C22—22no—
556C22—41no—
557C22—42no—
558C22—64no—
559C22—124no—
560C22—122no—
561C22—242no—
562C22—244no—
563C22—244no—
564C22C3×#88212yes—
565C22C22×C3×D1211yescharacteristic
566C4—22no—
567C4—64no—
568C4—122no—
569C4C3×#88211yescharacteristic
570C4C3×#88211yescharacteristic
571C4C2×C3×D3×D412yes—
572C3—21no—
573C3#7871711yescharacteristic, direct product factor
574C3C3×#302111yescharacteristic, complemented
575C2—21no—
576C2—22no—
577C2—42no—
578C2—124no—
579C2—242no—
580C2C2×C3×#88211yescharacteristic
subgroupG/H
centerC2×C3C2×#882
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#413C2
socleC2×C32C2×#120
hypercenterC3×#413D3
hypocenterC3C3×#3021
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#413

Complements

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

NH ≅ G/Ntype
1. C3×#5964C2semidirect
2. C3×#5830C2semidirect
3. C2×C3×#908C2semidirect
4. C32×#413C2semidirect
5. C3×#5139C2semidirect
6. C3×#4753C2semidirect
7. C3×#4747C2semidirect
9. #78717C3direct (N × H)
14. C3×#1007C22semidirect
16. C3×#1004C22semidirect
19. C22×C3×D12C22semidirect
20. C2×C32×M16C22semidirect
21. C3×#947C22semidirect
22. C3×#940C22semidirect
25. C3×#926C22semidirect
27. C3×#908C22semidirect
30. C32×#101C22semidirect
31. C32×#100C22semidirect
32. C3×#825C22semidirect
34. #5964C2×C3semidirect
37. #5830C2×C3semidirect
38. C2×#908C2×C3semidirect
43. C3×#413D3semidirect
44. C3×#413C2×C3semidirect
45. #5139C2×C3semidirect
46. #4753C2×C3semidirect
47. #4747C2×C3semidirect
70. C3×#235D4semidirect
76. C2×C3×D12D4semidirect
81. C2×C3×Dic6D4semidirect
107. #1007C22×C3semidirect
109. #1004C22×C3semidirect
112. C22×D12C22×C3semidirect
118. C2×C3×M16C2×D3semidirect
119. C2×C3×M16C22×C3semidirect
125. #947C22×C3semidirect
126. #940C22×C3semidirect
129. #926C22×C3semidirect
131. #908C22×C3semidirect
136. C3×#101C2×D3semidirect
137. C3×#101C22×C3semidirect
139. C3×#100C2×D3semidirect
140. C3×#100C22×C3semidirect
141. #825C22×C3semidirect
185. #413C3×D3semidirect
249. #235C3×D4semidirect
255. C2×D12C3×D4semidirect
260. C2×Dic6C3×D4semidirect
313. C2×M16C2×C3×D3semidirect
320. #101C2×C3×D3semidirect
321. #100C2×C3×D3semidirect
501. C32#3021semidirect
573. C3#78717direct (N × H)
574. C3C3×#3021semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C3×#58302
C2×C3×#9082
C32×#4132
C3×#51392
C3×#47532
C3×#47472
C3×#30213
#787173