Finite Group Databasebeta

← C3×#79214 #11867763

Subgroups — C3×#79214 #11867763

Order 1152.

1003 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic, complemented
2C2×C3×#907C211yesmaximal, characteristic, complemented
3C3×#5526C211yesmaximal, characteristic, complemented
4C3×#5510C211yesmaximal, characteristic
5C32×#428C211yesmaximal, characteristic, complemented
6C3×#5002C211yesmaximal, characteristic, complemented
7C2×C3×#810C211yesmaximal, characteristic, complemented
8C3×#4887C212yesmaximal, complemented
9C3×#4886C212yesmaximal, complemented
10C3×#4867C212yesmaximal, complemented
11C3×#4864C212yesmaximal, complemented
12C3×#3063—31nomaximal
13#79214C311yesmaximal, characteristic, direct product factor
14C2×C3×#235C2211yescharacteristic
15C2×C3×#235C2211yescharacteristic
16C22×C3×Dic6C2211yescharacteristic
17C2×C32×M16C2211yescharacteristic
18C32×#117C2211yescharacteristic
19C2×C32×C4⋊C4C2211yescharacteristic
20C3×#930—22no—
21C22×C3×C3⋊C8C2211yescharacteristic
22C3×#909—24no—
23C3×#907—24no—
24C2×C3×Dic12C2212yes—
25C2×C3×#204C2212yes—
26C2×C3×ΓL(1,𝔽25)C2212yes—
27C2×C3×D3×C8C2212yes—
28C3×#897—21no—
29C3×#897C2212yes—
30C3×#897C2212yes—
31C3×#897C2212yes—
32C3×#895—21no—
33C3×#895C2212yes—
34C3×#886—22no—
35C3×#875—21no—
36C3×#869—21no—
37C32×#107C2212yes—
38C32×#106C2212yes—
39C3×#810C2212yes—
40C3×#810C2212yes—
41C3×#810C2211yescharacteristic
42C3×#810C2211yescharacteristic
43C3×#809C2211yescharacteristic
44C3×#809C2211yescharacteristic
45C2×C3×#137—31no—
46C2×C3×#131—31no—
47C3×#537—31no—
48C3×#536—31no—
49C2×#909C2×C311yescharacteristic, complemented
50C2×#907C2×C311yescharacteristic, complemented
51#5526C2×C311yescharacteristic, complemented
52#5510C2×C311yescharacteristic
53C3×#470—32no—
54C3×#468—32no—
55C3×#467—32no—
56C3×#465—32no—
57C3×#428—21no—
58C3×#428D311yescharacteristic, complemented
59C3×#428C2×C311yescharacteristic, complemented
60C3×#416—31no—
61C2×C3×#103—31no—
62#5002C2×C311yescharacteristic, complemented
63C2×#810C2×C311yescharacteristic, complemented
64#4887C2×C312yescomplemented
65#4886C2×C312yescomplemented
66#4867C2×C312yescomplemented
67#4864C2×C312yescomplemented
68C22×C32×C4—21no—
69C22×C32×C4C2311yescharacteristic
70C2×C3×#67—21no—
71C2×C3×#67D412yes—
72C22×C3×Dic3—21no—
73C3×#235—22no—
74C3×#235—22no—
75C3×#235—22no—
76C3×#235—22no—
77C2×C3×D12—22no—
78C2×C3×D12D411yescharacteristic
79C2×C3×D12C2311yescharacteristic
80C2×C3×C4×D3—21no—
81C2×C3×C4×D3C2312yes—
82C2×C3×Dic6—22no—
83C2×C3×Dic6—22no—
84C2×C3×Dic6—21no—
85C2×C3×Dic6D411yescharacteristic
86C2×C3×Dic6C2312yes—
87C2×C3×Dic6C2311yescharacteristic
88C32×M16D414yescomplemented
89C2×C32×C8C2312yes—
90C32×C4⋊C4—21no—
91C32×C4⋊C4C2312yes—
92C32×C4⋊C4C2311yescharacteristic
93C32×C4⋊C4C2311yescharacteristic
94C32×C22⋊C4—21no—
95C32×C42—21no—
96C3×(C2×D3)⋊C4—22no—
97C3×(C2×D3)⋊C4—22no—
98C3×(C2×D3)⋊C4—22no—
99C3×(C2×D3)⋊C4—22no—
100C3×C4⋊Dic3—22no—
101C3×C4⋊Dic3—22no—
102C3×C4⋊Dic3—22no—
103C2×C3×C3⋊C8—22no—
104C2×C3×C3⋊C8C2311yescharacteristic
105C2×C3×C3⋊C8C2311yescharacteristic
106C3×Dic12—24no—
107C3×#204—24no—
108C3×ΓL(1,𝔽25)—24no—
109C3×D3×C8—24no—
110#3063—31noSylow
111C2×C3×#41—31no—
112C2×C3×#41—31no—
113C22×C3×Q8—31no—
114C2×#235C22×C311yescharacteristic
115C2×#235C22×C311yescharacteristic
116C22×Dic6C22×C311yescharacteristic
117C3×#137—64no—
118C2×C3×Q16—32no—
119C2×C3×SD16—32no—
120C3×#131—64no—
121C2×C3×M16—21no—
122C2×C3×M16—32no—
123C2×C3×M16C2×D311yescharacteristic
124C2×C3×M16C22×C311yescharacteristic
125C22×C3×C8—31no—
126C22×C3×C8—32no—
127C3×C4⋊Q8—61no—
128C3×#124—61no—
129C3×#123—62no—
130C3×C22⋊Q8—32no—
131C3×C22⋊Q8—32no—
132C3×C22⋊Q8—32no—
133C3×C22⋊Q8—61no—
134C3×#121—32no—
135C3×#121—61no—
136C3×#121—62no—
137C3×#117—21no—
138C3×#117C2×D311yescharacteristic
139C3×#117C22×C311yescharacteristic
140C2×C3×C4⋊C4—21no—
141C2×C3×C4⋊C4C2×D311yescharacteristic
142C2×C3×C4⋊C4C22×C311yescharacteristic
143#930—22no—
144C22×C3⋊C8C22×C311yescharacteristic
145#909—24no—
146#907—24no—
147C2×Dic12C22×C312yes—
148C2×#204C22×C312yes—
149C2×ΓL(1,𝔽25)C22×C312yes—
150C2×D3×C8C22×C312yes—
151#897—21no—
152#897C22×C312yes—
153#897C22×C312yes—
154#897C22×C312yes—
155#895—21no—
156#895C22×C312yes—
157#886—22no—
158#875—21no—
159#869—21no—
160C3×#107—22no—
161C3×#107C2×D312yes—
162C3×#107C22×C312yes—
163C3×#106—22no—
164C3×#106C2×D312yes—
165C3×#106C22×C312yes—
166C3×#103—32no—
167C3×#103—32no—
168C3×#103—31no—
169C3×#103—31no—
170C3×#102—31no—
171C3×#102—31no—
172#810C22×C312yes—
173#810C22×C311yescharacteristic
174#810C22×C312yes—
175#810C22×C311yescharacteristic
176#809C22×C311yescharacteristic
177#809C22×C311yescharacteristic
178C23×C32C2×D411yescharacteristic
179C22×C3×D3—21no—
180C22×C3×D3C2×D412yes—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4—81no—
185C2×C32×C4C2×D412yes—
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4#4112yes—
188C2×C32×C4C2411yescharacteristic
189C3×#67—24no—
190C3×#67—42no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3C2×D412yes—
195C3×D12—22no—
196C3×D12—21no—
197C3×D12—21no—
198C3×D12—82no—
199C3×C4×D3—22no—
200C3×C4×D3—22no—
201C3×C4×D3—42no—
202C3×Dic6—22no—
203C3×Dic6—22no—
204C3×Dic6—21no—
205C3×Dic6—22no—
206C3×Dic6—21no—
207C3×Dic6—42no—
208C32×C8C2×D414yes—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C2×#137—31no—
212C2×#131—31no—
213#537—31no—
214#536—31no—
215#470—32no—
216#468—32no—
217#467—32no—
218#465—32no—
219#428C3×D311yescharacteristic, complemented
220#416—31no—
221C2×#103—31no—
222C3×#41—62no—
223C3×#41—62no—
224C3×#41—62no—
225C3×#41—62no—
226C2×C3×Q8—31no—
227C2×C3×Q8—32no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—62no—
231C2×C3×Q8—62no—
232C2×C3×D4—32no—
233C2×C3×D4—31no—
234C2×C3×D4—31no—
235C2×C3×D4—62no—
236C2×C3×D4—61no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—32no—
241C22×C3×C4—41no—
242C22×C3×C4—61no—
243C22×C3×C4—61no—
244C22×C3×C4C22×D311yescharacteristic
245C22×C3×C4C23×C311yescharacteristic
246C2×#67—21no—
247C2×#67C3×D412yes—
248C22×Dic3—21no—
249#235—22no—
250#235—22no—
251#235—22no—
252#235—22no—
253C2×D12—22no—
254C2×D12C3×D411yescharacteristic
255C2×D12C23×C311yescharacteristic
256C2×C4×D3—21no—
257C2×C4×D3C23×C312yes—
258C2×Dic6—22no—
259C2×Dic6—22no—
260C2×Dic6—21no—
261C2×Dic6C3×D411yescharacteristic
262C2×Dic6C23×C312yes—
263C2×Dic6C23×C311yescharacteristic
264C3×Q16—64no—
265C3×SD16—64no—
266C3×M16—24no—
267C3×M16—64no—
268C3×M16D1214yescomplemented
269C3×M16C3×D414yescomplemented
270C2×C3×C8—22no—
271C2×C3×C8—31no—
272C2×C3×C8—31no—
273C2×C3×C8—62no—
274C2×C3×C8—64no—
275C2×C3×C8C22×D312yes—
276C2×C3×C8C23×C312yes—
277C3×C4⋊C4—21no—
278C3×C4⋊C4—21no—
279C3×C4⋊C4—21no—
280C3×C4⋊C4—22no—
281C3×C4⋊C4—21no—
282C3×C4⋊C4—41no—
283C3×C4⋊C4—62no—
284C3×C4⋊C4—62no—
285C3×C4⋊C4—62no—
286C3×C4⋊C4C22×D311yescharacteristic
287C3×C4⋊C4C22×D311yescharacteristic
288C3×C4⋊C4C22×D312yes—
289C3×C4⋊C4C23×C311yescharacteristic
290C3×C4⋊C4C23×C311yescharacteristic
291C3×C4⋊C4C23×C312yes—
292C3×C22⋊C4—21no—
293C3×C22⋊C4—21no—
294C3×C22⋊C4—41no—
295C3×C22⋊C4—62no—
296C3×C22⋊C4—62no—
297C3×C22⋊C4—62no—
298C3×C22⋊C4—62no—
299C3×C42—21no—
300C3×C42—21no—
301C3×C42—41no—
302(C2×D3)⋊C4—22no—
303(C2×D3)⋊C4—22no—
304(C2×D3)⋊C4—22no—
305(C2×D3)⋊C4—22no—
306C4⋊Dic3—22no—
307C4⋊Dic3—22no—
308C4⋊Dic3—22no—
309C2×C3⋊C8—22no—
310C2×C3⋊C8C23×C311yescharacteristic
311C2×C3⋊C8C23×C311yescharacteristic
312Dic12—24no—
313#204—24no—
314ΓL(1,𝔽25)—24no—
315D3×C8—24no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32#12112yes—
319C22×C32C22×D411yescharacteristic
320C2×C3×D3—22no—
321C2×C3×D3—22no—
322C2×C3×D3—22no—
323C2×C3×D3—41no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C32×C4—42no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4#12112yes—
330C32×C4C22×D411yescharacteristic
331C32×C4C2×#4111yescharacteristic
332C3×Dic3—22no—
333C3×Dic3—42no—
334C3×Dic3—41no—
335C2×#41—31no—
336C2×#41—31no—
337C22×Q8—31no—
338#137—64no—
339C2×Q16—32no—
340C2×SD16—32no—
341#131—64no—
342C2×M16—32no—
343C2×M16C2×C3×D311yescharacteristic
344C22×C8—31no—
345C22×C8—32no—
346C4⋊Q8—61no—
347#124—61no—
348#123—62no—
349C22⋊Q8—32no—
350C22⋊Q8—32no—
351C22⋊Q8—32no—
352C22⋊Q8—61no—
353#121—32no—
354#121—61no—
355#121—62no—
356#117C2×C3×D311yescharacteristic
357C2×C4⋊C4C2×C3×D311yescharacteristic
358#107C2×C3×D312yes—
359#106C2×C3×D312yes—
360#103—32no—
361#103—32no—
362#103—31no—
363#103—31no—
364#102—31no—
365#102—31no—
366C23×C3—21no—
367C23×C3—32no—
368C23×C3—61no—
369C23×C3D3×D411yescharacteristic
370C23×C3C2×C3×D411yescharacteristic
371C22×D3—21no—
372C22×D3C2×C3×D412yes—
373C3×Q8—62no—
374C3×Q8—62no—
375C3×Q8—61no—
376C3×Q8—62no—
377C3×Q8—61no—
378C3×Q8—122no—
379C3×D4—64no—
380C3×D4—62no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—122no—
384C3×D4—242no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
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—41no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—61no—
402C2×C3×C4—62no—
403C2×C3×C4—81no—
404C2×C3×C4—81no—
405C2×C3×C4—82no—
406C2×C3×C4—122no—
407C2×C3×C4—122no—
408C2×C3×C4C2×D1212yes—
409C2×C3×C4D3×D411yescharacteristic
410C2×C3×C4#23912yes—
411C2×C3×C4C2×C3×D412yes—
412C2×C3×C4C2×C3×D411yescharacteristic
413C2×C3×C4C3×#4112yes—
414C2×C3×C4C23×D311yescharacteristic
415C2×C3×C4C24×C311yescharacteristic
416#67—24no—
417#67—42no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—42no—
421C2×Dic3C2×C3×D412yes—
422D12—22no—
423D12—21no—
424D12—21no—
425D12—82no—
426C4×D3—22no—
427C4×D3—22no—
428C4×D3—42no—
429Dic6—22no—
430Dic6—22no—
431Dic6—21no—
432Dic6—22no—
433Dic6—21no—
434Dic6—42no—
435C3×C8—24no—
436C3×C8—61no—
437C3×C8—61no—
438C3×C8C2×D1214yes—
439C3×C8C2×C3×D414yes—
440C3⋊C8—21no—
441C3⋊C8—21no—
442C2×C32—22no—
443C2×C32C2×#12111yescharacteristic
444C2×C32#57711yescharacteristic
445C2×C32#57811yescharacteristic
446C3×D3—42no—
447C3×D3—81no—
448#41—62no—
449#41—62no—
450#41—62no—
451#41—62no—
452C2×Q8—31no—
453C2×Q8—32no—
454C2×Q8—31no—
455C2×Q8—61no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×D4—32no—
459C2×D4—31no—
460C2×D4—31no—
461C2×D4—62no—
462C2×D4—61no—
463C22×C4—21no—
464C22×C4—32no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4C22×C3×D311yescharacteristic
468Q16—64no—
469SD16—64no—
470M16—64no—
471M16C3×D1214yescomplemented
472C2×C8—31no—
473C2×C8—31no—
474C2×C8—62no—
475C2×C8—64no—
476C2×C8C22×C3×D312yes—
477C4⋊C4—21no—
478C4⋊C4—62no—
479C4⋊C4—62no—
480C4⋊C4—62no—
481C4⋊C4C22×C3×D311yescharacteristic
482C4⋊C4C22×C3×D311yescharacteristic
483C4⋊C4C22×C3×D312yes—
484C22⋊C4—21no—
485C22⋊C4—62no—
486C22⋊C4—62no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C42—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—62no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3—121no—
502C22×C3—121no—
503C22×C3—121no—
504C22×C3#89512yes—
505C22×C3C3×#12112yes—
506C22×C3C2×D3×D411yescharacteristic
507C22×C3C22×C3×D411yescharacteristic
508C2×D3—22no—
509C2×D3—22no—
510C2×D3—22no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—41no—
514C3×C4—22no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4—42no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—62no—
524C3×C4—82no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4—121no—
528C3×C4—122no—
529C3×C4#89512yes—
530C3×C4C3×#12112yes—
531C3×C4C22×D1211yescharacteristic
532C3×C4C2×#23911yescharacteristic
533C3×C4C22×C3×D411yescharacteristic
534C3×C4C2×C3×#4111yescharacteristic
535Dic3—22no—
536Dic3—42no—
537Dic3—41no—
538C32#306311yescharacteristic, complemented, Sylow
539C23—32no—
540C23—61no—
541C23C3×D3×D411yescharacteristic
542Q8—62no—
543Q8—62no—
544Q8—61no—
545Q8—62no—
546Q8—61no—
547Q8—122no—
548D4—64no—
549D4—62no—
550D4—61no—
551D4—61no—
552D4—122no—
553D4—242no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—62no—
562C2×C4—81no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4C2×C3×D1212yes—
566C2×C4C3×D3×D411yescharacteristic
567C2×C4C3×#23912yes—
568C2×C4C23×C3×D311yescharacteristic
569C8—61no—
570C8—61no—
571C8C2×C3×D1214yes—
572C2×C3—22no—
573C2×C3—22no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—42no—
578C2×C3—122no—
579C2×C3—241no—
580C2×C3C2×#89511yescharacteristic
581C2×C3#577511yescharacteristic
582C2×C3#577911yescharacteristic
583C2×C3C2×C3×#12111yescharacteristic
584C2×C3C3×#57711yescharacteristic
585C2×C3C3×#57811yescharacteristic
586D3—42no—
587D3—81no—
588C22—21no—
589C22—21no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22—121no—
594C22—121no—
595C22—121no—
596C22C3×#89512yes—
597C22C2×C3×D3×D411yescharacteristic
598C4—42no—
599C4—41no—
600C4—41no—
601C4—62no—
602C4—121no—
603C4—122no—
604C4C3×#89512yes—
605C4C22×C3×D1211yescharacteristic
606C4C2×C3×#23911yescharacteristic
607C3—21no—
608C3#7921411yescharacteristic, direct product factor
609C3C3×#306311yescharacteristic, complemented
610C2—22no—
611C2—122no—
612C2—241no—
613C2C2×C3×#89511yescharacteristic
614C2C3×#577511yescharacteristic
615C2C3×#577911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#428C2
socleC22×C32C22×D4
prime radicalC3×#5510C2
hypercenterC3×#428D3
hypocenterC3C3×#3063
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#428

Complements

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

NH ≅ G/Ntype
1. C2×C3×#909C2semidirect
2. C2×C3×#907C2semidirect
3. C3×#5526C2semidirect
5. C32×#428C2semidirect
6. C3×#5002C2semidirect
7. C2×C3×#810C2semidirect
8. C3×#4887C2semidirect
9. C3×#4886C2semidirect
10. C3×#4867C2semidirect
11. C3×#4864C2semidirect
13. #79214C3direct (N × H)
49. C2×#909C2×C3semidirect
50. C2×#907C2×C3semidirect
51. #5526C2×C3semidirect
58. C3×#428D3semidirect
59. C3×#428C2×C3semidirect
62. #5002C2×C3semidirect
63. C2×#810C2×C3semidirect
64. #4887C2×C3semidirect
65. #4886C2×C3semidirect
66. #4867C2×C3semidirect
67. #4864C2×C3semidirect
88. C32×M16D4semidirect
219. #428C3×D3semidirect
268. C3×M16D12semidirect
269. C3×M16C3×D4semidirect
471. M16C3×D12semidirect
538. C32#3063semidirect
608. C3#79214direct (N × H)
609. C3C3×#3063semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C2×C3×#9072
C3×#55262
C3×#55102
C32×#4282
C3×#50022
C2×C3×#8102
C3×#48872
C3×#48862
C3×#48672
C3×#48642
C3×#30633
#792143