Finite Group Databasebeta

← C3×#74685 #11807925

Subgroups — C3×#74685 #11807925

Order 1152.

1439 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#940C211yesmaximal, characteristic
2C32×#386C211yesmaximal, characteristic, complemented
3C3×#5220C211yesmaximal, characteristic, complemented
4C3×#5213C212yesmaximal, complemented
5C3×#4958C212yesmaximal, complemented
6C3×#4956C212yesmaximal, complemented
7C3×#4956C212yesmaximal, complemented
8C3×#4665C212yesmaximal, complemented
9C3×#4651C212yesmaximal, complemented
10C3×#2380—31nomaximal
11#74685C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C23×C32×C4C2211yescharacteristic
14C22×C3×#67C2212yes—
15C22×C3×#67C2212yes—
16C23×C3×Dic3C2211yescharacteristic
17C22×C3×D12C2211yescharacteristic
18C22×C3×C4×D3C2212yes—
19C2×C32×C22⋊C4C2211yescharacteristic
20C2×C32×C22⋊C4C2212yes—
21C2×C3×#217C2211yescharacteristic
22C2×C3×#217C2212yes—
23C3×#940C418yescomplemented
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×C4×Dic3C2211yescharacteristic
28C32×#95C2212yes—
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C2×C3×#127—31no—
35C2×#940C2×C311yescharacteristic
36C3×#386—21no—
37C3×#386—31no—
38C3×#386—32no—
39C3×#386—32no—
40C3×#386—32no—
41C3×#386—32no—
42C3×#386—32no—
43C3×#386D311yescharacteristic, complemented
44C3×#386C2×C311yescharacteristic, complemented
45C3×#383—32no—
46#5220C2×C311yescharacteristic, complemented
47#5213C2×C312yescomplemented
48#4958C2×C312yescomplemented
49#4956C2×C312yescomplemented
50#4956C2×C312yescomplemented
51#4665C2×C312yescomplemented
52#4651C2×C312yescomplemented
53C24×C32C2311yescharacteristic
54C24×C32C2311yescharacteristic
55C23×C3×D3C2312yes—
56C2×C32×D4—41no—
57C2×C32×D4—41no—
58C2×C32×D4C2×C414yescomplemented
59C22×C32×C4—42no—
60C22×C32×C4—41no—
61C22×C32×C4C2312yes—
62C22×C32×C4C2312yes—
63C22×C32×C4C2311yescharacteristic
64C2×C3×#67—42no—
65C2×C3×#67—42no—
66C2×C3×#67—42no—
67C2×C3×#67—42no—
68C2×C3×#67C2×C418yescomplemented
69C2×C3×#67C2×C418yescomplemented
70C22×C3×Dic3—41no—
71C22×C3×Dic3—42no—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2312yes—
75C2×C3×D12—41no—
76C2×C3×D12—41no—
77C2×C3×D12C2×C414yescomplemented
78C2×C3×C4×D3—42no—
79C2×C3×C4×D3—42no—
80C2×C3×C4×D3—42no—
81C32×C22⋊C4—42no—
82C32×C22⋊C4—41no—
83C3×#217—41no—
84C3×#217—42no—
85C3×(C2×D3)⋊C4—42no—
86C3×(C2×D3)⋊C4—42no—
87C3×(C2×D3)⋊C4—42no—
88C3×C4×Dic3C2×C414yescomplemented
89#2380—31noSylow
90C22×C3×D4—21no—
91C22×C3×D4—32no—
92C22×C3×D4—32no—
93C22×C3×D4—31no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C23×C3×C4—21no—
97C23×C3×C4—31no—
98C23×C3×C4—32no—
99C23×C3×C4C2×D311yescharacteristic
100C23×C3×C4C22×C311yescharacteristic
101C22×#67C22×C312yes—
102C22×#67C22×C312yes—
103C23×Dic3C22×C311yescharacteristic
104C22×D12C22×C311yescharacteristic
105C22×C4×D3C22×C312yes—
106C3×#127—38no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—32no—
110C2×C3×C22⋊C4—31no—
111C2×C3×C22⋊C4—32no—
112C2×C3×C22⋊C4—32no—
113C2×C3×C22⋊C4—32no—
114C2×C3×C22⋊C4C2×D312yes—
115C2×C3×C22⋊C4C2×D311yescharacteristic
116C2×C3×C22⋊C4C22×C311yescharacteristic
117C2×C3×C22⋊C4C22×C312yes—
118C2×C3×C42—31no—
119C2×#217C22×C311yescharacteristic
120C2×#217C22×C312yes—
121#940C3×C418yescomplemented
122C2×(C2×D3)⋊C4C22×C312yes—
123C2×(C2×D3)⋊C4C22×C312yes—
124C2×(C2×D3)⋊C4C22×C312yes—
125C2×C4×Dic3C22×C311yescharacteristic
126C3×#95—22no—
127C3×#95—32no—
128C3×#95—32no—
129C3×#95—32no—
130C3×#95—32no—
131C3×#95—32no—
132C3×#95C2×D312yes—
133C3×#95C22×C312yes—
134C3⋊#95C22×C312yes—
135C3⋊#95C22×C312yes—
136C3⋊#95C22×C312yes—
137C3⋊#95C22×C312yes—
138C3⋊#95C22×C312yes—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—22no—
142C23×C32—21no—
143C23×C32—22no—
144C23×C32—22no—
145C23×C32—21no—
146C23×C32—22no—
147C23×C32C22×C412yes—
148C23×C32C22×C412yes—
149C23×C32C2411yescharacteristic
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—22no—
156C22×C3×D3C22×C414yes—
157C32×D4—44no—
158C2×C32×C4—22no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—44no—
167C2×C32×C4C22×C412yes—
168C3×#67—48no—
169C3×#67—48no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—44no—
179C2×C3×Dic3C22×C414yes—
180C3×D12—44no—
181C3×C4×D3—48no—
182C2×#127—31no—
183#386—31no—
184#386—32no—
185#386—32no—
186#386—32no—
187#386—32no—
188#386—32no—
189#386C3×D311yescharacteristic, complemented
190#383—32no—
191C24×C3—21no—
192C24×C3—21no—
193C24×C3—32no—
194C24×C3C22×D311yescharacteristic
195C24×C3C22×D311yescharacteristic
196C24×C3C23×C311yescharacteristic
197C24×C3C23×C311yescharacteristic
198C23×D3C23×C312yes—
199C2×C3×D4—24no—
200C2×C3×D4—38no—
201C2×C3×D4—38no—
202C2×C3×D4—34no—
203C2×C3×D4—41no—
204C2×C3×D4—41no—
205C2×C3×D4—41no—
206C2×C3×D4—41no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4—121no—
210C2×C3×D4—121no—
211C2×C3×D4—122no—
212C2×C3×D4—122no—
213C2×C3×D4—122no—
214C2×C3×D4—122no—
215C2×C3×D4C4×D314yescomplemented
216C2×C3×D4C2×C3×C414yescomplemented
217C22×C3×C4—21no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—32no—
221C22×C3×C4—32no—
222C22×C3×C4—32no—
223C22×C3×C4—42no—
224C22×C3×C4—41no—
225C22×C3×C4—41no—
226C22×C3×C4—42no—
227C22×C3×C4—44no—
228C22×C3×C4—42no—
229C22×C3×C4—122no—
230C22×C3×C4—122no—
231C22×C3×C4—122no—
232C22×C3×C4—121no—
233C22×C3×C4—122no—
234C22×C3×C4C22×D312yes—
235C22×C3×C4C22×D312yes—
236C22×C3×C4C22×D311yescharacteristic
237C22×C3×C4C23×C312yes—
238C22×C3×C4C23×C312yes—
239C22×C3×C4C23×C311yescharacteristic
240C2×#67—42no—
241C2×#67—42no—
242C2×#67—42no—
243C2×#67—42no—
244C2×#67C2×C3×C418yescomplemented
245C2×#67C2×C3×C418yescomplemented
246C22×Dic3—41no—
247C22×Dic3—42no—
248C22×Dic3C23×C312yes—
249C22×Dic3C23×C312yes—
250C22×Dic3C23×C312yes—
251C2×D12—41no—
252C2×D12—41no—
253C2×D12C2×C3×C414yescomplemented
254C2×C4×D3—42no—
255C2×C4×D3—42no—
256C2×C4×D3—42no—
257C3×C22⋊C4—42no—
258C3×C22⋊C4—41no—
259C3×C22⋊C4—41no—
260C3×C22⋊C4—42no—
261C3×C22⋊C4—44no—
262C3×C22⋊C4—42no—
263C3×C22⋊C4—122no—
264C3×C22⋊C4—122no—
265C3×C22⋊C4—122no—
266C3×C22⋊C4—121no—
267C3×C22⋊C4—122no—
268C3×C42—34no—
269#217—41no—
270#217—42no—
271(C2×D3)⋊C4—42no—
272(C2×D3)⋊C4—42no—
273(C2×D3)⋊C4—42no—
274C4×Dic3C2×C3×C414yescomplemented
275C22×C32—22no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—22no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—42no—
282C22×C32—41no—
283C22×C32—41no—
284C22×C32—42no—
285C22×C32C23×C411yescharacteristic
286C22×C32H(5,𝔽2)12yes—
287C22×C32H(5,𝔽2)11yescharacteristic
288C22×C32H(5,𝔽2)11yescharacteristic
289C22×C32H(5,𝔽2)12yes—
290C2×C3×D3—24no—
291C2×C3×D3—24no—
292C2×C3×D3—24no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C32×C4—22no—
298C32×C4—42no—
299C32×C4—42no—
300C3×Dic3—24no—
301C3×Dic3—42no—
302C3×Dic3—42no—
303C22×D4—32no—
304C22×D4—32no—
305C22×D4—31no—
306C22×D4C2×C3×D311yescharacteristic
307C23×C4—31no—
308C23×C4—32no—
309C23×C4C2×C3×D311yescharacteristic
310#127—38no—
311C2×C22⋊C4—32no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—32no—
314C2×C22⋊C4—32no—
315C2×C22⋊C4—32no—
316C2×C22⋊C4C2×C3×D312yes—
317C2×C22⋊C4C2×C3×D311yescharacteristic
318C2×C42—31no—
319#95—32no—
320#95—32no—
321#95—32no—
322#95—32no—
323#95—32no—
324#95C2×C3×D312yes—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—21no—
339C23×C3—22no—
340C23×C3—21no—
341C23×C3—22no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—34no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—42no—
353C23×C3—62no—
354C23×C3—62no—
355C23×C3—62no—
356C23×C3—62no—
357C23×C3—62no—
358C23×C3—62no—
359C23×C3C2×C4×D312yes—
360C23×C3C2×C4×D312yes—
361C23×C3C22×C3×C412yes—
362C23×C3C22×C3×C412yes—
363C23×C3C23×D311yescharacteristic
364C23×C3C24×C311yescharacteristic
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—22no—
370C22×D3—22no—
371C22×D3C22×C3×C414yes—
372C3×D4—44no—
373C3×D4—44no—
374C3×D4—48no—
375C3×D4—124no—
376C3×D4—128no—
377C3×D4—128no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—34no—
397C2×C3×C4—42no—
398C2×C3×C4—42no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—42no—
402C2×C3×C4—42no—
403C2×C3×C4—42no—
404C2×C3×C4—48no—
405C2×C3×C4—44no—
406C2×C3×C4—44no—
407C2×C3×C4—62no—
408C2×C3×C4—62no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—62no—
412C2×C3×C4—62no—
413C2×C3×C4—62no—
414C2×C3×C4—62no—
415C2×C3×C4—128no—
416C2×C3×C4—124no—
417C2×C3×C4C2×C4×D312yes—
418C2×C3×C4C22×C3×C412yes—
419#67—48no—
420#67—48no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—44no—
430C2×Dic3C22×C3×C414yes—
431D12—44no—
432C4×D3—48no—
433C2×C32—42no—
434C2×C32—42no—
435C2×C32#51811yescharacteristic
436C2×C32#51812yes—
437C2×C32#56011yescharacteristic
438C2×C32#56011yescharacteristic
439C2×C32#56012yes—
440C3×D3—44no—
441C24—32no—
442C24C22×C3×D311yescharacteristic
443C24C22×C3×D311yescharacteristic
444C2×D4—38no—
445C2×D4—38no—
446C2×D4—34no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—121no—
450C2×D4—121no—
451C2×D4—122no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4C3×C4×D314yescomplemented
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—32no—
459C22×C4—42no—
460C22×C4—41no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—121no—
465C22×C4—122no—
466C22×C4C22×C3×D312yes—
467C22×C4C22×C3×D312yes—
468C22×C4C22×C3×D311yescharacteristic
469C22⋊C4—42no—
470C22⋊C4—41no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—122no—
474C22⋊C4—121no—
475C22⋊C4—122no—
476C42—34no—
477C22×C3—24no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—24no—
481C22×C3—24no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—22no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—44no—
492C22×C3—42no—
493C22×C3—44no—
494C22×C3—42no—
495C22×C3—44no—
496C22×C3—42no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—64no—
510C22×C3—64no—
511C22×C3—64no—
512C22×C3—81no—
513C22×C3—82no—
514C22×C3—81no—
515C22×C3—122no—
516C22×C3—122no—
517C22×C3—122no—
518C22×C3—122no—
519C22×C3C22×C4×D311yescharacteristic
520C22×C3#100411yescharacteristic
521C22×C3#100411yescharacteristic
522C22×C3#100412yes—
523C22×C3#100912yes—
524C22×C3C23×C3×C411yescharacteristic
525C22×C3C3×H(5,𝔽2)12yes—
526C22×C3C3×H(5,𝔽2)11yescharacteristic
527C22×C3C3×H(5,𝔽2)11yescharacteristic
528C22×C3C3×H(5,𝔽2)12yes—
529C2×D3—24no—
530C2×D3—24no—
531C2×D3—24no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C3×C4—22no—
537C3×C4—22no—
538C3×C4—42no—
539C3×C4—44no—
540C3×C4—42no—
541C3×C4—42no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—64no—
545C3×C4—82no—
546C3×C4—122no—
547C3×C4—122no—
548Dic3—24no—
549Dic3—42no—
550Dic3—42no—
551C32#238011yescharacteristic, complemented, Sylow
552C23—21no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—34no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—62no—
565C23—62no—
566C23—62no—
567C23C2×C3×C4×D312yes—
568C23C2×C3×C4×D312yes—
569C23C23×C3×D311yescharacteristic
570D4—44no—
571D4—124no—
572D4—128no—
573D4—128no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—34no—
583C2×C4—44no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—128no—
593C2×C4—124no—
594C2×C4C2×C3×C4×D312yes—
595C2×C3—22no—
596C2×C3—21no—
597C2×C3—22no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—44no—
601C2×C3—42no—
602C2×C3—42no—
603C2×C3—42no—
604C2×C3—42no—
605C2×C3—82no—
606C2×C3—124no—
607C2×C3#548611yescharacteristic
608C2×C3#554512yes—
609C2×C3#559511yescharacteristic
610C2×C3#562012yes—
611C2×C3#566611yescharacteristic
612C2×C3C3×#51812yes—
613C2×C3C3×#51811yescharacteristic
614C2×C3C3×#56011yescharacteristic
615C2×C3C3×#56012yes—
616C2×C3C3×#56011yescharacteristic
617D3—44no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—41no—
627C22—41no—
628C22—64no—
629C22—64no—
630C22—64no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22C22×C3×C4×D311yescharacteristic
636C22C3×#100412yes—
637C22C3×#100411yescharacteristic
638C22C3×#100411yescharacteristic
639C22C3×#100912yes—
640C4—22no—
641C4—42no—
642C4—42no—
643C4—64no—
644C4—122no—
645C4—122no—
646C3—21no—
647C3#7468511yescharacteristic, direct product factor
648C3C3×#238011yescharacteristic, complemented
649C2—42no—
650C2—42no—
651C2—124no—
652C2C3×#548611yescharacteristic
653C2C3×#554512yes—
654C2C3×#559511yescharacteristic
655C2C3×#562012yes—
656C2C3×#566611yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#386C2
socleC23×C32C24
prime radicalC2×C3×#940C2
cosocleC23×C3C24×C3
hypercenterC3×#386D3
hypocenterC3C3×#2380
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#386

Complements

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

NH ≅ G/Ntype
2. C32×#386C2semidirect
3. C3×#5220C2semidirect
4. C3×#5213C2semidirect
5. C3×#4958C2semidirect
6. C3×#4956C2semidirect
7. C3×#4956C2semidirect
8. C3×#4665C2semidirect
9. C3×#4651C2semidirect
11. #74685C3direct (N × H)
23. C3×#940C4semidirect
43. C3×#386D3semidirect
44. C3×#386C2×C3semidirect
46. #5220C2×C3semidirect
47. #5213C2×C3semidirect
48. #4958C2×C3semidirect
49. #4956C2×C3semidirect
50. #4956C2×C3semidirect
51. #4665C2×C3semidirect
52. #4651C2×C3semidirect
58. C2×C32×D4C2×C4semidirect
68. C2×C3×#67C2×C4semidirect
69. C2×C3×#67C2×C4semidirect
77. C2×C3×D12C2×C4semidirect
88. C3×C4×Dic3C2×C4semidirect
121. #940C3×C4semidirect
189. #386C3×D3semidirect
215. C2×C3×D4C4×D3semidirect
216. C2×C3×D4C2×C3×C4semidirect
244. C2×#67C2×C3×C4semidirect
245. C2×#67C2×C3×C4semidirect
253. C2×D12C2×C3×C4semidirect
274. C4×Dic3C2×C3×C4semidirect
455. C2×D4C3×C4×D3semidirect
551. C32#2380semidirect
647. C3#74685direct (N × H)
648. C3C3×#2380semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9402
C32×#3862
C3×#52202
C3×#52132
C3×#49582
C3×#49562
C3×#46652
C3×#46512
C3×#23803
#746853