Finite Group Databasebeta

← C3×#83768 #11857952

Subgroups — C3×#83768 #11857952

Order 1152.

1693 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic
2C22×C32×M16C211yesmaximal, characteristic, complemented
3C2×C3×#926C211yesmaximal, characteristic, complemented
4C3×#5133C214yesmaximal, complemented
5C3×#5131C212yesmaximal, complemented
6C3×#5120C212yesmaximal, complemented
7C3×#5113C214yesmaximal, complemented
8C3×#2896—31nomaximal
9#83768C311yesmaximal, characteristic, direct product factor
10C23×C32×C4C2211yescharacteristic
11C22×C3×#67—21no—
12C2×C3×#235—24no—
13C2×C3×#235—21no—
14C2×C3×#235—21no—
15C2×C3×#235C418yescomplemented
16C2×C3×#235C2212yes—
17C2×C3×#235C2212yes—
18C22×C3×D12C2211yescharacteristic
19C22×C3×C4×D3—21no—
20C22×C3×Dic6C2211yescharacteristic
21C2×C32×M16—24no—
22C2×C32×M16C2212yes—
23C2×C32×M16C2212yescomplemented
24C22×C32×C8C2212yescomplemented
25C2×C3×#217—21no—
26C3×#926—21no—
27C3×#926—21no—
28C3×#926C2212yes—
29C3×#926C2212yescomplemented
30C2×C3×C4⋊Dic3C2212yescomplemented
31C2×C3×C4×Dic3—21no—
32C3×#821C2218yescomplemented
33C3×#816C2218yescomplemented
34C22×C3×#41—31no—
35C22×#235C2×C311yescharacteristic
36C22×C3×M16—21no—
37C22×C3×M16D311yescharacteristic, complemented
38C22×C3×M16C2×C311yescharacteristic, complemented
39C2×C3×#117—31no—
40C2×#926C2×C311yescharacteristic, complemented
41C3×#419—32no—
42C3×#418—32no—
43C3×#417—34no—
44C3×#416—34no—
45#5133C2×C314yescomplemented
46#5131C2×C312yescomplemented
47#5120C2×C312yescomplemented
48#5113C2×C314yescomplemented
49C24×C32D411yescharacteristic
50C23×C3×D3—21no—
51C22×C32×C4C2×C414yes—
52C22×C32×C4D411yescharacteristic
53C22×C32×C4D411yescharacteristic
54C22×C32×C4D411yescharacteristic
55C22×C32×C4D414yes—
56C22×C32×C4C2311yescharacteristic
57C22×C32×C4C2311yescharacteristic
58C22×C32×C4C2311yescharacteristic
59C2×C3×#67—28no—
60C2×C3×#67—22no—
61C2×C3×#67—22no—
62C22×C3×Dic3—21no—
63C22×C3×Dic3—22no—
64C3×#235—28no—
65C3×#235—28no—
66C3×#235—24no—
67C3×#235—24no—
68C3×#235C2×C418yes—
69C3×#235C2×C418yescomplemented
70C2×C3×D12—21no—
71C2×C3×D12—22no—
72C2×C3×D12C2×C414yescomplemented
73C2×C3×D12C2312yes—
74C2×C3×C4×D3—28no—
75C2×C3×C4×D3—22no—
76C2×C3×C4×D3—22no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—22no—
79C2×C3×Dic6C2×C414yescomplemented
80C2×C3×Dic6C2312yes—
81C32×M16—24no—
82C32×M16—24no—
83C2×C32×C8—24no—
84C2×C32×C8C2314yes—
85C3×#217—22no—
86C3×#217—22no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3C2314yes—
89C3×C4×Dic3—22no—
90C3×C4×Dic3—22no—
91#2896—31noSylow
92C2×C3×#41—38no—
93C2×C3×#41—32no—
94C2×C3×#41—32no—
95C2×C3×#41—61no—
96C2×C3×#41—61no—
97C2×C3×#41—64no—
98C22×C3×Q8—31no—
99C22×C3×D4—31no—
100C22×C3×D4—61no—
101C23×C3×C4—21no—
102C23×C3×C4—61no—
103C23×C3×C4C2×D311yescharacteristic
104C23×C3×C4C22×C311yescharacteristic
105C22×#67—21no—
106C2×#235—21no—
107C2×#235—21no—
108C2×#235—24no—
109C2×#235C3×C418yescomplemented
110C2×#235C22×C312yes—
111C2×#235C22×C312yes—
112C22×D12C22×C311yescharacteristic
113C22×C4×D3—21no—
114C22×Dic6C22×C311yescharacteristic
115C2×C3×M16—28no—
116C2×C3×M16—24no—
117C2×C3×M16—24no—
118C2×C3×M16—22no—
119C2×C3×M16—22no—
120C2×C3×M16C2×D312yes—
121C2×C3×M16C2×D312yescomplemented
122C2×C3×M16C22×C312yes—
123C2×C3×M16C22×C312yescomplemented
124C22×C3×C8—22no—
125C22×C3×C8C2×D312yescomplemented
126C22×C3×C8C22×C312yescomplemented
127C3×#117—32no—
128C3×#117—32no—
129C3×#117—61no—
130C3×#117—61no—
131C2×C3×C4⋊C4—32no—
132C2×C3×C22⋊C4—61no—
133C2×C3×C42—61no—
134C2×#217—21no—
135#926—21no—
136#926—21no—
137#926C22×C312yes—
138#926C22×C312yescomplemented
139C2×C4⋊Dic3C22×C312yescomplemented
140C2×C4×Dic3—21no—
141C3×#103—38no—
142C3×#102—38no—
143#821C22×C318yescomplemented
144#816C22×C318yescomplemented
145C23×C32—42no—
146C23×C32C22⋊C414yes—
147C23×C32C2×D411yescharacteristic
148C23×C32C2×D411yescharacteristic
149C23×C32C2×D411yescharacteristic
150C22×C3×D3—24no—
151C22×C3×D3—22no—
152C22×C3×D3—44no—
153C2×C32×C4C22⋊C414yes—
154C2×C32×C4C22⋊C414yes—
155C2×C32×C4C22⋊C414yes—
156C2×C32×C4C22×C412yes—
157C2×C32×C4C22×C412yes—
158C2×C32×C4C22×C412yes—
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D412yes—
163C2×C32×C4C2×D412yes—
164C2×C32×C4C2×D412yes—
165C2×C32×C4C2411yescharacteristic
166C3×#67—28no—
167C3×#67—28no—
168C2×C3×Dic3—24no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—24no—
171C2×C3×Dic3—44no—
172C3×D12—24no—
173C3×D12—22no—
174C3×D12C22×C414yes—
175C3×C4×D3—28no—
176C3×C4×D3—28no—
177C3×Dic6—24no—
178C3×Dic6—22no—
179C3×Dic6C22×C414yes—
180C32×C8—24no—
181C22×#41—31no—
182C22×M16C3×D311yescharacteristic, complemented
183C2×#117—31no—
184#419—32no—
185#418—32no—
186#417—34no—
187#416—34no—
188C24×C3—21no—
189C24×C3—61no—
190C24×C3D1211yescharacteristic
191C24×C3C3×D411yescharacteristic
192C23×D3—21no—
193C3×#41—38no—
194C3×#41—38no—
195C3×#41—68no—
196C3×#41—68no—
197C3×#41—64no—
198C3×#41—64no—
199C2×C3×Q8—34no—
200C2×C3×Q8—32no—
201C2×C3×Q8—61no—
202C2×C3×Q8—62no—
203C2×C3×D4—34no—
204C2×C3×D4—32no—
205C2×C3×D4—68no—
206C2×C3×D4—62no—
207C2×C3×D4—62no—
208C2×C3×D4—61no—
209C2×C3×D4—62no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—24no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—61no—
219C22×C3×C4—68no—
220C22×C3×C4—62no—
221C22×C3×C4—62no—
222C22×C3×C4—62no—
223C22×C3×C4C4×D314yes—
224C22×C3×C4D1211yescharacteristic
225C22×C3×C4D1211yescharacteristic
226C22×C3×C4D1211yescharacteristic
227C22×C3×C4#6714yes—
228C22×C3×C4C2×C3×C414yes—
229C22×C3×C4C3×D411yescharacteristic
230C22×C3×C4C3×D411yescharacteristic
231C22×C3×C4C3×D411yescharacteristic
232C22×C3×C4C3×D414yes—
233C22×C3×C4C22×D311yescharacteristic
234C22×C3×C4C22×D311yescharacteristic
235C22×C3×C4C22×D311yescharacteristic
236C22×C3×C4C23×C311yescharacteristic
237C22×C3×C4C23×C311yescharacteristic
238C22×C3×C4C23×C311yescharacteristic
239C2×#67—28no—
240C2×#67—22no—
241C2×#67—22no—
242C22×Dic3—21no—
243C22×Dic3—22no—
244#235—28no—
245#235—28no—
246#235—24no—
247#235—24no—
248#235C2×C3×C418yes—
249#235C2×C3×C418yescomplemented
250C2×D12—21no—
251C2×D12—22no—
252C2×D12C2×C3×C414yescomplemented
253C2×D12C23×C312yes—
254C2×C4×D3—28no—
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×Dic6—21no—
258C2×Dic6—22no—
259C2×Dic6C2×C3×C414yescomplemented
260C2×Dic6C23×C312yes—
261C3×M16—28no—
262C3×M16—28no—
263C3×M16—24no—
264C3×M16—24no—
265C3×M16—24no—
266C3×M16—24no—
267C2×C3×C8—28no—
268C2×C3×C8—24no—
269C2×C3×C8—24no—
270C2×C3×C8—24no—
271C2×C3×C8C22×D314yes—
272C2×C3×C8C23×C314yes—
273C3×C4⋊C4—34no—
274C3×C4⋊C4—62no—
275C3×C22⋊C4—62no—
276C3×C22⋊C4—62no—
277C3×C42—62no—
278C3×C42—62no—
279#217—22no—
280#217—22no—
281C4⋊Dic3—22no—
282C4⋊Dic3C23×C314yes—
283C4×Dic3—22no—
284C4×Dic3—22no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32—22no—
288C22×C32—44no—
289C22×C32C2×C22⋊C412yes—
290C22×C32C2×C22⋊C412yes—
291C22×C32C2×C22⋊C412yes—
292C22×C32C22×D411yescharacteristic
293C2×C3×D3—24no—
294C2×C3×D3—48no—
295C2×C3×D3—44no—
296C32×C4C2×C22⋊C412yes—
297C32×C4C2×C22⋊C412yes—
298C32×C4C2×C22⋊C412yes—
299C32×C4C23×C411yescharacteristic
300C32×C4C22×D411yescharacteristic
301C3×Dic3—24no—
302C3×Dic3—44no—
303C2×#41—38no—
304C2×#41—32no—
305C2×#41—32no—
306C2×#41—61no—
307C2×#41—61no—
308C2×#41—64no—
309C22×Q8—31no—
310C22×D4—31no—
311C22×D4—61no—
312C23×C4—61no—
313C23×C4C2×C3×D311yescharacteristic
314C2×M16—24no—
315C2×M16C2×C3×D312yes—
316C2×M16C2×C3×D312yescomplemented
317C22×C8C2×C3×D312yescomplemented
318#117—32no—
319#117—32no—
320#117—61no—
321#117—61no—
322C2×C4⋊C4—32no—
323C2×C22⋊C4—61no—
324C2×C42—61no—
325#103—38no—
326#102—38no—
327C23×C3—24no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—44no—
334C23×C3—64no—
335C23×C3—62no—
336C23×C3—124no—
337C23×C3(C2×D3)⋊C414yes—
338C23×C3C3×C22⋊C414yes—
339C23×C3C2×D1211yescharacteristic
340C23×C3C2×D1211yescharacteristic
341C23×C3C2×D1211yescharacteristic
342C23×C3C2×C3×D411yescharacteristic
343C23×C3C2×C3×D411yescharacteristic
344C23×C3C2×C3×D411yescharacteristic
345C22×D3—24no—
346C22×D3—22no—
347C22×D3—44no—
348C3×Q8—34no—
349C3×Q8—64no—
350C3×Q8—62no—
351C3×D4—34no—
352C3×D4—68no—
353C3×D4—68no—
354C3×D4—64no—
355C3×D4—62no—
356C2×C3×C4—21no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—64no—
370C2×C3×C4—62no—
371C2×C3×C4—68no—
372C2×C3×C4—68no—
373C2×C3×C4—64no—
374C2×C3×C4—124no—
375C2×C3×C4(C2×D3)⋊C414yes—
376C2×C3×C4(C2×D3)⋊C414yes—
377C2×C3×C4(C2×D3)⋊C414yes—
378C2×C3×C4C3×C22⋊C414yes—
379C2×C3×C4C3×C22⋊C414yes—
380C2×C3×C4C3×C22⋊C414yes—
381C2×C3×C4C2×C4×D312yes—
382C2×C3×C4C2×C4×D312yes—
383C2×C3×C4C2×C4×D312yes—
384C2×C3×C4C2×D1211yescharacteristic
385C2×C3×C4C2×D1211yescharacteristic
386C2×C3×C4C2×D1211yescharacteristic
387C2×C3×C4C2×#6712yes—
388C2×C3×C4C2×#6712yes—
389C2×C3×C4C2×#6712yes—
390C2×C3×C4C22×C3×C412yes—
391C2×C3×C4C22×C3×C412yes—
392C2×C3×C4C22×C3×C412yes—
393C2×C3×C4C2×C3×D411yescharacteristic
394C2×C3×C4C2×C3×D411yescharacteristic
395C2×C3×C4C2×C3×D411yescharacteristic
396C2×C3×C4C2×C3×D412yes—
397C2×C3×C4C2×C3×D412yes—
398C2×C3×C4C2×C3×D412yes—
399C2×C3×C4C23×D311yescharacteristic
400C2×C3×C4C24×C311yescharacteristic
401#67—28no—
402#67—28no—
403C2×Dic3—24no—
404C2×Dic3—22no—
405C2×Dic3—24no—
406C2×Dic3—44no—
407D12—24no—
408D12—22no—
409D12C22×C3×C414yes—
410C4×D3—28no—
411C4×D3—28no—
412Dic6—24no—
413Dic6—22no—
414Dic6C22×C3×C414yes—
415C3×C8—28no—
416C3×C8—24no—
417C3×C8—24no—
418C2×C32—22no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32C22×C22⋊C411yescharacteristic
422C2×C32#57512yes—
423C3×D3—44no—
424C24—61no—
425C24C3×D1211yescharacteristic
426#41—38no—
427#41—38no—
428#41—68no—
429#41—68no—
430#41—64no—
431#41—64no—
432C2×Q8—32no—
433C2×Q8—34no—
434C2×Q8—61no—
435C2×Q8—62no—
436C2×D4—32no—
437C2×D4—34no—
438C2×D4—68no—
439C2×D4—62no—
440C2×D4—62no—
441C2×D4—61no—
442C2×D4—62no—
443C22×C4—61no—
444C22×C4—68no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4C3×C4×D314yes—
449C22×C4C3×D1211yescharacteristic
450C22×C4C3×D1211yescharacteristic
451C22×C4C3×D1211yescharacteristic
452C22×C4C3×#6714yes—
453C22×C4C22×C3×D311yescharacteristic
454C22×C4C22×C3×D311yescharacteristic
455C22×C4C22×C3×D311yescharacteristic
456M16—24no—
457M16—24no—
458C2×C8—24no—
459C2×C8C22×C3×D314yes—
460C4⋊C4—34no—
461C4⋊C4—62no—
462C22⋊C4—62no—
463C22⋊C4—62no—
464C42—62no—
465C42—62no—
466C22×C3—24no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—48no—
482C22×C3—64no—
483C22×C3—128no—
484C22×C3—124no—
485C22×C3C2×(C2×D3)⋊C412yes—
486C22×C3C2×(C2×D3)⋊C412yes—
487C22×C3C2×(C2×D3)⋊C412yes—
488C22×C3C2×C3×C22⋊C412yes—
489C22×C3C2×C3×C22⋊C412yes—
490C22×C3C2×C3×C22⋊C412yes—
491C22×C3C22×D1211yescharacteristic
492C22×C3C22×C3×D411yescharacteristic
493C2×D3—24no—
494C2×D3—48no—
495C2×D3—44no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—64no—
502C3×C4—124no—
503C3×C4C2×(C2×D3)⋊C412yes—
504C3×C4C2×(C2×D3)⋊C412yes—
505C3×C4C2×(C2×D3)⋊C412yes—
506C3×C4C2×C3×C22⋊C412yes—
507C3×C4C2×C3×C22⋊C412yes—
508C3×C4C2×C3×C22⋊C412yes—
509C3×C4C22×C4×D311yescharacteristic
510C3×C4C22×#6711yescharacteristic
511C3×C4C23×C3×C411yescharacteristic
512C3×C4C22×C3×D411yescharacteristic
513Dic3—24no—
514Dic3—44no—
515C32#289611yescharacteristic, complemented, Sylow
516C23—42no—
517C23—64no—
518C23—62no—
519C23—124no—
520C23C3×(C2×D3)⋊C414yes—
521C23C2×C3×D1211yescharacteristic
522C23C2×C3×D1211yescharacteristic
523C23C2×C3×D1211yescharacteristic
524Q8—34no—
525Q8—64no—
526Q8—62no—
527D4—34no—
528D4—68no—
529D4—68no—
530D4—64no—
531D4—62no—
532C2×C4—64no—
533C2×C4—62no—
534C2×C4—68no—
535C2×C4—68no—
536C2×C4—64no—
537C2×C4—124no—
538C2×C4C3×(C2×D3)⋊C414yes—
539C2×C4C3×(C2×D3)⋊C414yes—
540C2×C4C3×(C2×D3)⋊C414yes—
541C2×C4C2×C3×C4×D312yes—
542C2×C4C2×C3×C4×D312yes—
543C2×C4C2×C3×C4×D312yes—
544C2×C4C2×C3×D1211yescharacteristic
545C2×C4C2×C3×D1211yescharacteristic
546C2×C4C2×C3×D1211yescharacteristic
547C2×C4C2×C3×#6712yes—
548C2×C4C2×C3×#6712yes—
549C2×C4C2×C3×#6712yes—
550C2×C4C23×C3×D311yescharacteristic
551C8—24no—
552C2×C3—24no—
553C2×C3—22no—
554C2×C3—22no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—22no—
558C2×C3—22no—
559C2×C3—22no—
560C2×C3—21no—
561C2×C3—42no—
562C2×C3—42no—
563C2×C3—124no—
564C2×C3#574812yes—
565C2×C3C22×(C2×D3)⋊C411yescharacteristic
566C2×C3C22×C3×C22⋊C411yescharacteristic
567C2×C3C3×#57512yes—
568D3—44no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—64no—
574C22—128no—
575C22—124no—
576C22C2×C3×(C2×D3)⋊C412yes—
577C22C2×C3×(C2×D3)⋊C412yes—
578C22C2×C3×(C2×D3)⋊C412yes—
579C22C22×C3×D1211yescharacteristic
580C4—64no—
581C4—124no—
582C4C2×C3×(C2×D3)⋊C412yes—
583C4C2×C3×(C2×D3)⋊C412yes—
584C4C2×C3×(C2×D3)⋊C412yes—
585C4C22×C3×C4×D311yescharacteristic
586C4C22×C3×#6711yescharacteristic
587C3—21no—
588C3#8376811yescharacteristic, direct product factor
589C3C3×#289611yescharacteristic, complemented
590C2—22no—
591C2—22no—
592C2—22no—
593C2—124no—
594C2C3×#574812yes—
595C2C22×C3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D12
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC22×C32×M16C2
socleC22×C32C22×D4
prime radicalC22×C3×#235C2
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×M16D3
hypocenterC3C3×#2896
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C22×C3×M16

Complements

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

NH ≅ G/Ntype
2. C22×C32×M16C2semidirect
3. C2×C3×#926C2semidirect
4. C3×#5133C2semidirect
5. C3×#5131C2semidirect
6. C3×#5120C2semidirect
7. C3×#5113C2semidirect
9. #83768C3direct (N × H)
15. C2×C3×#235C4semidirect
23. C2×C32×M16C22semidirect
24. C22×C32×C8C22semidirect
29. C3×#926C22semidirect
30. C2×C3×C4⋊Dic3C22semidirect
32. C3×#821C22semidirect
33. C3×#816C22semidirect
37. C22×C3×M16D3semidirect
38. C22×C3×M16C2×C3semidirect
40. C2×#926C2×C3semidirect
45. #5133C2×C3semidirect
46. #5131C2×C3semidirect
47. #5120C2×C3semidirect
48. #5113C2×C3semidirect
69. C3×#235C2×C4semidirect
72. C2×C3×D12C2×C4semidirect
79. C2×C3×Dic6C2×C4semidirect
109. C2×#235C3×C4semidirect
121. C2×C3×M16C2×D3semidirect
123. C2×C3×M16C22×C3semidirect
125. C22×C3×C8C2×D3semidirect
126. C22×C3×C8C22×C3semidirect
138. #926C22×C3semidirect
139. C2×C4⋊Dic3C22×C3semidirect
143. #821C22×C3semidirect
144. #816C22×C3semidirect
182. C22×M16C3×D3semidirect
249. #235C2×C3×C4semidirect
252. C2×D12C2×C3×C4semidirect
259. C2×Dic6C2×C3×C4semidirect
316. C2×M16C2×C3×D3semidirect
317. C22×C8C2×C3×D3semidirect
515. C32#2896semidirect
588. C3#83768direct (N × H)
589. C3C3×#2896semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C22×C32×M162
C2×C3×#9262
C3×#51332
C3×#51312
C3×#51202
C3×#51132
C3×#28963
#837683