Finite Group Databasebeta

← C3×#78917 #11868643

Subgroups — C3×#78917 #11868643

Order 1152.

975 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic
2C3×#5517C211yesmaximal, characteristic, complemented
3C32×#418C211yesmaximal, characteristic
4C3×#5215C211yesmaximal, characteristic
5C3×#5121C211yesmaximal, characteristic, complemented
6C3×#5118C211yesmaximal, characteristic, complemented
7C3×#4995C211yesmaximal, characteristic, complemented
8C3×#4766C212yesmaximal, complemented
9C3×#4763C212yesmaximal, complemented
10C3×#4761C212yesmaximal, complemented
11C3×#4759C212yesmaximal, complemented
12C3×#3077—31nomaximal
13#78917C311yesmaximal, characteristic, direct product factor
14C22×C32×D4C2211yescharacteristic
15C23×C3×Dic3—21no—
16C2×C32×M16C2211yescharacteristic
17C32×#117C2211yescharacteristic
18C2×C3×#217—21no—
19C3×D4×Dic3—21no—
20C3×D4×Dic3—21no—
21C3×D4×Dic3C2212yes—
22C3×D4×Dic3C2212yes—
23C3×#926C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27C3×#890C2211yescharacteristic, complemented
28C3×#890C2211yescharacteristic, complemented
29C3×#888C2211yescharacteristic, complemented
30C3×#888C2211yescharacteristic, complemented
31C3×#878—21no—
32C3×#878—21no—
33C3×C4×Dic6—21no—
34C3×C4×Dic6—21no—
35C32×#102C2212yes—
36C32×#102C2212yes—
37C3×#832C2212yes—
38C3×#832C2212yes—
39C3×#818C2212yescomplemented
40C3×#817C2212yescomplemented
41C3×Dic3⋊C8C2212yescomplemented
42C3×Dic3⋊C8C2212yescomplemented
43C3×#808C2212yescomplemented
44C3×#807C2212yescomplemented
45C3×#533—31no—
46C2×C3×C4×D4—31no—
47C2×D4×Dic3C2×C311yescharacteristic
48#5517C2×C311yescharacteristic, complemented
49C3×#478—32no—
50C3×#478—32no—
51C3×#476—32no—
52C3×#474—32no—
53C3×#428—31no—
54C3×#428—31no—
55C3×#423—31no—
56C3×#418—21no—
57C3×#418—31no—
58C3×#418D311yescharacteristic
59C3×#418C2×C311yescharacteristic
60#5215C2×C311yescharacteristic
61#5121C2×C311yescharacteristic, complemented
62#5118C2×C311yescharacteristic, complemented
63#4995C2×C311yescharacteristic, complemented
64#4766C2×C312yescomplemented
65#4763C2×C312yescomplemented
66#4761C2×C312yescomplemented
67#4759C2×C312yescomplemented
68C24×C32—21no—
69C2×C32×D4—21no—
70C2×C32×D4—22no—
71C2×C32×D4Q814yescomplemented
72C2×C32×D4C2312yes—
73C22×C32×C4C2311yescharacteristic
74C22×C3×Dic3—21no—
75C22×C3×Dic3—22no—
76C22×C3×Dic3—22no—
77C22×C3×Dic3—44no—
78C22×C3×Dic3D411yescharacteristic
79C22×C3×Dic3D411yescharacteristic
80C2×C3×Dic6—21no—
81C2×C3×Dic6—21no—
82C32×M16—22no—
83C2×C32×C8C2312yes—
84C32×C4⋊C4C2311yescharacteristic
85C32×C4⋊C4C2311yescharacteristic
86C32×C22⋊C4—21no—
87C32×C42—21no—
88C3×#217—21no—
89C3×#217—22no—
90C3×#217—22no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3C2311yescharacteristic
95C3×C4⋊Dic3C2311yescharacteristic
96C3×C4⋊Dic3C2312yes—
97C3×Dic3⋊C4—21no—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3D411yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C3×C4×Dic3C2311yescharacteristic
106C3×#208—22no—
107C2×C3×C3⋊C8C2312yes—
108#3077—31noSylow
109C22×C3×D4—21no—
110C22×C3×D4C2×D311yescharacteristic
111C22×C3×D4C22×C311yescharacteristic
112C23×C3×C4—61no—
113C23×Dic3—21no—
114C2×C3×M16—21no—
115C2×C3×M16—31no—
116C2×C3×M16C2×D311yescharacteristic
117C2×C3×M16C22×C311yescharacteristic
118C3×C4⋊Q8—31no—
119C3×C4⋊Q8—31no—
120C3×#125—31no—
121C3×#125—31no—
122C3×C22⋊Q8—61no—
123C3×C22⋊Q8—61no—
124C3×C4×Q8—61no—
125C3×C4×Q8—61no—
126C3×C4×D4—32no—
127C3×C4×D4—32no—
128C3×C4×D4—61no—
129C3×C4×D4—61no—
130C3×#117—21no—
131C3×#117—31no—
132C3×#117C2×D311yescharacteristic
133C3×#117C22×C311yescharacteristic
134C2×C3×C4⋊C4—31no—
135C2×C3×C22⋊C4—61no—
136C2×C3×C42—31no—
137C2×#217—21no—
138D4×Dic3—21no—
139D4×Dic3—21no—
140D4×Dic3C22×C312yes—
141D4×Dic3C22×C312yes—
142#926C22×C311yescharacteristic
143C2×C4⋊Dic3C22×C311yescharacteristic
144C2×C4×Dic3C22×C311yescharacteristic
145C2×#208C22×C311yescharacteristic
146#890C22×C311yescharacteristic, complemented
147#890C22×C311yescharacteristic, complemented
148#888C22×C311yescharacteristic, complemented
149#888C22×C311yescharacteristic, complemented
150#878—21no—
151#878—21no—
152C4×Dic6—21no—
153C4×Dic6—21no—
154C3×#107—32no—
155C3×#107—32no—
156C3×#106—32no—
157C3×#106—32no—
158C3×C4⋊C8—32no—
159C3×C4⋊C8—32no—
160C3×#102—22no—
161C3×#102—22no—
162C3×#102—32no—
163C3×#102—32no—
164C3×#102C2×D312yes—
165C3×#102C2×D312yes—
166C3×#102C22×C312yes—
167C3×#102C22×C312yes—
168#832C22×C312yes—
169#832C22×C312yes—
170#818C22×C312yescomplemented
171#817C22×C312yescomplemented
172Dic3⋊C8C22×C312yescomplemented
173Dic3⋊C8C22×C312yescomplemented
174#808C22×C312yescomplemented
175#807C22×C312yescomplemented
176C23×C32—24no—
177C23×C32—22no—
178C23×C32—42no—
179C23×C32—42no—
180C23×C32C2×D411yescharacteristic
181C32×D4—24no—
182C32×D4—22no—
183C32×D4C2×Q814yes—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2×Q812yes—
188C2×C32×C4#4112yes—
189C2×C32×C4C2411yescharacteristic
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—44no—
196C2×C3×Dic3—42no—
197C2×C3×Dic3—44no—
198C2×C3×Dic3C2×D411yescharacteristic
199C2×C3×Dic3C2×D411yescharacteristic
200C2×C3×Dic3C2×D411yescharacteristic
201C2×C3×Dic3C2×D411yescharacteristic
202C3×Dic6—42no—
203C3×Dic6—42no—
204C32×C8—22no—
205C3×C3⋊C8—22no—
206#533—31no—
207C2×C4×D4—31no—
208#478—32no—
209#478—32no—
210#476—32no—
211#474—32no—
212#428—31no—
213#428—31no—
214#423—31no—
215#418—31no—
216#418C3×D311yescharacteristic
217C24×C3—21no—
218C24×C3—21no—
219C24×C3—41no—
220C2×C3×Q8—61no—
221C2×C3×Q8—61no—
222C2×C3×D4—21no—
223C2×C3×D4—22no—
224C2×C3×D4—21no—
225C2×C3×D4—22no—
226C2×C3×D4—24no—
227C2×C3×D4—22no—
228C2×C3×D4—41no—
229C2×C3×D4—42no—
230C2×C3×D4Dic614yescomplemented
231C2×C3×D4C3×Q814yescomplemented
232C2×C3×D4C22×D312yes—
233C2×C3×D4C23×C312yes—
234C22×C3×C4—21no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—62no—
238C22×C3×C4—62no—
239C22×C3×C4—61no—
240C22×C3×C4—124no—
241C22×C3×C4C22×D311yescharacteristic
242C22×C3×C4C23×C311yescharacteristic
243C22×Dic3—21no—
244C22×Dic3—22no—
245C22×Dic3—22no—
246C22×Dic3—44no—
247C22×Dic3C3×D411yescharacteristic
248C22×Dic3C3×D411yescharacteristic
249C2×Dic6—21no—
250C2×Dic6—21no—
251C3×M16—22no—
252C3×M16—22no—
253C3×M16—42no—
254C3×M16—62no—
255C2×C3×C8—22no—
256C2×C3×C8—32no—
257C2×C3×C8C22×D312yes—
258C2×C3×C8C23×C312yes—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—32no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4—61no—
270C3×C4⋊C4—61no—
271C3×C4⋊C4C22×D311yescharacteristic
272C3×C4⋊C4C22×D311yescharacteristic
273C3×C4⋊C4C23×C311yescharacteristic
274C3×C4⋊C4C23×C311yescharacteristic
275C3×C22⋊C4—21no—
276C3×C22⋊C4—21no—
277C3×C22⋊C4—41no—
278C3×C22⋊C4—62no—
279C3×C22⋊C4—62no—
280C3×C22⋊C4—61no—
281C3×C42—21no—
282C3×C42—21no—
283C3×C42—31no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—31no—
287C3×C42—41no—
288C3×C42—61no—
289#217—21no—
290#217—22no—
291#217—22no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294C4⋊Dic3—21no—
295C4⋊Dic3C23×C312yes—
296C4⋊Dic3C23×C311yescharacteristic
297C4⋊Dic3C23×C311yescharacteristic
298Dic3⋊C4—21no—
299Dic3⋊C4—21no—
300Dic3⋊C4—21no—
301Dic3⋊C4—21no—
302C4×Dic3—21no—
303C4×Dic3C3×D411yescharacteristic
304C4×Dic3C3×D411yescharacteristic
305C4×Dic3C23×C311yescharacteristic
306C4×Dic3C23×C311yescharacteristic
307#208—22no—
308C2×C3⋊C8C23×C312yes—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—24no—
312C22×C32—48no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32C22⋊Q812yes—
316C22×C32C22×D411yescharacteristic
317C32×C4—41no—
318C32×C4—41no—
319C32×C4C22⋊Q812yes—
320C32×C4C22×Q811yescharacteristic
321C32×C4C2×#4111yescharacteristic
322C3×Dic3—21no—
323C3×Dic3—21no—
324C3×Dic3—21no—
325C3×Dic3—41no—
326C3×Dic3—42no—
327C3×Dic3—41no—
328C3×Dic3AGL(1,ℤ/8)12yes—
329C22×D4C2×C3×D311yescharacteristic
330C23×C4—61no—
331C2×M16—31no—
332C2×M16C2×C3×D311yescharacteristic
333C4⋊Q8—31no—
334C4⋊Q8—31no—
335#125—31no—
336#125—31no—
337C22⋊Q8—61no—
338C22⋊Q8—61no—
339C4×Q8—61no—
340C4×Q8—61no—
341C4×D4—32no—
342C4×D4—32no—
343C4×D4—61no—
344C4×D4—61no—
345#117—31no—
346#117C2×C3×D311yescharacteristic
347C2×C4⋊C4—31no—
348C2×C22⋊C4—61no—
349C2×C42—31no—
350#107—32no—
351#107—32no—
352#106—32no—
353#106—32no—
354C4⋊C8—32no—
355C4⋊C8—32no—
356#102—32no—
357#102—32no—
358#102C2×C3×D312yes—
359#102C2×C3×D312yes—
360C23×C3—24no—
361C23×C3—24no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—21no—
365C23×C3—44no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3—42no—
369C23×C3—42no—
370C23×C3—42no—
371C23×C3—82no—
372C23×C3—82no—
373C23×C3D3×D411yescharacteristic
374C23×C3C2×C3×D411yescharacteristic
375C3×Q8—122no—
376C3×Q8—122no—
377C3×D4—24no—
378C3×D4—24no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—24no—
382C3×D4—44no—
383C3×D4—42no—
384C3×D4C2×Dic614yes—
385C3×D4C2×C3×Q814yes—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—31no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—62no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4—62no—
404C2×C3×C4—124no—
405C2×C3×C4—121no—
406C2×C3×C4—124no—
407C2×C3×C4—122no—
408C2×C3×C4C2×Dic612yes—
409C2×C3×C4D3×D411yescharacteristic
410C2×C3×C4#23712yes—
411C2×C3×C4C2×C3×D411yescharacteristic
412C2×C3×C4C2×C3×Q812yes—
413C2×C3×C4C3×#4112yes—
414C2×C3×C4C23×D311yescharacteristic
415C2×C3×C4C24×C311yescharacteristic
416C2×Dic3—22no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—41no—
421C2×Dic3—44no—
422C2×Dic3—42no—
423C2×Dic3—44no—
424C2×Dic3C2×C3×D411yescharacteristic
425C2×Dic3C2×C3×D411yescharacteristic
426C2×Dic3C2×C3×D411yescharacteristic
427C2×Dic3C2×C3×D411yescharacteristic
428Dic6—42no—
429Dic6—42no—
430C3×C8—22no—
431C3×C8—22no—
432C3×C8—42no—
433C3×C8—62no—
434C3⋊C8—22no—
435C2×C32—22no—
436C2×C32—44no—
437C2×C32C2×C22⋊Q811yescharacteristic
438C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C32#57511yescharacteristic
440C24—21no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4C3×Dic614yescomplemented
446C2×D4C22×C3×D312yes—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—61no—
452C22×C4—124no—
453C22×C4C22×C3×D311yescharacteristic
454M16—22no—
455M16—62no—
456C2×C8—32no—
457C2×C8C22×C3×D312yes—
458C4⋊C4—31no—
459C4⋊C4—31no—
460C4⋊C4—32no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4C22×C3×D311yescharacteristic
469C4⋊C4C22×C3×D311yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—62no—
472C22⋊C4—62no—
473C22⋊C4—61no—
474C42—21no—
475C42—31no—
476C42—31no—
477C42—31no—
478C42—31no—
479C42—61no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—22no—
487C22×C3—21no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—44no—
491C22×C3—48no—
492C22×C3—48no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—88no—
500C22×C3#87812yes—
501C22×C3C3×C22⋊Q812yes—
502C22×C3C2×D3×D411yescharacteristic
503C22×C3C22×C3×D411yescharacteristic
504C3×C4—22no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—32no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—61no—
513C3×C4—61no—
514C3×C4—61no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4—122no—
518C3×C4—121no—
519C3×C4—121no—
520C3×C4#87812yes—
521C3×C4C3×C22⋊Q812yes—
522C3×C4C22×Dic611yescharacteristic
523C3×C4C2×#23711yescharacteristic
524C3×C4C22×C3×Q811yescharacteristic
525C3×C4C2×C3×#4111yescharacteristic
526Dic3—21no—
527Dic3—21no—
528Dic3—21no—
529Dic3—41no—
530Dic3—42no—
531Dic3—41no—
532Dic3C3×AGL(1,ℤ/8)12yes—
533C32#307711yescharacteristic, complemented, Sylow
534C23—24no—
535C23—22no—
536C23—42no—
537C23—42no—
538C23C3×D3×D411yescharacteristic
539Q8—122no—
540Q8—122no—
541D4—24no—
542D4—22no—
543D4C2×C3×Dic614yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—124no—
555C2×C4—121no—
556C2×C4—124no—
557C2×C4—122no—
558C2×C4C2×C3×Dic612yes—
559C2×C4C3×D3×D411yescharacteristic
560C2×C4C3×#23712yes—
561C2×C4C23×C3×D311yescharacteristic
562C8—22no—
563C8—62no—
564C2×C3—22no—
565C2×C3—22no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—42no—
570C2×C3—44no—
571C2×C3—44no—
572C2×C3—84no—
573C2×C3C2×#87811yescharacteristic
574C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
575C2×C3#577311yescharacteristic
576C2×C3C2×C3×C22⋊Q811yescharacteristic
577C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
578C2×C3C3×#57511yescharacteristic
579C22—21no—
580C22—21no—
581C22—24no—
582C22—48no—
583C22—42no—
584C22—42no—
585C22C3×#87812yes—
586C22C2×C3×D3×D411yescharacteristic
587C4—32no—
588C4—41no—
589C4—41no—
590C4—61no—
591C4—61no—
592C4—61no—
593C4—122no—
594C4—121no—
595C4—121no—
596C4C3×#87812yes—
597C4C22×C3×Dic611yescharacteristic
598C4C2×C3×#23711yescharacteristic
599C3—21no—
600C3#7891711yescharacteristic, direct product factor
601C3C3×#307711yescharacteristic, complemented
602C2—22no—
603C2—44no—
604C2C2×C3×#87811yescharacteristic
605C2C3×D3×AGL(1,ℤ/8)11yescharacteristic
606C2C3×#577311yescharacteristic
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×#3077
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×#5517C2semidirect
5. C3×#5121C2semidirect
6. C3×#5118C2semidirect
7. C3×#4995C2semidirect
8. C3×#4766C2semidirect
9. C3×#4763C2semidirect
10. C3×#4761C2semidirect
11. C3×#4759C2semidirect
13. #78917C3direct (N × H)
27. C3×#890C22semidirect
28. C3×#890C22semidirect
29. C3×#888C22semidirect
30. C3×#888C22semidirect
39. C3×#818C22semidirect
40. C3×#817C22semidirect
41. C3×Dic3⋊C8C22semidirect
42. C3×Dic3⋊C8C22semidirect
43. C3×#808C22semidirect
44. C3×#807C22semidirect
48. #5517C2×C3semidirect
61. #5121C2×C3semidirect
62. #5118C2×C3semidirect
63. #4995C2×C3semidirect
64. #4766C2×C3semidirect
65. #4763C2×C3semidirect
66. #4761C2×C3semidirect
67. #4759C2×C3semidirect
71. C2×C32×D4Q8semidirect
146. #890C22×C3semidirect
147. #890C22×C3semidirect
148. #888C22×C3semidirect
149. #888C22×C3semidirect
170. #818C22×C3semidirect
171. #817C22×C3semidirect
172. Dic3⋊C8C22×C3semidirect
173. Dic3⋊C8C22×C3semidirect
174. #808C22×C3semidirect
175. #807C22×C3semidirect
230. C2×C3×D4Dic6semidirect
231. C2×C3×D4C3×Q8semidirect
445. C2×D4C3×Dic6semidirect
533. C32#3077semidirect
600. C3#78917direct (N × H)
601. C3C3×#3077semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic32
C3×#55172
C32×#4182
C3×#52152
C3×#51212
C3×#51182
C3×#49952
C3×#47662
C3×#47632
C3×#47612
C3×#47592
C3×#30773
#789173