Finite Group Databasebeta

← C3×#83423 #11867766

Subgroups — C3×#83423 #11867766

Order 1152.

1091 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#537C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C2×C3×#951C211yesmaximal, characteristic, complemented
4C2×C3×#948C211yesmaximal, characteristic, complemented
5C3×#5233C211yesmaximal, characteristic, complemented
6C3×#5052C212yesmaximal, complemented
7C3×#5051C212yesmaximal, complemented
8C3×#5042C212yesmaximal, complemented
9C3×#5041C212yesmaximal, complemented
10C2×C3×#810C211yesmaximal, characteristic, complemented
11C3×#4964C211yesmaximal, characteristic, complemented
12C3×#3063—31nomaximal
13#83423C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2211yescharacteristic
15C2×C32×#41C2211yescharacteristic
16C22×C3×Dic6C2211yescharacteristic
17C32×C22⋊Q8—21no—
18C32×C22⋊Q8C2212yes—
19C32×#121—21no—
20C32×#121—22no—
21C32×#121C2212yes—
22C2×C32×C4⋊C4C2211yescharacteristic
23C3×#951—24no—
24C3×#948—24no—
25C3×#944—21no—
26C2×C3×#216C2212yes—
27C3×#936—21no—
28C3×#935—22no—
29C2×C3×#214C2212yes—
30C3×#926C2211yescharacteristic
31C3×#924C2212yes—
32C3×#924C2212yes—
33C2×C3×#208C2212yes—
34C2×C3×#208C2211yescharacteristic
35C22×C3×C3⋊C8C2211yescharacteristic
36C22×C3×C3⋊C8C2212yes—
37C3×#835C2211yescharacteristic
38C3×#835C2211yescharacteristic
39C3×#832C2211yescharacteristic
40C3×#832C2211yescharacteristic
41C3×#810C2212yes—
42C3×#810C2212yes—
43C3×#808C2212yes—
44C3×#807C2212yes—
45C2×C3×#137—31no—
46C2×C3×#131—31no—
47C3×#537—21no—
48C3×#537D311yescharacteristic
49C3×#537C2×C311yescharacteristic
50C3×#536—31no—
51#5825C2×C311yescharacteristic, complemented
52C2×#951C2×C311yescharacteristic, complemented
53C2×#948C2×C311yescharacteristic, complemented
54C3×#470—32no—
55C3×#468—32no—
56C3×#467—32no—
57C3×#465—32no—
58C3×#428—31no—
59C3×#416—31no—
60C2×C3×#103—31no—
61#5233C2×C311yescharacteristic, complemented
62#5052C2×C312yescomplemented
63#5051C2×C312yescomplemented
64#5042C2×C312yescomplemented
65#5041C2×C312yescomplemented
66C2×#810C2×C311yescharacteristic, complemented
67#4964C2×C311yescharacteristic, complemented
68C32×#41—22no—
69C32×#41—22no—
70C32×#41—22no—
71C32×#41—22no—
72C2×C32×Q8D411yescharacteristic
73C2×C32×Q8C2311yescharacteristic
74C2×C32×D4—21no—
75C2×C32×D4—22no—
76C2×C32×D4D412yes—
77C2×C32×D4D411yescharacteristic
78C2×C32×D4C2311yescharacteristic
79C22×C32×C4—21no—
80C22×C32×C4—21no—
81C22×C32×C4C2312yes—
82C22×C32×C4C2311yescharacteristic
83C22×C3×Dic3—21no—
84C2×C3×Dic6—22no—
85C2×C3×Dic6—22no—
86C2×C3×Dic6—21no—
87C2×C3×Dic6C2312yes—
88C32×C4⋊C4—21no—
89C32×C4⋊C4—22no—
90C32×C4⋊C4C2312yes—
91C32×C22⋊C4—22no—
92C32×C22⋊C4—22no—
93C3×#217—22no—
94C3×#217—22no—
95C3×#217—21no—
96C3×#216—24no—
97C3×#214—24no—
98C3×C4⋊Dic3C2311yescharacteristic
99C3×C4⋊Dic3C2311yescharacteristic
100C3×Dic3⋊C4—22no—
101C3×Dic3⋊C4—22no—
102C3×C4×Dic3—21no—
103C3×#208—24no—
104C3×#208D414yescomplemented
105C2×C3×C3⋊C8—22no—
106C2×C3×C3⋊C8—24no—
107C2×C3×C3⋊C8C2311yescharacteristic
108C2×C3×C3⋊C8C2311yescharacteristic
109C2×C3×C3⋊C8C2312yes—
110#3063—31noSylow
111C2×C3×#41—21no—
112C2×C3×#41—21no—
113C2×C3×#41C2×D311yescharacteristic
114C2×C3×#41C2×D311yescharacteristic
115C2×C3×#41C22×C311yescharacteristic
116C2×C3×#41C22×C311yescharacteristic
117C22×C3×Q8—31no—
118C22×Dic6C22×C311yescharacteristic
119C3×#137—64no—
120C2×C3×Q16—32no—
121C2×C3×SD16—32no—
122C3×#131—64no—
123C2×C3×M16—31no—
124C2×C3×M16—32no—
125C22×C3×C8—31no—
126C22×C3×C8—32no—
127C3×C4⋊Q8—61no—
128C3×#124—61no—
129C3×#123—62no—
130C3×C22⋊Q8—22no—
131C3×C22⋊Q8—21no—
132C3×C22⋊Q8—21no—
133C3×C22⋊Q8—22no—
134C3×C22⋊Q8—32no—
135C3×C22⋊Q8—32no—
136C3×C22⋊Q8C2×D312yes—
137C3×C22⋊Q8C22×C312yes—
138C3×#121—22no—
139C3×#121—24no—
140C3×#121—21no—
141C3×#121—22no—
142C3×#121—21no—
143C3×#121—22no—
144C3×#121—22no—
145C3×#121C2×D312yes—
146C3×#121C22×C312yes—
147C3×#117—31no—
148C2×C3×C4⋊C4—21no—
149C2×C3×C4⋊C4C2×D311yescharacteristic
150C2×C3×C4⋊C4C22×C311yescharacteristic
151#951—24no—
152#948—24no—
153#944—21no—
154C2×#216C22×C312yes—
155#936—21no—
156#935—22no—
157C2×#214C22×C312yes—
158#926C22×C311yescharacteristic
159#924C22×C312yes—
160#924C22×C312yes—
161C2×#208C22×C312yes—
162C2×#208C22×C311yescharacteristic
163C22×C3⋊C8C22×C311yescharacteristic
164C22×C3⋊C8C22×C312yes—
165C3×#107—32no—
166C3×#106—32no—
167C3×#103—31no—
168C3×#103—31no—
169C3×#103—32no—
170C3×#103—32no—
171C3×#102—31no—
172C3×#102—31no—
173#835C22×C311yescharacteristic
174#835C22×C311yescharacteristic
175#832C22×C311yescharacteristic
176#832C22×C311yescharacteristic
177#810C22×C312yes—
178#810C22×C312yes—
179#808C22×C312yes—
180#807C22×C312yes—
181C23×C32—21no—
182C23×C32C2×D412yes—
183C23×C32C2×D411yescharacteristic
184C32×Q8—22no—
185C32×Q8—21no—
186C32×Q8—21no—
187C32×D4—24no—
188C32×D4—22no—
189C32×D4—21no—
190C32×D4—21no—
191C32×D4—42no—
192C32×D4—82no—
193C2×C32×C4—22no—
194C2×C32×C4—21no—
195C2×C32×C4—22no—
196C2×C32×C4—22no—
197C2×C32×C4—42no—
198C2×C32×C4—81no—
199C2×C32×C4C2×D412yes—
200C2×C32×C4C2×D412yes—
201C2×C32×C4C2×D411yescharacteristic
202C2×C32×C4#4112yes—
203C2×C32×C4C2411yescharacteristic
204C2×C3×Dic3—21no—
205C2×C3×Dic3—21no—
206C2×C3×Dic3—22no—
207C2×C3×Dic3—42no—
208C3×Dic6—22no—
209C3×Dic6—22no—
210C3×Dic6—42no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8C2×D414yes—
214C2×#137—31no—
215C2×#131—31no—
216#537C3×D311yescharacteristic
217#536—31no—
218#470—32no—
219#468—32no—
220#467—32no—
221#465—32no—
222#428—31no—
223#416—31no—
224C2×#103—31no—
225C3×#41—22no—
226C3×#41—22no—
227C3×#41—22no—
228C3×#41—22no—
229C3×#41—22no—
230C3×#41—22no—
231C3×#41—22no—
232C3×#41—22no—
233C3×#41—42no—
234C3×#41—42no—
235C3×#41—42no—
236C3×#41—42no—
237C2×C3×Q8—21no—
238C2×C3×Q8—21no—
239C2×C3×Q8—32no—
240C2×C3×Q8—61no—
241C2×C3×Q8—62no—
242C2×C3×Q8—62no—
243C2×C3×Q8#6711yescharacteristic
244C2×C3×Q8C3×D411yescharacteristic
245C2×C3×Q8C22×D311yescharacteristic
246C2×C3×Q8C23×C311yescharacteristic
247C2×C3×D4—22no—
248C2×C3×D4—21no—
249C2×C3×D4—21no—
250C2×C3×D4—24no—
251C2×C3×D4—22no—
252C2×C3×D4—22no—
253C2×C3×D4—22no—
254C2×C3×D4—21no—
255C2×C3×D4—21no—
256C2×C3×D4#6712yes—
257C2×C3×D4#6711yescharacteristic
258C2×C3×D4C3×D412yes—
259C2×C3×D4C3×D411yescharacteristic
260C2×C3×D4C22×D311yescharacteristic
261C2×C3×D4C23×C311yescharacteristic
262C22×C3×C4—22no—
263C22×C3×C4—22no—
264C22×C3×C4—21no—
265C22×C3×C4—21no—
266C22×C3×C4—21no—
267C22×C3×C4—21no—
268C22×C3×C4—21no—
269C22×C3×C4—22no—
270C22×C3×C4—61no—
271C22×C3×C4C22×D312yes—
272C22×C3×C4C22×D311yescharacteristic
273C22×C3×C4C23×C312yes—
274C22×C3×C4C23×C311yescharacteristic
275C22×Dic3—21no—
276C2×Dic6—22no—
277C2×Dic6—22no—
278C2×Dic6—21no—
279C2×Dic6C23×C312yes—
280C3×Q16—64no—
281C3×SD16—64no—
282C3×M16—34no—
283C3×M16—64no—
284C2×C3×C8—32no—
285C2×C3×C8—31no—
286C2×C3×C8—31no—
287C2×C3×C8—64no—
288C2×C3×C8—62no—
289C3×C4⋊C4—22no—
290C3×C4⋊C4—24no—
291C3×C4⋊C4—21no—
292C3×C4⋊C4—22no—
293C3×C4⋊C4—21no—
294C3×C4⋊C4—22no—
295C3×C4⋊C4—22no—
296C3×C4⋊C4—31no—
297C3×C4⋊C4—31no—
298C3×C4⋊C4—62no—
299C3×C4⋊C4—62no—
300C3×C4⋊C4C22×D312yes—
301C3×C4⋊C4C23×C312yes—
302C3×C22⋊C4—24no—
303C3×C22⋊C4—24no—
304C3×C22⋊C4—22no—
305C3×C22⋊C4—22no—
306C3×C22⋊C4—22no—
307C3×C22⋊C4—22no—
308C3×C22⋊C4—61no—
309C3×C22⋊C4—62no—
310C3×C22⋊C4—62no—
311C3×C42—61no—
312#217—22no—
313#217—22no—
314#217—21no—
315#216—24no—
316#214—24no—
317C4⋊Dic3C23×C311yescharacteristic
318C4⋊Dic3C23×C311yescharacteristic
319Dic3⋊C4—22no—
320Dic3⋊C4—22no—
321C4×Dic3—21no—
322#208—24no—
323#208C3×D414yescomplemented
324C2×C3⋊C8—22no—
325C2×C3⋊C8—24no—
326C2×C3⋊C8C23×C311yescharacteristic
327C2×C3⋊C8C23×C311yescharacteristic
328C2×C3⋊C8C23×C312yes—
329C22×C32—21no—
330C22×C32—21no—
331C22×C32—22no—
332C22×C32—22no—
333C22×C32—22no—
334C22×C32—41no—
335C22×C32—41no—
336C22×C32—41no—
337C22×C32#12112yes—
338C22×C32C22×D411yescharacteristic
339C32×C4—22no—
340C32×C4—41no—
341C32×C4—42no—
342C32×C4#12112yes—
343C32×C4C22×D411yescharacteristic
344C32×C4C2×#4111yescharacteristic
345C3×Dic3—42no—
346C3×Dic3—41no—
347C3×Dic3—41no—
348C2×#41C2×C3×D311yescharacteristic
349C2×#41C2×C3×D311yescharacteristic
350C22×Q8—31no—
351#137—64no—
352C2×Q16—32no—
353C2×SD16—32no—
354#131—64no—
355C2×M16—31no—
356C2×M16—32no—
357C22×C8—31no—
358C22×C8—32no—
359C4⋊Q8—61no—
360#124—61no—
361#123—62no—
362C22⋊Q8—21no—
363C22⋊Q8—32no—
364C22⋊Q8—32no—
365C22⋊Q8C2×C3×D312yes—
366#121—21no—
367#121—22no—
368#121C2×C3×D312yes—
369#117—31no—
370C2×C4⋊C4C2×C3×D311yescharacteristic
371#107—32no—
372#106—32no—
373#103—31no—
374#103—31no—
375#103—32no—
376#103—32no—
377#102—31no—
378#102—31no—
379C23×C3—22no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—22no—
384C23×C3C2×#6712yes—
385C23×C3C2×#6711yescharacteristic
386C23×C3C2×C3×D412yes—
387C23×C3C2×C3×D411yescharacteristic
388C3×Q8—22no—
389C3×Q8—22no—
390C3×Q8—21no—
391C3×Q8—21no—
392C3×Q8—21no—
393C3×Q8—21no—
394C3×Q8—42no—
395C3×Q8—41no—
396C3×Q8—41no—
397C3×Q8—62no—
398C3×Q8—62no—
399C3×Q8—122no—
400C3×D4—24no—
401C3×D4—24no—
402C3×D4—22no—
403C3×D4—22no—
404C3×D4—21no—
405C3×D4—21no—
406C3×D4—21no—
407C3×D4—21no—
408C3×D4—42no—
409C3×D4—41no—
410C3×D4—41no—
411C3×D4—44no—
412C3×D4—44no—
413C3×D4—42no—
414C3×D4—42no—
415C3×D4—82no—
416C3×D4—82no—
417C3×D4—84no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—24no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—21no—
426C2×C3×C4—22no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—21no—
432C2×C3×C4—22no—
433C2×C3×C4—42no—
434C2×C3×C4—42no—
435C2×C3×C4—44no—
436C2×C3×C4—42no—
437C2×C3×C4—42no—
438C2×C3×C4—61no—
439C2×C3×C4—61no—
440C2×C3×C4—62no—
441C2×C3×C4—81no—
442C2×C3×C4—81no—
443C2×C3×C4—82no—
444C2×C3×C4—122no—
445C2×C3×C4D3×D412yes—
446C2×C3×C4#23712yes—
447C2×C3×C4C2×#6712yes—
448C2×C3×C4C2×#6711yescharacteristic
449C2×C3×C4C2×C3×D412yes—
450C2×C3×C4C2×C3×D412yes—
451C2×C3×C4C2×C3×D411yescharacteristic
452C2×C3×C4C3×#4112yes—
453C2×C3×C4C23×D311yescharacteristic
454C2×C3×C4C24×C311yescharacteristic
455C2×Dic3—21no—
456C2×Dic3—21no—
457C2×Dic3—22no—
458C2×Dic3—42no—
459Dic6—22no—
460Dic6—22no—
461Dic6—42no—
462C3×C8—34no—
463C3×C8—61no—
464C3×C8—61no—
465C3⋊C8—21no—
466C3⋊C8—21no—
467C3⋊C8C2×C3×D414yes—
468C2×C32—22no—
469C2×C32—42no—
470C2×C32—81no—
471C2×C32C2×#12111yescharacteristic
472C2×C32#57711yescharacteristic
473C2×C32#57811yescharacteristic
474#41—22no—
475#41—22no—
476#41—22no—
477#41—22no—
478C2×Q8—32no—
479C2×Q8—61no—
480C2×Q8—62no—
481C2×Q8—62no—
482C2×Q8C3×#6711yescharacteristic
483C2×Q8C22×C3×D311yescharacteristic
484C2×D4—21no—
485C2×D4—22no—
486C2×D4C3×#6712yes—
487C2×D4C3×#6711yescharacteristic
488C2×D4C22×C3×D311yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—61no—
492C22×C4C22×C3×D312yes—
493C22×C4C22×C3×D311yescharacteristic
494Q16—64no—
495SD16—64no—
496M16—34no—
497M16—64no—
498C2×C8—32no—
499C2×C8—31no—
500C2×C8—31no—
501C2×C8—64no—
502C2×C8—62no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—31no—
506C4⋊C4—31no—
507C4⋊C4—62no—
508C4⋊C4—62no—
509C4⋊C4C22×C3×D312yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—61no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C42—61no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—22no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—21no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3#93912yes—
543C22×C3C3×#12112yes—
544C22×C3C22×#6711yescharacteristic
545C22×C3C22×C3×D411yescharacteristic
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—22no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—42no—
552C3×C4—42no—
553C3×C4—44no—
554C3×C4—42no—
555C3×C4—41no—
556C3×C4—42no—
557C3×C4—41no—
558C3×C4—122no—
559C3×C4—121no—
560C3×C4—121no—
561C3×C4#93912yes—
562C3×C4C3×#12112yes—
563C3×C4C2×D3×D411yescharacteristic
564C3×C4C2×#23711yescharacteristic
565C3×C4C22×C3×D411yescharacteristic
566C3×C4C2×C3×#4111yescharacteristic
567Dic3—41no—
568Dic3—41no—
569Dic3—42no—
570C32#306311yescharacteristic, complemented, Sylow
571C23—21no—
572C23C2×C3×#6712yes—
573C23C2×C3×#6711yescharacteristic
574Q8—22no—
575Q8—21no—
576Q8—21no—
577Q8—62no—
578Q8—62no—
579Q8—122no—
580D4—24no—
581D4—22no—
582D4—21no—
583D4—21no—
584D4—42no—
585D4—82no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—42no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—81no—
595C2×C4—122no—
596C2×C4C3×D3×D412yes—
597C2×C4C3×#23712yes—
598C2×C4C2×C3×#6712yes—
599C2×C4C2×C3×#6711yescharacteristic
600C2×C4C23×C3×D311yescharacteristic
601C8—34no—
602C8—61no—
603C8—61no—
604C2×C3—22no—
605C2×C3—22no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—42no—
610C2×C3—42no—
611C2×C3—42no—
612C2×C3—82no—
613C2×C3—82no—
614C2×C3—81no—
615C2×C3—81no—
616C2×C3C2×#93911yescharacteristic
617C2×C3#583611yescharacteristic
618C2×C3#583811yescharacteristic
619C2×C3C2×C3×#12111yescharacteristic
620C2×C3C3×#57711yescharacteristic
621C2×C3C3×#57811yescharacteristic
622C22—21no—
623C22—21no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22C3×#93912yes—
631C22C22×C3×#6711yescharacteristic
632C4—22no—
633C4—41no—
634C4—42no—
635C4—122no—
636C4—121no—
637C4—121no—
638C4C3×#93912yes—
639C4C2×C3×D3×D411yescharacteristic
640C4C2×C3×#23711yescharacteristic
641C3—21no—
642C3#8342311yescharacteristic, direct product factor
643C3C3×#306311yescharacteristic, complemented
644C2—22no—
645C2—42no—
646C2—81no—
647C2C2×C3×#93911yescharacteristic
648C2C3×#583611yescharacteristic
649C2C3×#583811yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC32×#537C2
socleC22×C32C22×D4
hypercenterC3×#537D3
hypocenterC3C3×#3063
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C3×#537

Complements

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

NH ≅ G/Ntype
2. C3×#5825C2semidirect
3. C2×C3×#951C2semidirect
4. C2×C3×#948C2semidirect
5. C3×#5233C2semidirect
6. C3×#5052C2semidirect
7. C3×#5051C2semidirect
8. C3×#5042C2semidirect
9. C3×#5041C2semidirect
10. C2×C3×#810C2semidirect
11. C3×#4964C2semidirect
13. #83423C3direct (N × H)
51. #5825C2×C3semidirect
52. C2×#951C2×C3semidirect
53. C2×#948C2×C3semidirect
61. #5233C2×C3semidirect
62. #5052C2×C3semidirect
63. #5051C2×C3semidirect
64. #5042C2×C3semidirect
65. #5041C2×C3semidirect
66. C2×#810C2×C3semidirect
67. #4964C2×C3semidirect
104. C3×#208D4semidirect
323. #208C3×D4semidirect
570. C32#3063semidirect
642. C3#83423direct (N × H)
643. C3C3×#3063semidirect

Maximal subgroups

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

subgroupindex
C32×#5372
C3×#58252
C2×C3×#9512
C2×C3×#9482
C3×#52332
C3×#50522
C3×#50512
C3×#50422
C3×#50412
C2×C3×#8102
C3×#49642
C3×#30633
#834233