Finite Group Databasebeta

← C3×#78918 #11868941

Subgroups — C3×#78918 #11868941

Order 1152.

891 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×#5798C211yesmaximal, characteristic
2C3×#5519C211yesmaximal, characteristic, complemented
3C32×#418C211yesmaximal, characteristic
4C2×C3×#832C211yesmaximal, characteristic
5C3×#5121C211yesmaximal, characteristic, complemented
6C3×#5119C211yesmaximal, characteristic, complemented
7C3×#4998C211yesmaximal, characteristic, complemented
8C3×#4766C212yesmaximal, complemented
9C3×#4763C212yesmaximal, complemented
10C3×#4761C212yesmaximal, complemented
11C3×#4759C212yesmaximal, complemented
12C3×#3081—31nomaximal
13#78918C311yesmaximal, characteristic, direct product factor
14C22×C32×D4C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C32×#117C2211yescharacteristic
17C2×C3×#217—22no—
18C3×D4×Dic3—21no—
19C3×D4×Dic3—21no—
20C3×D4×Dic3C2212yes—
21C3×D4×Dic3C2212yes—
22C3×#926C2211yescharacteristic
23C3×#926C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#890C2212yescomplemented
27C3×#888C2212yescomplemented
28C3×#878—22no—
29C3×#870—21no—
30C3×#869—21no—
31C32×#102C2212yes—
32C32×#102C2212yes—
33C3×#832C2212yes—
34C3×#832C2212yes—
35C3×#818C2212yescomplemented
36C3×#817C2212yescomplemented
37C3×Dic3⋊C8C2212yescomplemented
38C3×Dic3⋊C8C2212yescomplemented
39C3×#808C2211yescharacteristic, complemented
40C3×#808C2211yescharacteristic, complemented
41C3×#807C2211yescharacteristic, complemented
42C3×#807C2211yescharacteristic, complemented
43C3×#544—31no—
44C3×#518—31no—
45#5798C2×C311yescharacteristic
46#5519C2×C311yescharacteristic, complemented
47C3×#478—32no—
48C3×#478—32no—
49C3×#476—32no—
50C3×#474—32no—
51C3×#428—31no—
52C3×#427—31no—
53C3×#424—31no—
54C3×#418—21no—
55C3×#418D311yescharacteristic
56C3×#418C2×C311yescharacteristic
57C2×C3×#102—31no—
58C2×#832C2×C311yescharacteristic
59#5121C2×C311yescharacteristic, complemented
60#5119C2×C311yescharacteristic, complemented
61#4998C2×C311yescharacteristic, complemented
62#4766C2×C312yescomplemented
63#4763C2×C312yescomplemented
64#4761C2×C312yescomplemented
65#4759C2×C312yescomplemented
66C24×C32—21no—
67C2×C32×D4—21no—
68C2×C32×D4—22no—
69C2×C32×D4Q814yescomplemented
70C2×C32×D4C2312yes—
71C22×C32×C4C2311yescharacteristic
72C22×C3×Dic3—21no—
73C22×C3×Dic3—22no—
74C22×C3×Dic3—22no—
75C2×C3×Dic6—22no—
76C32×M16—22no—
77C2×C32×C8C2312yes—
78C32×C4⋊C4C2311yescharacteristic
79C32×C4⋊C4C2311yescharacteristic
80C32×C22⋊C4—21no—
81C32×C42—21no—
82C3×#217—21no—
83C3×#217—22no—
84C3×#217—22no—
85C3×#217D412yes—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3D411yescharacteristic
89C3×C4⋊Dic3D411yescharacteristic
90C3×C4⋊Dic3C2312yes—
91C3×C4⋊Dic3C2311yescharacteristic
92C3×C4⋊Dic3C2311yescharacteristic
93C3×Dic3⋊C4—22no—
94C3×Dic3⋊C4—22no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2312yes—
97C2×C3×C3⋊C8—22no—
98C2×C3×C3⋊C8C2311yescharacteristic
99C2×C3×C3⋊C8C2311yescharacteristic
100#3081—31noSylow
101C22×C3×D4—21no—
102C22×C3×D4C2×D311yescharacteristic
103C22×C3×D4C22×C311yescharacteristic
104C2×C3×M16—21no—
105C2×C3×M16C2×D311yescharacteristic
106C2×C3×M16C22×C311yescharacteristic
107C22×C3×C8—31no—
108C3×C4⋊Q8—32no—
109C3×C4⋊Q8—61no—
110C3×#125—32no—
111C3×#125—61no—
112C3×C22⋊Q8—62no—
113C3×C4×D4—32no—
114C3×C4×D4—32no—
115C3×C4×D4—61no—
116C3×C4×D4—61no—
117C3×#117—21no—
118C3×#117—31no—
119C3×#117—31no—
120C3×#117C2×D311yescharacteristic
121C3×#117C22×C311yescharacteristic
122C2×C3×C4⋊C4—31no—
123C2×C3×C22⋊C4—62no—
124C2×#217—22no—
125D4×Dic3—21no—
126D4×Dic3—21no—
127D4×Dic3C22×C312yes—
128D4×Dic3C22×C312yes—
129#926C22×C311yescharacteristic
130#926C22×C311yescharacteristic
131C2×C4⋊Dic3C22×C311yescharacteristic
132C22×C3⋊C8C22×C311yescharacteristic
133#890C22×C312yescomplemented
134#888C22×C312yescomplemented
135#878—22no—
136#870—21no—
137#869—21no—
138C3×#107—32no—
139C3×#107—31no—
140C3×#107—31no—
141C3×#106—32no—
142C3×#106—31no—
143C3×#106—31no—
144C3×C4⋊C8—32no—
145C3×C4⋊C8—32no—
146C3×#102—22no—
147C3×#102—22no—
148C3×#102—32no—
149C3×#102—32no—
150C3×#102C2×D312yes—
151C3×#102C2×D312yes—
152C3×#102C22×C312yes—
153C3×#102C22×C312yes—
154#832C22×C312yes—
155#832C22×C312yes—
156#818C22×C312yescomplemented
157#817C22×C312yescomplemented
158Dic3⋊C8C22×C312yescomplemented
159Dic3⋊C8C22×C312yescomplemented
160#808C22×C311yescharacteristic, complemented
161#808C22×C311yescharacteristic, complemented
162#807C22×C311yescharacteristic, complemented
163#807C22×C311yescharacteristic, complemented
164C23×C32—22no—
165C23×C32—24no—
166C23×C32—81no—
167C23×C32—81no—
168C23×C32C2×D411yescharacteristic
169C32×D4—24no—
170C32×D4—22no—
171C32×D4C2×Q814yes—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×Q812yes—
176C2×C32×C4#4112yes—
177C2×C32×C4C2411yescharacteristic
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—82no—
182C2×C3×Dic3—82no—
183C2×C3×Dic3—81no—
184C2×C3×Dic3C2×D412yes—
185C2×C3×Dic3C2×D412yes—
186C3×Dic6—82no—
187C32×C8—22no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—21no—
190#544—31no—
191#518—31no—
192#478—32no—
193#478—32no—
194#476—32no—
195#474—32no—
196#428—31no—
197#427—31no—
198#424—31no—
199#418C3×D311yescharacteristic
200C2×#102—31no—
201C24×C3—21no—
202C24×C3—21no—
203C24×C3—41no—
204C2×C3×Q8—62no—
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4—21no—
208C2×C3×D4—22no—
209C2×C3×D4—24no—
210C2×C3×D4—22no—
211C2×C3×D4—41no—
212C2×C3×D4—42no—
213C2×C3×D4Dic614yescomplemented
214C2×C3×D4C3×Q814yescomplemented
215C2×C3×D4C22×D312yes—
216C2×C3×D4C23×C312yes—
217C22×C3×C4—21no—
218C22×C3×C4—62no—
219C22×C3×C4—62no—
220C22×C3×C4—61no—
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C23×C311yescharacteristic
223C22×Dic3—21no—
224C22×Dic3—22no—
225C22×Dic3—22no—
226C2×Dic6—22no—
227C3×M16—22no—
228C3×M16—22no—
229C3×M16—42no—
230C2×C3×C8—22no—
231C2×C3×C8—31no—
232C2×C3×C8—31no—
233C2×C3×C8—62no—
234C2×C3×C8C22×D312yes—
235C2×C3×C8C23×C312yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4—62no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4C22×D311yescharacteristic
248C3×C4⋊C4C22×D311yescharacteristic
249C3×C4⋊C4C23×C311yescharacteristic
250C3×C4⋊C4C23×C311yescharacteristic
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—32no—
254C3×C22⋊C4—41no—
255C3×C22⋊C4—62no—
256C3×C22⋊C4—62no—
257C3×C22⋊C4—61no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—32no—
261C3×C42—41no—
262C3×C42—61no—
263#217—21no—
264#217—22no—
265#217—22no—
266#217C3×D412yes—
267C4⋊Dic3—21no—
268C4⋊Dic3—22no—
269C4⋊Dic3C3×D411yescharacteristic
270C4⋊Dic3C3×D411yescharacteristic
271C4⋊Dic3C23×C311yescharacteristic
272C4⋊Dic3C23×C311yescharacteristic
273C4⋊Dic3C23×C312yes—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276C4×Dic3—21no—
277C4×Dic3C23×C312yes—
278C2×C3⋊C8—22no—
279C2×C3⋊C8C23×C311yescharacteristic
280C2×C3⋊C8C23×C311yescharacteristic
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—24no—
284C22×C32—42no—
285C22×C32—42no—
286C22×C32—84no—
287C22×C32C22⋊Q812yes—
288C22×C32C22×D411yescharacteristic
289C32×C4—41no—
290C32×C4—41no—
291C32×C4C22⋊Q812yes—
292C32×C4C22×Q811yescharacteristic
293C32×C4C2×#4111yescharacteristic
294C3×Dic3—22no—
295C3×Dic3—22no—
296C3×Dic3—42no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C22×D4C2×C3×D311yescharacteristic
300C2×M16C2×C3×D311yescharacteristic
301C22×C8—31no—
302C4⋊Q8—32no—
303C4⋊Q8—61no—
304#125—32no—
305#125—61no—
306C22⋊Q8—62no—
307C4×D4—32no—
308C4×D4—32no—
309C4×D4—61no—
310C4×D4—61no—
311#117—31no—
312#117—31no—
313#117C2×C3×D311yescharacteristic
314C2×C4⋊C4—31no—
315C2×C22⋊C4—62no—
316#107—32no—
317#107—31no—
318#107—31no—
319#106—32no—
320#106—31no—
321#106—31no—
322C4⋊C8—32no—
323C4⋊C8—32no—
324#102—32no—
325#102—32no—
326#102C2×C3×D312yes—
327#102C2×C3×D312yes—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—24no—
331C23×C3—24no—
332C23×C3—21no—
333C23×C3—42no—
334C23×C3—44no—
335C23×C3—81no—
336C23×C3—81no—
337C23×C3—81no—
338C23×C3—81no—
339C23×C3—82no—
340C23×C3—82no—
341C23×C3D3×D411yescharacteristic
342C23×C3C2×C3×D411yescharacteristic
343C3×Q8—242no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—24no—
349C3×D4—44no—
350C3×D4—42no—
351C3×D4C2×Dic614yes—
352C3×D4C2×C3×Q814yes—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—32no—
362C2×C3×C4—32no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—62no—
368C2×C3×C4—242no—
369C2×C3×C4—242no—
370C2×C3×C4—241no—
371C2×C3×C4C2×Dic612yes—
372C2×C3×C4D3×D411yescharacteristic
373C2×C3×C4#23712yes—
374C2×C3×C4C2×C3×D411yescharacteristic
375C2×C3×C4C2×C3×Q812yes—
376C2×C3×C4C3×#4112yes—
377C2×C3×C4C23×D311yescharacteristic
378C2×C3×C4C24×C311yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—82no—
383C2×Dic3—82no—
384C2×Dic3—81no—
385C2×Dic3C2×C3×D412yes—
386C2×Dic3C2×C3×D412yes—
387Dic6—82no—
388C3×C8—22no—
389C3×C8—22no—
390C3×C8—42no—
391C3×C8—61no—
392C3×C8—61no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C2×C32—22no—
396C2×C32—44no—
397C2×C32C2×C22⋊Q811yescharacteristic
398C2×C32#57611yescharacteristic
399C2×C32#57711yescharacteristic
400C24—21no—
401C2×Q8—62no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4C3×Dic614yescomplemented
405C2×D4C22×C3×D312yes—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4C22×C3×D311yescharacteristic
410M16—22no—
411C2×C8—31no—
412C2×C8—31no—
413C2×C8—62no—
414C2×C8C22×C3×D312yes—
415C4⋊C4—31no—
416C4⋊C4—31no—
417C4⋊C4—32no—
418C4⋊C4—31no—
419C4⋊C4—31no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4C22×C3×D311yescharacteristic
425C4⋊C4C22×C3×D311yescharacteristic
426C22⋊C4—21no—
427C22⋊C4—32no—
428C22⋊C4—62no—
429C22⋊C4—62no—
430C22⋊C4—61no—
431C42—21no—
432C42—32no—
433C42—61no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—24no—
439C22×C3—24no—
440C22×C3—22no—
441C22×C3—21no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—44no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—84no—
452C22×C3—84no—
453C22×C3—88no—
454C22×C3#87812yes—
455C22×C3C3×C22⋊Q812yes—
456C22×C3C2×D3×D411yescharacteristic
457C22×C3C22×C3×D411yescharacteristic
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—41no—
462C3×C4—41no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—62no—
466C3×C4—62no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4—121no—
470C3×C4—122no—
471C3×C4—121no—
472C3×C4#87812yes—
473C3×C4C3×C22⋊Q812yes—
474C3×C4C22×Dic611yescharacteristic
475C3×C4C2×#23711yescharacteristic
476C3×C4C22×C3×Q811yescharacteristic
477C3×C4C2×C3×#4111yescharacteristic
478Dic3—22no—
479Dic3—22no—
480Dic3—42no—
481Dic3—41no—
482Dic3—41no—
483C32#308111yescharacteristic, complemented, Sylow
484C23—22no—
485C23—24no—
486C23—81no—
487C23—81no—
488C23C3×D3×D411yescharacteristic
489Q8—242no—
490D4—24no—
491D4—22no—
492D4C2×C3×Dic614yes—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—242no—
501C2×C4—242no—
502C2×C4—241no—
503C2×C4C2×C3×Dic612yes—
504C2×C4C3×D3×D411yescharacteristic
505C2×C4C3×#23712yes—
506C2×C4C23×C3×D311yescharacteristic
507C8—22no—
508C8—61no—
509C8—61no—
510C2×C3—22no—
511C2×C3—22no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3—44no—
517C2×C3—44no—
518C2×C3—84no—
519C2×C3C2×#87811yescharacteristic
520C2×C3#577411yescharacteristic
521C2×C3#577511yescharacteristic
522C2×C3C2×C3×C22⋊Q811yescharacteristic
523C2×C3C3×#57611yescharacteristic
524C2×C3C3×#57711yescharacteristic
525C22—21no—
526C22—21no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—84no—
531C22C3×#87812yes—
532C22C2×C3×D3×D411yescharacteristic
533C4—41no—
534C4—41no—
535C4—62no—
536C4—62no—
537C4—121no—
538C4—122no—
539C4—121no—
540C4C3×#87812yes—
541C4C22×C3×Dic611yescharacteristic
542C4C2×C3×#23711yescharacteristic
543C3—21no—
544C3#7891811yescharacteristic, direct product factor
545C3C3×#308111yescharacteristic, complemented
546C2—22no—
547C2—44no—
548C2C2×C3×#87811yescharacteristic
549C2C3×#577411yescharacteristic
550C2C3×#577511yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#418C2
socleC22×C32C22×D4
prime radicalC22×C32×D4C22
hypercenterC3×#418D3
hypocenterC3C3×#3081
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#418

Complements

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

NH ≅ G/Ntype
2. C3×#5519C2semidirect
5. C3×#5121C2semidirect
6. C3×#5119C2semidirect
7. C3×#4998C2semidirect
8. C3×#4766C2semidirect
9. C3×#4763C2semidirect
10. C3×#4761C2semidirect
11. C3×#4759C2semidirect
13. #78918C3direct (N × H)
26. C3×#890C22semidirect
27. C3×#888C22semidirect
35. C3×#818C22semidirect
36. C3×#817C22semidirect
37. C3×Dic3⋊C8C22semidirect
38. C3×Dic3⋊C8C22semidirect
39. C3×#808C22semidirect
40. C3×#808C22semidirect
41. C3×#807C22semidirect
42. C3×#807C22semidirect
46. #5519C2×C3semidirect
59. #5121C2×C3semidirect
60. #5119C2×C3semidirect
61. #4998C2×C3semidirect
62. #4766C2×C3semidirect
63. #4763C2×C3semidirect
64. #4761C2×C3semidirect
65. #4759C2×C3semidirect
69. C2×C32×D4Q8semidirect
133. #890C22×C3semidirect
134. #888C22×C3semidirect
156. #818C22×C3semidirect
157. #817C22×C3semidirect
158. Dic3⋊C8C22×C3semidirect
159. Dic3⋊C8C22×C3semidirect
160. #808C22×C3semidirect
161. #808C22×C3semidirect
162. #807C22×C3semidirect
163. #807C22×C3semidirect
213. C2×C3×D4Dic6semidirect
214. C2×C3×D4C3×Q8semidirect
404. C2×D4C3×Dic6semidirect
483. C32#3081semidirect
544. C3#78918direct (N × H)
545. C3C3×#3081semidirect

Maximal subgroups

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

subgroupindex
C3×#57982
C3×#55192
C32×#4182
C2×C3×#8322
C3×#51212
C3×#51192
C3×#49982
C3×#47662
C3×#47632
C3×#47612
C3×#47592
C3×#30813
#789183