Finite Group Databasebeta

← C3×#78821 #11860588

Subgroups — C3×#78821 #11860588

Order 1152.

1019 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×#5827C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic
3C3×#5516C211yesmaximal, characteristic, complemented
4C32×#416C211yesmaximal, characteristic
5C2×C3×#816C211yesmaximal, characteristic, complemented
6C3×#5101C211yesmaximal, characteristic, complemented
7C3×#4995C211yesmaximal, characteristic, complemented
8C3×#4790C212yesmaximal, complemented
9C3×#4789C212yesmaximal, complemented
10C3×#4758C212yesmaximal, complemented
11C3×#4757C212yesmaximal, complemented
12C3×#2947—31nomaximal
13#78821C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C22×C32×C8C2211yescharacteristic
17C32×#117C2211yescharacteristic
18C3×#951C418yescomplemented
19C3×Q8×Dic3C2212yes—
20C2×C3×#216C2212yes—
21C3×D4×Dic3—22no—
22C3×D4×Dic3C2212yes—
23C2×C3×#214C2212yes—
24C3×#926—21no—
25C3×#926C2211yescharacteristic
26C2×C3×C4⋊Dic3—21no—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×#208C2211yescharacteristic
29C3×#887C2212yes—
30C3×#887C2212yes—
31C3×#877—22no—
32C3×C4×Dic6—21no—
33C3×C4×Dic6—21no—
34C32×#103C2211yescharacteristic
35C32×#103C2211yescharacteristic
36C32×#102C2211yescharacteristic
37C32×#102C2211yescharacteristic
38C3×#816C2212yes—
39C3×#816C2212yes—
40C3×#815C2212yes—
41C3×C8×Dic3C2212yes—
42C3×#808C2212yes—
43C3×#807C2212yes—
44C2×C3×#137—31no—
45C3×#520—31no—
46C3×#519—31no—
47#5827C2×C311yescharacteristic
48C2×#951C2×C311yescharacteristic
49#5516C2×C311yescharacteristic, complemented
50C3×#441—32no—
51C3×#440—32no—
52C3×C4×Q16—32no—
53C3×C4×SD16—32no—
54C3×#428—31no—
55C3×#416—21no—
56C3×#416D311yescharacteristic
57C3×#416C2×C311yescharacteristic
58C2×C3×#103—31no—
59C3×#405—31no—
60C2×#816C2×C311yescharacteristic, complemented
61#5101C2×C311yescharacteristic, complemented
62#4995C2×C311yescharacteristic, complemented
63#4790C2×C312yescomplemented
64#4789C2×C312yescomplemented
65#4758C2×C312yescomplemented
66#4757C2×C312yescomplemented
67C32×#41—22no—
68C32×#41C2×C414yes—
69C2×C32×Q8C2311yescharacteristic
70C2×C32×D4—21no—
71C2×C32×D4C2311yescharacteristic
72C22×C32×C4—21no—
73C22×C32×C4C2311yescharacteristic
74C22×C3×Dic3—21no—
75C22×C3×Dic3—21no—
76C22×C3×Dic3—22no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—22no—
79C2×C3×Dic6C2×C414yescomplemented
80C2×C3×Dic6C2312yes—
81C2×C32×C8—22no—
82C2×C32×C8C2311yescharacteristic
83C2×C32×C8C2311yescharacteristic
84C32×C4⋊C4C2311yescharacteristic
85C32×C4⋊C4C2311yescharacteristic
86C32×C22⋊C4—21no—
87C32×C42—21no—
88C3×#217—22no—
89C3×#217D412yes—
90C3×#216C2×C418yescomplemented
91C3×#214C2×C418yescomplemented
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—22no—
94C3×C4⋊Dic3D411yescharacteristic
95C3×C4⋊Dic3D411yescharacteristic
96C3×C4⋊Dic3C2312yes—
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—22no—
99C3×C4×Dic3—22no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3—22no—
102C3×C4×Dic3C2312yes—
103C3×#208C2×C414yescomplemented
104C2×C3×C3⋊C8C2312yes—
105#2947—31noSylow
106C2×C3×#41—21no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C22×C311yescharacteristic
109C22×C3×Q8—31no—
110C22×Dic6C22×C311yescharacteristic
111C3×#137—38no—
112C2×C3×Q16—32no—
113C2×C3×SD16—32no—
114C2×C3×M16—31no—
115C22×C3×C8—21no—
116C22×C3×C8C2×D311yescharacteristic
117C22×C3×C8C22×C311yescharacteristic
118C3×C4×Q8—32no—
119C3×C4×Q8—32no—
120C3×C4×Q8—32no—
121C3×C4×Q8—61no—
122C3×C4×Q8—61no—
123C3×C4×D4—32no—
124C3×C4×D4—62no—
125C3×#117—21no—
126C3×#117—31no—
127C3×#117—62no—
128C3×#117—61no—
129C3×#117C2×D311yescharacteristic
130C3×#117C22×C311yescharacteristic
131C2×C3×C4⋊C4—31no—
132C2×C3×C4⋊C4—61no—
133#951C3×C418yescomplemented
134Q8×Dic3C22×C312yes—
135C2×#216C22×C312yes—
136D4×Dic3—22no—
137D4×Dic3C22×C312yes—
138C2×#214C22×C312yes—
139#926—21no—
140#926C22×C311yescharacteristic
141C2×C4⋊Dic3—21no—
142C2×C4⋊Dic3C22×C311yescharacteristic
143C2×#208C22×C311yescharacteristic
144#887C22×C312yes—
145#887C22×C312yes—
146#877—22no—
147C4×Dic6—21no—
148C4×Dic6—21no—
149C3×#107—32no—
150C3×#106—32no—
151C3×#103—21no—
152C3×#103—21no—
153C3×#103—32no—
154C3×#103—32no—
155C3×#103C2×D311yescharacteristic
156C3×#103C2×D311yescharacteristic
157C3×#103C22×C311yescharacteristic
158C3×#103C22×C311yescharacteristic
159C3×#102—21no—
160C3×#102—21no—
161C3×#102C2×D311yescharacteristic
162C3×#102C2×D311yescharacteristic
163C3×#102C22×C311yescharacteristic
164C3×#102C22×C311yescharacteristic
165C3×#97—32no—
166C3×C4×C8—32no—
167#816C22×C312yes—
168#816C22×C312yes—
169#815C22×C312yes—
170C8×Dic3C22×C312yes—
171#808C22×C312yes—
172#807C22×C312yes—
173C23×C32—21no—
174C23×C32C2×D411yescharacteristic
175C32×Q8—21no—
176C32×Q8C22×C412yes—
177C32×D4—24no—
178C32×D4—21no—
179C32×D4C22×C412yes—
180C2×C32×C4—21no—
181C2×C32×C4—24no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4C22×C412yes—
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4#4112yes—
187C2×C32×C4C2411yescharacteristic
188C2×C3×Dic3—24no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—82no—
192C2×C3×Dic3—81no—
193C2×C3×Dic3C2×D412yes—
194C2×C3×Dic3C2×D412yes—
195C3×Dic6—24no—
196C3×Dic6—22no—
197C3×Dic6C22×C414yes—
198C32×C8—21no—
199C32×C8—21no—
200C3×C3⋊C8C22×C414yes—
201C2×#137—31no—
202#520—31no—
203#519—31no—
204#441—32no—
205#440—32no—
206C4×Q16—32no—
207C4×SD16—32no—
208#428—31no—
209#416C3×D311yescharacteristic
210C2×#103—31no—
211#405—31no—
212C3×#41—22no—
213C3×#41—22no—
214C3×#41—24no—
215C3×#41—42no—
216C3×#41C4×D314yes—
217C3×#41C2×C3×C414yes—
218C2×C3×Q8—21no—
219C2×C3×Q8—34no—
220C2×C3×Q8—32no—
221C2×C3×Q8—61no—
222C2×C3×Q8—62no—
223C2×C3×Q8C22×D311yescharacteristic
224C2×C3×Q8C23×C311yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—41no—
229C2×C3×D4C22×D311yescharacteristic
230C2×C3×D4C23×C311yescharacteristic
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—41no—
235C22×C3×C4—62no—
236C22×C3×C4—61no—
237C22×C3×C4—61no—
238C22×C3×C4C22×D311yescharacteristic
239C22×C3×C4C23×C311yescharacteristic
240C22×Dic3—21no—
241C22×Dic3—21no—
242C22×Dic3—22no—
243C2×Dic6—21no—
244C2×Dic6—22no—
245C2×Dic6C2×C3×C414yescomplemented
246C2×Dic6C23×C312yes—
247C3×Q16—38no—
248C3×SD16—38no—
249C3×M16—34no—
250C2×C3×C8—22no—
251C2×C3×C8—22no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—32no—
255C2×C3×C8—42no—
256C2×C3×C8C22×D311yescharacteristic
257C2×C3×C8C22×D311yescharacteristic
258C2×C3×C8C23×C311yescharacteristic
259C2×C3×C8C23×C311yescharacteristic
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—31no—
264C3×C4⋊C4—32no—
265C3×C4⋊C4—62no—
266C3×C4⋊C4—62no—
267C3×C4⋊C4—62no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4C22×D311yescharacteristic
270C3×C4⋊C4C22×D311yescharacteristic
271C3×C4⋊C4C23×C311yescharacteristic
272C3×C4⋊C4C23×C311yescharacteristic
273C3×C22⋊C4—21no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—32no—
276C3×C22⋊C4—41no—
277C3×C22⋊C4—62no—
278C3×C42—21no—
279C3×C42—21no—
280C3×C42—32no—
281C3×C42—41no—
282C3×C42—62no—
283C3×C42—62no—
284C3×C42—62no—
285#217—22no—
286#217C3×D412yes—
287#216C2×C3×C418yescomplemented
288#214C2×C3×C418yescomplemented
289C4⋊Dic3—21no—
290C4⋊Dic3—22no—
291C4⋊Dic3C3×D411yescharacteristic
292C4⋊Dic3C3×D411yescharacteristic
293C4⋊Dic3C23×C312yes—
294Dic3⋊C4—22no—
295Dic3⋊C4—22no—
296C4×Dic3—22no—
297C4×Dic3—22no—
298C4×Dic3—22no—
299C4×Dic3C23×C312yes—
300#208C2×C3×C414yescomplemented
301C2×C3⋊C8C23×C312yes—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—22no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32C4×D412yes—
308C22×C32C22×D411yescharacteristic
309C32×C4—22no—
310C32×C4—41no—
311C32×C4—41no—
312C32×C4C4×D412yes—
313C32×C4C23×C411yescharacteristic
314C32×C4C2×#4111yescharacteristic
315C3×Dic3—24no—
316C3×Dic3—22no—
317C3×Dic3—22no—
318C3×Dic3—42no—
319C2×#41C2×C3×D311yescharacteristic
320C22×Q8—31no—
321#137—38no—
322C2×Q16—32no—
323C2×SD16—32no—
324C2×M16—31no—
325C22×C8C2×C3×D311yescharacteristic
326C4×Q8—32no—
327C4×Q8—32no—
328C4×Q8—32no—
329C4×Q8—61no—
330C4×Q8—61no—
331C4×D4—32no—
332C4×D4—62no—
333#117—31no—
334#117—62no—
335#117—61no—
336#117C2×C3×D311yescharacteristic
337C2×C4⋊C4—31no—
338C2×C4⋊C4—61no—
339#107—32no—
340#106—32no—
341#103—32no—
342#103—32no—
343#103C2×C3×D311yescharacteristic
344#103C2×C3×D311yescharacteristic
345#102C2×C3×D311yescharacteristic
346#102C2×C3×D311yescharacteristic
347#97—32no—
348C4×C8—32no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—41no—
353C23×C3D3×D411yescharacteristic
354C23×C3C2×C3×D411yescharacteristic
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—22no—
358C3×Q8—34no—
359C3×Q8—41no—
360C3×Q8—64no—
361C3×Q8—62no—
362C3×Q8C2×C4×D312yes—
363C3×Q8C22×C3×C412yes—
364C3×D4—24no—
365C3×D4—24no—
366C3×D4—21no—
367C3×D4—21no—
368C3×D4—22no—
369C3×D4—41no—
370C3×D4—44no—
371C3×D4C2×C4×D312yes—
372C3×D4C22×C3×C412yes—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—32no—
386C2×C3×C4—32no—
387C2×C3×C4—44no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—64no—
392C2×C3×C4—62no—
393C2×C3×C4—62no—
394C2×C3×C4—242no—
395C2×C3×C4—241no—
396C2×C3×C4C2×C4×D312yes—
397C2×C3×C4D3×D411yescharacteristic
398C2×C3×C4#23712yes—
399C2×C3×C4C22×C3×C412yes—
400C2×C3×C4C2×C3×D411yescharacteristic
401C2×C3×C4C3×#4112yes—
402C2×C3×C4C23×D311yescharacteristic
403C2×C3×C4C24×C311yescharacteristic
404C2×Dic3—24no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—82no—
408C2×Dic3—81no—
409C2×Dic3C2×C3×D412yes—
410C2×Dic3C2×C3×D412yes—
411Dic6—24no—
412Dic6—22no—
413Dic6C22×C3×C414yes—
414C3×C8—21no—
415C3×C8—21no—
416C3×C8—21no—
417C3×C8—21no—
418C3×C8—34no—
419C3×C8—41no—
420C3×C8—41no—
421C3⋊C8C22×C3×C414yes—
422C2×C32—22no—
423C2×C32—42no—
424C2×C32C2×C4×D411yescharacteristic
425C2×C32#57711yescharacteristic
426C2×C32#57811yescharacteristic
427#41—22no—
428#41C3×C4×D314yes—
429C2×Q8—32no—
430C2×Q8—34no—
431C2×Q8—61no—
432C2×Q8—62no—
433C2×Q8C22×C3×D311yescharacteristic
434C2×D4—21no—
435C2×D4C22×C3×D311yescharacteristic
436C22×C4—21no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4C22×C3×D311yescharacteristic
441Q16—38no—
442SD16—38no—
443M16—34no—
444C2×C8—22no—
445C2×C8—32no—
446C2×C8C22×C3×D311yescharacteristic
447C2×C8C22×C3×D311yescharacteristic
448C4⋊C4—31no—
449C4⋊C4—31no—
450C4⋊C4—32no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—61no—
455C4⋊C4C22×C3×D311yescharacteristic
456C4⋊C4C22×C3×D311yescharacteristic
457C22⋊C4—21no—
458C22⋊C4—32no—
459C22⋊C4—62no—
460C42—21no—
461C42—32no—
462C42—62no—
463C42—62no—
464C42—62no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—21no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3#67⋊C412yes—
483C22×C3C3×C4×D412yes—
484C22×C3C2×D3×D411yescharacteristic
485C22×C3C22×C3×D411yescharacteristic
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—64no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—122no—
502C3×C4#67⋊C412yes—
503C3×C4C3×C4×D412yes—
504C3×C4C22×C4×D311yescharacteristic
505C3×C4C2×#23711yescharacteristic
506C3×C4C23×C3×C411yescharacteristic
507C3×C4C2×C3×#4111yescharacteristic
508Dic3—22no—
509Dic3—22no—
510Dic3—24no—
511Dic3—42no—
512C32#294711yescharacteristic, complemented, Sylow
513C23—21no—
514C23C3×D3×D411yescharacteristic
515Q8—21no—
516Q8—34no—
517Q8—64no—
518Q8—62no—
519Q8C2×C3×C4×D312yes—
520D4—24no—
521D4—21no—
522D4C2×C3×C4×D312yes—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—64no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—242no—
533C2×C4—241no—
534C2×C4C2×C3×C4×D312yes—
535C2×C4C3×D3×D411yescharacteristic
536C2×C4C3×#23712yes—
537C2×C4C23×C3×D311yescharacteristic
538C8—21no—
539C8—21no—
540C8—34no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—42no—
547C2×C3—42no—
548C2×C3—42no—
549C2×C3—82no—
550C2×C3C2×#67⋊C411yescharacteristic
551C2×C3#577011yescharacteristic
552C2×C3#577111yescharacteristic
553C2×C3C2×C3×C4×D411yescharacteristic
554C2×C3C3×#57711yescharacteristic
555C2×C3C3×#57811yescharacteristic
556C22—21no—
557C22—21no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22C3×#67⋊C412yes—
562C22C2×C3×D3×D411yescharacteristic
563C4—22no—
564C4—41no—
565C4—41no—
566C4—62no—
567C4—62no—
568C4—64no—
569C4—122no—
570C4C3×#67⋊C412yes—
571C4C22×C3×C4×D311yescharacteristic
572C4C2×C3×#23711yescharacteristic
573C3—21no—
574C3#7882111yescharacteristic, direct product factor
575C3C3×#294711yescharacteristic, complemented
576C2—22no—
577C2—42no—
578C2C2×C3×#67⋊C411yescharacteristic
579C2C3×#577011yescharacteristic
580C2C3×#577111yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC32×#416C2
socleC22×C32C22×D4
prime radicalC2×C32×#41C22
cosocleC2×C3×C4C24×C3
hypercenterC3×#416D3
hypocenterC3C3×#2947
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#416

Complements

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

NH ≅ G/Ntype
3. C3×#5516C2semidirect
5. C2×C3×#816C2semidirect
6. C3×#5101C2semidirect
7. C3×#4995C2semidirect
8. C3×#4790C2semidirect
9. C3×#4789C2semidirect
10. C3×#4758C2semidirect
11. C3×#4757C2semidirect
13. #78821C3direct (N × H)
18. C3×#951C4semidirect
49. #5516C2×C3semidirect
60. C2×#816C2×C3semidirect
61. #5101C2×C3semidirect
62. #4995C2×C3semidirect
63. #4790C2×C3semidirect
64. #4789C2×C3semidirect
65. #4758C2×C3semidirect
66. #4757C2×C3semidirect
79. C2×C3×Dic6C2×C4semidirect
90. C3×#216C2×C4semidirect
91. C3×#214C2×C4semidirect
103. C3×#208C2×C4semidirect
133. #951C3×C4semidirect
245. C2×Dic6C2×C3×C4semidirect
287. #216C2×C3×C4semidirect
288. #214C2×C3×C4semidirect
300. #208C2×C3×C4semidirect
512. C32#2947semidirect
574. C3#78821direct (N × H)
575. C3C3×#2947semidirect

Maximal subgroups

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

subgroupindex
C3×#58272
C2×C3×#9512
C3×#55162
C32×#4162
C2×C3×#8162
C3×#51012
C3×#49952
C3×#47902
C3×#47892
C3×#47582
C3×#47572
C3×#29473
#788213