Finite Group Databasebeta

← C3×#78869 #11860917

Subgroups — C3×#78869 #11860917

Order 1152.

1049 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5827C211yesmaximal, characteristic, complemented
2C2×C3×#950C211yesmaximal, characteristic
3C3×#5504C211yesmaximal, characteristic, complemented
4C32×#417C211yesmaximal, characteristic, complemented
5C3×#5133C211yesmaximal, characteristic, complemented
6C3×#5122C211yesmaximal, characteristic, complemented
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C3×#4789C212yesmaximal, complemented
9C3×#4788C212yesmaximal, complemented
10C3×#4785C212yesmaximal, complemented
11C3×#4758C212yesmaximal, complemented
12C3×#2952—31nomaximal
13#78869C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2211yescharacteristic
15C2×C3×#235C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C3×#950C418yescomplemented
19C3×Q8×Dic3C2212yes—
20C2×C3×#216C2211yescharacteristic
21C2×C3×#215C2211yescharacteristic
22C3×D4×Dic3—22no—
23C3×D4×Dic3C2212yes—
24C2×C3×#214C2211yescharacteristic
25C2×C3×#213C2211yescharacteristic
26C3×C4×#67—22no—
27C3×#926—21no—
28C3×#926C2211yescharacteristic
29C2×C3×C4⋊Dic3—21no—
30C2×C3×C4⋊Dic3C2211yescharacteristic
31C22×C3×C3⋊C8C2211yescharacteristic
32C3×#893C2212yes—
33C3×#892—21no—
34C3×D3×C4⋊C4—21no—
35C3×#887C2212yes—
36C32×#103C2212yes—
37C32×#102C2212yes—
38C3×#821C2212yes—
39C3×#816C2212yes—
40C3×#815C2212yes—
41C3×C8×Dic3C2212yes—
42C3×#808C2212yes—
43C3×#808C2212yes—
44C2×C3×#135—31no—
45C3×#520—31no—
46C3×#520—31no—
47#5827C2×C311yescharacteristic, complemented
48C2×#950C2×C311yescharacteristic
49#5504C2×C311yescharacteristic, complemented
50C3×#442—32no—
51C3×#441—32no—
52C3×C4×SD16—32no—
53C3×C4×SD16—32no—
54C2×C3×#106—31no—
55C3×#417—21no—
56C3×#417—31no—
57C3×#417D311yescharacteristic, complemented
58C3×#417C2×C311yescharacteristic, complemented
59C3×#405—31no—
60#5133C2×C311yescharacteristic, complemented
61#5122C2×C311yescharacteristic, complemented
62C2×#808C2×C311yescharacteristic, complemented
63#4789C2×C312yescomplemented
64#4788C2×C312yescomplemented
65#4785C2×C312yescomplemented
66#4758C2×C312yescomplemented
67C32×#41—22no—
68C32×#41C2×C414yescomplemented
69C2×C32×Q8C2311yescharacteristic
70C2×C32×D4—21no—
71C2×C32×D4C2311yescharacteristic
72C22×C32×C4—21no—
73C22×C32×C4—21no—
74C22×C32×C4C2311yescharacteristic
75C2×C3×#67—21no—
76C22×C3×Dic3—21no—
77C22×C3×Dic3—22no—
78C3×#235—22no—
79C3×#235C2×C414yescomplemented
80C2×C3×D12C2311yescharacteristic
81C2×C3×C4×D3—21no—
82C2×C3×C4×D3—22no—
83C2×C3×Dic6C2311yescharacteristic
84C32×M16—22no—
85C2×C32×C8C2312yes—
86C32×C4⋊C4—21no—
87C32×C4⋊C4C2312yes—
88C3×#217—22no—
89C3×#217D412yes—
90C3×#216C2×C414yescomplemented
91C3×#215C2×C414yescomplemented
92C3×#214C2×C414yescomplemented
93C3×#213C2×C414yescomplemented
94C3×(C2×D3)⋊C4—22no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—22no—
97C3×C4⋊Dic3D411yescharacteristic
98C3×C4⋊Dic3D411yescharacteristic
99C3×C4⋊Dic3C2312yes—
100C3×Dic3⋊C4—22no—
101C3×C4×Dic3—22no—
102C3×C4×Dic3—22no—
103C3×C4×Dic3C2312yes—
104C2×C3×C3⋊C8C2×C414yescomplemented
105C2×C3×C3⋊C8C2311yescharacteristic
106C2×C3×C3⋊C8C2311yescharacteristic
107#2952—31noSylow
108C2×C3×#41—21no—
109C2×C3×#41—31no—
110C2×C3×#41C2×D311yescharacteristic
111C2×C3×#41C22×C311yescharacteristic
112C2×#235C22×C311yescharacteristic
113C3×#135—38no—
114C2×C3×Q16—31no—
115C2×C3×SD16—31no—
116C2×C3×SD16—31no—
117C2×C3×D8—31no—
118C2×C3×M16—21no—
119C2×C3×M16C2×D311yescharacteristic
120C2×C3×M16C22×C311yescharacteristic
121C22×C3×C8—31no—
122C3×C4×Q8—32no—
123C3×C4×Q8—32no—
124C3×C4×D4—32no—
125C3×C4×D4—32no—
126C3×C4×D4—62no—
127C3×C4×D4—62no—
128C3×#117—31no—
129C3×#117—61no—
130C3×#117—61no—
131C2×C3×C4⋊C4—21no—
132C2×C3×C4⋊C4—31no—
133C2×C3×C4⋊C4—61no—
134C2×C3×C4⋊C4—61no—
135C2×C3×C4⋊C4C2×D311yescharacteristic
136C2×C3×C4⋊C4C22×C311yescharacteristic
137#950C3×C418yescomplemented
138Q8×Dic3C22×C312yes—
139C2×#216C22×C311yescharacteristic
140C2×#215C22×C311yescharacteristic
141D4×Dic3—22no—
142D4×Dic3C22×C312yes—
143C2×#214C22×C311yescharacteristic
144C2×#213C22×C311yescharacteristic
145C4×#67—22no—
146#926—21no—
147#926C22×C311yescharacteristic
148C2×C4⋊Dic3—21no—
149C2×C4⋊Dic3C22×C311yescharacteristic
150C22×C3⋊C8C22×C311yescharacteristic
151#893C22×C312yes—
152#892—21no—
153D3×C4⋊C4—21no—
154#887C22×C312yes—
155C3×#106—32no—
156C3×#106—32no—
157C3×#103—22no—
158C3×#103—32no—
159C3×#103C2×D312yes—
160C3×#103C22×C312yes—
161C3×#102—22no—
162C3×#102—32no—
163C3×#102C2×D312yes—
164C3×#102C22×C312yes—
165C3×#97—32no—
166C3×C4×C8—32no—
167#821C22×C312yes—
168#816C22×C312yes—
169#815C22×C312yes—
170C8×Dic3C22×C312yes—
171#808C22×C312yes—
172#808C22×C312yes—
173C23×C32—21no—
174C23×C32C2×D411yescharacteristic
175C22×C3×D3—21no—
176C32×Q8—21no—
177C32×Q8C22×C412yes—
178C32×D4—24no—
179C32×D4—21no—
180C32×D4C22×C412yes—
181C2×C32×C4—21no—
182C2×C32×C4—24no—
183C2×C32×C4—22no—
184C2×C32×C4—81no—
185C2×C32×C4C22×C412yes—
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4#4112yes—
188C2×C32×C4C2411yescharacteristic
189C3×#67—24no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—82no—
193C2×C3×Dic3—81no—
194C2×C3×Dic3C2×D412yes—
195C2×C3×Dic3C2×D412yes—
196C3×D12—21no—
197C3×D12C22×C412yes—
198C3×C4×D3—24no—
199C3×C4×D3—82no—
200C3×Dic6—21no—
201C3×Dic6C22×C412yes—
202C32×C8—22no—
203C3×C3⋊C8C22×C412yes—
204C3×C3⋊C8C22×C412yes—
205C2×#135—31no—
206#520—31no—
207#520—31no—
208#442—32no—
209#441—32no—
210C4×SD16—32no—
211C4×SD16—32no—
212C2×#106—31no—
213#417—31no—
214#417C3×D311yescharacteristic, complemented
215#405—31no—
216C3×#41—22no—
217C3×#41—22no—
218C3×#41—24no—
219C3×#41—34no—
220C3×#41—42no—
221C3×#41—62no—
222C3×#41C4×D314yescomplemented
223C3×#41C2×C3×C414yescomplemented
224C2×C3×Q8—21no—
225C2×C3×Q8—31no—
226C2×C3×Q8C22×D311yescharacteristic
227C2×C3×Q8C23×C311yescharacteristic
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—31no—
232C2×C3×D4—41no—
233C2×C3×D4—61no—
234C2×C3×D4C22×D311yescharacteristic
235C2×C3×D4C23×C311yescharacteristic
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—41no—
242C22×C3×C4—41no—
243C22×C3×C4—62no—
244C22×C3×C4—62no—
245C22×C3×C4—61no—
246C22×C3×C4—61no—
247C22×C3×C4C22×D311yescharacteristic
248C22×C3×C4C23×C311yescharacteristic
249C2×#67—21no—
250C22×Dic3—21no—
251C22×Dic3—22no—
252#235—22no—
253#235C2×C3×C414yescomplemented
254C2×D12C23×C311yescharacteristic
255C2×C4×D3—21no—
256C2×C4×D3—22no—
257C2×Dic6C23×C311yescharacteristic
258C3×Q16—34no—
259C3×SD16—34no—
260C3×SD16—34no—
261C3×D8—34no—
262C3×M16—22no—
263C3×M16—22no—
264C3×M16—42no—
265C2×C3×C8—22no—
266C2×C3×C8—34no—
267C2×C3×C8—31no—
268C2×C3×C8—31no—
269C2×C3×C8C22×D312yes—
270C2×C3×C8C23×C312yes—
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—31no—
275C3×C4⋊C4—31no—
276C3×C4⋊C4—32no—
277C3×C4⋊C4—41no—
278C3×C4⋊C4—62no—
279C3×C4⋊C4—62no—
280C3×C4⋊C4—61no—
281C3×C4⋊C4C22×D312yes—
282C3×C4⋊C4C23×C312yes—
283C3×C22⋊C4—32no—
284C3×C22⋊C4—62no—
285C3×C22⋊C4—62no—
286C3×C42—32no—
287C3×C42—62no—
288C3×C42—62no—
289#217—22no—
290#217C3×D412yes—
291#216C2×C3×C414yescomplemented
292#215C2×C3×C414yescomplemented
293#214C2×C3×C414yescomplemented
294#213C2×C3×C414yescomplemented
295(C2×D3)⋊C4—22no—
296C4⋊Dic3—21no—
297C4⋊Dic3—22no—
298C4⋊Dic3C3×D411yescharacteristic
299C4⋊Dic3C3×D411yescharacteristic
300C4⋊Dic3C23×C312yes—
301Dic3⋊C4—22no—
302C4×Dic3—22no—
303C4×Dic3—22no—
304C4×Dic3C23×C312yes—
305C2×C3⋊C8C2×C3×C414yescomplemented
306C2×C3⋊C8C23×C311yescharacteristic
307C2×C3⋊C8C23×C311yescharacteristic
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—22no—
311C22×C32—41no—
312C22×C32—41no—
313C22×C32C4×D412yes—
314C22×C32C22×D411yescharacteristic
315C2×C3×D3—22no—
316C2×C3×D3—41no—
317C2×C3×D3—41no—
318C32×C4—22no—
319C32×C4—42no—
320C32×C4C4×D412yes—
321C32×C4C23×C411yescharacteristic
322C32×C4C2×#4111yescharacteristic
323C3×Dic3—22no—
324C3×Dic3—22no—
325C3×Dic3—22no—
326C3×Dic3—42no—
327C2×#41—31no—
328C2×#41C2×C3×D311yescharacteristic
329#135—38no—
330C2×Q16—31no—
331C2×SD16—31no—
332C2×SD16—31no—
333C2×D8—31no—
334C2×M16C2×C3×D311yescharacteristic
335C22×C8—31no—
336C4×Q8—32no—
337C4×Q8—32no—
338C4×D4—32no—
339C4×D4—32no—
340C4×D4—62no—
341C4×D4—62no—
342#117—31no—
343#117—61no—
344#117—61no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4C2×C3×D311yescharacteristic
349#106—32no—
350#106—32no—
351#103—32no—
352#103C2×C3×D312yes—
353#102—32no—
354#102C2×C3×D312yes—
355#97—32no—
356C4×C8—32no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—41no—
361C23×C3—61no—
362C23×C3D3×D411yescharacteristic
363C23×C3C2×C3×D411yescharacteristic
364C22×D3—21no—
365C3×Q8—21no—
366C3×Q8—21no—
367C3×Q8—22no—
368C3×Q8—32no—
369C3×Q8—41no—
370C3×Q8—61no—
371C3×Q8C2×C4×D312yes—
372C3×Q8C22×C3×C412yes—
373C3×D4—24no—
374C3×D4—24no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—22no—
378C3×D4—32no—
379C3×D4—41no—
380C3×D4—44no—
381C3×D4—64no—
382C3×D4—61no—
383C3×D4C2×C4×D312yes—
384C3×D4C22×C3×C412yes—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—24no—
388C2×C3×C4—24no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—32no—
396C2×C3×C4—32no—
397C2×C3×C4—44no—
398C2×C3×C4—41no—
399C2×C3×C4—42no—
400C2×C3×C4—61no—
401C2×C3×C4—64no—
402C2×C3×C4—62no—
403C2×C3×C4—81no—
404C2×C3×C4—81no—
405C2×C3×C4—82no—
406C2×C3×C4—242no—
407C2×C3×C4—242no—
408C2×C3×C4—241no—
409C2×C3×C4C2×C4×D312yes—
410C2×C3×C4D3×D411yescharacteristic
411C2×C3×C4#23712yes—
412C2×C3×C4C22×C3×C412yes—
413C2×C3×C4C2×C3×D411yescharacteristic
414C2×C3×C4C3×#4112yes—
415C2×C3×C4C23×D311yescharacteristic
416C2×C3×C4C24×C311yescharacteristic
417#67—24no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—82no—
421C2×Dic3—81no—
422C2×Dic3C2×C3×D412yes—
423C2×Dic3C2×C3×D412yes—
424D12—21no—
425D12C22×C3×C412yes—
426C4×D3—24no—
427C4×D3—82no—
428Dic6—21no—
429Dic6C22×C3×C412yes—
430C3×C8—22no—
431C3×C8—22no—
432C3×C8—32no—
433C3×C8—32no—
434C3×C8—42no—
435C3⋊C8C22×C3×C412yes—
436C3⋊C8C22×C3×C412yes—
437C2×C32—22no—
438C2×C32—42no—
439C2×C32C2×C4×D411yescharacteristic
440C2×C32#57711yescharacteristic
441C2×C32#57711yescharacteristic
442C3×D3—42no—
443#41—22no—
444#41—34no—
445#41—62no—
446#41C3×C4×D314yescomplemented
447C2×Q8—31no—
448C2×Q8C22×C3×D311yescharacteristic
449C2×D4—21no—
450C2×D4—31no—
451C2×D4—61no—
452C2×D4C22×C3×D311yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4C22×C3×D311yescharacteristic
460Q16—34no—
461SD16—34no—
462SD16—34no—
463D8—34no—
464M16—22no—
465C2×C8—31no—
466C2×C8—31no—
467C2×C8—34no—
468C2×C8C22×C3×D312yes—
469C4⋊C4—21no—
470C4⋊C4—31no—
471C4⋊C4—31no—
472C4⋊C4—32no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C4⋊C4—61no—
476C4⋊C4C22×C3×D312yes—
477C22⋊C4—32no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C42—32no—
481C42—62no—
482C42—62no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—62no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—121no—
502C22×C3—121no—
503C22×C3#67⋊C412yes—
504C22×C3C3×C4×D412yes—
505C22×C3C2×D3×D411yescharacteristic
506C22×C3C22×C3×D411yescharacteristic
507C2×D3—22no—
508C2×D3—41no—
509C2×D3—41no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—22no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—62no—
519C3×C4—62no—
520C3×C4—62no—
521C3×C4—82no—
522C3×C4—122no—
523C3×C4#67⋊C412yes—
524C3×C4C3×C4×D412yes—
525C3×C4C22×C4×D311yescharacteristic
526C3×C4C2×#23711yescharacteristic
527C3×C4C23×C3×C411yescharacteristic
528C3×C4C2×C3×#4111yescharacteristic
529Dic3—22no—
530Dic3—22no—
531Dic3—22no—
532Dic3—42no—
533C32#295211yescharacteristic, complemented, Sylow
534C23—21no—
535C23—61no—
536C23C3×D3×D411yescharacteristic
537Q8—21no—
538Q8—32no—
539Q8—61no—
540Q8C2×C3×C4×D312yes—
541D4—24no—
542D4—21no—
543D4—32no—
544D4—64no—
545D4—61no—
546D4C2×C3×C4×D312yes—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—32no—
551C2×C4—32no—
552C2×C4—61no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—81no—
556C2×C4—242no—
557C2×C4—242no—
558C2×C4—241no—
559C2×C4C2×C3×C4×D312yes—
560C2×C4C3×D3×D411yescharacteristic
561C2×C4C3×#23712yes—
562C2×C4C23×C3×D311yescharacteristic
563C8—22no—
564C8—32no—
565C8—32no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—42no—
572C2×C3—42no—
573C2×C3—42no—
574C2×C3—82no—
575C2×C3—122no—
576C2×C3C2×#67⋊C411yescharacteristic
577C2×C3#577511yescharacteristic
578C2×C3#577811yescharacteristic
579C2×C3C2×C3×C4×D411yescharacteristic
580C2×C3C3×#57711yescharacteristic
581C2×C3C3×#57711yescharacteristic
582D3—42no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—62no—
589C22—121no—
590C22—121no—
591C22C3×#67⋊C412yes—
592C22C2×C3×D3×D411yescharacteristic
593C4—22no—
594C4—42no—
595C4—62no—
596C4—62no—
597C4—62no—
598C4—122no—
599C4C3×#67⋊C412yes—
600C4C22×C3×C4×D311yescharacteristic
601C4C2×C3×#23711yescharacteristic
602C3—21no—
603C3#7886911yescharacteristic, direct product factor
604C3C3×#295211yescharacteristic, complemented
605C2—22no—
606C2—42no—
607C2—122no—
608C2C2×C3×#67⋊C411yescharacteristic
609C2C3×#577511yescharacteristic
610C2C3×#577811yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#417C2
socleC22×C32C22×D4
prime radicalC2×C3×#950C2
cosocleC2×C3×C4C24×C3
hypercenterC3×#417D3
hypocenterC3C3×#2952
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#417

Complements

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

NH ≅ G/Ntype
1. C3×#5827C2semidirect
3. C3×#5504C2semidirect
4. C32×#417C2semidirect
5. C3×#5133C2semidirect
6. C3×#5122C2semidirect
7. C2×C3×#808C2semidirect
8. C3×#4789C2semidirect
9. C3×#4788C2semidirect
10. C3×#4785C2semidirect
11. C3×#4758C2semidirect
13. #78869C3direct (N × H)
18. C3×#950C4semidirect
47. #5827C2×C3semidirect
49. #5504C2×C3semidirect
57. C3×#417D3semidirect
58. C3×#417C2×C3semidirect
60. #5133C2×C3semidirect
61. #5122C2×C3semidirect
62. C2×#808C2×C3semidirect
63. #4789C2×C3semidirect
64. #4788C2×C3semidirect
65. #4785C2×C3semidirect
66. #4758C2×C3semidirect
68. C32×#41C2×C4semidirect
79. C3×#235C2×C4semidirect
90. C3×#216C2×C4semidirect
91. C3×#215C2×C4semidirect
92. C3×#214C2×C4semidirect
93. C3×#213C2×C4semidirect
104. C2×C3×C3⋊C8C2×C4semidirect
137. #950C3×C4semidirect
214. #417C3×D3semidirect
222. C3×#41C4×D3semidirect
223. C3×#41C2×C3×C4semidirect
253. #235C2×C3×C4semidirect
291. #216C2×C3×C4semidirect
292. #215C2×C3×C4semidirect
293. #214C2×C3×C4semidirect
294. #213C2×C3×C4semidirect
305. C2×C3⋊C8C2×C3×C4semidirect
446. #41C3×C4×D3semidirect
533. C32#2952semidirect
603. C3#78869direct (N × H)
604. C3C3×#2952semidirect

Maximal subgroups

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

subgroupindex
C3×#58272
C2×C3×#9502
C3×#55042
C32×#4172
C3×#51332
C3×#51222
C2×C3×#8082
C3×#47892
C3×#47882
C3×#47852
C3×#47582
C3×#29523
#788693