Finite Group Databasebeta

← C3×#78720 #11868096

Subgroups — C3×#78720 #11868096

Order 1152.

1007 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5822C211yesmaximal, characteristic, complemented
2C2×C3×#949C211yesmaximal, characteristic, complemented
3C2×C3×#908C211yesmaximal, characteristic, complemented
4C3×#5745C211yesmaximal, characteristic, complemented
5C32×#413C211yesmaximal, characteristic, complemented
6C2×C3×#823C211yesmaximal, characteristic, complemented
7C3×#5124C211yesmaximal, characteristic, complemented
8C3×#4754C214yesmaximal, complemented
9C3×#4749C214yesmaximal, complemented
10C3×#3068—31nomaximal
11#78720C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×D12C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C3×#951—21no—
18C3×#950—22no—
19C3×#949—21no—
20C3×#949—22no—
21C3×#949—22no—
22C3×#942C2212yescomplemented
23C2×C3×#215C2212yescomplemented
24C3×#932C2212yescomplemented
25C2×C3×#213C2212yescomplemented
26C2×C3×#208C2211yescharacteristic
27C2×C3×#208C2211yescharacteristic
28C3×#908—22no—
29C3×#908C2214yes—
30C3×#907C2214yescomplemented
31C3×D3×M16—22no—
32C2×C3×D24—21no—
33C2×C3×#204—21no—
34C3×#901—22no—
35C32×#101C2212yescomplemented
36C32×#100C2212yescomplemented
37C3×#823C2214yescomplemented
38C3×#822C2214yescomplemented
39C3×#575—31no—
40C2×C3×AGL(1,ℤ/8)—31no—
41C2×C3×AGL(1,ℤ/8)—31no—
42C3×#568—31no—
43#5822C2×C311yescharacteristic, complemented
44C2×#949C2×C311yescharacteristic, complemented
45C2×#908C2×C311yescharacteristic, complemented
46#5745C2×C311yescharacteristic, complemented
47C3×#472—34no—
48C3×#471—34no—
49C3×#430—31no—
50C3×#413—21no—
51C3×#413D311yescharacteristic, complemented
52C3×#413C2×C311yescharacteristic, complemented
53C2×C3×#100—31no—
54C2×#823C2×C311yescharacteristic, complemented
55#5124C2×C311yescharacteristic, complemented
56#4754C2×C314yescomplemented
57#4749C2×C314yescomplemented
58C23×C3×D3—21no—
59C32×#41—21no—
60C32×#41—21no—
61C32×#41—21no—
62C32×#41—21no—
63C2×C32×Q8C2311yescharacteristic
64C2×C32×D4—21no—
65C2×C32×D4C2311yescharacteristic
66C22×C32×C4—21no—
67C22×C32×C4C2311yescharacteristic
68C2×C3×#67—21no—
69C3×#235—21no—
70C3×#235—21no—
71C3×#235D412yes—
72C3×#235C2312yes—
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12—21no—
76C2×C3×D12—21no—
77C2×C3×D12—21no—
78C2×C3×D12D411yescharacteristic
79C2×C3×D12C2312yes—
80C2×C3×C4×D3—21no—
81C2×C3×Dic6D411yescharacteristic
82C32×M16—21no—
83C32×M16—21no—
84C32×M16C2312yes—
85C32×M16C2312yes—
86C2×C32×C8—21no—
87C2×C32×C8—21no—
88C3×#216—22no—
89C3×#215—22no—
90C3×#215—24no—
91C3×#214—22no—
92C3×#213—22no—
93C3×#213—24no—
94C3×#208—21no—
95C3×#208D412yes—
96C3×#208D412yes—
97C3×#208C2312yes—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2312yes—
100C3×D24—24no—
101C3×#204—24no—
102C3×ΓL(1,𝔽25)—24no—
103C3×D3×C8—24no—
104#3068—31noSylow
105C2×C3×#41—21no—
106C2×C3×#41—31no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C22×C311yescharacteristic
109C22×C3×D4—31no—
110C2×#235C22×C311yescharacteristic
111C22×D12C22×C311yescharacteristic
112C3×#137—32no—
113C3×#137—61no—
114C3×AGL(1,ℤ/8)—32no—
115C3×AGL(1,ℤ/8)—34no—
116C3×AGL(1,ℤ/8)—62no—
117C3×AGL(1,ℤ/8)—62no—
118C3×AGL(1,ℤ/8)—62no—
119C3×AGL(1,ℤ/8)—61no—
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
136#951—21no—
137#950—22no—
138#949—21no—
139#949—22no—
140#949—22no—
141#942C22×C312yescomplemented
142C2×#215C22×C312yescomplemented
143#932C22×C312yescomplemented
144C2×#213C22×C312yescomplemented
145C2×#208C22×C311yescharacteristic
146C2×#208C22×C311yescharacteristic
147#908—22no—
148#908C22×C314yes—
149#907C22×C314yescomplemented
150D3×M16—22no—
151C2×D24—21no—
152C2×#204—21no—
153#901—22no—
154C3×#108—34no—
155C3×#101—22no—
156C3×#101C2×D312yescomplemented
157C3×#101C22×C312yescomplemented
158C3×#100—22no—
159C3×#100—34no—
160C3×#100C2×D312yescomplemented
161C3×#100C22×C312yescomplemented
162#823C22×C314yescomplemented
163#822C22×C314yescomplemented
164C23×C32—21no—
165C23×C32#4111yescharacteristic
166C22×C3×D3—21no—
167C22×C3×D3—22no—
168C22×C3×D3—41no—
169C22×C3×D3—41no—
170C22×C3×D3—82no—
171C32×Q8—21no—
172C32×Q8—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—41no—
176C32×D4—41no—
177C2×C32×C4—21no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2×D411yescharacteristic
183C2×C32×C4C2×D411yescharacteristic
184C2×C32×C4#4111yescharacteristic
185C2×C32×C4C2411yescharacteristic
186C3×#67—22no—
187C3×#67—22no—
188C2×C3×Dic3—21no—
189C3×D12—21no—
190C3×D12—22no—
191C3×D12—22no—
192C3×D12—41no—
193C3×D12—41no—
194C3×D12C2×D412yes—
195C3×C4×D3—22no—
196C3×C4×D3—22no—
197C3×Dic6—21no—
198C3×Dic6C2×D412yes—
199C32×C8—22no—
200C32×C8—22no—
201C3×C3⋊C8—22no—
202C3×C3⋊C8C2×D414yes—
203#575—31no—
204C2×AGL(1,ℤ/8)—31no—
205C2×AGL(1,ℤ/8)—31no—
206#568—31no—
207#472—34no—
208#471—34no—
209#430—31no—
210#413C3×D311yescharacteristic, complemented
211C2×#100—31no—
212C24×C3—61no—
213C23×D3—21no—
214C3×#41—21no—
215C3×#41—21no—
216C3×#41—21no—
217C3×#41—21no—
218C3×#41—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—32no—
223C3×#41—32no—
224C3×#41—41no—
225C3×#41—41no—
226C3×#41—41no—
227C3×#41—41no—
228C3×#41—61no—
229C3×#41—61no—
230C2×C3×Q8—21no—
231C2×C3×Q8—31no—
232C2×C3×Q8C22×D311yescharacteristic
233C2×C3×Q8C23×C311yescharacteristic
234C2×C3×D4—22no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—31no—
239C2×C3×D4—32no—
240C2×C3×D4—61no—
241C2×C3×D4—61no—
242C2×C3×D4—61no—
243C2×C3×D4—61no—
244C2×C3×D4—61no—
245C2×C3×D4—61no—
246C2×C3×D4C22×D311yescharacteristic
247C2×C3×D4C23×C311yescharacteristic
248C22×C3×C4—22no—
249C22×C3×C4—21no—
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—61no—
253C22×C3×C4C22×D311yescharacteristic
254C22×C3×C4C23×C311yescharacteristic
255C2×#67—21no—
256#235—21no—
257#235—21no—
258#235C3×D412yes—
259#235C23×C312yes—
260C2×D12—21no—
261C2×D12—21no—
262C2×D12—21no—
263C2×D12—21no—
264C2×D12—21no—
265C2×D12C3×D411yescharacteristic
266C2×D12C23×C312yes—
267C2×C4×D3—21no—
268C2×Dic6C3×D411yescharacteristic
269C3×Q16—62no—
270C3×SD16—64no—
271C3×SD16—64no—
272C3×SD16—62no—
273C3×SD16—62no—
274C3×D8—64no—
275C3×D8—64no—
276C3×D8—62no—
277C3×M16—21no—
278C3×M16—21no—
279C3×M16—21no—
280C3×M16—21no—
281C3×M16—22no—
282C3×M16—22no—
283C3×M16—32no—
284C3×M16—32no—
285C3×M16—32no—
286C3×M16—41no—
287C3×M16—41no—
288C3×M16—64no—
289C3×M16—61no—
290C3×M16C22×D312yes—
291C3×M16C22×D312yes—
292C3×M16C23×C312yes—
293C3×M16C23×C312yes—
294C2×C3×C8—21no—
295C2×C3×C8—21no—
296C2×C3×C8—21no—
297C2×C3×C8—21no—
298C2×C3×C8—32no—
299C2×C3×C8—41no—
300C2×C3×C8—41no—
301C2×C3×C8—64no—
302C2×C3×C8—61no—
303#216—22no—
304#215—22no—
305#215—24no—
306#214—22no—
307#213—22no—
308#213—24no—
309#208—21no—
310#208C3×D412yes—
311#208C3×D412yes—
312#208C23×C312yes—
313C2×C3⋊C8—21no—
314C2×C3⋊C8C23×C312yes—
315D24—24no—
316#204—24no—
317ΓL(1,𝔽25)—24no—
318D3×C8—24no—
319C22×C32—41no—
320C22×C32—41no—
321C22×C32—81no—
322C22×C32#12111yescharacteristic
323C22×C32#12111yescharacteristic
324C22×C32C2×#4111yescharacteristic
325C2×C3×D3—22no—
326C2×C3×D3—42no—
327C2×C3×D3—81no—
328C2×C3×D3—82no—
329C2×C3×D3—82no—
330C2×C3×D3—82no—
331C32×C4—41no—
332C32×C4#12111yescharacteristic
333C32×C4#12111yescharacteristic
334C32×C4C22×D411yescharacteristic
335C32×C4C22×D411yescharacteristic
336C3×Dic3—22no—
337C2×#41—31no—
338C2×#41C2×C3×D311yescharacteristic
339C22×D4—31no—
340#137—32no—
341#137—61no—
342AGL(1,ℤ/8)—32no—
343AGL(1,ℤ/8)—34no—
344AGL(1,ℤ/8)—62no—
345AGL(1,ℤ/8)—62no—
346AGL(1,ℤ/8)—62no—
347AGL(1,ℤ/8)—61no—
348#135—62no—
349C2×SD16—32no—
350C2×SD16—61no—
351C2×D8—32no—
352C2×D8—61no—
353#131—34no—
354#131—62no—
355C2×M16—31no—
356C2×M16—31no—
357C2×M16—62no—
358C2×M16C2×C3×D311yescharacteristic
359C2×M16C2×C3×D311yescharacteristic
360#108—34no—
361#101C2×C3×D312yescomplemented
362#100—34no—
363#100C2×C3×D312yescomplemented
364C23×C3—22no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—61no—
369C23×C3—62no—
370C23×C3—121no—
371C23×C3—121no—
372C23×C3—242no—
373C23×C3#23511yescharacteristic
374C23×C3C3×#4111yescharacteristic
375C22×D3—21no—
376C22×D3—22no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—82no—
380C3×Q8—21no—
381C3×Q8—21no—
382C3×Q8—21no—
383C3×Q8—21no—
384C3×Q8—32no—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—61no—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—32no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—42no—
396C3×D4—42no—
397C3×D4—41no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4—41no—
401C3×D4—62no—
402C3×D4—62no—
403C3×D4—61no—
404C3×D4—62no—
405C3×D4—62no—
406C3×D4—121no—
407C3×D4—121no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—42no—
418C2×C3×C4—42no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—61no—
424C2×C3×C4—62no—
425C2×C3×C4—62no—
426C2×C3×C4#23511yescharacteristic
427C2×C3×C4D3×D411yescharacteristic
428C2×C3×C4D3×D411yescharacteristic
429C2×C3×C4D3×D411yescharacteristic
430C2×C3×C4D3×D411yescharacteristic
431C2×C3×C4C2×C3×D411yescharacteristic
432C2×C3×C4C2×C3×D411yescharacteristic
433C2×C3×C4C2×C3×D411yescharacteristic
434C2×C3×C4C2×C3×D411yescharacteristic
435C2×C3×C4C3×#4111yescharacteristic
436C2×C3×C4C23×D311yescharacteristic
437C2×C3×C4C24×C311yescharacteristic
438#67—22no—
439#67—22no—
440C2×Dic3—21no—
441D12—21no—
442D12—22no—
443D12—22no—
444D12—41no—
445D12—41no—
446D12C2×C3×D412yes—
447C4×D3—22no—
448C4×D3—22no—
449Dic6—21no—
450Dic6C2×C3×D412yes—
451C3×C8—22no—
452C3×C8—22no—
453C3×C8—22no—
454C3×C8—22no—
455C3×C8—34no—
456C3×C8—42no—
457C3×C8—42no—
458C3×C8—62no—
459C3⋊C8—22no—
460C3⋊C8C2×C3×D414yes—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—81no—
465C2×C32C2×#12111yescharacteristic
466C3×D3—42no—
467C3×D3—82no—
468C24—61no—
469#41—21no—
470#41—21no—
471#41—21no—
472#41—21no—
473#41—32no—
474#41—32no—
475#41—61no—
476#41—61no—
477C2×Q8—31no—
478C2×Q8C22×C3×D311yescharacteristic
479C2×D4—21no—
480C2×D4—32no—
481C2×D4—31no—
482C2×D4—61no—
483C2×D4—61no—
484C2×D4—61no—
485C2×D4—61no—
486C2×D4—61no—
487C2×D4—61no—
488C2×D4C22×C3×D311yescharacteristic
489C22×C4—21no—
490C22×C4—61no—
491C22×C4C22×C3×D311yescharacteristic
492Q16—62no—
493SD16—64no—
494SD16—64no—
495SD16—62no—
496SD16—62no—
497D8—64no—
498D8—64no—
499D8—62no—
500M16—21no—
501M16—21no—
502M16—32no—
503M16—32no—
504M16—32no—
505M16—64no—
506M16—61no—
507M16C22×C3×D312yes—
508M16C22×C3×D312yes—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—32no—
512C2×C8—64no—
513C2×C8—61no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—42no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—62no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—82no—
527C22×C3—122no—
528C22×C3—241no—
529C22×C3—242no—
530C22×C3—242no—
531C22×C3—242no—
532C22×C3#88411yescharacteristic
533C22×C3#88411yescharacteristic
534C22×C3C3×#12111yescharacteristic
535C22×C3C3×#12111yescharacteristic
536C22×C3C2×#23511yescharacteristic
537C22×C3C2×C3×#4111yescharacteristic
538C2×D3—22no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—82no—
542C2×D3—82no—
543C2×D3—82no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—42no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—62no—
552C3×C4#88411yescharacteristic
553C3×C4#88411yescharacteristic
554C3×C4C3×#12111yescharacteristic
555C3×C4C3×#12111yescharacteristic
556C3×C4C2×D3×D411yescharacteristic
557C3×C4C2×D3×D411yescharacteristic
558C3×C4C22×C3×D411yescharacteristic
559C3×C4C22×C3×D411yescharacteristic
560Dic3—22no—
561C32#306811yescharacteristic, complemented, Sylow
562C23—21no—
563C23—61no—
564C23—62no—
565C23—121no—
566C23—121no—
567C23—242no—
568C23C3×#23511yescharacteristic
569Q8—21no—
570Q8—21no—
571Q8—32no—
572Q8—61no—
573D4—21no—
574D4—21no—
575D4—32no—
576D4—41no—
577D4—41no—
578D4—62no—
579D4—62no—
580D4—61no—
581D4—62no—
582D4—62no—
583D4—121no—
584D4—121no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—61no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4C3×#23511yescharacteristic
592C2×C4C3×D3×D411yescharacteristic
593C2×C4C3×D3×D411yescharacteristic
594C2×C4C3×D3×D411yescharacteristic
595C2×C4C3×D3×D411yescharacteristic
596C2×C4C23×C3×D311yescharacteristic
597C8—22no—
598C8—22no—
599C8—34no—
600C8—62no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—41no—
611C2×C3—82no—
612C2×C3—81no—
613C2×C3—81no—
614C2×C3—122no—
615C2×C3—242no—
616C2×C3C2×#88411yescharacteristic
617C2×C3C2×C3×#12111yescharacteristic
618D3—42no—
619D3—82no—
620C22—41no—
621C22—41no—
622C22—62no—
623C22—81no—
624C22—122no—
625C22—241no—
626C22—242no—
627C22—242no—
628C22—242no—
629C22C3×#88411yescharacteristic
630C22C3×#88411yescharacteristic
631C22C2×C3×#23511yescharacteristic
632C4—41no—
633C4—62no—
634C4C3×#88411yescharacteristic
635C4C3×#88411yescharacteristic
636C4C2×C3×D3×D411yescharacteristic
637C4C2×C3×D3×D411yescharacteristic
638C3—21no—
639C3#7872011yescharacteristic, direct product factor
640C3C3×#306811yescharacteristic, complemented
641C2—21no—
642C2—21no—
643C2—21no—
644C2—81no—
645C2—122no—
646C2—242no—
647C2C2×C3×#88411yescharacteristic
subgroupG/H
centerC2×C3C2×#884
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#413C2
socleC2×C32C2×#121
hypercenterC3×#413D3
hypocenterC3C3×#3068
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#413

Complements

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

NH ≅ G/Ntype
1. C3×#5822C2semidirect
2. C2×C3×#949C2semidirect
3. C2×C3×#908C2semidirect
4. C3×#5745C2semidirect
5. C32×#413C2semidirect
6. C2×C3×#823C2semidirect
7. C3×#5124C2semidirect
8. C3×#4754C2semidirect
9. C3×#4749C2semidirect
11. #78720C3direct (N × H)
22. C3×#942C22semidirect
23. C2×C3×#215C22semidirect
24. C3×#932C22semidirect
25. C2×C3×#213C22semidirect
30. C3×#907C22semidirect
35. C32×#101C22semidirect
36. C32×#100C22semidirect
37. C3×#823C22semidirect
38. C3×#822C22semidirect
43. #5822C2×C3semidirect
44. C2×#949C2×C3semidirect
45. C2×#908C2×C3semidirect
46. #5745C2×C3semidirect
51. C3×#413D3semidirect
52. C3×#413C2×C3semidirect
54. C2×#823C2×C3semidirect
55. #5124C2×C3semidirect
56. #4754C2×C3semidirect
57. #4749C2×C3semidirect
141. #942C22×C3semidirect
142. C2×#215C22×C3semidirect
143. #932C22×C3semidirect
144. C2×#213C22×C3semidirect
149. #907C22×C3semidirect
156. C3×#101C2×D3semidirect
157. C3×#101C22×C3semidirect
160. C3×#100C2×D3semidirect
161. C3×#100C22×C3semidirect
162. #823C22×C3semidirect
163. #822C22×C3semidirect
210. #413C3×D3semidirect
361. #101C2×C3×D3semidirect
363. #100C2×C3×D3semidirect
561. C32#3068semidirect
639. C3#78720direct (N × H)
640. C3C3×#3068semidirect

Maximal subgroups

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

subgroupindex
C3×#58222
C2×C3×#9492
C2×C3×#9082
C3×#57452
C32×#4132
C2×C3×#8232
C3×#51242
C3×#47542
C3×#47492
C3×#30683
#787203