Finite Group Databasebeta

← C3×#78703 #11868084

Subgroups — C3×#78703 #11868084

Order 1152.

1025 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#932C211yesmaximal, characteristic, complemented
2C2×C3×#932C211yesmaximal, characteristic, complemented
3C3×#5748C211yesmaximal, characteristic, complemented
4C3×#5745C211yesmaximal, characteristic, complemented
5C2×C32×#100C211yesmaximal, characteristic, complemented
6C3×#5132C211yesmaximal, characteristic, complemented
7C3×#5124C211yesmaximal, characteristic, complemented
8C3×#4749C214yesmaximal, complemented
9C3×#4748C214yesmaximal, complemented
10C3×#3068—31nomaximal
11#78703C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C3×#235C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C2×C3×#214—21no—
18C2×C3×#214C2212yescomplemented
19C3×#932—22no—
20C3×#932—22no—
21C3×#932—22no—
22C3×#932C2214yes—
23C2×C3×#213—21no—
24C2×C3×#213C2212yescomplemented
25C2×C3×#208C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27C3×#909—21no—
28C3×#909C2212yescomplemented
29C3×#908—21no—
30C3×#908C2212yescomplemented
31C3×#907C2214yescomplemented
32C3×D3×M16—22no—
33C3×#904—22no—
34C3×#901—22no—
35C32×#100C2214yescomplemented
36C3×#824C2212yescomplemented
37C3×#823C2212yescomplemented
38C3×#822C2214yescomplemented
39C3×#575—31no—
40C2×C3×AGL(1,ℤ/8)—31no—
41C2×C3×AGL(1,ℤ/8)—31no—
42C3×#568—31no—
43C2×#932C2×C311yescharacteristic, complemented
44C2×#932C2×C311yescharacteristic, complemented
45#5748C2×C311yescharacteristic, complemented
46#5745C2×C311yescharacteristic, complemented
47C3×#472—34no—
48C3×#471—34no—
49C3×#430—31no—
50C3×#413—31no—
51C2×C3×#100—21no—
52C2×C3×#100D311yescharacteristic, complemented
53C2×C3×#100C2×C311yescharacteristic, complemented
54#5132C2×C311yescharacteristic, complemented
55#5124C2×C311yescharacteristic, complemented
56#4749C2×C314yescomplemented
57#4748C2×C314yescomplemented
58C24×C32—21no—
59C2×C32×D4—21no—
60C2×C32×D4—21no—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—21no—
67C2×C3×#67—21no—
68C3×#235—21no—
69C3×#235—21no—
70C3×#235—21no—
71C3×#235—21no—
72C3×#235—21no—
73C3×#235—21no—
74C3×#235D412yes—
75C3×#235C2312yes—
76C2×C3×D12D411yescharacteristic
77C2×C3×D12C2311yescharacteristic
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3—21no—
80C2×C3×Dic6D411yescharacteristic
81C2×C3×Dic6C2311yescharacteristic
82C32×M16—21no—
83C32×M16—21no—
84C32×M16C2312yes—
85C32×M16C2312yes—
86C2×C32×C8—21no—
87C2×C32×C8—21no—
88C3×#214—24no—
89C3×#214—24no—
90C3×#213—24no—
91C3×#213—24no—
92C3×#208—21no—
93C3×#208D412yes—
94C3×#208D412yes—
95C3×#208C2312yes—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2312yes—
98C3×Dic12—22no—
99C3×D24—22no—
100C3×#204—22no—
101C3×#204—22no—
102C3×ΓL(1,𝔽25)—24no—
103C3×D3×C8—24no—
104#3068—31noSylow
105C2×C3×#41—31no—
106C2×C3×#41—31no—
107C22×C3×D4—21no—
108C22×C3×D4C2×D311yescharacteristic
109C22×C3×D4C22×C311yescharacteristic
110C2×#235C22×C311yescharacteristic
111C2×#235C22×C311yescharacteristic
112C3×#137—32no—
113C3×#137—61no—
114C3×AGL(1,ℤ/8)—34no—
115C3×AGL(1,ℤ/8)—32no—
116C3×AGL(1,ℤ/8)—61no—
117C3×AGL(1,ℤ/8)—62no—
118C3×AGL(1,ℤ/8)—62no—
119C3×AGL(1,ℤ/8)—62no—
120C3×#135—62no—
121C2×C3×SD16—32no—
122C2×C3×SD16—61no—
123C2×C3×D8—32no—
124C2×C3×D8—61no—
125C3×#131—34no—
126C3×#131—62no—
127C2×C3×M16—21no—
128C2×C3×M16—21no—
129C2×C3×M16—31no—
130C2×C3×M16—31no—
131C2×C3×M16—62no—
132C2×C3×M16C2×D311yescharacteristic
133C2×C3×M16C2×D311yescharacteristic
134C2×C3×M16C22×C311yescharacteristic
135C2×C3×M16C22×C311yescharacteristic
136C2×#214—21no—
137C2×#214C22×C312yescomplemented
138#932—22no—
139#932—22no—
140#932—22no—
141#932C22×C314yes—
142C2×#213—21no—
143C2×#213C22×C312yescomplemented
144C2×#208C22×C311yescharacteristic
145C2×#208C22×C311yescharacteristic
146#909—21no—
147#909C22×C312yescomplemented
148#908—21no—
149#908C22×C312yescomplemented
150#907C22×C314yescomplemented
151D3×M16—22no—
152#904—22no—
153#901—22no—
154C3×#108—34no—
155C3×#101—32no—
156C3×#100—24no—
157C3×#100—32no—
158C3×#100C2×D314yescomplemented
159C3×#100C22×C314yescomplemented
160#824C22×C312yescomplemented
161#823C22×C312yescomplemented
162#822C22×C314yescomplemented
163C23×C32—22no—
164C23×C32—41no—
165C23×C32—41no—
166C23×C32—82no—
167C23×C32#4111yescharacteristic
168C22×C3×D3—21no—
169C22×C3×D3—21no—
170C32×D4—22no—
171C32×D4—22no—
172C32×D4—41no—
173C32×D4—41no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4#4111yescharacteristic
179C2×C32×C4C2411yescharacteristic
180C3×#67—22no—
181C3×#67—22no—
182C3×#67—41no—
183C3×#67—41no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12C2×D412yes—
190C3×C4×D3—22no—
191C3×C4×D3—22no—
192C3×C4×D3—41no—
193C3×C4×D3—41no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6C2×D412yes—
198C32×C8—22no—
199C32×C8—22no—
200C3×C3⋊C8—22no—
201C3×C3⋊C8C2×D414yes—
202#575—31no—
203C2×AGL(1,ℤ/8)—31no—
204C2×AGL(1,ℤ/8)—31no—
205#568—31no—
206#472—34no—
207#471—34no—
208#430—31no—
209#413—31no—
210C2×#100C3×D311yescharacteristic, complemented
211C24×C3—22no—
212C24×C3—21no—
213C24×C3—21no—
214C3×#41—32no—
215C3×#41—32no—
216C3×#41—61no—
217C3×#41—61no—
218C3×#41—61no—
219C3×#41—61no—
220C3×#41—61no—
221C3×#41—61no—
222C2×C3×Q8—31no—
223C2×C3×Q8—31no—
224C2×C3×D4—22no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—22no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—41no—
239C2×C3×D4—41no—
240C2×C3×D4—41no—
241C2×C3×D4—41no—
242C2×C3×D4—61no—
243C2×C3×D4—61no—
244C2×C3×D4C22×D312yes—
245C2×C3×D4C23×C312yes—
246C22×C3×C4—21no—
247C22×C3×C4—61no—
248C22×C3×C4—61no—
249C22×C3×C4C22×D311yescharacteristic
250C22×C3×C4C23×C311yescharacteristic
251C2×#67—21no—
252C2×#67—21no—
253#235—21no—
254#235—21no—
255#235—21no—
256#235—21no—
257#235—21no—
258#235—21no—
259#235C3×D412yes—
260#235C23×C312yes—
261C2×D12C3×D411yescharacteristic
262C2×D12C23×C311yescharacteristic
263C2×C4×D3—21no—
264C2×C4×D3—21no—
265C2×Dic6C3×D411yescharacteristic
266C2×Dic6C23×C311yescharacteristic
267C3×Q16—62no—
268C3×SD16—62no—
269C3×SD16—62no—
270C3×SD16—64no—
271C3×SD16—64no—
272C3×D8—62no—
273C3×D8—64no—
274C3×D8—64no—
275C3×M16—21no—
276C3×M16—21no—
277C3×M16—21no—
278C3×M16—21no—
279C3×M16—22no—
280C3×M16—22no—
281C3×M16—32no—
282C3×M16—32no—
283C3×M16—32no—
284C3×M16—41no—
285C3×M16—41no—
286C3×M16—64no—
287C3×M16—61no—
288C3×M16C22×D312yes—
289C3×M16C22×D312yes—
290C3×M16C23×C312yes—
291C3×M16C23×C312yes—
292C2×C3×C8—21no—
293C2×C3×C8—21no—
294C2×C3×C8—21no—
295C2×C3×C8—21no—
296C2×C3×C8—32no—
297C2×C3×C8—41no—
298C2×C3×C8—41no—
299C2×C3×C8—64no—
300C2×C3×C8—61no—
301#214—24no—
302#214—24no—
303#213—24no—
304#213—24no—
305#208—21no—
306#208C3×D412yes—
307#208C3×D412yes—
308#208C23×C312yes—
309C2×C3⋊C8—21no—
310C2×C3⋊C8C23×C312yes—
311Dic12—22no—
312D24—22no—
313#204—22no—
314#204—22no—
315ΓL(1,𝔽25)—24no—
316D3×C8—24no—
317C22×C32—41no—
318C22×C32—42no—
319C22×C32—82no—
320C22×C32—82no—
321C22×C32—82no—
322C22×C32#12111yescharacteristic
323C22×C32#12111yescharacteristic
324C22×C32C2×#4111yescharacteristic
325C2×C3×D3—22no—
326C2×C3×D3—41no—
327C2×C3×D3—81no—
328C2×C3×D3—81no—
329C32×C4#12111yescharacteristic
330C32×C4#12111yescharacteristic
331C32×C4C22×D411yescharacteristic
332C32×C4C22×D411yescharacteristic
333C3×Dic3—22no—
334C3×Dic3—41no—
335C2×#41—31no—
336C2×#41—31no—
337C22×D4C2×C3×D311yescharacteristic
338#137—32no—
339#137—61no—
340AGL(1,ℤ/8)—34no—
341AGL(1,ℤ/8)—32no—
342AGL(1,ℤ/8)—61no—
343AGL(1,ℤ/8)—62no—
344AGL(1,ℤ/8)—62no—
345AGL(1,ℤ/8)—62no—
346#135—62no—
347C2×SD16—32no—
348C2×SD16—61no—
349C2×D8—32no—
350C2×D8—61no—
351#131—34no—
352#131—62no—
353C2×M16—31no—
354C2×M16—31no—
355C2×M16—62no—
356C2×M16C2×C3×D311yescharacteristic
357C2×M16C2×C3×D311yescharacteristic
358#108—34no—
359#101—32no—
360#100—32no—
361#100C2×C3×D314yescomplemented
362C23×C3—24no—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—21no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—61no—
373C23×C3—61no—
374C23×C3—82no—
375C23×C3—82no—
376C23×C3—84no—
377C23×C3#23511yescharacteristic
378C23×C3C3×#4111yescharacteristic
379C22×D3—21no—
380C22×D3—21no—
381C3×Q8—32no—
382C3×Q8—61no—
383C3×Q8—61no—
384C3×Q8—61no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—32no—
390C3×D4—42no—
391C3×D4—42no—
392C3×D4—42no—
393C3×D4—42no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—62no—
399C3×D4—62no—
400C3×D4—61no—
401C3×D4—61no—
402C3×D4—61no—
403C3×D4—121no—
404C3×D4—121no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—61no—
412C2×C3×C4—62no—
413C2×C3×C4—62no—
414C2×C3×C4—61no—
415C2×C3×C4—121no—
416C2×C3×C4—121no—
417C2×C3×C4#23511yescharacteristic
418C2×C3×C4D3×D411yescharacteristic
419C2×C3×C4D3×D411yescharacteristic
420C2×C3×C4D3×D411yescharacteristic
421C2×C3×C4D3×D411yescharacteristic
422C2×C3×C4C2×C3×D411yescharacteristic
423C2×C3×C4C2×C3×D411yescharacteristic
424C2×C3×C4C2×C3×D411yescharacteristic
425C2×C3×C4C2×C3×D411yescharacteristic
426C2×C3×C4C3×#4111yescharacteristic
427C2×C3×C4C23×D311yescharacteristic
428C2×C3×C4C24×C311yescharacteristic
429#67—22no—
430#67—22no—
431#67—41no—
432#67—41no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435D12—21no—
436D12—21no—
437D12—21no—
438D12C2×C3×D412yes—
439C4×D3—22no—
440C4×D3—22no—
441C4×D3—41no—
442C4×D3—41no—
443Dic6—21no—
444Dic6—21no—
445Dic6—21no—
446Dic6C2×C3×D412yes—
447C3×C8—22no—
448C3×C8—22no—
449C3×C8—22no—
450C3×C8—22no—
451C3×C8—34no—
452C3×C8—42no—
453C3×C8—42no—
454C3×C8—62no—
455C3⋊C8—22no—
456C3⋊C8C2×C3×D414yes—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—82no—
461C2×C32C2×#12111yescharacteristic
462C3×D3—42no—
463C3×D3—81no—
464C24—21no—
465#41—32no—
466#41—32no—
467#41—61no—
468#41—61no—
469#41—61no—
470#41—61no—
471#41—61no—
472#41—61no—
473C2×Q8—31no—
474C2×Q8—31no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—31no—
481C2×D4—31no—
482C2×D4—61no—
483C2×D4—61no—
484C2×D4C22×C3×D312yes—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4C22×C3×D311yescharacteristic
488Q16—62no—
489SD16—62no—
490SD16—62no—
491SD16—64no—
492SD16—64no—
493D8—62no—
494D8—64no—
495D8—64no—
496M16—21no—
497M16—21no—
498M16—32no—
499M16—32no—
500M16—32no—
501M16—64no—
502M16—61no—
503M16C22×C3×D312yes—
504M16C22×C3×D312yes—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—32no—
508C2×C8—64no—
509C2×C8—61no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—44no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—62no—
520C22×C3—82no—
521C22×C3—82no—
522C22×C3—82no—
523C22×C3—82no—
524C22×C3—84no—
525C22×C3—82no—
526C22×C3—82no—
527C22×C3—84no—
528C22×C3—84no—
529C22×C3—121no—
530C22×C3—241no—
531C22×C3—241no—
532C22×C3#88411yescharacteristic
533C22×C3#88411yescharacteristic
534C22×C3C3×#12111yescharacteristic
535C22×C3C3×#12111yescharacteristic
536C22×C3C2×#23511yescharacteristic
537C22×C3C2×C3×#4111yescharacteristic
538C2×D3—22no—
539C2×D3—41no—
540C2×D3—81no—
541C2×D3—81no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—62no—
547C3×C4—121no—
548C3×C4#88411yescharacteristic
549C3×C4#88411yescharacteristic
550C3×C4C3×#12111yescharacteristic
551C3×C4C3×#12111yescharacteristic
552C3×C4C2×D3×D411yescharacteristic
553C3×C4C2×D3×D411yescharacteristic
554C3×C4C22×C3×D411yescharacteristic
555C3×C4C22×C3×D411yescharacteristic
556Dic3—22no—
557Dic3—41no—
558C32#306811yescharacteristic, complemented, Sylow
559C23—22no—
560C23—41no—
561C23—41no—
562C23—61no—
563C23—61no—
564C23—82no—
565C23C3×#23511yescharacteristic
566Q8—32no—
567Q8—61no—
568Q8—61no—
569Q8—61no—
570D4—22no—
571D4—22no—
572D4—32no—
573D4—41no—
574D4—41no—
575D4—62no—
576D4—62no—
577D4—61no—
578D4—61no—
579D4—61no—
580D4—121no—
581D4—121no—
582C2×C4—61no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—61no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4C3×#23511yescharacteristic
589C2×C4C3×D3×D411yescharacteristic
590C2×C4C3×D3×D411yescharacteristic
591C2×C4C3×D3×D411yescharacteristic
592C2×C4C3×D3×D411yescharacteristic
593C2×C4C23×C3×D311yescharacteristic
594C8—22no—
595C8—22no—
596C8—34no—
597C8—62no—
598C2×C3—22no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3—84no—
609C2×C3—82no—
610C2×C3—82no—
611C2×C3—122no—
612C2×C3—241no—
613C2×C3C2×#88411yescharacteristic
614C2×C3C2×C3×#12111yescharacteristic
615D3—42no—
616D3—81no—
617C22—41no—
618C22—42no—
619C22—62no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—121no—
624C22—241no—
625C22—241no—
626C22C3×#88411yescharacteristic
627C22C3×#88411yescharacteristic
628C22C2×C3×#23511yescharacteristic
629C4—62no—
630C4—121no—
631C4C3×#88411yescharacteristic
632C4C3×#88411yescharacteristic
633C4C2×C3×D3×D411yescharacteristic
634C4C2×C3×D3×D411yescharacteristic
635C3—21no—
636C3#7870311yescharacteristic, direct product factor
637C3C3×#306811yescharacteristic, complemented
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—122no—
643C2—241no—
644C2C2×C3×#88411yescharacteristic
subgroupG/H
centerC2×C3C2×#884
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#100C2
socleC2×C32C2×#121
hypercenterC2×C3×#100D3
hypocenterC3C3×#3068
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#100

Complements

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

NH ≅ G/Ntype
1. C2×C3×#932C2semidirect
2. C2×C3×#932C2semidirect
3. C3×#5748C2semidirect
4. C3×#5745C2semidirect
5. C2×C32×#100C2semidirect
6. C3×#5132C2semidirect
7. C3×#5124C2semidirect
8. C3×#4749C2semidirect
9. C3×#4748C2semidirect
11. #78703C3direct (N × H)
18. C2×C3×#214C22semidirect
24. C2×C3×#213C22semidirect
28. C3×#909C22semidirect
30. C3×#908C22semidirect
31. C3×#907C22semidirect
35. C32×#100C22semidirect
36. C3×#824C22semidirect
37. C3×#823C22semidirect
38. C3×#822C22semidirect
43. C2×#932C2×C3semidirect
44. C2×#932C2×C3semidirect
45. #5748C2×C3semidirect
46. #5745C2×C3semidirect
52. C2×C3×#100D3semidirect
53. C2×C3×#100C2×C3semidirect
54. #5132C2×C3semidirect
55. #5124C2×C3semidirect
56. #4749C2×C3semidirect
57. #4748C2×C3semidirect
137. C2×#214C22×C3semidirect
143. C2×#213C22×C3semidirect
147. #909C22×C3semidirect
149. #908C22×C3semidirect
150. #907C22×C3semidirect
158. C3×#100C2×D3semidirect
159. C3×#100C22×C3semidirect
160. #824C22×C3semidirect
161. #823C22×C3semidirect
162. #822C22×C3semidirect
210. C2×#100C3×D3semidirect
361. #100C2×C3×D3semidirect
558. C32#3068semidirect
636. C3#78703direct (N × H)
637. C3C3×#3068semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9322
C3×#57482
C3×#57452
C2×C32×#1002
C3×#51322
C3×#51242
C3×#47492
C3×#47482
C3×#30683
#787033