Finite Group Databasebeta

← C3×#78712 #11868161

Subgroups — C3×#78712 #11868161

Order 1152.

965 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#942C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic, complemented
3C3×#5748C211yesmaximal, characteristic
4C3×#5745C211yesmaximal, characteristic, complemented
5C2×C32×#101C211yesmaximal, characteristic, complemented
6C3×#5132C211yesmaximal, characteristic, complemented
7C3×#5124C211yesmaximal, characteristic, complemented
8C3×#4755C214yesmaximal, complemented
9C3×#4754C214yesmaximal, complemented
10C3×#3069—31nomaximal
11#78712C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C3×#235C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C2×C3×#216—21no—
18C2×C3×#216C2212yescomplemented
19C3×#942—22no—
20C3×#942—22no—
21C3×#942—22no—
22C3×#942C2214yes—
23C2×C3×#215—21no—
24C2×C3×#215C2212yescomplemented
25C2×C3×#208C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27C3×#909—21no—
28C3×#909C2212yes—
29C3×#908—21no—
30C3×#908C2212yes—
31C3×#907C2214yes—
32C3×D3×M16—22no—
33C3×#904—22no—
34C3×#901—22no—
35C32×#101C2214yescomplemented
36C3×#824C2212yescomplemented
37C3×#823C2212yescomplemented
38C3×#822C2214yescomplemented
39C3×#575—31no—
40C2×C3×#137—31no—
41C2×C3×#137—31no—
42C3×#568—31no—
43C2×#942C2×C311yescharacteristic, complemented
44C2×#942C2×C311yescharacteristic, complemented
45#5748C2×C311yescharacteristic
46#5745C2×C311yescharacteristic, complemented
47C3×#473—34no—
48C3×#471—34no—
49C3×#430—31no—
50C3×#413—31no—
51C2×C3×#101—21no—
52C2×C3×#101D311yescharacteristic, complemented
53C2×C3×#101C2×C311yescharacteristic, complemented
54#5132C2×C311yescharacteristic, complemented
55#5124C2×C311yescharacteristic, complemented
56#4755C2×C314yescomplemented
57#4754C2×C314yescomplemented
58C2×C32×Q8—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8—21no—
61C2×C32×Q8—21no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2312yes—
64C22×C32×C4—21no—
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×#216—24no—
89C3×#216—24no—
90C3×#215—24no—
91C3×#215—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#3069—31noSylow
105C2×C3×#41—31no—
106C2×C3×#41—31no—
107C22×C3×Q8—21no—
108C22×C3×Q8C2×D311yescharacteristic
109C22×C3×Q8C22×C311yescharacteristic
110C2×#235C22×C311yescharacteristic
111C2×#235C22×C311yescharacteristic
112C3×#137—34no—
113C3×#137—32no—
114C3×#137—61no—
115C3×#137—62no—
116C3×#137—62no—
117C3×#137—62no—
118C3×AGL(1,ℤ/8)—32no—
119C3×AGL(1,ℤ/8)—61no—
120C3×#135—62no—
121C2×C3×Q16—32no—
122C2×C3×Q16—61no—
123C2×C3×SD16—32no—
124C2×C3×SD16—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×#216—21no—
137C2×#216C22×C312yescomplemented
138#942—22no—
139#942—22no—
140#942—22no—
141#942C22×C314yes—
142C2×#215—21no—
143C2×#215C22×C312yescomplemented
144C2×#208C22×C311yescharacteristic
145C2×#208C22×C311yescharacteristic
146#909—21no—
147#909C22×C312yes—
148#908—21no—
149#908C22×C312yes—
150#907C22×C314yes—
151D3×M16—22no—
152#904—22no—
153#901—22no—
154C3×#108—34no—
155C3×#101—24no—
156C3×#101—32no—
157C3×#101C2×D314yescomplemented
158C3×#101C22×C314yescomplemented
159C3×#100—32no—
160#824C22×C312yescomplemented
161#823C22×C312yescomplemented
162#822C22×C314yescomplemented
163C23×C32#4111yescharacteristic
164C22×C3×D3—21no—
165C22×C3×D3—21no—
166C32×Q8—22no—
167C32×Q8—22no—
168C32×Q8—41no—
169C32×Q8—41no—
170C2×C32×C4—22no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4#4111yescharacteristic
178C2×C32×C4C2411yescharacteristic
179C3×#67—22no—
180C3×#67—22no—
181C3×#67—41no—
182C3×#67—41no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12C2×D412yes—
189C3×C4×D3—22no—
190C3×C4×D3—22no—
191C3×C4×D3—41no—
192C3×C4×D3—41no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6C2×D412yes—
197C32×C8—22no—
198C32×C8—22no—
199C3×C3⋊C8—22no—
200C3×C3⋊C8C2×D414yes—
201#575—31no—
202C2×#137—31no—
203C2×#137—31no—
204#568—31no—
205#473—34no—
206#471—34no—
207#430—31no—
208#413—31no—
209C2×#101C3×D311yescharacteristic, complemented
210C3×#41—32no—
211C3×#41—32no—
212C3×#41—61no—
213C3×#41—61no—
214C3×#41—61no—
215C3×#41—61no—
216C3×#41—61no—
217C3×#41—61no—
218C2×C3×Q8—22no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—22no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—41no—
233C2×C3×Q8—41no—
234C2×C3×Q8—41no—
235C2×C3×Q8—41no—
236C2×C3×Q8C22×D312yes—
237C2×C3×Q8C23×C312yes—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—61no—
241C2×C3×D4—61no—
242C22×C3×C4—22no—
243C22×C3×C4—21no—
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4—61no—
247C22×C3×C4—61no—
248C22×C3×C4C22×D311yescharacteristic
249C22×C3×C4C23×C311yescharacteristic
250C2×#67—21no—
251C2×#67—21no—
252#235—21no—
253#235—21no—
254#235—21no—
255#235—21no—
256#235—21no—
257#235—21no—
258#235C3×D412yes—
259#235C23×C312yes—
260C2×D12C3×D411yescharacteristic
261C2×D12C23×C311yescharacteristic
262C2×C4×D3—21no—
263C2×C4×D3—21no—
264C2×Dic6C3×D411yescharacteristic
265C2×Dic6C23×C311yescharacteristic
266C3×Q16—62no—
267C3×Q16—64no—
268C3×Q16—64no—
269C3×SD16—62no—
270C3×SD16—62no—
271C3×SD16—64no—
272C3×SD16—64no—
273C3×D8—62no—
274C3×M16—21no—
275C3×M16—21no—
276C3×M16—21no—
277C3×M16—21no—
278C3×M16—22no—
279C3×M16—22no—
280C3×M16—32no—
281C3×M16—32no—
282C3×M16—32no—
283C3×M16—41no—
284C3×M16—41no—
285C3×M16—64no—
286C3×M16—61no—
287C3×M16C22×D312yes—
288C3×M16C22×D312yes—
289C3×M16C23×C312yes—
290C3×M16C23×C312yes—
291C2×C3×C8—21no—
292C2×C3×C8—21no—
293C2×C3×C8—21no—
294C2×C3×C8—21no—
295C2×C3×C8—32no—
296C2×C3×C8—41no—
297C2×C3×C8—41no—
298C2×C3×C8—64no—
299C2×C3×C8—61no—
300#216—24no—
301#216—24no—
302#215—24no—
303#215—24no—
304#208—21no—
305#208C3×D412yes—
306#208C3×D412yes—
307#208C23×C312yes—
308C2×C3⋊C8—21no—
309C2×C3⋊C8C23×C312yes—
310Dic12—22no—
311D24—22no—
312#204—22no—
313#204—22no—
314ΓL(1,𝔽25)—24no—
315D3×C8—24no—
316C22×C32—41no—
317C22×C32#12111yescharacteristic
318C22×C32#12111yescharacteristic
319C22×C32C2×#4111yescharacteristic
320C2×C3×D3—22no—
321C2×C3×D3—41no—
322C2×C3×D3—81no—
323C2×C3×D3—81no—
324C32×C4—42no—
325C32×C4#12111yescharacteristic
326C32×C4#12111yescharacteristic
327C32×C4C22×D411yescharacteristic
328C32×C4C22×D411yescharacteristic
329C3×Dic3—22no—
330C3×Dic3—41no—
331C2×#41—31no—
332C2×#41—31no—
333C22×Q8C2×C3×D311yescharacteristic
334#137—34no—
335#137—32no—
336#137—61no—
337#137—62no—
338#137—62no—
339#137—62no—
340AGL(1,ℤ/8)—32no—
341AGL(1,ℤ/8)—61no—
342#135—62no—
343C2×Q16—32no—
344C2×Q16—61no—
345C2×SD16—32no—
346C2×SD16—61no—
347#131—34no—
348#131—62no—
349C2×M16—31no—
350C2×M16—31no—
351C2×M16—62no—
352C2×M16C2×C3×D311yescharacteristic
353C2×M16C2×C3×D311yescharacteristic
354#108—34no—
355#101—32no—
356#101C2×C3×D314yescomplemented
357#100—32no—
358C23×C3—21no—
359C23×C3—61no—
360C23×C3—61no—
361C23×C3#23511yescharacteristic
362C23×C3C3×#4111yescharacteristic
363C22×D3—21no—
364C22×D3—21no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—32no—
370C3×Q8—42no—
371C3×Q8—42no—
372C3×Q8—42no—
373C3×Q8—42no—
374C3×Q8—41no—
375C3×Q8—41no—
376C3×Q8—41no—
377C3×Q8—41no—
378C3×Q8—61no—
379C3×Q8—61no—
380C3×Q8—61no—
381C3×D4—32no—
382C3×D4—62no—
383C3×D4—62no—
384C3×D4—61no—
385C3×D4—61no—
386C3×D4—61no—
387C3×D4—121no—
388C3×D4—121no—
389C2×C3×C4—24no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—42no—
399C2×C3×C4—42no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—61no—
405C2×C3×C4—62no—
406C2×C3×C4—62no—
407C2×C3×C4—61no—
408C2×C3×C4—121no—
409C2×C3×C4—121no—
410C2×C3×C4#23511yescharacteristic
411C2×C3×C4D3×D411yescharacteristic
412C2×C3×C4D3×D411yescharacteristic
413C2×C3×C4D3×D411yescharacteristic
414C2×C3×C4D3×D411yescharacteristic
415C2×C3×C4C2×C3×D411yescharacteristic
416C2×C3×C4C2×C3×D411yescharacteristic
417C2×C3×C4C2×C3×D411yescharacteristic
418C2×C3×C4C2×C3×D411yescharacteristic
419C2×C3×C4C3×#4111yescharacteristic
420C2×C3×C4C23×D311yescharacteristic
421C2×C3×C4C24×C311yescharacteristic
422#67—22no—
423#67—22no—
424#67—41no—
425#67—41no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428D12—21no—
429D12—21no—
430D12—21no—
431D12C2×C3×D412yes—
432C4×D3—22no—
433C4×D3—22no—
434C4×D3—41no—
435C4×D3—41no—
436Dic6—21no—
437Dic6—21no—
438Dic6—21no—
439Dic6C2×C3×D412yes—
440C3×C8—22no—
441C3×C8—22no—
442C3×C8—22no—
443C3×C8—22no—
444C3×C8—34no—
445C3×C8—42no—
446C3×C8—42no—
447C3×C8—62no—
448C3⋊C8—22no—
449C3⋊C8C2×C3×D414yes—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32C2×#12111yescharacteristic
454C3×D3—42no—
455C3×D3—81no—
456#41—32no—
457#41—32no—
458#41—61no—
459#41—61no—
460#41—61no—
461#41—61no—
462#41—61no—
463#41—61no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—31no—
470C2×Q8—31no—
471C2×Q8C22×C3×D312yes—
472C2×D4—31no—
473C2×D4—31no—
474C2×D4—61no—
475C2×D4—61no—
476C22×C4—21no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4C22×C3×D311yescharacteristic
480Q16—62no—
481Q16—64no—
482Q16—64no—
483SD16—62no—
484SD16—62no—
485SD16—64no—
486SD16—64no—
487D8—62no—
488M16—21no—
489M16—21no—
490M16—32no—
491M16—32no—
492M16—32no—
493M16—64no—
494M16—61no—
495M16C22×C3×D312yes—
496M16C22×C3×D312yes—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—32no—
500C2×C8—64no—
501C2×C8—61no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—62no—
509C22×C3—121no—
510C22×C3—241no—
511C22×C3—241no—
512C22×C3#88411yescharacteristic
513C22×C3#88411yescharacteristic
514C22×C3C3×#12111yescharacteristic
515C22×C3C3×#12111yescharacteristic
516C22×C3C2×#23511yescharacteristic
517C22×C3C2×C3×#4111yescharacteristic
518C2×D3—22no—
519C2×D3—41no—
520C2×D3—81no—
521C2×D3—81no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—44no—
527C3×C4—42no—
528C3×C4—42no—
529C3×C4—62no—
530C3×C4—121no—
531C3×C4#88411yescharacteristic
532C3×C4#88411yescharacteristic
533C3×C4C3×#12111yescharacteristic
534C3×C4C3×#12111yescharacteristic
535C3×C4C2×D3×D411yescharacteristic
536C3×C4C2×D3×D411yescharacteristic
537C3×C4C22×C3×D411yescharacteristic
538C3×C4C22×C3×D411yescharacteristic
539Dic3—22no—
540Dic3—41no—
541C32#306911yescharacteristic, complemented, Sylow
542C23—61no—
543C23—61no—
544C23C3×#23511yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—32no—
548Q8—41no—
549Q8—41no—
550Q8—61no—
551Q8—61no—
552Q8—61no—
553D4—32no—
554D4—62no—
555D4—62no—
556D4—61no—
557D4—61no—
558D4—61no—
559D4—121no—
560D4—121no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—61no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4C3×#23511yescharacteristic
571C2×C4C3×D3×D411yescharacteristic
572C2×C4C3×D3×D411yescharacteristic
573C2×C4C3×D3×D411yescharacteristic
574C2×C4C3×D3×D411yescharacteristic
575C2×C4C23×C3×D311yescharacteristic
576C8—22no—
577C8—22no—
578C8—34no—
579C8—62no—
580C2×C3—22no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—41no—
589C2×C3—41no—
590C2×C3—122no—
591C2×C3—241no—
592C2×C3C2×#88411yescharacteristic
593C2×C3C2×C3×#12111yescharacteristic
594D3—42no—
595D3—81no—
596C22—41no—
597C22—62no—
598C22—121no—
599C22—241no—
600C22—241no—
601C22C3×#88411yescharacteristic
602C22C3×#88411yescharacteristic
603C22C2×C3×#23511yescharacteristic
604C4—42no—
605C4—62no—
606C4—121no—
607C4C3×#88411yescharacteristic
608C4C3×#88411yescharacteristic
609C4C2×C3×D3×D411yescharacteristic
610C4C2×C3×D3×D411yescharacteristic
611C3—21no—
612C3#7871211yescharacteristic, direct product factor
613C3C3×#306911yescharacteristic, complemented
614C2—21no—
615C2—21no—
616C2—21no—
617C2—122no—
618C2—241no—
619C2C2×C3×#88411yescharacteristic
subgroupG/H
centerC2×C3C2×#884
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#101C2
socleC2×C32C2×#121
prime radicalC3×#5748C2
hypercenterC2×C3×#101D3
hypocenterC3C3×#3069
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
1. C2×C3×#942C2semidirect
2. C2×C3×#942C2semidirect
4. C3×#5745C2semidirect
5. C2×C32×#101C2semidirect
6. C3×#5132C2semidirect
7. C3×#5124C2semidirect
8. C3×#4755C2semidirect
9. C3×#4754C2semidirect
11. #78712C3direct (N × H)
18. C2×C3×#216C22semidirect
24. C2×C3×#215C22semidirect
35. C32×#101C22semidirect
36. C3×#824C22semidirect
37. C3×#823C22semidirect
38. C3×#822C22semidirect
43. C2×#942C2×C3semidirect
44. C2×#942C2×C3semidirect
46. #5745C2×C3semidirect
52. C2×C3×#101D3semidirect
53. C2×C3×#101C2×C3semidirect
54. #5132C2×C3semidirect
55. #5124C2×C3semidirect
56. #4755C2×C3semidirect
57. #4754C2×C3semidirect
137. C2×#216C22×C3semidirect
143. C2×#215C22×C3semidirect
157. C3×#101C2×D3semidirect
158. C3×#101C22×C3semidirect
160. #824C22×C3semidirect
161. #823C22×C3semidirect
162. #822C22×C3semidirect
209. C2×#101C3×D3semidirect
356. #101C2×C3×D3semidirect
541. C32#3069semidirect
612. C3#78712direct (N × H)
613. C3C3×#3069semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9422
C3×#57482
C3×#57452
C2×C32×#1012
C3×#51322
C3×#51242
C3×#47552
C3×#47542
C3×#30693
#787123