Finite Group Databasebeta

← C3×#78678 #11864417

Subgroups — C3×#78678 #11864417

Order 1152.

1543 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic, complemented
2C3×#5830C211yesmaximal, characteristic, complemented
3C2×C3×#908C211yesmaximal, characteristic, complemented
4C22×C3×#204C211yesmaximal, characteristic, complemented
5C32×#408C211yesmaximal, characteristic, complemented
6C3×#5120C211yesmaximal, characteristic, complemented
7C3×#5113C211yesmaximal, characteristic, complemented
8C3×#4738C214yesmaximal, complemented
9C3×#4734C214yesmaximal, complemented
10C3×#3010—31nomaximal
11#78678C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C3×#1010—24no—
14C3×#1010—24no—
15C2×C3×D3×Q8—21no—
16C2×C3×#237—21no—
17C2×C3×#237—22no—
18C2×C3×#235C2211yescharacteristic
19C2×C3×#235C2212yescomplemented
20C22×C3×D12C2211yescharacteristic
21C22×C3×Dic6C2211yescharacteristic
22C22×C3×Dic6C2212yescomplemented
23C2×C32×M16C2211yescharacteristic
24C22×C32×C8C2211yescharacteristic
25C3×#947—21no—
26C3×#940—21no—
27C3×#937—22no—
28C3×#930C2214yes—
29C3×#926C2211yescharacteristic
30C3×#908—24no—
31C2×C3×D24C2212yes—
32C2×C3×#204—24no—
33C2×C3×#204C2212yes—
34C2×C3×#204C2214yes—
35C32×C22⋊C8C2214yescomplemented
36C3×#821C2212yescomplemented
37C3×#816C2214yescomplemented
38C3×#816C2212yescomplemented
39C2×C3×2−1+4—31no—
40C2×#1010C2×C311yescharacteristic, complemented
41C2×C3×AGL(1,ℤ/8)—31no—
42C22×C3×SD16—31no—
43C3×#535—31no—
44#5830C2×C311yescharacteristic, complemented
45C2×#908C2×C311yescharacteristic, complemented
46C22×#204C2×C311yescharacteristic, complemented
47C3×#449—34no—
48C3×#448—34no—
49C3×#419—31no—
50C3×#416—31no—
51C3×#408—21no—
52C3×#408D311yescharacteristic, complemented
53C3×#408C2×C311yescharacteristic, complemented
54#5120C2×C311yescharacteristic, complemented
55#5113C2×C311yescharacteristic, complemented
56#4738C2×C314yescomplemented
57#4734C2×C314yescomplemented
58C23×C3×D3—21no—
59C32×#41—22no—
60C32×#41—22no—
61C2×C32×Q8D411yescharacteristic
62C2×C32×D4D412yes—
63C2×C32×D4D411yescharacteristic
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—21no—
67C2×C3×#67—22no—
68C2×C3×#67—24no—
69C22×C3×Dic3—21no—
70C22×C3×Dic3—22no—
71C3×D3×Q8—28no—
72C3×#237—216no—
73C3×#237—28no—
74C3×#235—22no—
75C3×#235—24no—
76C3×#235—24no—
77C3×#235D414yescomplemented
78C2×C3×D12—22no—
79C2×C3×D12—22no—
80C2×C3×D12—21no—
81C2×C3×D12C2312yes—
82C2×C3×D12C2311yescharacteristic
83C2×C3×C4×D3—21no—
84C2×C3×C4×D3—22no—
85C2×C3×Dic6—21no—
86C2×C3×Dic6—22no—
87C2×C3×Dic6—22no—
88C2×C3×Dic6—24no—
89C2×C3×Dic6—24no—
90C2×C3×Dic6D414yescomplemented
91C2×C3×Dic6C2312yes—
92C2×C3×Dic6C2311yescharacteristic
93C32×M16—22no—
94C2×C32×C8—22no—
95C2×C32×C8C2312yes—
96C2×C32×C8C2312yes—
97C3×#217—21no—
98C3×(C2×D3)⋊C4—24no—
99C3×C4⋊Dic3C2312yes—
100C3×C4×Dic3—21no—
101C3×D24—24no—
102C3×#204—28no—
103C3×#204—24no—
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—62no—
111C2×C3×#41—61no—
112C2×C3×#41C2×D311yescharacteristic
113C2×C3×#41C22×C311yescharacteristic
114C22×C3×Q8—31no—
115C22×C3×Q8—32no—
116C22×C3×Q8—61no—
117C22×C3×D4—31no—
118#1010—24no—
119#1010—24no—
120C2×D3×Q8—21no—
121C2×#237—21no—
122C2×#237—22no—
123C2×#235C22×C311yescharacteristic
124C2×#235C22×C312yescomplemented
125C22×D12C22×C311yescharacteristic
126C22×Dic6C22×C311yescharacteristic
127C22×Dic6C22×C312yescomplemented
128C3×AGL(1,ℤ/8)—64no—
129C2×C3×SD16—34no—
130C2×C3×SD16—32no—
131C2×C3×SD16—64no—
132C2×C3×D8—32no—
133C2×C3×M16—21no—
134C2×C3×M16C2×D311yescharacteristic
135C2×C3×M16C22×C311yescharacteristic
136C22×C3×C8—21no—
137C22×C3×C8C2×D311yescharacteristic
138C22×C3×C8C22×C311yescharacteristic
139C3×#127—61no—
140C3×#124—61no—
141C3×#121—34no—
142C3×#120—62no—
143C3×#117—31no—
144#947—21no—
145#940—21no—
146#937—22no—
147#930C22×C314yes—
148#926C22×C311yescharacteristic
149#908—24no—
150C2×D24C22×C312yes—
151C2×#204—24no—
152C2×#204C22×C314yes—
153C2×#204C22×C312yes—
154C3×#103—32no—
155C3×#103—34no—
156C3×#102—32no—
157C3×C22⋊C8—24no—
158C3×C22⋊C8C2×D314yescomplemented
159C3×C22⋊C8C22×C314yescomplemented
160#821C22×C312yescomplemented
161#816C22×C314yescomplemented
162#816C22×C312yescomplemented
163C23×C32C2×D412yes—
164C23×C32C2×D411yescharacteristic
165C22×C3×D3—21no—
166C22×C3×D3—22no—
167C22×C3×D3—42no—
168C22×C3×D3—82no—
169C32×Q8—22no—
170C32×D4—24no—
171C32×D4—22no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4C2×D412yes—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2411yescharacteristic
179C3×#67—28no—
180C3×#67—24no—
181C3×#67—84no—
182C2×C3×Dic3—28no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—22no—
187C3×D12—21no—
188C3×D12—22no—
189C3×D12—22no—
190C3×D12—42no—
191C3×D12C2×D412yes—
192C3×C4×D3—28no—
193C3×C4×D3—24no—
194C3×Dic6—28no—
195C3×Dic6—24no—
196C3×Dic6—22no—
197C3×Dic6—21no—
198C3×Dic6C2×D412yes—
199C3×Dic6C2×D414yes—
200C32×C8—22no—
201C32×C8—22no—
202C2×2−1+4—31no—
203C2×AGL(1,ℤ/8)—31no—
204C22×SD16—31no—
205#535—31no—
206#449—34no—
207#448—34no—
208#419—31no—
209#416—31no—
210#408C3×D311yescharacteristic, complemented
211C24×C3—61no—
212C23×D3—21no—
213C3×#41—22no—
214C3×#41—22no—
215C3×#41—22no—
216C3×#41—22no—
217C3×#41—34no—
218C3×#41—42no—
219C3×#41—42no—
220C3×#41—64no—
221C3×#41—64no—
222C3×#41—616no—
223C3×#41—68no—
224C3×#41—62no—
225C2×C3×Q8—21no—
226C2×C3×Q8—34no—
227C2×C3×Q8—32no—
228C2×C3×Q8—31no—
229C2×C3×Q8—64no—
230C2×C3×Q8—64no—
231C2×C3×Q8—68no—
232C2×C3×Q8—62no—
233C2×C3×Q8—61no—
234C2×C3×Q8—62no—
235C2×C3×Q8D1211yescharacteristic
236C2×C3×Q8C3×D411yescharacteristic
237C2×C3×D4—22no—
238C2×C3×D4—21no—
239C2×C3×D4—31no—
240C2×C3×D4—32no—
241C2×C3×D4—62no—
242C2×C3×D4—61no—
243C2×C3×D4—64no—
244C2×C3×D4—61no—
245C2×C3×D4—62no—
246C2×C3×D4—62no—
247C2×C3×D4D1212yes—
248C2×C3×D4D1211yescharacteristic
249C2×C3×D4C3×D412yes—
250C2×C3×D4C3×D411yescharacteristic
251C22×C3×C4—21no—
252C22×C3×C4—22no—
253C22×C3×C4—62no—
254C22×C3×C4—61no—
255C22×C3×C4—62no—
256C22×C3×C4—61no—
257C22×C3×C4C22×D312yes—
258C22×C3×C4C22×D311yescharacteristic
259C22×C3×C4C23×C312yes—
260C22×C3×C4C23×C311yescharacteristic
261C2×#67—21no—
262C2×#67—22no—
263C2×#67—24no—
264C22×Dic3—21no—
265C22×Dic3—22no—
266D3×Q8—28no—
267#237—216no—
268#237—28no—
269#235—22no—
270#235—24no—
271#235—24no—
272#235C3×D414yescomplemented
273C2×D12—22no—
274C2×D12—22no—
275C2×D12—21no—
276C2×D12C23×C312yes—
277C2×D12C23×C311yescharacteristic
278C2×C4×D3—21no—
279C2×C4×D3—22no—
280C2×Dic6—21no—
281C2×Dic6—22no—
282C2×Dic6—22no—
283C2×Dic6—24no—
284C2×Dic6—24no—
285C2×Dic6C3×D414yescomplemented
286C2×Dic6C23×C312yes—
287C2×Dic6C23×C311yescharacteristic
288C3×SD16—68no—
289C3×SD16—64no—
290C3×D8—64no—
291C3×M16—22no—
292C3×M16—22no—
293C3×M16—42no—
294C2×C3×C8—22no—
295C2×C3×C8—22no—
296C2×C3×C8—22no—
297C2×C3×C8—22no—
298C2×C3×C8—42no—
299C2×C3×C8C22×D312yes—
300C2×C3×C8C22×D312yes—
301C2×C3×C8C23×C312yes—
302C2×C3×C8C23×C312yes—
303C3×C4⋊C4—32no—
304C3×C22⋊C4—61no—
305C3×C22⋊C4—64no—
306C3×C42—61no—
307#217—21no—
308(C2×D3)⋊C4—24no—
309C4⋊Dic3C23×C312yes—
310C4×Dic3—21no—
311D24—24no—
312#204—28no—
313#204—24no—
314C22×C32—22no—
315C22×C32—22no—
316C22×C32—24no—
317C22×C32#12012yes—
318C22×C32C22×D411yescharacteristic
319C2×C3×D3—22no—
320C2×C3×D3—42no—
321C2×C3×D3—42no—
322C2×C3×D3—44no—
323C2×C3×D3—84no—
324C32×C4—22no—
325C32×C4#12012yes—
326C32×C4C22×D411yescharacteristic
327C32×C4C22×D411yescharacteristic
328C3×Dic3—22no—
329C3×Dic3—24no—
330C3×Dic3—42no—
3312−1+4—64no—
3322−1+4—64no—
333C2×#41—31no—
334C2×#41—32no—
335C2×#41—62no—
336C2×#41—61no—
337C2×#41C2×C3×D311yescharacteristic
338C22×Q8—31no—
339C22×Q8—32no—
340C22×Q8—61no—
341C22×D4—31no—
342AGL(1,ℤ/8)—64no—
343C2×SD16—34no—
344C2×SD16—32no—
345C2×SD16—64no—
346C2×D8—32no—
347C2×M16C2×C3×D311yescharacteristic
348C22×C8C2×C3×D311yescharacteristic
349#127—61no—
350#124—61no—
351#121—34no—
352#120—62no—
353#117—31no—
354#103—32no—
355#103—34no—
356#102—32no—
357C22⋊C8C2×C3×D314yescomplemented
358C23×C3—21no—
359C23×C3—22no—
360C23×C3—61no—
361C23×C3—62no—
362C23×C3—122no—
363C23×C3—242no—
364C23×C3C2×D1212yes—
365C23×C3C2×D1211yescharacteristic
366C23×C3C2×C3×D412yes—
367C23×C3C2×C3×D411yescharacteristic
368C22×D3—21no—
369C22×D3—22no—
370C22×D3—42no—
371C22×D3—82no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—34no—
375C3×Q8—32no—
376C3×Q8—42no—
377C3×Q8—68no—
378C3×Q8—64no—
379C3×Q8—62no—
380C3×Q8—61no—
381C3×D4—24no—
382C3×D4—24no—
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—32no—
386C3×D4—42no—
387C3×D4—44no—
388C3×D4—68no—
389C3×D4—64no—
390C3×D4—61no—
391C3×D4—62no—
392C3×D4—62no—
393C3×D4—122no—
394C3×D4—244no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—68no—
407C2×C3×C4—68no—
408C2×C3×C4—64no—
409C2×C3×C4—62no—
410C2×C3×C4—61no—
411C2×C3×C4—64no—
412C2×C3×C4—62no—
413C2×C3×C4C2×D1212yes—
414C2×C3×C4C2×D1211yescharacteristic
415C2×C3×C4D3×D412yes—
416C2×C3×C4D3×D412yes—
417C2×C3×C4C2×C3×D412yes—
418C2×C3×C4C2×C3×D411yescharacteristic
419C2×C3×C4C2×C3×D412yes—
420C2×C3×C4C2×C3×D412yes—
421C2×C3×C4C23×D311yescharacteristic
422C2×C3×C4C24×C311yescharacteristic
423#67—28no—
424#67—24no—
425#67—84no—
426C2×Dic3—28no—
427C2×Dic3—24no—
428C2×Dic3—22no—
429C2×Dic3—21no—
430C2×Dic3—22no—
431D12—21no—
432D12—22no—
433D12—22no—
434D12—42no—
435D12C2×C3×D412yes—
436C4×D3—28no—
437C4×D3—24no—
438Dic6—28no—
439Dic6—24no—
440Dic6—22no—
441Dic6—21no—
442Dic6C2×C3×D412yes—
443Dic6C2×C3×D414yes—
444C3×C8—22no—
445C3×C8—22no—
446C3×C8—22no—
447C3×C8—22no—
448C3×C8—42no—
449C3×C8—42no—
450C2×C32—22no—
451C2×C32—42no—
452C2×C32C2×#12011yescharacteristic
453C2×C32#57712yes—
454C3×D3—42no—
455C3×D3—82no—
456C24—61no—
457#41—22no—
458#41—22no—
459#41—34no—
460#41—64no—
461#41—64no—
462#41—616no—
463#41—68no—
464#41—62no—
465C2×Q8—32no—
466C2×Q8—31no—
467C2×Q8—34no—
468C2×Q8—64no—
469C2×Q8—64no—
470C2×Q8—68no—
471C2×Q8—62no—
472C2×Q8—61no—
473C2×Q8—62no—
474C2×Q8C3×D1211yescharacteristic
475C2×D4—31no—
476C2×D4—32no—
477C2×D4—62no—
478C2×D4—61no—
479C2×D4—64no—
480C2×D4—61no—
481C2×D4—62no—
482C2×D4—62no—
483C2×D4C3×D1212yes—
484C2×D4C3×D1211yescharacteristic
485C22×C4—62no—
486C22×C4—61no—
487C22×C4—62no—
488C22×C4—61no—
489C22×C4C22×C3×D312yes—
490C22×C4C22×C3×D311yescharacteristic
491SD16—68no—
492SD16—64no—
493D8—64no—
494M16—22no—
495C2×C8—22no—
496C2×C8C22×C3×D312yes—
497C2×C8C22×C3×D312yes—
498C4⋊C4—32no—
499C22⋊C4—61no—
500C22⋊C4—64no—
501C42—61no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—24no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—21no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—44no—
513C22×C3—62no—
514C22×C3—122no—
515C22×C3—122no—
516C22×C3—124no—
517C22×C3—244no—
518C22×C3#88212yes—
519C22×C3C3×#12012yes—
520C22×C3C22×D1211yescharacteristic
521C22×C3C22×C3×D411yescharacteristic
522C2×D3—22no—
523C2×D3—42no—
524C2×D3—44no—
525C2×D3—42no—
526C2×D3—84no—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—42no—
533C3×C4—64no—
534C3×C4—62no—
535C3×C4—122no—
536C3×C4#88212yes—
537C3×C4C3×#12012yes—
538C3×C4C2×D3×D411yescharacteristic
539C3×C4C2×D3×D411yescharacteristic
540C3×C4C22×C3×D411yescharacteristic
541C3×C4C22×C3×D411yescharacteristic
542Dic3—22no—
543Dic3—24no—
544Dic3—42no—
545C32#301011yescharacteristic, complemented, Sylow
546C23—61no—
547C23—62no—
548C23—122no—
549C23—242no—
550C23C2×C3×D1212yes—
551C23C2×C3×D1211yescharacteristic
552Q8—22no—
553Q8—34no—
554Q8—32no—
555Q8—68no—
556Q8—64no—
557Q8—62no—
558Q8—61no—
559D4—24no—
560D4—22no—
561D4—32no—
562D4—68no—
563D4—64no—
564D4—61no—
565D4—62no—
566D4—62no—
567D4—122no—
568D4—244no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—68no—
572C2×C4—68no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4C2×C3×D1212yes—
579C2×C4C2×C3×D1211yescharacteristic
580C2×C4C3×D3×D412yes—
581C2×C4C3×D3×D412yes—
582C2×C4C23×C3×D311yescharacteristic
583C8—22no—
584C8—22no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—22no—
589C2×C3—42no—
590C2×C3—42no—
591C2×C3—42no—
592C2×C3—82no—
593C2×C3—122no—
594C2×C3—242no—
595C2×C3C2×#88211yescharacteristic
596C2×C3#575112yes—
597C2×C3C2×C3×#12011yescharacteristic
598C2×C3C3×#57712yes—
599D3—42no—
600D3—82no—
601C22—22no—
602C22—22no—
603C22—24no—
604C22—62no—
605C22—122no—
606C22—122no—
607C22—124no—
608C22—244no—
609C22C3×#88212yes—
610C22C22×C3×D1211yescharacteristic
611C4—22no—
612C4—64no—
613C4—62no—
614C4—122no—
615C4C3×#88212yes—
616C4C2×C3×D3×D411yescharacteristic
617C4C2×C3×D3×D411yescharacteristic
618C3—21no—
619C3#7867811yescharacteristic, direct product factor
620C3C3×#301011yescharacteristic, complemented
621C2—22no—
622C2—42no—
623C2—122no—
624C2—242no—
625C2C2×C3×#88211yescharacteristic
626C2C3×#575112yes—
subgroupG/H
centerC22×C3C22×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#408C2
socleC22×C32C22×D4
hypercenterC3×#408D3
hypocenterC3C3×#3010
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C3×#408

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1010C2semidirect
2. C3×#5830C2semidirect
3. C2×C3×#908C2semidirect
4. C22×C3×#204C2semidirect
5. C32×#408C2semidirect
6. C3×#5120C2semidirect
7. C3×#5113C2semidirect
8. C3×#4738C2semidirect
9. C3×#4734C2semidirect
11. #78678C3direct (N × H)
19. C2×C3×#235C22semidirect
22. C22×C3×Dic6C22semidirect
35. C32×C22⋊C8C22semidirect
36. C3×#821C22semidirect
37. C3×#816C22semidirect
38. C3×#816C22semidirect
40. C2×#1010C2×C3semidirect
44. #5830C2×C3semidirect
45. C2×#908C2×C3semidirect
46. C22×#204C2×C3semidirect
52. C3×#408D3semidirect
53. C3×#408C2×C3semidirect
54. #5120C2×C3semidirect
55. #5113C2×C3semidirect
56. #4738C2×C3semidirect
57. #4734C2×C3semidirect
77. C3×#235D4semidirect
90. C2×C3×Dic6D4semidirect
124. C2×#235C22×C3semidirect
127. C22×Dic6C22×C3semidirect
158. C3×C22⋊C8C2×D3semidirect
159. C3×C22⋊C8C22×C3semidirect
160. #821C22×C3semidirect
161. #816C22×C3semidirect
162. #816C22×C3semidirect
210. #408C3×D3semidirect
272. #235C3×D4semidirect
285. C2×Dic6C3×D4semidirect
357. C22⋊C8C2×C3×D3semidirect
545. C32#3010semidirect
619. C3#78678direct (N × H)
620. 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
C3×#58302
C2×C3×#9082
C22×C3×#2042
C32×#4082
C3×#51202
C3×#51132
C3×#47382
C3×#47342
C3×#30103
#786783