Finite Group Databasebeta

← C3×#78671 #11864340

Subgroups — C3×#78671 #11864340

Order 1152.

1583 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×#1009C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C2×C3×#909C211yesmaximal, characteristic, complemented
4C22×C3×#204C211yesmaximal, characteristic, complemented
5C32×#408C211yesmaximal, characteristic, complemented
6C3×#5131C211yesmaximal, characteristic, complemented
7C3×#5113C211yesmaximal, characteristic, complemented
8C3×#4733C214yesmaximal, complemented
9C3×#4731C214yesmaximal, complemented
10C3×#3009—31nomaximal
11#78671C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C3×#1009—24no—
14C3×#1009—24no—
15C2×C3×#239—21no—
16C2×C3×D3×D4—22no—
17C2×C3×D3×D4—21no—
18C2×C3×#235C2212yes—
19C2×C3×#235C2211yescharacteristic
20C22×C3×D12C2212yes—
21C22×C3×D12C2211yescharacteristic
22C22×C3×Dic6C2211yescharacteristic, complemented
23C2×C32×M16C2211yescharacteristic, complemented
24C22×C32×C8C2211yescharacteristic, complemented
25C3×#944—21no—
26C3×#936—21no—
27C3×#935—22no—
28C3×#926C2211yescharacteristic, complemented
29C3×#924C2214yescomplemented
30C3×#909—24no—
31C2×C3×Dic12C2212yescomplemented
32C2×C3×#204—24no—
33C2×C3×#204C2212yescomplemented
34C2×C3×#204C2214yescomplemented
35C32×C22⋊C8C2214yescomplemented
36C3×#821C2212yescomplemented
37C3×#821C2214yescomplemented
38C3×#816C2212yescomplemented
39C2×C3×H(5,𝔽2)—31no—
40C2×#1009C2×C311yescharacteristic
41C2×C3×#137—31no—
42C22×C3×SD16—31no—
43C3×#536—31no—
44#5825C2×C311yescharacteristic, complemented
45C2×#909C2×C311yescharacteristic, complemented
46C22×#204C2×C311yescharacteristic, complemented
47C3×#452—34no—
48C3×#450—34no—
49C3×#418—31no—
50C3×#416—31no—
51C3×#408—21no—
52C3×#408D311yescharacteristic, complemented
53C3×#408C2×C311yescharacteristic, complemented
54#5131C2×C311yescharacteristic, complemented
55#5113C2×C311yescharacteristic, complemented
56#4733C2×C314yescomplemented
57#4731C2×C314yescomplemented
58C23×C3×D3—22no—
59C23×C3×D3—21no—
60C32×#41—22no—
61C32×#41—22no—
62C2×C32×Q8D411yescharacteristic
63C2×C32×D4D412yes—
64C2×C32×D4D411yescharacteristic
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C2×C3×#67—22no—
68C2×C3×#67—21no—
69C22×C3×Dic3—21no—
70C3×#239—28no—
71C3×D3×D4—216no—
72C3×D3×D4—28no—
73C3×#235—22no—
74C3×#235—24no—
75C3×#235—24no—
76C3×#235D414yes—
77C2×C3×D12—22no—
78C2×C3×D12—21no—
79C2×C3×D12—22no—
80C2×C3×D12—24no—
81C2×C3×D12—24no—
82C2×C3×D12D414yes—
83C2×C3×D12C2311yescharacteristic
84C2×C3×D12C2312yes—
85C2×C3×C4×D3—22no—
86C2×C3×C4×D3—21no—
87C2×C3×Dic6—22no—
88C2×C3×Dic6—22no—
89C2×C3×Dic6—21no—
90C2×C3×Dic6C2312yes—
91C2×C3×Dic6C2311yescharacteristic
92C32×M16—22no—
93C2×C32×C8—22no—
94C2×C32×C8C2312yes—
95C2×C32×C8C2312yes—
96C3×#217—24no—
97C3×#217—21no—
98C3×C4⋊Dic3C2312yes—
99C3×Dic3⋊C4—24no—
100C3×C4×Dic3—21no—
101C3×Dic12—24no—
102C3×#204—28no—
103C3×#204—24no—
104#3009—31noSylow
105C3×H(5,𝔽2)—64no—
106C3×H(5,𝔽2)—64no—
107C2×C3×#41—21no—
108C2×C3×#41—31no—
109C2×C3×#41—32no—
110C2×C3×#41—61no—
111C2×C3×#41C2×D311yescharacteristic
112C2×C3×#41C22×C311yescharacteristic
113C22×C3×Q8—31no—
114C22×C3×D4—31no—
115C22×C3×D4—32no—
116C22×C3×D4—62no—
117C22×C3×D4—61no—
118#1009—24no—
119#1009—24no—
120C2×#239—21no—
121C2×D3×D4—22no—
122C2×D3×D4—21no—
123C2×#235C22×C312yes—
124C2×#235C22×C311yescharacteristic
125C22×D12C22×C312yes—
126C22×D12C22×C311yescharacteristic
127C22×Dic6C22×C311yescharacteristic, complemented
128C3×#137—64no—
129C2×C3×Q16—32no—
130C2×C3×SD16—34no—
131C2×C3×SD16—32no—
132C2×C3×SD16—64no—
133C2×C3×M16—21no—
134C2×C3×M16C2×D311yescharacteristic, complemented
135C2×C3×M16C22×C311yescharacteristic, complemented
136C22×C3×C8—21no—
137C22×C3×C8C2×D311yescharacteristic, complemented
138C22×C3×C8C22×C311yescharacteristic, complemented
139C3×C4⋊Q8—61no—
140C3×#124—61no—
141C3×#123—62no—
142C3×C22⋊Q8—34no—
143C3×#117—31no—
144#944—21no—
145#936—21no—
146#935—22no—
147#926C22×C311yescharacteristic, complemented
148#924C22×C314yescomplemented
149#909—24no—
150C2×Dic12C22×C312yescomplemented
151C2×#204—24no—
152C2×#204C22×C314yescomplemented
153C2×#204C22×C312yescomplemented
154C3×#103—32no—
155C3×#102—32no—
156C3×#102—34no—
157C3×C22⋊C8—24no—
158C3×C22⋊C8C2×D314yescomplemented
159C3×C22⋊C8C22×C314yescomplemented
160#821C22×C314yescomplemented
161#821C22×C312yescomplemented
162#816C22×C312yescomplemented
163C23×C32C2×D412yes—
164C23×C32C2×D411yescharacteristic
165C22×C3×D3—22no—
166C22×C3×D3—21no—
167C22×C3×D3—28no—
168C22×C3×D3—24no—
169C22×C3×D3—82no—
170C22×C3×D3—84no—
171C32×Q8—22no—
172C32×D4—24no—
173C32×D4—22no—
174C2×C32×C4—22no—
175C2×C32×C4—22no—
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2×D412yes—
179C2×C32×C4C2×D412yes—
180C2×C32×C4C2411yescharacteristic
181C3×#67—28no—
182C3×#67—24no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—42no—
187C3×D12—28no—
188C3×D12—24no—
189C3×D12—21no—
190C3×D12—22no—
191C3×D12C2×D412yes—
192C3×D12C2×D414yes—
193C3×C4×D3—28no—
194C3×C4×D3—24no—
195C3×Dic6—21no—
196C3×Dic6—22no—
197C3×Dic6—22no—
198C3×Dic6—42no—
199C3×Dic6C2×D412yes—
200C32×C8—22no—
201C32×C8—22no—
202C2×H(5,𝔽2)—31no—
203C2×#137—31no—
204C22×SD16—31no—
205#536—31no—
206#452—34no—
207#450—34no—
208#418—31no—
209#416—31no—
210#408C3×D311yescharacteristic, complemented
211C24×C3—62no—
212C24×C3—61no—
213C23×D3—22no—
214C23×D3—21no—
215C3×#41—22no—
216C3×#41—22no—
217C3×#41—22no—
218C3×#41—22no—
219C3×#41—34no—
220C3×#41—42no—
221C3×#41—42no—
222C3×#41—64no—
223C3×#41—64no—
224C3×#41—68no—
225C3×#41—62no—
226C2×C3×Q8—21no—
227C2×C3×Q8—31no—
228C2×C3×Q8—32no—
229C2×C3×Q8—61no—
230C2×C3×Q8—62no—
231C2×C3×Q8—62no—
232C2×C3×Q8D1211yescharacteristic
233C2×C3×Q8C3×D411yescharacteristic
234C2×C3×D4—22no—
235C2×C3×D4—21no—
236C2×C3×D4—34no—
237C2×C3×D4—32no—
238C2×C3×D4—31no—
239C2×C3×D4—64no—
240C2×C3×D4—64no—
241C2×C3×D4—616no—
242C2×C3×D4—68no—
243C2×C3×D4—62no—
244C2×C3×D4—61no—
245C2×C3×D4—62no—
246C2×C3×D4—61no—
247C2×C3×D4—62no—
248C2×C3×D4D1212yes—
249C2×C3×D4D1211yescharacteristic
250C2×C3×D4C3×D412yes—
251C2×C3×D4C3×D411yescharacteristic
252C22×C3×C4—21no—
253C22×C3×C4—22no—
254C22×C3×C4—62no—
255C22×C3×C4—61no—
256C22×C3×C4—61no—
257C22×C3×C4C22×D312yes—
258C22×C3×C4C22×D311yescharacteristic
259C22×C3×C4C23×C312yes—
260C22×C3×C4C23×C311yescharacteristic
261C2×#67—22no—
262C2×#67—21no—
263C22×Dic3—21no—
264#239—28no—
265D3×D4—216no—
266D3×D4—28no—
267#235—22no—
268#235—24no—
269#235—24no—
270#235C3×D414yes—
271C2×D12—22no—
272C2×D12—21no—
273C2×D12—22no—
274C2×D12—24no—
275C2×D12—24no—
276C2×D12C3×D414yes—
277C2×D12C23×C311yescharacteristic
278C2×D12C23×C312yes—
279C2×C4×D3—22no—
280C2×C4×D3—21no—
281C2×Dic6—22no—
282C2×Dic6—22no—
283C2×Dic6—21no—
284C2×Dic6C23×C312yes—
285C2×Dic6C23×C311yescharacteristic
286C3×Q16—64no—
287C3×SD16—68no—
288C3×SD16—64no—
289C3×M16—22no—
290C3×M16—22no—
291C3×M16—42no—
292C2×C3×C8—22no—
293C2×C3×C8—22no—
294C2×C3×C8—22no—
295C2×C3×C8—22no—
296C2×C3×C8—42no—
297C2×C3×C8C22×D312yes—
298C2×C3×C8C22×D312yes—
299C2×C3×C8C23×C312yes—
300C2×C3×C8C23×C312yes—
301C3×C4⋊C4—32no—
302C3×C4⋊C4—64no—
303C3×C22⋊C4—61no—
304C3×C22⋊C4—64no—
305C3×C42—61no—
306#217—24no—
307#217—21no—
308C4⋊Dic3C23×C312yes—
309Dic3⋊C4—24no—
310C4×Dic3—21no—
311Dic12—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—24no—
320C2×C3×D3—22no—
321C2×C3×D3—44no—
322C2×C3×D3—42no—
323C2×C3×D3—84no—
324C2×C3×D3—88no—
325C32×C4—22no—
326C32×C4#12012yes—
327C32×C4C22×D411yescharacteristic
328C32×C4C22×D411yescharacteristic
329C3×Dic3—22no—
330C3×Dic3—42no—
331C3×Dic3—42no—
332H(5,𝔽2)—64no—
333H(5,𝔽2)—64no—
334C2×#41—31no—
335C2×#41—32no—
336C2×#41—61no—
337C2×#41C2×C3×D311yescharacteristic
338C22×Q8—31no—
339C22×D4—31no—
340C22×D4—32no—
341C22×D4—62no—
342C22×D4—61no—
343#137—64no—
344C2×Q16—32no—
345C2×SD16—34no—
346C2×SD16—32no—
347C2×SD16—64no—
348C2×M16C2×C3×D311yescharacteristic, complemented
349C22×C8C2×C3×D311yescharacteristic, complemented
350C4⋊Q8—61no—
351#124—61no—
352#123—62no—
353C22⋊Q8—34no—
354#117—31no—
355#103—32no—
356#102—32no—
357#102—34no—
358C22⋊C8C2×C3×D314yescomplemented
359C23×C3—21no—
360C23×C3—22no—
361C23×C3—68no—
362C23×C3—62no—
363C23×C3—61no—
364C23×C3—64no—
365C23×C3—242no—
366C23×C3—244no—
367C23×C3C2×D1212yes—
368C23×C3C2×D1211yescharacteristic
369C23×C3C2×C3×D412yes—
370C23×C3C2×C3×D411yescharacteristic
371C22×D3—22no—
372C22×D3—21no—
373C22×D3—28no—
374C22×D3—24no—
375C22×D3—82no—
376C22×D3—84no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—32no—
380C3×Q8—42no—
381C3×Q8—61no—
382C3×Q8—62no—
383C3×Q8—62no—
384C3×Q8—122no—
385C3×D4—24no—
386C3×D4—24no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—34no—
390C3×D4—32no—
391C3×D4—42no—
392C3×D4—44no—
393C3×D4—68no—
394C3×D4—68no—
395C3×D4—64no—
396C3×D4—64no—
397C3×D4—61no—
398C3×D4—62no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—42no—
409C2×C3×C4—42no—
410C2×C3×C4—68no—
411C2×C3×C4—61no—
412C2×C3×C4—64no—
413C2×C3×C4—62no—
414C2×C3×C4—62no—
415C2×C3×C4—122no—
416C2×C3×C4C2×D1212yes—
417C2×C3×C4C2×D1211yescharacteristic
418C2×C3×C4D3×D412yes—
419C2×C3×C4D3×D412yes—
420C2×C3×C4C2×C3×D412yes—
421C2×C3×C4C2×C3×D411yescharacteristic
422C2×C3×C4C2×C3×D412yes—
423C2×C3×C4C2×C3×D412yes—
424C2×C3×C4C23×D311yescharacteristic
425C2×C3×C4C24×C311yescharacteristic
426#67—28no—
427#67—24no—
428C2×Dic3—21no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—42no—
432D12—28no—
433D12—24no—
434D12—21no—
435D12—22no—
436D12C2×C3×D412yes—
437D12C2×C3×D414yes—
438C4×D3—28no—
439C4×D3—24no—
440Dic6—21no—
441Dic6—22no—
442Dic6—22no—
443Dic6—42no—
444Dic6C2×C3×D412yes—
445C3×C8—22no—
446C3×C8—22no—
447C3×C8—22no—
448C3×C8—22no—
449C3×C8—42no—
450C3×C8—42no—
451C2×C32—22no—
452C2×C32—42no—
453C2×C32C2×#12011yescharacteristic
454C2×C32#57712yes—
455C3×D3—44no—
456C3×D3—42no—
457C24—62no—
458C24—61no—
459#41—22no—
460#41—22no—
461#41—34no—
462#41—64no—
463#41—64no—
464#41—68no—
465#41—62no—
466C2×Q8—31no—
467C2×Q8—32no—
468C2×Q8—61no—
469C2×Q8—62no—
470C2×Q8—62no—
471C2×Q8C3×D1211yescharacteristic
472C2×D4—32no—
473C2×D4—31no—
474C2×D4—34no—
475C2×D4—64no—
476C2×D4—64no—
477C2×D4—616no—
478C2×D4—68no—
479C2×D4—62no—
480C2×D4—61no—
481C2×D4—62no—
482C2×D4—61no—
483C2×D4—62no—
484C2×D4C3×D1212yes—
485C2×D4C3×D1211yescharacteristic
486C22×C4—62no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4C22×C3×D312yes—
490C22×C4C22×C3×D311yescharacteristic
491Q16—64no—
492SD16—68no—
493SD16—64no—
494M16—22no—
495C2×C8—22no—
496C2×C8C22×C3×D312yes—
497C2×C8C22×C3×D312yes—
498C4⋊C4—32no—
499C4⋊C4—64no—
500C22⋊C4—61no—
501C22⋊C4—64no—
502C42—61no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—24no—
506C22×C3—22no—
507C22×C3—24no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—21no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—44no—
514C22×C3—64no—
515C22×C3—62no—
516C22×C3—124no—
517C22×C3—122no—
518C22×C3—244no—
519C22×C3—248no—
520C22×C3#88212yes—
521C22×C3C3×#12012yes—
522C22×C3C22×D1211yescharacteristic
523C22×C3C22×C3×D411yescharacteristic
524C2×D3—24no—
525C2×D3—22no—
526C2×D3—44no—
527C2×D3—42no—
528C2×D3—84no—
529C2×D3—88no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—42no—
536C3×C4—62no—
537C3×C4—122no—
538C3×C4—122no—
539C3×C4#88212yes—
540C3×C4C3×#12012yes—
541C3×C4C2×D3×D411yescharacteristic
542C3×C4C2×D3×D411yescharacteristic
543C3×C4C22×C3×D411yescharacteristic
544C3×C4C22×C3×D411yescharacteristic
545Dic3—22no—
546Dic3—42no—
547Dic3—42no—
548C32#300911yescharacteristic, complemented, Sylow
549C23—68no—
550C23—62no—
551C23—61no—
552C23—64no—
553C23—242no—
554C23—244no—
555C23C2×C3×D1212yes—
556C23C2×C3×D1211yescharacteristic
557Q8—22no—
558Q8—32no—
559Q8—61no—
560Q8—62no—
561Q8—62no—
562Q8—122no—
563D4—24no—
564D4—22no—
565D4—34no—
566D4—32no—
567D4—68no—
568D4—68no—
569D4—64no—
570D4—64no—
571D4—61no—
572D4—62no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—68no—
576C2×C4—61no—
577C2×C4—64no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—122no—
581C2×C4C2×C3×D1212yes—
582C2×C4C2×C3×D1211yescharacteristic
583C2×C4C3×D3×D412yes—
584C2×C4C3×D3×D412yes—
585C2×C4C23×C3×D311yescharacteristic
586C8—22no—
587C8—22no—
588C2×C3—22no—
589C2×C3—22no—
590C2×C3—21no—
591C2×C3—22no—
592C2×C3—42no—
593C2×C3—42no—
594C2×C3—42no—
595C2×C3—82no—
596C2×C3—124no—
597C2×C3—122no—
598C2×C3C2×#88211yescharacteristic
599C2×C3#575112yes—
600C2×C3C2×C3×#12011yescharacteristic
601C2×C3C3×#57712yes—
602D3—44no—
603D3—42no—
604C22—22no—
605C22—22no—
606C22—24no—
607C22—64no—
608C22—62no—
609C22—124no—
610C22—122no—
611C22—244no—
612C22—248no—
613C22C3×#88212yes—
614C22C22×C3×D1211yescharacteristic
615C4—22no—
616C4—62no—
617C4—122no—
618C4—122no—
619C4C3×#88212yes—
620C4C2×C3×D3×D411yescharacteristic
621C4C2×C3×D3×D411yescharacteristic
622C3—21no—
623C3#7867111yescharacteristic, direct product factor
624C3C3×#300911yescharacteristic, complemented
625C2—22no—
626C2—42no—
627C2—124no—
628C2—122no—
629C2C2×C3×#88211yescharacteristic
630C2C3×#575112yes—
subgroupG/H
centerC22×C3C22×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#408C2
socleC22×C32C22×D4
prime radicalC2×C3×#1009C2
hypercenterC3×#408D3
hypocenterC3C3×#3009
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
2. C3×#5825C2semidirect
3. C2×C3×#909C2semidirect
4. C22×C3×#204C2semidirect
5. C32×#408C2semidirect
6. C3×#5131C2semidirect
7. C3×#5113C2semidirect
8. C3×#4733C2semidirect
9. C3×#4731C2semidirect
11. #78671C3direct (N × H)
22. C22×C3×Dic6C22semidirect
23. C2×C32×M16C22semidirect
24. C22×C32×C8C22semidirect
28. C3×#926C22semidirect
29. C3×#924C22semidirect
31. C2×C3×Dic12C22semidirect
33. C2×C3×#204C22semidirect
34. C2×C3×#204C22semidirect
35. C32×C22⋊C8C22semidirect
36. C3×#821C22semidirect
37. C3×#821C22semidirect
38. C3×#816C22semidirect
44. #5825C2×C3semidirect
45. C2×#909C2×C3semidirect
46. C22×#204C2×C3semidirect
52. C3×#408D3semidirect
53. C3×#408C2×C3semidirect
54. #5131C2×C3semidirect
55. #5113C2×C3semidirect
56. #4733C2×C3semidirect
57. #4731C2×C3semidirect
127. C22×Dic6C22×C3semidirect
134. C2×C3×M16C2×D3semidirect
135. C2×C3×M16C22×C3semidirect
137. C22×C3×C8C2×D3semidirect
138. C22×C3×C8C22×C3semidirect
147. #926C22×C3semidirect
148. #924C22×C3semidirect
150. C2×Dic12C22×C3semidirect
152. C2×#204C22×C3semidirect
153. C2×#204C22×C3semidirect
158. C3×C22⋊C8C2×D3semidirect
159. C3×C22⋊C8C22×C3semidirect
160. #821C22×C3semidirect
161. #821C22×C3semidirect
162. #816C22×C3semidirect
210. #408C3×D3semidirect
348. C2×M16C2×C3×D3semidirect
349. C22×C8C2×C3×D3semidirect
358. C22⋊C8C2×C3×D3semidirect
548. C32#3009semidirect
623. C3#78671direct (N × H)
624. C3C3×#3009semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10092
C3×#58252
C2×C3×#9092
C22×C3×#2042
C32×#4082
C3×#51312
C3×#51132
C3×#47332
C3×#47312
C3×#30093
#786713