Finite Group Databasebeta

← C3×#78841 #11867769

Subgroups — C3×#78841 #11867769

Order 1152.

1023 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5833C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic, complemented
3C2×C3×#901C211yesmaximal, characteristic, complemented
4C3×#5526C211yesmaximal, characteristic, complemented
5C32×#416C211yesmaximal, characteristic, complemented
6C2×C3×#816C211yesmaximal, characteristic, complemented
7C3×#4995C211yesmaximal, characteristic, complemented
8C3×#4816C212yesmaximal, complemented
9C3×#4813C212yesmaximal, complemented
10C3×#4780C212yesmaximal, complemented
11C3×#4778C212yesmaximal, complemented
12C3×#3063—31nomaximal
13#78841C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2211yescharacteristic
15C2×C3×#235C2211yescharacteristic
16C22×C3×Dic6C2211yescharacteristic
17C22×C32×C8C2211yescharacteristic
18C32×#117C2211yescharacteristic
19C3×#951—24no—
20C3×#946C2212yes—
21C2×C3×#216C2212yes—
22C3×#939—22no—
23C3×#938C2212yes—
24C2×C3×#214C2212yes—
25C3×#930—21no—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C3×#924—21no—
28C2×C3×#208C2211yescharacteristic
29C3×#901—24no—
30C2×C3×ΓL(1,𝔽25)C2212yes—
31C2×C3×D3×C8C2212yes—
32C3×#897C2212yes—
33C3×#897C2212yes—
34C3×#886—22no—
35C3×#875—21no—
36C3×#869—21no—
37C32×#103C2211yescharacteristic
38C32×#103C2211yescharacteristic
39C32×#102C2211yescharacteristic
40C32×#102C2211yescharacteristic
41C3×#816C2212yes—
42C3×#816C2212yes—
43C3×#808C2212yes—
44C3×#807C2212yes—
45C2×C3×#137—31no—
46C2×C3×#131—31no—
47C3×#537—31no—
48C3×#536—31no—
49#5833C2×C311yescharacteristic
50C2×#951C2×C311yescharacteristic, complemented
51C2×#901C2×C311yescharacteristic, complemented
52#5526C2×C311yescharacteristic, complemented
53C3×#470—32no—
54C3×#468—32no—
55C3×#467—32no—
56C3×#465—32no—
57C3×#428—31no—
58C3×#416—21no—
59C3×#416D311yescharacteristic, complemented
60C3×#416C2×C311yescharacteristic, complemented
61C2×C3×#103—31no—
62C2×#816C2×C311yescharacteristic, complemented
63#4995C2×C311yescharacteristic, complemented
64#4816C2×C312yescomplemented
65#4813C2×C312yescomplemented
66#4780C2×C312yescomplemented
67#4778C2×C312yescomplemented
68C32×#41—22no—
69C32×#41—22no—
70C2×C32×Q8C2311yescharacteristic
71C2×C32×D4—21no—
72C2×C32×D4C2311yescharacteristic
73C22×C32×C4—21no—
74C22×C32×C4C2311yescharacteristic
75C2×C3×#67—22no—
76C2×C3×#67D412yes—
77C22×C3×Dic3—21no—
78C22×C3×Dic3—21no—
79C3×#235—22no—
80C3×#235—22no—
81C2×C3×D12D411yescharacteristic
82C2×C3×C4×D3C2312yes—
83C2×C3×Dic6—22no—
84C2×C3×Dic6—22no—
85C2×C3×Dic6—21no—
86C2×C3×Dic6D411yescharacteristic
87C2×C3×Dic6C2312yes—
88C2×C32×C8—22no—
89C2×C32×C8C2311yescharacteristic
90C2×C32×C8C2311yescharacteristic
91C32×C4⋊C4C2311yescharacteristic
92C32×C4⋊C4C2311yescharacteristic
93C32×C22⋊C4—21no—
94C32×C42—21no—
95C3×#217—22no—
96C3×#216—24no—
97C3×#214—24no—
98C3×(C2×D3)⋊C4—22no—
99C3×(C2×D3)⋊C4—22no—
100C3×(C2×D3)⋊C4—22no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—22no—
103C3×C4⋊Dic3—22no—
104C3×C4⋊Dic3C2312yes—
105C3×Dic3⋊C4—22no—
106C3×#208D414yescomplemented
107C2×C3×C3⋊C8C2312yes—
108C3×ΓL(1,𝔽25)—24no—
109C3×D3×C8—24no—
110#3063—31noSylow
111C2×C3×#41—21no—
112C2×C3×#41—31no—
113C2×C3×#41C2×D311yescharacteristic
114C2×C3×#41C22×C311yescharacteristic
115C22×C3×Q8—31no—
116C2×#235C22×C311yescharacteristic
117C22×Dic6C22×C311yescharacteristic
118C3×#137—64no—
119C2×C3×Q16—32no—
120C2×C3×SD16—32no—
121C3×#131—64no—
122C2×C3×M16—31no—
123C2×C3×M16—32no—
124C22×C3×C8—21no—
125C22×C3×C8—32no—
126C22×C3×C8C2×D311yescharacteristic
127C22×C3×C8C22×C311yescharacteristic
128C3×C4⋊Q8—61no—
129C3×#124—61no—
130C3×#123—62no—
131C3×C22⋊Q8—32no—
132C3×C22⋊Q8—32no—
133C3×C22⋊Q8—32no—
134C3×C22⋊Q8—61no—
135C3×#121—32no—
136C3×#121—61no—
137C3×#121—62no—
138C3×#117—21no—
139C3×#117C2×D311yescharacteristic
140C3×#117C22×C311yescharacteristic
141C2×C3×C4⋊C4—31no—
142#951—24no—
143#946C22×C312yes—
144C2×#216C22×C312yes—
145#939—22no—
146#938C22×C312yes—
147C2×#214C22×C312yes—
148#930—21no—
149C2×C4⋊Dic3C22×C311yescharacteristic
150#924—21no—
151C2×#208C22×C311yescharacteristic
152#901—24no—
153C2×ΓL(1,𝔽25)C22×C312yes—
154C2×D3×C8C22×C312yes—
155#897C22×C312yes—
156#897C22×C312yes—
157#886—22no—
158#875—21no—
159#869—21no—
160C3×#107—32no—
161C3×#106—32no—
162C3×#103—21no—
163C3×#103—21no—
164C3×#103—32no—
165C3×#103—32no—
166C3×#103C2×D311yescharacteristic
167C3×#103C2×D311yescharacteristic
168C3×#103C22×C311yescharacteristic
169C3×#103C22×C311yescharacteristic
170C3×#102—21no—
171C3×#102—21no—
172C3×#102C2×D311yescharacteristic
173C3×#102C2×D311yescharacteristic
174C3×#102C22×C311yescharacteristic
175C3×#102C22×C311yescharacteristic
176#816C22×C312yes—
177#816C22×C312yes—
178#808C22×C312yes—
179#807C22×C312yes—
180C23×C32—21no—
181C23×C32C2×D411yescharacteristic
182C22×C3×D3C2×D412yes—
183C32×Q8—21no—
184C32×Q8—21no—
185C32×D4—21no—
186C32×D4—21no—
187C32×D4—42no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—42no—
192C2×C32×C4C2×D411yescharacteristic
193C2×C32×C4C2×D412yes—
194C2×C32×C4#4112yes—
195C2×C32×C4C2411yescharacteristic
196C3×#67—24no—
197C3×#67—82no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—42no—
201C2×C3×Dic3—81no—
202C2×C3×Dic3C2×D412yes—
203C3×D12—22no—
204C3×C4×D3—22no—
205C3×C4×D3—22no—
206C3×Dic6—22no—
207C3×Dic6—22no—
208C3×Dic6—22no—
209C3×Dic6—42no—
210C32×C8—21no—
211C32×C8—21no—
212C3×C3⋊C8C2×D414yes—
213C2×#137—31no—
214C2×#131—31no—
215#537—31no—
216#536—31no—
217#470—32no—
218#468—32no—
219#467—32no—
220#465—32no—
221#428—31no—
222#416C3×D311yescharacteristic, complemented
223C2×#103—31no—
224C3×#41—24no—
225C3×#41—22no—
226C3×#41—22no—
227C3×#41—22no—
228C3×#41—22no—
229C3×#41—42no—
230C3×#41—62no—
231C3×#41—62no—
232C2×C3×Q8—21no—
233C2×C3×Q8—31no—
234C2×C3×Q8—32no—
235C2×C3×Q8—61no—
236C2×C3×Q8—62no—
237C2×C3×Q8—62no—
238C2×C3×Q8C22×D311yescharacteristic
239C2×C3×Q8C23×C311yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—32no—
244C2×C3×D4—31no—
245C2×C3×D4—41no—
246C2×C3×D4—62no—
247C2×C3×D4C22×D311yescharacteristic
248C2×C3×D4C23×C311yescharacteristic
249C22×C3×C4—21no—
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—32no—
253C22×C3×C4—41no—
254C22×C3×C4—61no—
255C22×C3×C4—61no—
256C22×C3×C4C22×D311yescharacteristic
257C22×C3×C4C23×C311yescharacteristic
258C2×#67—22no—
259C2×#67C3×D412yes—
260C22×Dic3—21no—
261C22×Dic3—21no—
262#235—22no—
263#235—22no—
264C2×D12C3×D411yescharacteristic
265C2×C4×D3C23×C312yes—
266C2×Dic6—22no—
267C2×Dic6—22no—
268C2×Dic6—21no—
269C2×Dic6C3×D411yescharacteristic
270C2×Dic6C23×C312yes—
271C3×Q16—64no—
272C3×SD16—64no—
273C3×M16—34no—
274C3×M16—64no—
275C2×C3×C8—22no—
276C2×C3×C8—22no—
277C2×C3×C8—21no—
278C2×C3×C8—21no—
279C2×C3×C8—32no—
280C2×C3×C8—42no—
281C2×C3×C8—64no—
282C2×C3×C8C22×D311yescharacteristic
283C2×C3×C8C22×D311yescharacteristic
284C2×C3×C8C23×C311yescharacteristic
285C2×C3×C8C23×C311yescharacteristic
286C3×C4⋊C4—21no—
287C3×C4⋊C4—21no—
288C3×C4⋊C4—32no—
289C3×C4⋊C4—62no—
290C3×C4⋊C4—62no—
291C3×C4⋊C4—62no—
292C3×C4⋊C4—61no—
293C3×C4⋊C4C22×D311yescharacteristic
294C3×C4⋊C4C22×D311yescharacteristic
295C3×C4⋊C4C23×C311yescharacteristic
296C3×C4⋊C4C23×C311yescharacteristic
297C3×C22⋊C4—21no—
298C3×C22⋊C4—21no—
299C3×C22⋊C4—41no—
300C3×C22⋊C4—62no—
301C3×C22⋊C4—62no—
302C3×C22⋊C4—62no—
303C3×C22⋊C4—62no—
304C3×C42—21no—
305C3×C42—21no—
306C3×C42—41no—
307#217—22no—
308#216—24no—
309#214—24no—
310(C2×D3)⋊C4—22no—
311(C2×D3)⋊C4—22no—
312(C2×D3)⋊C4—22no—
313C4⋊Dic3—21no—
314C4⋊Dic3—22no—
315C4⋊Dic3—22no—
316C4⋊Dic3C23×C312yes—
317Dic3⋊C4—22no—
318#208C3×D414yescomplemented
319C2×C3⋊C8C23×C312yes—
320ΓL(1,𝔽25)—24no—
321D3×C8—24no—
322C22×C32—21no—
323C22×C32—21no—
324C22×C32—41no—
325C22×C32—41no—
326C22×C32—41no—
327C22×C32#12112yes—
328C22×C32C22×D411yescharacteristic
329C2×C3×D3—22no—
330C2×C3×D3—22no—
331C2×C3×D3—22no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4#12112yes—
336C32×C4C22×D411yescharacteristic
337C32×C4C2×#4111yescharacteristic
338C3×Dic3—22no—
339C3×Dic3—42no—
340C3×Dic3—42no—
341C2×#41—31no—
342C2×#41C2×C3×D311yescharacteristic
343C22×Q8—31no—
344#137—64no—
345C2×Q16—32no—
346C2×SD16—32no—
347#131—64no—
348C2×M16—31no—
349C2×M16—32no—
350C22×C8—32no—
351C22×C8C2×C3×D311yescharacteristic
352C4⋊Q8—61no—
353#124—61no—
354#123—62no—
355C22⋊Q8—32no—
356C22⋊Q8—32no—
357C22⋊Q8—32no—
358C22⋊Q8—61no—
359#121—32no—
360#121—61no—
361#121—62no—
362#117C2×C3×D311yescharacteristic
363C2×C4⋊C4—31no—
364#107—32no—
365#106—32no—
366#103—32no—
367#103—32no—
368#103C2×C3×D311yescharacteristic
369#103C2×C3×D311yescharacteristic
370#102C2×C3×D311yescharacteristic
371#102C2×C3×D311yescharacteristic
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—32no—
376C23×C3—41no—
377C23×C3D3×D411yescharacteristic
378C23×C3C2×C3×D411yescharacteristic
379C22×D3C2×C3×D412yes—
380C3×Q8—22no—
381C3×Q8—21no—
382C3×Q8—21no—
383C3×Q8—21no—
384C3×Q8—21no—
385C3×Q8—41no—
386C3×Q8—62no—
387C3×Q8—62no—
388C3×Q8—62no—
389C3×Q8—122no—
390C3×D4—22no—
391C3×D4—21no—
392C3×D4—21no—
393C3×D4—21no—
394C3×D4—21no—
395C3×D4—41no—
396C3×D4—44no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—64no—
400C3×D4—62no—
401C3×D4—242no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—22no—
412C2×C3×C4—32no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—44no—
417C2×C3×C4—42no—
418C2×C3×C4—42no—
419C2×C3×C4—62no—
420C2×C3×C4—62no—
421C2×C3×C4—62no—
422C2×C3×C4—62no—
423C2×C3×C4—122no—
424C2×C3×C4—241no—
425C2×C3×C4#23512yes—
426C2×C3×C4D3×D411yescharacteristic
427C2×C3×C4D3×D412yes—
428C2×C3×C4C2×C3×D411yescharacteristic
429C2×C3×C4C2×C3×D412yes—
430C2×C3×C4C3×#4112yes—
431C2×C3×C4C23×D311yescharacteristic
432C2×C3×C4C24×C311yescharacteristic
433#67—24no—
434#67—82no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—42no—
438C2×Dic3—81no—
439C2×Dic3C2×C3×D412yes—
440D12—22no—
441C4×D3—22no—
442C4×D3—22no—
443Dic6—22no—
444Dic6—22no—
445Dic6—22no—
446Dic6—42no—
447C3×C8—21no—
448C3×C8—21no—
449C3×C8—21no—
450C3×C8—21no—
451C3×C8—34no—
452C3×C8—41no—
453C3×C8—41no—
454C3⋊C8C2×C3×D414yes—
455C2×C32—22no—
456C2×C32—81no—
457C2×C32C2×#12111yescharacteristic
458C2×C32#57711yescharacteristic
459C2×C32#57811yescharacteristic
460C3×D3—42no—
461#41—22no—
462#41—22no—
463#41—62no—
464#41—62no—
465C2×Q8—31no—
466C2×Q8—32no—
467C2×Q8—61no—
468C2×Q8—62no—
469C2×Q8—62no—
470C2×Q8C22×C3×D311yescharacteristic
471C2×D4—21no—
472C2×D4—32no—
473C2×D4—31no—
474C2×D4—62no—
475C2×D4C22×C3×D311yescharacteristic
476C22×C4—21no—
477C22×C4—32no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4C22×C3×D311yescharacteristic
481Q16—64no—
482SD16—64no—
483M16—34no—
484M16—64no—
485C2×C8—22no—
486C2×C8—32no—
487C2×C8—64no—
488C2×C8C22×C3×D311yescharacteristic
489C2×C8C22×C3×D311yescharacteristic
490C4⋊C4—32no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—62no—
494C4⋊C4—61no—
495C4⋊C4C22×C3×D311yescharacteristic
496C4⋊C4C22×C3×D311yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—62no—
499C22⋊C4—62no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C42—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—42no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—62no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3#88412yes—
524C22×C3C3×#12112yes—
525C22×C3C2×D3×D411yescharacteristic
526C22×C3C22×C3×D411yescharacteristic
527C2×D3—22no—
528C2×D3—22no—
529C2×D3—22no—
530C3×C4—22no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—62no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4—122no—
544C3×C4—122no—
545C3×C4#88412yes—
546C3×C4C3×#12112yes—
547C3×C4C2×#23511yescharacteristic
548C3×C4C2×D3×D411yescharacteristic
549C3×C4C22×C3×D411yescharacteristic
550C3×C4C2×C3×#4111yescharacteristic
551Dic3—22no—
552Dic3—42no—
553Dic3—42no—
554C32#306311yescharacteristic, complemented, Sylow
555C23—21no—
556C23—32no—
557C23C3×D3×D411yescharacteristic
558Q8—21no—
559Q8—21no—
560Q8—62no—
561Q8—62no—
562Q8—62no—
563Q8—122no—
564D4—21no—
565D4—21no—
566D4—42no—
567D4—64no—
568D4—62no—
569D4—242no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—32no—
574C2×C4—42no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—122no—
580C2×C4—241no—
581C2×C4C3×#23512yes—
582C2×C4C3×D3×D411yescharacteristic
583C2×C4C3×D3×D412yes—
584C2×C4C23×C3×D311yescharacteristic
585C8—21no—
586C8—21no—
587C8—34no—
588C2×C3—22no—
589C2×C3—22no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—42no—
594C2×C3—82no—
595C2×C3—81no—
596C2×C3—81no—
597C2×C3—122no—
598C2×C3C2×#88411yescharacteristic
599C2×C3#577011yescharacteristic
600C2×C3#577111yescharacteristic
601C2×C3C2×C3×#12111yescharacteristic
602C2×C3C3×#57711yescharacteristic
603C2×C3C3×#57811yescharacteristic
604D3—42no—
605C22—21no—
606C22—21no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—62no—
611C22—62no—
612C22—62no—
613C22C3×#88412yes—
614C22C2×C3×D3×D411yescharacteristic
615C4—41no—
616C4—41no—
617C4—41no—
618C4—62no—
619C4—122no—
620C4—122no—
621C4C3×#88412yes—
622C4C2×C3×#23511yescharacteristic
623C4C2×C3×D3×D411yescharacteristic
624C3—21no—
625C3#7884111yescharacteristic, direct product factor
626C3C3×#306311yescharacteristic, complemented
627C2—22no—
628C2—81no—
629C2—122no—
630C2C2×C3×#88411yescharacteristic
631C2C3×#577011yescharacteristic
632C2C3×#577111yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#416C2
socleC22×C32C22×D4
prime radicalC3×#5833C2
hypercenterC3×#416D3
hypocenterC3C3×#3063
LCSG ▷ C2×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
2. C2×C3×#951C2semidirect
3. C2×C3×#901C2semidirect
4. C3×#5526C2semidirect
5. C32×#416C2semidirect
6. C2×C3×#816C2semidirect
7. C3×#4995C2semidirect
8. C3×#4816C2semidirect
9. C3×#4813C2semidirect
10. C3×#4780C2semidirect
11. C3×#4778C2semidirect
13. #78841C3direct (N × H)
50. C2×#951C2×C3semidirect
51. C2×#901C2×C3semidirect
52. #5526C2×C3semidirect
59. C3×#416D3semidirect
60. C3×#416C2×C3semidirect
62. C2×#816C2×C3semidirect
63. #4995C2×C3semidirect
64. #4816C2×C3semidirect
65. #4813C2×C3semidirect
66. #4780C2×C3semidirect
67. #4778C2×C3semidirect
106. C3×#208D4semidirect
222. #416C3×D3semidirect
318. #208C3×D4semidirect
554. C32#3063semidirect
625. C3#78841direct (N × H)
626. C3C3×#3063semidirect

Maximal subgroups

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

subgroupindex
C3×#58332
C2×C3×#9512
C2×C3×#9012
C3×#55262
C32×#4162
C2×C3×#8162
C3×#49952
C3×#48162
C3×#48132
C3×#47802
C3×#47782
C3×#30633
#788413