Finite Group Databasebeta

← C3×#78836 #11864420

Subgroups — C3×#78836 #11864420

Order 1152.

1543 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic, complemented
2C2×C3×#949C211yesmaximal, characteristic, complemented
3C22×C3×#204C211yesmaximal, characteristic, complemented
4C3×#5525C211yesmaximal, characteristic, complemented
5C32×#416C211yesmaximal, characteristic, complemented
6C3×#5110C211yesmaximal, characteristic, complemented
7C3×#5004C211yesmaximal, characteristic, complemented
8C3×#4806C214yesmaximal, complemented
9C3×#4781C214yesmaximal, complemented
10C3×#3010—31nomaximal
11#78836C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C3×#1010—24no—
14C3×#1010—24no—
15C2×C3×D3×Q8C2212yescomplemented
16C2×C3×#237—22no—
17C2×C3×#237C2212yescomplemented
18C2×C3×#235—21no—
19C2×C3×#235C2211yescharacteristic
20C22×C3×D12C2211yescharacteristic
21C22×C3×Dic6—21no—
22C22×C3×Dic6C2211yescharacteristic
23C22×C32×C8C2211yescharacteristic
24C32×#117C2211yescharacteristic
25C3×#949—24no—
26C2×C3×#215C2212yes—
27C2×C3×#213C2212yes—
28C2×C3×#208C2211yescharacteristic
29C2×C3×#204—24no—
30C2×C3×#204C2214yes—
31C3×#895C2214yes—
32C3×#882—22no—
33C3×#875—21no—
34C3×#874—21no—
35C32×#103C2212yescomplemented
36C32×#102C2212yescomplemented
37C3×(C2×D3)⋊C8C2214yescomplemented
38C3×#810C2214yescomplemented
39C2×C3×2−1+4—31no—
40C2×#1010C2×C311yescharacteristic, complemented
41C2×C3×AGL(1,ℤ/8)—31no—
42C22×C3×SD16—31no—
43C3×#535—31no—
44C2×#949C2×C311yescharacteristic, complemented
45C22×#204C2×C311yescharacteristic, complemented
46#5525C2×C311yescharacteristic, complemented
47C3×#449—34no—
48C3×#448—34no—
49C3×#419—31no—
50C3×#416—21no—
51C3×#416D311yescharacteristic, complemented
52C3×#416C2×C311yescharacteristic, complemented
53C3×#408—31no—
54#5110C2×C311yescharacteristic, complemented
55#5004C2×C311yescharacteristic, complemented
56#4806C2×C314yescomplemented
57#4781C2×C314yescomplemented
58C23×C3×D3—21no—
59C32×#41—22no—
60C32×#41D414yescomplemented
61C2×C32×Q8C2311yescharacteristic
62C2×C32×D4—21no—
63C2×C32×D4C2311yescharacteristic
64C22×C32×C4—21no—
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—22no—
67C2×C3×#67D412yes—
68C22×C3×Dic3—21no—
69C22×C3×Dic3—22no—
70C3×D3×Q8—24no—
71C3×D3×Q8—24no—
72C3×#237—216no—
73C3×#237—24no—
74C3×#237—24no—
75C3×#235—28no—
76C3×#235—22no—
77C3×#235—22no—
78C2×C3×D12—22no—
79C2×C3×D12—22no—
80C2×C3×D12—21no—
81C2×C3×D12—24no—
82C2×C3×D12D411yescharacteristic
83C2×C3×D12C2312yes—
84C2×C3×C4×D3—22no—
85C2×C3×C4×D3C2312yes—
86C2×C3×Dic6—28no—
87C2×C3×Dic6—21no—
88C2×C3×Dic6—22no—
89C2×C3×Dic6—22no—
90C2×C3×Dic6D411yescharacteristic
91C2×C3×Dic6D414yescomplemented
92C2×C3×Dic6C2312yes—
93C2×C32×C8—22no—
94C2×C32×C8C2312yes—
95C32×C4⋊C4C2312yes—
96C32×C22⋊C4—21no—
97C32×C42—21no—
98C3×#215—24no—
99C3×#213—24no—
100C3×(C2×D3)⋊C4—24no—
101C3×#208—22no—
102C2×C3×C3⋊C8C2312yes—
103C3×#204—28no—
104#3010—31noSylow
105C3×2−1+4—64no—
106C3×2−1+4—64no—
107C2×C3×#41—21no—
108C2×C3×#41—31no—
109C2×C3×#41—32no—
110C2×C3×#41—61no—
111C2×C3×#41—62no—
112C2×C3×#41C2×D311yescharacteristic
113C2×C3×#41C22×C311yescharacteristic
114C22×C3×Q8—32no—
115C22×C3×Q8—31no—
116C22×C3×Q8—61no—
117C22×C3×D4—31no—
118#1010—24no—
119#1010—24no—
120C2×D3×Q8C22×C312yescomplemented
121C2×#237—22no—
122C2×#237C22×C312yescomplemented
123C2×#235—21no—
124C2×#235C22×C311yescharacteristic
125C22×D12C22×C311yescharacteristic
126C22×Dic6—21no—
127C22×Dic6C22×C311yescharacteristic
128C3×AGL(1,ℤ/8)—64no—
129C2×C3×SD16—32no—
130C2×C3×SD16—34no—
131C2×C3×SD16—64no—
132C2×C3×D8—32no—
133C2×C3×M16—31no—
134C22×C3×C8—21no—
135C22×C3×C8C2×D311yescharacteristic
136C22×C3×C8C22×C311yescharacteristic
137C3×#127—61no—
138C3×#124—61no—
139C3×#121—34no—
140C3×#120—62no—
141C3×#117—21no—
142C3×#117C2×D311yescharacteristic
143C3×#117C22×C311yescharacteristic
144#949—24no—
145C2×#215C22×C312yes—
146C2×#213C22×C312yes—
147C2×#208C22×C311yescharacteristic
148C2×#204—24no—
149C2×#204C22×C314yes—
150#895C22×C314yes—
151#882—22no—
152#875—21no—
153#874—21no—
154C3×#103—22no—
155C3×#103—34no—
156C3×#103C2×D312yescomplemented
157C3×#103C22×C312yescomplemented
158C3×#102—22no—
159C3×#102C2×D312yescomplemented
160C3×#102C22×C312yescomplemented
161C3×C22⋊C8—34no—
162(C2×D3)⋊C8C22×C314yescomplemented
163#810C22×C314yescomplemented
164C23×C32—21no—
165C23×C32C2×D411yescharacteristic
166C22×C3×D3—22no—
167C22×C3×D3—42no—
168C22×C3×D3—82no—
169C22×C3×D3C2×D412yes—
170C32×Q8—21no—
171C32×Q8C2×D412yes—
172C32×D4—24no—
173C32×D4—21no—
174C32×D4C2×D412yes—
175C2×C32×C4—21no—
176C2×C32×C4—24no—
177C2×C32×C4—22no—
178C2×C32×C4C2×D412yes—
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2×D412yes—
181C2×C32×C4C2411yescharacteristic
182C3×#67—28no—
183C3×#67—24no—
184C2×C3×Dic3—28no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3C2×D412yes—
188C3×D12—22no—
189C3×D12—22no—
190C3×D12—22no—
191C3×D12—42no—
192C3×D12—84no—
193C3×C4×D3—28no—
194C3×C4×D3—22no—
195C3×C4×D3—22no—
196C3×Dic6—28no—
197C3×Dic6—22no—
198C3×Dic6—24no—
199C3×Dic6—22no—
200C3×Dic6C2×D414yes—
201C32×C8—22no—
202C3×C3⋊C8—22no—
203C2×2−1+4—31no—
204C2×AGL(1,ℤ/8)—31no—
205C22×SD16—31no—
206#535—31no—
207#449—34no—
208#448—34no—
209#419—31no—
210#416C3×D311yescharacteristic, complemented
211#408—31no—
212C24×C3—61no—
213C23×D3—21no—
214C3×#41—22no—
215C3×#41—22no—
216C3×#41—24no—
217C3×#41—42no—
218C3×#41—68no—
219C3×#41—64no—
220C3×#41—616no—
221C3×#41—64no—
222C3×#41—62no—
223C3×#41—62no—
224C3×#41D1214yescomplemented
225C3×#41C3×D414yescomplemented
226C2×C3×Q8—21no—
227C2×C3×Q8—31no—
228C2×C3×Q8—34no—
229C2×C3×Q8—32no—
230C2×C3×Q8—62no—
231C2×C3×Q8—68no—
232C2×C3×Q8—64no—
233C2×C3×Q8—64no—
234C2×C3×Q8—62no—
235C2×C3×Q8—61no—
236C2×C3×Q8C22×D311yescharacteristic
237C2×C3×Q8C23×C311yescharacteristic
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—32no—
242C2×C3×D4—31no—
243C2×C3×D4—32no—
244C2×C3×D4—41no—
245C2×C3×D4—64no—
246C2×C3×D4—62no—
247C2×C3×D4—62no—
248C2×C3×D4—62no—
249C2×C3×D4—61no—
250C2×C3×D4C22×D311yescharacteristic
251C2×C3×D4C23×C311yescharacteristic
252C22×C3×C4—21no—
253C22×C3×C4—21no—
254C22×C3×C4—21no—
255C22×C3×C4—32no—
256C22×C3×C4—41no—
257C22×C3×C4—62no—
258C22×C3×C4—62no—
259C22×C3×C4—61no—
260C22×C3×C4C22×D311yescharacteristic
261C22×C3×C4C23×C311yescharacteristic
262C2×#67—22no—
263C2×#67C3×D412yes—
264C22×Dic3—21no—
265C22×Dic3—22no—
266D3×Q8—24no—
267D3×Q8—24no—
268#237—216no—
269#237—24no—
270#237—24no—
271#235—28no—
272#235—22no—
273#235—22no—
274C2×D12—22no—
275C2×D12—22no—
276C2×D12—21no—
277C2×D12—24no—
278C2×D12C3×D411yescharacteristic
279C2×D12C23×C312yes—
280C2×C4×D3—22no—
281C2×C4×D3C23×C312yes—
282C2×Dic6—28no—
283C2×Dic6—21no—
284C2×Dic6—22no—
285C2×Dic6—22no—
286C2×Dic6C3×D411yescharacteristic
287C2×Dic6C3×D414yescomplemented
288C2×Dic6C23×C312yes—
289C3×SD16—68no—
290C3×SD16—64no—
291C3×D8—64no—
292C3×M16—62no—
293C2×C3×C8—22no—
294C2×C3×C8—22no—
295C2×C3×C8—22no—
296C2×C3×C8—32no—
297C2×C3×C8—42no—
298C2×C3×C8C22×D312yes—
299C2×C3×C8C23×C312yes—
300C3×C4⋊C4—22no—
301C3×C4⋊C4C22×D312yes—
302C3×C4⋊C4C23×C312yes—
303C3×C22⋊C4—21no—
304C3×C22⋊C4—21no—
305C3×C22⋊C4—41no—
306C3×C22⋊C4—64no—
307C3×C42—21no—
308C3×C42—21no—
309C3×C42—41no—
310#215—24no—
311#213—24no—
312(C2×D3)⋊C4—24no—
313#208—22no—
314C2×C3⋊C8C23×C312yes—
315#204—28no—
316C22×C32—22no—
317C22×C32—22no—
318C22×C32—42no—
319C22×C32#12012yes—
320C22×C32C22×D411yescharacteristic
321C2×C3×D3—22no—
322C2×C3×D3—24no—
323C2×C3×D3—42no—
324C2×C3×D3—44no—
325C2×C3×D3—84no—
326C32×C4—22no—
327C32×C4—42no—
328C32×C4#12012yes—
329C32×C4C22×D411yescharacteristic
330C32×C4C22×D411yescharacteristic
331C3×Dic3—22no—
332C3×Dic3—24no—
3332−1+4—64no—
3342−1+4—64no—
335C2×#41—31no—
336C2×#41—32no—
337C2×#41—61no—
338C2×#41—62no—
339C2×#41C2×C3×D311yescharacteristic
340C22×Q8—32no—
341C22×Q8—31no—
342C22×Q8—61no—
343C22×D4—31no—
344AGL(1,ℤ/8)—64no—
345C2×SD16—32no—
346C2×SD16—34no—
347C2×SD16—64no—
348C2×D8—32no—
349C2×M16—31no—
350C22×C8C2×C3×D311yescharacteristic
351#127—61no—
352#124—61no—
353#121—34no—
354#120—62no—
355#117C2×C3×D311yescharacteristic
356#103—34no—
357#103C2×C3×D312yescomplemented
358#102C2×C3×D312yescomplemented
359C22⋊C8—34no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—32no—
364C23×C3—41no—
365C23×C3—62no—
366C23×C3—122no—
367C23×C3—242no—
368C23×C3D3×D411yescharacteristic
369C23×C3C2×C3×D411yescharacteristic
370C22×D3—22no—
371C22×D3—42no—
372C22×D3—82no—
373C22×D3C2×C3×D412yes—
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—22no—
377C3×Q8—34no—
378C3×Q8—41no—
379C3×Q8—68no—
380C3×Q8—62no—
381C3×Q8—64no—
382C3×Q8—62no—
383C3×Q8C2×D1212yes—
384C3×Q8C2×C3×D412yes—
385C3×D4—24no—
386C3×D4—24no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—22no—
390C3×D4—41no—
391C3×D4—44no—
392C3×D4—68no—
393C3×D4—64no—
394C3×D4—62no—
395C3×D4—62no—
396C3×D4—62no—
397C3×D4—122no—
398C3×D4—244no—
399C3×D4C2×D1212yes—
400C3×D4C2×C3×D412yes—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—24no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—32no—
412C2×C3×C4—44no—
413C2×C3×C4—41no—
414C2×C3×C4—42no—
415C2×C3×C4—68no—
416C2×C3×C4—68no—
417C2×C3×C4—64no—
418C2×C3×C4—62no—
419C2×C3×C4—62no—
420C2×C3×C4—62no—
421C2×C3×C4C2×D1212yes—
422C2×C3×C4D3×D411yescharacteristic
423C2×C3×C4D3×D412yes—
424C2×C3×C4C2×C3×D412yes—
425C2×C3×C4C2×C3×D411yescharacteristic
426C2×C3×C4C2×C3×D412yes—
427C2×C3×C4C23×D311yescharacteristic
428C2×C3×C4C24×C311yescharacteristic
429#67—28no—
430#67—24no—
431C2×Dic3—28no—
432C2×Dic3—24no—
433C2×Dic3—22no—
434C2×Dic3C2×C3×D412yes—
435D12—22no—
436D12—22no—
437D12—22no—
438D12—42no—
439D12—84no—
440C4×D3—28no—
441C4×D3—22no—
442C4×D3—22no—
443Dic6—28no—
444Dic6—22no—
445Dic6—24no—
446Dic6—22no—
447Dic6C2×C3×D414yes—
448C3×C8—22no—
449C3×C8—22no—
450C3×C8—42no—
451C3×C8—62no—
452C3⋊C8—22no—
453C2×C32—22no—
454C2×C32—42no—
455C2×C32C2×#12011yescharacteristic
456C2×C32#57712yes—
457C3×D3—42no—
458C3×D3—82no—
459C24—61no—
460#41—22no—
461#41—68no—
462#41—64no—
463#41—616no—
464#41—64no—
465#41—62no—
466#41—62no—
467#41C3×D1214yescomplemented
468C2×Q8—31no—
469C2×Q8—32no—
470C2×Q8—34no—
471C2×Q8—62no—
472C2×Q8—68no—
473C2×Q8—64no—
474C2×Q8—64no—
475C2×Q8—62no—
476C2×Q8—61no—
477C2×Q8C22×C3×D311yescharacteristic
478C2×D4—21no—
479C2×D4—32no—
480C2×D4—32no—
481C2×D4—31no—
482C2×D4—64no—
483C2×D4—62no—
484C2×D4—62no—
485C2×D4—62no—
486C2×D4—61no—
487C2×D4C22×C3×D311yescharacteristic
488C22×C4—21no—
489C22×C4—32no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—61no—
493C22×C4C22×C3×D311yescharacteristic
494SD16—68no—
495SD16—64no—
496D8—64no—
497M16—62no—
498C2×C8—22no—
499C2×C8—32no—
500C2×C8C22×C3×D312yes—
501C4⋊C4C22×C3×D312yes—
502C22⋊C4—21no—
503C22⋊C4—64no—
504C42—21no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—21no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—62no—
516C22×C3—64no—
517C22×C3—82no—
518C22×C3—122no—
519C22×C3—124no—
520C22×C3—244no—
521C22×C3#88212yes—
522C22×C3C3×#12012yes—
523C22×C3C2×D3×D411yescharacteristic
524C22×C3C22×C3×D411yescharacteristic
525C2×D3—24no—
526C2×D3—22no—
527C2×D3—44no—
528C2×D3—42no—
529C2×D3—84no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—42no—
538C3×C4—62no—
539C3×C4—64no—
540C3×C4—82no—
541C3×C4#88212yes—
542C3×C4C3×#12012yes—
543C3×C4C22×D1211yescharacteristic
544C3×C4C2×D3×D411yescharacteristic
545C3×C4C22×C3×D411yescharacteristic
546C3×C4C22×C3×D411yescharacteristic
547Dic3—22no—
548Dic3—24no—
549C32#301011yescharacteristic, complemented, Sylow
550C23—21no—
551C23—32no—
552C23—62no—
553C23—122no—
554C23—242no—
555C23C3×D3×D411yescharacteristic
556Q8—21no—
557Q8—34no—
558Q8—68no—
559Q8—62no—
560Q8—64no—
561Q8—62no—
562Q8C2×C3×D1212yes—
563D4—24no—
564D4—21no—
565D4—68no—
566D4—64no—
567D4—62no—
568D4—62no—
569D4—62no—
570D4—122no—
571D4—244no—
572D4C2×C3×D1212yes—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—32no—
577C2×C4—68no—
578C2×C4—68no—
579C2×C4—64no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4C2×C3×D1212yes—
584C2×C4C3×D3×D411yescharacteristic
585C2×C4C3×D3×D412yes—
586C2×C4C23×C3×D311yescharacteristic
587C8—22no—
588C8—62no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—22no—
593C2×C3—42no—
594C2×C3—42no—
595C2×C3—42no—
596C2×C3—82no—
597C2×C3—122no—
598C2×C3—242no—
599C2×C3C2×#88211yescharacteristic
600C2×C3#577012yes—
601C2×C3C2×C3×#12011yescharacteristic
602C2×C3C3×#57712yes—
603D3—42no—
604D3—82no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—62no—
609C22—64no—
610C22—122no—
611C22—124no—
612C22—244no—
613C22C3×#88212yes—
614C22C2×C3×D3×D411yescharacteristic
615C4—22no—
616C4—42no—
617C4—62no—
618C4—64no—
619C4C3×#88212yes—
620C4C22×C3×D1211yescharacteristic
621C4C2×C3×D3×D411yescharacteristic
622C3—21no—
623C3#7883611yescharacteristic, direct product factor
624C3C3×#301011yescharacteristic, complemented
625C2—22no—
626C2—42no—
627C2—122no—
628C2—242no—
629C2C2×C3×#88211yescharacteristic
630C2C3×#577012yes—
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#416C2
socleC22×C32C22×D4
hypercenterC3×#416D3
hypocenterC3C3×#3010
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#416

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1010C2semidirect
2. C2×C3×#949C2semidirect
3. C22×C3×#204C2semidirect
4. C3×#5525C2semidirect
5. C32×#416C2semidirect
6. C3×#5110C2semidirect
7. C3×#5004C2semidirect
8. C3×#4806C2semidirect
9. C3×#4781C2semidirect
11. #78836C3direct (N × H)
15. C2×C3×D3×Q8C22semidirect
17. C2×C3×#237C22semidirect
35. C32×#103C22semidirect
36. C32×#102C22semidirect
37. C3×(C2×D3)⋊C8C22semidirect
38. C3×#810C22semidirect
40. C2×#1010C2×C3semidirect
44. C2×#949C2×C3semidirect
45. C22×#204C2×C3semidirect
46. #5525C2×C3semidirect
51. C3×#416D3semidirect
52. C3×#416C2×C3semidirect
54. #5110C2×C3semidirect
55. #5004C2×C3semidirect
56. #4806C2×C3semidirect
57. #4781C2×C3semidirect
60. C32×#41D4semidirect
91. C2×C3×Dic6D4semidirect
120. C2×D3×Q8C22×C3semidirect
122. C2×#237C22×C3semidirect
156. C3×#103C2×D3semidirect
157. C3×#103C22×C3semidirect
159. C3×#102C2×D3semidirect
160. C3×#102C22×C3semidirect
162. (C2×D3)⋊C8C22×C3semidirect
163. #810C22×C3semidirect
210. #416C3×D3semidirect
224. C3×#41D12semidirect
225. C3×#41C3×D4semidirect
287. C2×Dic6C3×D4semidirect
357. #103C2×C3×D3semidirect
358. #102C2×C3×D3semidirect
467. #41C3×D12semidirect
549. C32#3010semidirect
623. C3#78836direct (N × H)
624. C3C3×#3010semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C2×C3×#9492
C22×C3×#2042
C3×#55252
C32×#4162
C3×#51102
C3×#50042
C3×#48062
C3×#47812
C3×#30103
#788363