Finite Group Databasebeta

← C3×#83948 #11864408

Subgroups — C3×#83948 #11864408

Order 1152.

1613 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic, complemented
2C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3C3×#5805C211yesmaximal, characteristic, complemented
4C22×C3×#214C211yesmaximal, characteristic, complemented
5C3×#5233C211yesmaximal, characteristic, complemented
6C3×#5170C214yesmaximal, complemented
7C3×#5158C214yesmaximal, complemented
8C3×#5129C211yesmaximal, characteristic, complemented
9C3×#5120C211yesmaximal, characteristic, complemented
10C3×#3010—31nomaximal
11#83948C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C22×C32×D4C2211yescharacteristic
14C3×#1010—24no—
15C3×#1010—24no—
16C2×C3×D3×Q8C2212yescomplemented
17C2×C3×#237—22no—
18C2×C3×#237C2212yescomplemented
19C2×C3×#235—21no—
20C2×C3×#235C2211yescharacteristic
21C22×C3×Dic6—21no—
22C22×C3×Dic6C2211yescharacteristic
23C32×AGL(1,ℤ/8)—24no—
24C2×C32×SD16C2212yes—
25C2×C32×D8C2212yes—
26C2×C32×M16C2211yescharacteristic
27C3×#953—22no—
28C3×#940—21no—
29C3×#938C2214yes—
30C3×#936—21no—
31C2×C3×#214—24no—
32C2×C3×#214C2214yes—
33C3×#926C2211yescharacteristic
34C22×C3×C3⋊C8C2211yescharacteristic
35C3×#835C2212yescomplemented
36C3×#832C2212yescomplemented
37C3×(C2×D3)⋊C8C2214yescomplemented
38C3×#816C2214yescomplemented
39C2×C3×2−1+4—31no—
40C2×#1010C2×C311yescharacteristic, complemented
41C2×C3×AGL(1,ℤ/8)—21no—
42C2×C3×AGL(1,ℤ/8)D311yescharacteristic, complemented
43C2×C3×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
44C22×C3×SD16—31no—
45C3×#535—31no—
46#5805C2×C311yescharacteristic, complemented
47C22×#214C2×C311yescharacteristic, complemented
48C3×#449—34no—
49C3×#448—34no—
50C3×#419—31no—
51C3×#416—31no—
52C3×#408—31no—
53#5233C2×C311yescharacteristic, complemented
54#5170C2×C314yescomplemented
55#5158C2×C314yescomplemented
56#5129C2×C311yescharacteristic, complemented
57#5120C2×C311yescharacteristic, complemented
58C24×C32—21no—
59C32×#41—22no—
60C32×#41D414yescomplemented
61C2×C32×Q8C2311yescharacteristic
62C2×C32×D4—21no—
63C2×C32×D4—22no—
64C2×C32×D4—22no—
65C2×C32×D4—21no—
66C2×C32×D4C2312yes—
67C2×C32×D4C2311yescharacteristic
68C22×C32×C4—21no—
69C22×C32×C4C2311yescharacteristic
70C2×C3×#67—22no—
71C2×C3×#67—24no—
72C2×C3×#67D412yes—
73C22×C3×Dic3—21no—
74C22×C3×Dic3—22no—
75C3×D3×Q8—24no—
76C3×D3×Q8—24no—
77C3×#237—216no—
78C3×#237—24no—
79C3×#237—24no—
80C3×#235—28no—
81C3×#235—22no—
82C3×#235—22no—
83C2×C3×D12D411yescharacteristic
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—
93C32×SD16—24no—
94C32×D8—24no—
95C32×M16—22no—
96C2×C32×C8C2312yes—
97C3×#217—21no—
98C3×#217—24no—
99C3×#214—28no—
100C3×C4⋊Dic3C2312yes—
101C3×C4×Dic3—21no—
102C2×C3×C3⋊C8—22no—
103C2×C3×C3⋊C8C2312yes—
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—21no—
118C22×C3×D4C2×D311yescharacteristic
119C22×C3×D4C22×C311yescharacteristic
120#1010—24no—
121#1010—24no—
122C2×D3×Q8C22×C312yescomplemented
123C2×#237—22no—
124C2×#237C22×C312yescomplemented
125C2×#235—21no—
126C2×#235C22×C311yescharacteristic
127C22×Dic6—21no—
128C22×Dic6C22×C311yescharacteristic
129C3×AGL(1,ℤ/8)—28no—
130C3×AGL(1,ℤ/8)—24no—
131C3×AGL(1,ℤ/8)—24no—
132C2×C3×SD16—22no—
133C2×C3×SD16—34no—
134C2×C3×SD16—64no—
135C2×C3×SD16C2×D312yes—
136C2×C3×SD16C22×C312yes—
137C2×C3×D8—22no—
138C2×C3×D8C2×D312yes—
139C2×C3×D8C22×C312yes—
140C2×C3×M16—21no—
141C2×C3×M16C2×D311yescharacteristic
142C2×C3×M16C22×C311yescharacteristic
143C22×C3×C8—31no—
144C3×#127—61no—
145C3×#124—61no—
146C3×#121—34no—
147C3×#120—62no—
148C3×#117—31no—
149#953—22no—
150#940—21no—
151#938C22×C314yes—
152#936—21no—
153C2×#214—24no—
154C2×#214C22×C314yes—
155#926C22×C311yescharacteristic
156C22×C3⋊C8C22×C311yescharacteristic
157C3×#103—32no—
158C3×#103—34no—
159C3×#102—32no—
160C3×C22⋊C8—34no—
161#835C22×C312yescomplemented
162#832C22×C312yescomplemented
163(C2×D3)⋊C8C22×C314yescomplemented
164#816C22×C314yescomplemented
165C23×C32—21no—
166C23×C32—22no—
167C23×C32—42no—
168C23×C32—82no—
169C23×C32C2×D411yescharacteristic
170C22×C3×D3C2×D412yes—
171C32×Q8—21no—
172C32×Q8C2×D412yes—
173C32×D4—22no—
174C32×D4—24no—
175C32×D4—21no—
176C32×D4—22no—
177C32×D4—42no—
178C32×D4C2×D412yes—
179C2×C32×C4—21no—
180C2×C32×C4—24no—
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2×D412yes—
183C2×C32×C4C2×D412yes—
184C2×C32×C4C2411yescharacteristic
185C3×#67—28no—
186C3×#67—24no—
187C3×#67—84no—
188C2×C3×Dic3—28no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3C2×D412yes—
193C3×D12—22no—
194C3×C4×D3—28no—
195C3×C4×D3—22no—
196C3×C4×D3—22no—
197C3×Dic6—28no—
198C3×Dic6—22no—
199C3×Dic6—24no—
200C3×Dic6—22no—
201C3×Dic6C2×D414yes—
202C32×C8—22no—
203C3×C3⋊C8—22no—
204C2×2−1+4—31no—
205C2×AGL(1,ℤ/8)C3×D311yescharacteristic, complemented
206C22×SD16—31no—
207#535—31no—
208#449—34no—
209#448—34no—
210#419—31no—
211#416—31no—
212#408—31no—
213C24×C3—21no—
214C24×C3—21no—
215C24×C3—41no—
216C3×#41—22no—
217C3×#41—22no—
218C3×#41—24no—
219C3×#41—42no—
220C3×#41—68no—
221C3×#41—64no—
222C3×#41—616no—
223C3×#41—64no—
224C3×#41—62no—
225C3×#41—62no—
226C3×#41#6714yescomplemented
227C3×#41C3×D414yescomplemented
228C2×C3×Q8—21no—
229C2×C3×Q8—31no—
230C2×C3×Q8—34no—
231C2×C3×Q8—32no—
232C2×C3×Q8—62no—
233C2×C3×Q8—68no—
234C2×C3×Q8—64no—
235C2×C3×Q8—64no—
236C2×C3×Q8—61no—
237C2×C3×Q8—62no—
238C2×C3×Q8C22×D311yescharacteristic
239C2×C3×Q8C23×C311yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—24no—
243C2×C3×D4—22no—
244C2×C3×D4—21no—
245C2×C3×D4—22no—
246C2×C3×D4—22no—
247C2×C3×D4—22no—
248C2×C3×D4—21no—
249C2×C3×D4—21no—
250C2×C3×D4—22no—
251C2×C3×D4—32no—
252C2×C3×D4—31no—
253C2×C3×D4—42no—
254C2×C3×D4—41no—
255C2×C3×D4—41no—
256C2×C3×D4—62no—
257C2×C3×D4—64no—
258C2×C3×D4C22×D312yes—
259C2×C3×D4C22×D311yescharacteristic
260C2×C3×D4C23×C312yes—
261C2×C3×D4C23×C311yescharacteristic
262C22×C3×C4—21no—
263C22×C3×C4—21no—
264C22×C3×C4—21no—
265C22×C3×C4—32no—
266C22×C3×C4—41no—
267C22×C3×C4—62no—
268C22×C3×C4—62no—
269C22×C3×C4—61no—
270C22×C3×C4C22×D311yescharacteristic
271C22×C3×C4C23×C311yescharacteristic
272C2×#67—22no—
273C2×#67—24no—
274C2×#67C3×D412yes—
275C22×Dic3—21no—
276C22×Dic3—22no—
277D3×Q8—24no—
278D3×Q8—24no—
279#237—216no—
280#237—24no—
281#237—24no—
282#235—28no—
283#235—22no—
284#235—22no—
285C2×D12C3×D411yescharacteristic
286C2×C4×D3—22no—
287C2×C4×D3C23×C312yes—
288C2×Dic6—28no—
289C2×Dic6—21no—
290C2×Dic6—22no—
291C2×Dic6—22no—
292C2×Dic6C3×D411yescharacteristic
293C2×Dic6C3×D414yescomplemented
294C2×Dic6C23×C312yes—
295C3×SD16—28no—
296C3×SD16—24no—
297C3×SD16—24no—
298C3×SD16—68no—
299C3×D8—28no—
300C3×D8—24no—
301C3×D8—24no—
302C3×M16—24no—
303C3×M16—22no—
304C3×M16—22no—
305C2×C3×C8—22no—
306C2×C3×C8—32no—
307C2×C3×C8—62no—
308C2×C3×C8C22×D312yes—
309C2×C3×C8C23×C312yes—
310C3×C4⋊C4—32no—
311C3×C22⋊C4—64no—
312C3×C22⋊C4—61no—
313C3×C42—61no—
314#217—21no—
315#217—24no—
316#214—28no—
317C4⋊Dic3C23×C312yes—
318C4×Dic3—21no—
319C2×C3⋊C8—22no—
320C2×C3⋊C8C23×C312yes—
321C22×C32—22no—
322C22×C32—22no—
323C22×C32—42no—
324C22×C32—42no—
325C22×C32—44no—
326C22×C32—84no—
327C22×C32#12012yes—
328C22×C32C22×D411yescharacteristic
329C2×C3×D3—22no—
330C2×C3×D3—24no—
331C32×C4—22no—
332C32×C4#12012yes—
333C32×C4C22×D411yescharacteristic
334C32×C4C22×D411yescharacteristic
335C3×Dic3—22no—
336C3×Dic3—24no—
337C3×Dic3—42no—
3382−1+4—64no—
3392−1+4—64no—
340C2×#41—31no—
341C2×#41—32no—
342C2×#41—61no—
343C2×#41—62no—
344C2×#41C2×C3×D311yescharacteristic
345C22×Q8—32no—
346C22×Q8—31no—
347C22×Q8—61no—
348C22×D4C2×C3×D311yescharacteristic
349AGL(1,ℤ/8)—24no—
350C2×SD16—34no—
351C2×SD16—64no—
352C2×SD16C2×C3×D312yes—
353C2×D8C2×C3×D312yes—
354C2×M16C2×C3×D311yescharacteristic
355C22×C8—31no—
356#127—61no—
357#124—61no—
358#121—34no—
359#120—62no—
360#117—31no—
361#103—32no—
362#103—34no—
363#102—32no—
364C22⋊C8—34no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—22no—
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—32no—
371C23×C3—41no—
372C23×C3—42no—
373C23×C3—44no—
374C23×C3—42no—
375C23×C3—42no—
376C23×C3—82no—
377C23×C3—82no—
378C23×C3—84no—
379C23×C3D3×D411yescharacteristic
380C23×C3C2×C3×D411yescharacteristic
381C22×D3C2×C3×D412yes—
382C3×Q8—21no—
383C3×Q8—21no—
384C3×Q8—22no—
385C3×Q8—34no—
386C3×Q8—41no—
387C3×Q8—68no—
388C3×Q8—62no—
389C3×Q8—64no—
390C3×Q8—62no—
391C3×Q8C2×#6712yes—
392C3×Q8C2×C3×D412yes—
393C3×D4—24no—
394C3×D4—24no—
395C3×D4—24no—
396C3×D4—21no—
397C3×D4—22no—
398C3×D4—22no—
399C3×D4—21no—
400C3×D4—22no—
401C3×D4—22no—
402C3×D4—22no—
403C3×D4—42no—
404C3×D4—41no—
405C3×D4—44no—
406C3×D4—44no—
407C3×D4—42no—
408C3×D4—42no—
409C3×D4—68no—
410C3×D4—64no—
411C3×D4—62no—
412C3×D4—244no—
413C3×D4C2×#6712yes—
414C3×D4C2×C3×D412yes—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—24no—
418C2×C3×C4—24no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—32no—
424C2×C3×C4—44no—
425C2×C3×C4—41no—
426C2×C3×C4—68no—
427C2×C3×C4—68no—
428C2×C3×C4—64no—
429C2×C3×C4—62no—
430C2×C3×C4—62no—
431C2×C3×C4—62no—
432C2×C3×C4—62no—
433C2×C3×C4D3×D411yescharacteristic
434C2×C3×C4D3×D412yes—
435C2×C3×C4C2×#6712yes—
436C2×C3×C4C2×C3×D411yescharacteristic
437C2×C3×C4C2×C3×D412yes—
438C2×C3×C4C2×C3×D412yes—
439C2×C3×C4C23×D311yescharacteristic
440C2×C3×C4C24×C311yescharacteristic
441#67—28no—
442#67—24no—
443#67—84no—
444C2×Dic3—28no—
445C2×Dic3—24no—
446C2×Dic3—22no—
447C2×Dic3—22no—
448C2×Dic3C2×C3×D412yes—
449D12—22no—
450C4×D3—28no—
451C4×D3—22no—
452C4×D3—22no—
453Dic6—28no—
454Dic6—22no—
455Dic6—24no—
456Dic6—22no—
457Dic6C2×C3×D414yes—
458C3×C8—24no—
459C3×C8—22no—
460C3×C8—22no—
461C3×C8—62no—
462C3⋊C8—22no—
463C2×C32—22no—
464C2×C32—42no—
465C2×C32—82no—
466C2×C32C2×#12011yescharacteristic
467C2×C32#57712yes—
468C3×D3—42no—
469C24—21no—
470#41—22no—
471#41—68no—
472#41—64no—
473#41—616no—
474#41—64no—
475#41—62no—
476#41—62no—
477#41C3×#6714yescomplemented
478C2×Q8—31no—
479C2×Q8—32no—
480C2×Q8—34no—
481C2×Q8—62no—
482C2×Q8—68no—
483C2×Q8—64no—
484C2×Q8—64no—
485C2×Q8—61no—
486C2×Q8—62no—
487C2×Q8C22×C3×D311yescharacteristic
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—32no—
493C2×D4—31no—
494C2×D4—62no—
495C2×D4—64no—
496C2×D4C22×C3×D312yes—
497C2×D4C22×C3×D311yescharacteristic
498C22×C4—21no—
499C22×C4—32no—
500C22×C4—62no—
501C22×C4—62no—
502C22×C4—61no—
503C22×C4C22×C3×D311yescharacteristic
504SD16—24no—
505SD16—68no—
506D8—24no—
507M16—22no—
508C2×C8—32no—
509C2×C8—62no—
510C2×C8C22×C3×D312yes—
511C4⋊C4—32no—
512C22⋊C4—64no—
513C22⋊C4—61no—
514C42—61no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—21no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—44no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—44no—
527C22×C3—44no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—64no—
531C22×C3—62no—
532C22×C3—84no—
533C22×C3—82no—
534C22×C3—84no—
535C22×C3—84no—
536C22×C3—88no—
537C22×C3#93712yes—
538C22×C3C3×#12012yes—
539C22×C3C2×D3×D411yescharacteristic
540C22×C3C22×C3×D411yescharacteristic
541C2×D3—24no—
542C2×D3—22no—
543C3×C4—22no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—42no—
549C3×C4—62no—
550C3×C4—64no—
551C3×C4—122no—
552C3×C4#93712yes—
553C3×C4C3×#12012yes—
554C3×C4C2×D3×D411yescharacteristic
555C3×C4C22×#6711yescharacteristic
556C3×C4C22×C3×D411yescharacteristic
557C3×C4C22×C3×D411yescharacteristic
558Dic3—22no—
559Dic3—24no—
560Dic3—42no—
561C32#301011yescharacteristic, complemented, Sylow
562C23—21no—
563C23—22no—
564C23—32no—
565C23—42no—
566C23—82no—
567C23C3×D3×D411yescharacteristic
568Q8—21no—
569Q8—34no—
570Q8—68no—
571Q8—62no—
572Q8—64no—
573Q8—62no—
574Q8C2×C3×#6712yes—
575D4—24no—
576D4—21no—
577D4—22no—
578D4—22no—
579D4—42no—
580D4—68no—
581D4—64no—
582D4—62no—
583D4—244no—
584D4C2×C3×#6712yes—
585C2×C4—21no—
586C2×C4—24no—
587C2×C4—32no—
588C2×C4—68no—
589C2×C4—68no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4C3×D3×D411yescharacteristic
596C2×C4C3×D3×D412yes—
597C2×C4C2×C3×#6712yes—
598C2×C4C23×C3×D311yescharacteristic
599C8—22no—
600C8—62no—
601C2×C3—22no—
602C2×C3—22no—
603C2×C3—22no—
604C2×C3—21no—
605C2×C3—42no—
606C2×C3—42no—
607C2×C3—42no—
608C2×C3—82no—
609C2×C3—84no—
610C2×C3—82no—
611C2×C3—82no—
612C2×C3—122no—
613C2×C3#577512yes—
614C2×C3C2×#93711yescharacteristic
615C2×C3C2×C3×#12011yescharacteristic
616C2×C3C3×#57712yes—
617D3—42no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—64no—
624C22—62no—
625C22—84no—
626C22C3×#93712yes—
627C22C2×C3×D3×D411yescharacteristic
628C4—22no—
629C4—62no—
630C4—64no—
631C4—122no—
632C4C3×#93712yes—
633C4C2×C3×D3×D411yescharacteristic
634C4C22×C3×#6711yescharacteristic
635C3—21no—
636C3#8394811yescharacteristic, direct product factor
637C3C3×#301011yescharacteristic, complemented
638C2—22no—
639C2—42no—
640C2—82no—
641C2—122no—
642C2C3×#577512yes—
643C2C2×C3×#93711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×AGL(1,ℤ/8)C2
socleC22×C32C22×D4
hypercenterC2×C3×AGL(1,ℤ/8)D3
hypocenterC3C3×#3010
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1010C2semidirect
2. C2×C32×AGL(1,ℤ/8)C2semidirect
3. C3×#5805C2semidirect
4. C22×C3×#214C2semidirect
5. C3×#5233C2semidirect
6. C3×#5170C2semidirect
7. C3×#5158C2semidirect
8. C3×#5129C2semidirect
9. C3×#5120C2semidirect
11. #83948C3direct (N × H)
16. C2×C3×D3×Q8C22semidirect
18. C2×C3×#237C22semidirect
35. C3×#835C22semidirect
36. C3×#832C22semidirect
37. C3×(C2×D3)⋊C8C22semidirect
38. C3×#816C22semidirect
40. C2×#1010C2×C3semidirect
42. C2×C3×AGL(1,ℤ/8)D3semidirect
43. C2×C3×AGL(1,ℤ/8)C2×C3semidirect
46. #5805C2×C3semidirect
47. C22×#214C2×C3semidirect
53. #5233C2×C3semidirect
54. #5170C2×C3semidirect
55. #5158C2×C3semidirect
56. #5129C2×C3semidirect
57. #5120C2×C3semidirect
60. C32×#41D4semidirect
91. C2×C3×Dic6D4semidirect
122. C2×D3×Q8C22×C3semidirect
124. C2×#237C22×C3semidirect
161. #835C22×C3semidirect
162. #832C22×C3semidirect
163. (C2×D3)⋊C8C22×C3semidirect
164. #816C22×C3semidirect
205. C2×AGL(1,ℤ/8)C3×D3semidirect
226. C3×#41#67semidirect
227. C3×#41C3×D4semidirect
293. C2×Dic6C3×D4semidirect
477. #41C3×#67semidirect
561. C32#3010semidirect
636. C3#83948direct (N × H)
637. 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×C32×AGL(1,ℤ/8)2
C3×#58052
C22×C3×#2142
C3×#52332
C3×#51702
C3×#51582
C3×#51292
C3×#51202
C3×#30103
#839483