Finite Group Databasebeta

← C3×#78721 #11868173

Subgroups — C3×#78721 #11868173

Order 1152.

967 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#951C211yesmaximal, characteristic, complemented
2C3×#5822C211yesmaximal, characteristic
3C2×C3×#909C211yesmaximal, characteristic, complemented
4C3×#5745C211yesmaximal, characteristic, complemented
5C32×#413C211yesmaximal, characteristic, complemented
6C2×C3×#824C211yesmaximal, characteristic, complemented
7C3×#5124C211yesmaximal, characteristic, complemented
8C3×#4755C214yesmaximal, complemented
9C3×#4748C214yesmaximal, complemented
10C3×#3069—31nomaximal
11#78721C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C32×M16C2211yescharacteristic
17C3×#951—21no—
18C3×#951—22no—
19C3×#951—22no—
20C3×#950—22no—
21C3×#949—21no—
22C2×C3×#216C2212yescomplemented
23C3×#942C2212yes—
24C2×C3×#214C2212yescomplemented
25C3×#932C2212yes—
26C2×C3×#208C2211yescharacteristic
27C2×C3×#208C2211yescharacteristic
28C3×#909—22no—
29C3×#909C2214yes—
30C3×#907C2214yes—
31C3×D3×M16—22no—
32C2×C3×Dic12—21no—
33C2×C3×#204—21no—
34C3×#901—22no—
35C32×#101C2212yescomplemented
36C32×#100C2212yescomplemented
37C3×#824C2214yescomplemented
38C3×#822C2214yescomplemented
39C3×#575—31no—
40C2×C3×#137—31no—
41C2×C3×#137—31no—
42C3×#568—31no—
43C2×#951C2×C311yescharacteristic, complemented
44#5822C2×C311yescharacteristic
45C2×#909C2×C311yescharacteristic, complemented
46#5745C2×C311yescharacteristic, complemented
47C3×#473—34no—
48C3×#471—34no—
49C3×#430—31no—
50C3×#413—21no—
51C3×#413D311yescharacteristic, complemented
52C3×#413C2×C311yescharacteristic, complemented
53C2×C3×#101—31no—
54C2×#824C2×C311yescharacteristic, complemented
55#5124C2×C311yescharacteristic, complemented
56#4755C2×C314yescomplemented
57#4748C2×C314yescomplemented
58C32×#41—21no—
59C32×#41—21no—
60C32×#41—21no—
61C32×#41—21no—
62C2×C32×Q8C2311yescharacteristic
63C2×C32×D4—21no—
64C2×C32×D4C2311yescharacteristic
65C22×C32×C4—21no—
66C22×C32×C4C2311yescharacteristic
67C2×C3×#67—21no—
68C22×C3×Dic3—21no—
69C3×#235—21no—
70C3×#235—21no—
71C3×#235D412yes—
72C3×#235C2312yes—
73C2×C3×D12D411yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6D411yescharacteristic
81C2×C3×Dic6C2312yes—
82C32×M16—21no—
83C32×M16—21no—
84C32×M16C2312yes—
85C32×M16C2312yes—
86C2×C32×C8—21no—
87C2×C32×C8—21no—
88C3×#216—22no—
89C3×#216—24no—
90C3×#215—22no—
91C3×#214—22no—
92C3×#214—24no—
93C3×#213—22no—
94C3×#208—21no—
95C3×#208D412yes—
96C3×#208D412yes—
97C3×#208C2312yes—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2312yes—
100C3×Dic12—24no—
101C3×#204—24no—
102C3×ΓL(1,𝔽25)—24no—
103C3×D3×C8—24no—
104#3069—31noSylow
105C2×C3×#41—21no—
106C2×C3×#41—31no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C22×C311yescharacteristic
109C22×C3×Q8—31no—
110C2×#235C22×C311yescharacteristic
111C22×Dic6C22×C311yescharacteristic
112C3×#137—32no—
113C3×#137—34no—
114C3×#137—62no—
115C3×#137—62no—
116C3×#137—62no—
117C3×#137—61no—
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
136#951—21no—
137#951—22no—
138#951—22no—
139#950—22no—
140#949—21no—
141C2×#216C22×C312yescomplemented
142#942C22×C312yes—
143C2×#214C22×C312yescomplemented
144#932C22×C312yes—
145C2×#208C22×C311yescharacteristic
146C2×#208C22×C311yescharacteristic
147#909—22no—
148#909C22×C314yes—
149#907C22×C314yes—
150D3×M16—22no—
151C2×Dic12—21no—
152C2×#204—21no—
153#901—22no—
154C3×#108—34no—
155C3×#101—22no—
156C3×#101—34no—
157C3×#101C2×D312yescomplemented
158C3×#101C22×C312yescomplemented
159C3×#100—22no—
160C3×#100C2×D312yescomplemented
161C3×#100C22×C312yescomplemented
162#824C22×C314yescomplemented
163#822C22×C314yescomplemented
164C23×C32—21no—
165C23×C32#4111yescharacteristic
166C22×C3×D3—21no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×D4—21no—
170C32×D4—21no—
171C32×D4—41no—
172C32×D4—41no—
173C2×C32×C4—21no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4#4111yescharacteristic
181C2×C32×C4C2411yescharacteristic
182C3×#67—22no—
183C3×#67—22no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—41no—
188C3×D12—21no—
189C3×D12C2×D412yes—
190C3×C4×D3—22no—
191C3×C4×D3—22no—
192C3×Dic6—21no—
193C3×Dic6—22no—
194C3×Dic6—22no—
195C3×Dic6—41no—
196C3×Dic6—41no—
197C3×Dic6C2×D412yes—
198C32×C8—22no—
199C32×C8—22no—
200C3×C3⋊C8—22no—
201C3×C3⋊C8C2×D414yes—
202#575—31no—
203C2×#137—31no—
204C2×#137—31no—
205#568—31no—
206#473—34no—
207#471—34no—
208#430—31no—
209#413C3×D311yescharacteristic, complemented
210C2×#101—31no—
211C3×#41—21no—
212C3×#41—21no—
213C3×#41—21no—
214C3×#41—21no—
215C3×#41—21no—
216C3×#41—21no—
217C3×#41—21no—
218C3×#41—21no—
219C3×#41—32no—
220C3×#41—32no—
221C3×#41—41no—
222C3×#41—41no—
223C3×#41—41no—
224C3×#41—41no—
225C3×#41—61no—
226C3×#41—61no—
227C2×C3×Q8—21no—
228C2×C3×Q8—31no—
229C2×C3×Q8—32no—
230C2×C3×Q8—61no—
231C2×C3×Q8—61no—
232C2×C3×Q8—61no—
233C2×C3×Q8—61no—
234C2×C3×Q8—61no—
235C2×C3×Q8C22×D311yescharacteristic
236C2×C3×Q8C23×C311yescharacteristic
237C2×C3×D4—22no—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—31no—
242C2×C3×D4—61no—
243C2×C3×D4C22×D311yescharacteristic
244C2×C3×D4C23×C311yescharacteristic
245C22×C3×C4—22no—
246C22×C3×C4—21no—
247C22×C3×C4—21no—
248C22×C3×C4—21no—
249C22×C3×C4—61no—
250C22×C3×C4—61no—
251C22×C3×C4C22×D311yescharacteristic
252C22×C3×C4C23×C311yescharacteristic
253C2×#67—21no—
254C22×Dic3—21no—
255#235—21no—
256#235—21no—
257#235C3×D412yes—
258#235C23×C312yes—
259C2×D12C3×D411yescharacteristic
260C2×C4×D3—21no—
261C2×Dic6—21no—
262C2×Dic6—21no—
263C2×Dic6—21no—
264C2×Dic6—21no—
265C2×Dic6—21no—
266C2×Dic6C3×D411yescharacteristic
267C2×Dic6C23×C312yes—
268C3×Q16—64no—
269C3×Q16—64no—
270C3×Q16—62no—
271C3×SD16—64no—
272C3×SD16—64no—
273C3×SD16—62no—
274C3×SD16—62no—
275C3×D8—62no—
276C3×M16—21no—
277C3×M16—21no—
278C3×M16—21no—
279C3×M16—21no—
280C3×M16—22no—
281C3×M16—22no—
282C3×M16—32no—
283C3×M16—32no—
284C3×M16—32no—
285C3×M16—41no—
286C3×M16—41no—
287C3×M16—64no—
288C3×M16—61no—
289C3×M16C22×D312yes—
290C3×M16C22×D312yes—
291C3×M16C23×C312yes—
292C3×M16C23×C312yes—
293C2×C3×C8—21no—
294C2×C3×C8—21no—
295C2×C3×C8—21no—
296C2×C3×C8—21no—
297C2×C3×C8—32no—
298C2×C3×C8—41no—
299C2×C3×C8—41no—
300C2×C3×C8—64no—
301C2×C3×C8—61no—
302#216—22no—
303#216—24no—
304#215—22no—
305#214—22no—
306#214—24no—
307#213—22no—
308#208—21no—
309#208C3×D412yes—
310#208C3×D412yes—
311#208C23×C312yes—
312C2×C3⋊C8—21no—
313C2×C3⋊C8C23×C312yes—
314Dic12—24no—
315#204—24no—
316ΓL(1,𝔽25)—24no—
317D3×C8—24no—
318C22×C32—41no—
319C22×C32—41no—
320C22×C32—81no—
321C22×C32#12111yescharacteristic
322C22×C32#12111yescharacteristic
323C22×C32C2×#4111yescharacteristic
324C2×C3×D3—22no—
325C2×C3×D3—81no—
326C32×C4—41no—
327C32×C4#12111yescharacteristic
328C32×C4#12111yescharacteristic
329C32×C4C22×D411yescharacteristic
330C32×C4C22×D411yescharacteristic
331C3×Dic3—22no—
332C3×Dic3—42no—
333C2×#41—31no—
334C2×#41C2×C3×D311yescharacteristic
335C22×Q8—31no—
336#137—32no—
337#137—34no—
338#137—62no—
339#137—62no—
340#137—62no—
341#137—61no—
342AGL(1,ℤ/8)—32no—
343AGL(1,ℤ/8)—61no—
344#135—62no—
345C2×Q16—32no—
346C2×Q16—61no—
347C2×SD16—32no—
348C2×SD16—61no—
349#131—34no—
350#131—62no—
351C2×M16—31no—
352C2×M16—31no—
353C2×M16—62no—
354C2×M16C2×C3×D311yescharacteristic
355C2×M16C2×C3×D311yescharacteristic
356#108—34no—
357#101—34no—
358#101C2×C3×D312yescomplemented
359#100C2×C3×D312yescomplemented
360C23×C3—22no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—61no—
365C23×C3#23511yescharacteristic
366C23×C3C3×#4111yescharacteristic
367C22×D3—21no—
368C3×Q8—21no—
369C3×Q8—21no—
370C3×Q8—21no—
371C3×Q8—21no—
372C3×Q8—32no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×Q8—61no—
376C3×Q8—62no—
377C3×Q8—62no—
378C3×Q8—121no—
379C3×Q8—121no—
380C3×D4—21no—
381C3×D4—21no—
382C3×D4—21no—
383C3×D4—21no—
384C3×D4—32no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—42no—
388C3×D4—42no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—62no—
394C3×D4—62no—
395C3×D4—61no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—61no—
412C2×C3×C4—62no—
413C2×C3×C4—62no—
414C2×C3×C4—62no—
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—
431C2×Dic3—21no—
432C2×Dic3—22no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435D12—21no—
436D12C2×C3×D412yes—
437C4×D3—22no—
438C4×D3—22no—
439Dic6—21no—
440Dic6—22no—
441Dic6—22no—
442Dic6—41no—
443Dic6—41no—
444Dic6C2×C3×D412yes—
445C3×C8—22no—
446C3×C8—22no—
447C3×C8—22no—
448C3×C8—22no—
449C3×C8—34no—
450C3×C8—42no—
451C3×C8—42no—
452C3×C8—62no—
453C3⋊C8—22no—
454C3⋊C8C2×C3×D414yes—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—81no—
459C2×C32C2×#12111yescharacteristic
460C3×D3—42no—
461#41—21no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—32no—
466#41—32no—
467#41—61no—
468#41—61no—
469C2×Q8—32no—
470C2×Q8—31no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8C22×C3×D311yescharacteristic
477C2×D4—21no—
478C2×D4—31no—
479C2×D4—61no—
480C2×D4C22×C3×D311yescharacteristic
481C22×C4—21no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4C22×C3×D311yescharacteristic
485Q16—64no—
486Q16—64no—
487Q16—62no—
488SD16—64no—
489SD16—64no—
490SD16—62no—
491SD16—62no—
492D8—62no—
493M16—21no—
494M16—21no—
495M16—32no—
496M16—32no—
497M16—32no—
498M16—64no—
499M16—61no—
500M16C22×C3×D312yes—
501M16C22×C3×D312yes—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—32no—
505C2×C8—64no—
506C2×C8—61no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—42no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—62no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—82no—
520C22×C3—241no—
521C22×C3#88411yescharacteristic
522C22×C3#88411yescharacteristic
523C22×C3C3×#12111yescharacteristic
524C22×C3C3×#12111yescharacteristic
525C22×C3C2×#23511yescharacteristic
526C22×C3C2×C3×#4111yescharacteristic
527C2×D3—22no—
528C2×D3—81no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—62no—
537C3×C4—122no—
538C3×C4#88411yescharacteristic
539C3×C4#88411yescharacteristic
540C3×C4C3×#12111yescharacteristic
541C3×C4C3×#12111yescharacteristic
542C3×C4C2×D3×D411yescharacteristic
543C3×C4C2×D3×D411yescharacteristic
544C3×C4C22×C3×D411yescharacteristic
545C3×C4C22×C3×D411yescharacteristic
546Dic3—22no—
547Dic3—42no—
548C32#306911yescharacteristic, complemented, Sylow
549C23—21no—
550C23—61no—
551C23C3×#23511yescharacteristic
552Q8—21no—
553Q8—21no—
554Q8—32no—
555Q8—61no—
556Q8—62no—
557Q8—62no—
558Q8—121no—
559Q8—121no—
560D4—21no—
561D4—21no—
562D4—32no—
563D4—41no—
564D4—41no—
565D4—62no—
566D4—62no—
567D4—61no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—61no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4C3×#23511yescharacteristic
578C2×C4C3×D3×D411yescharacteristic
579C2×C4C3×D3×D411yescharacteristic
580C2×C4C3×D3×D411yescharacteristic
581C2×C4C3×D3×D411yescharacteristic
582C2×C4C23×C3×D311yescharacteristic
583C8—22no—
584C8—22no—
585C8—34no—
586C8—62no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—41no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—82no—
598C2×C3—81no—
599C2×C3—81no—
600C2×C3—122no—
601C2×C3C2×#88411yescharacteristic
602C2×C3C2×C3×#12111yescharacteristic
603D3—42no—
604C22—41no—
605C22—41no—
606C22—62no—
607C22—81no—
608C22—241no—
609C22C3×#88411yescharacteristic
610C22C3×#88411yescharacteristic
611C22C2×C3×#23511yescharacteristic
612C4—41no—
613C4—62no—
614C4—122no—
615C4C3×#88411yescharacteristic
616C4C3×#88411yescharacteristic
617C4C2×C3×D3×D411yescharacteristic
618C4C2×C3×D3×D411yescharacteristic
619C3—21no—
620C3#7872111yescharacteristic, direct product factor
621C3C3×#306911yescharacteristic, complemented
622C2—21no—
623C2—21no—
624C2—21no—
625C2—81no—
626C2—122no—
627C2C2×C3×#88411yescharacteristic
subgroupG/H
centerC2×C3C2×#884
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#413C2
socleC2×C32C2×#121
prime radicalC3×#5822C2
hypercenterC3×#413D3
hypocenterC3C3×#3069
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. C2×C3×#951C2semidirect
3. C2×C3×#909C2semidirect
4. C3×#5745C2semidirect
5. C32×#413C2semidirect
6. C2×C3×#824C2semidirect
7. C3×#5124C2semidirect
8. C3×#4755C2semidirect
9. C3×#4748C2semidirect
11. #78721C3direct (N × H)
22. C2×C3×#216C22semidirect
24. C2×C3×#214C22semidirect
35. C32×#101C22semidirect
36. C32×#100C22semidirect
37. C3×#824C22semidirect
38. C3×#822C22semidirect
43. C2×#951C2×C3semidirect
45. C2×#909C2×C3semidirect
46. #5745C2×C3semidirect
51. C3×#413D3semidirect
52. C3×#413C2×C3semidirect
54. C2×#824C2×C3semidirect
55. #5124C2×C3semidirect
56. #4755C2×C3semidirect
57. #4748C2×C3semidirect
141. C2×#216C22×C3semidirect
143. C2×#214C22×C3semidirect
157. C3×#101C2×D3semidirect
158. C3×#101C22×C3semidirect
160. C3×#100C2×D3semidirect
161. C3×#100C22×C3semidirect
162. #824C22×C3semidirect
163. #822C22×C3semidirect
209. #413C3×D3semidirect
358. #101C2×C3×D3semidirect
359. #100C2×C3×D3semidirect
548. C32#3069semidirect
620. C3#78721direct (N × H)
621. C3C3×#3069semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9512
C3×#58222
C2×C3×#9092
C3×#57452
C32×#4132
C2×C3×#8242
C3×#51242
C3×#47552
C3×#47482
C3×#30693
#787213