Finite Group Databasebeta

← C3×#73417 #11805745

Subgroups — C3×#73417 #11805745

Order 1152.

1265 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C32×#376C211yesmaximal, characteristic, complemented
3C3×#4940C212yesmaximal, complemented
4C3×C4×(C2×D3)⋊C4C212yesmaximal, complemented
5C3×#4937C211yesmaximal, characteristic, complemented
6C3×#4667C211yesmaximal, characteristic, complemented
7C3×#4666C211yesmaximal, characteristic, complemented
8C3×#4665C212yesmaximal
9C3×#4664C211yesmaximal, characteristic, complemented
10C3×D3×#95C211yesmaximal, characteristic, complemented
11C3×#4654C212yesmaximal, complemented
12C3×#2353—31nomaximal
13#73417C311yesmaximal, characteristic, direct product factor
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3C2212yes—
16C22×C3×C4×D3C2211yescharacteristic
17C22×C3×C4×D3C2211yescharacteristic
18C2×C32×C42C2212yes—
19C2×C32×C42C2211yescharacteristic
20C2×C3×(C2×D3)⋊C4C2212yes—
21C2×C3×(C2×D3)⋊C4C2212yes—
22C2×C3×(C2×D3)⋊C4C2212yes—
23C2×C3×(C2×D3)⋊C4C2211yescharacteristic
24C2×C3×(C2×D3)⋊C4C2211yescharacteristic
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×Dic3⋊C4C2212yes—
27C2×C3×C4×Dic3C2212yes—
28C3×C4×D12C418yescomplemented
29C32×#95C2212yes—
30C32×#95C2211yescharacteristic
31C32×#95C2211yescharacteristic
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2211yescharacteristic
36C3×C3⋊#95C2211yescharacteristic
37C3×C3⋊#95C2212yes—
38C2×C3×C4×D4—31no—
39C2×C4×D12C2×C311yescharacteristic
40C3×#388—32no—
41C3×#387—32no—
42C3×#386—32no—
43C3×#383—31no—
44C3×#381—31no—
45C3×#380—31no—
46C3×C4×C22⋊C4—32no—
47C3×C4×C22⋊C4—31no—
48C3×#376—21no—
49C3×#376D311yescharacteristic, complemented
50C3×#376C2×C311yescharacteristic, complemented
51C2×C3×#95—31no—
52#4940C2×C312yescomplemented
53C4×(C2×D3)⋊C4C2×C312yescomplemented
54#4937C2×C311yescharacteristic, complemented
55#4667C2×C311yescharacteristic, complemented
56#4666C2×C311yescharacteristic, complemented
57#4665C2×C312yes—
58#4664C2×C311yescharacteristic, complemented
59D3×#95C2×C311yescharacteristic, complemented
60#4654C2×C312yescomplemented
61C23×C3×D3C2311yescharacteristic
62C23×C3×D3C2311yescharacteristic
63C22×C32×C4C2312yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2312yes—
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C2×C3×D12—22no—
73C2×C3×D12—22no—
74C2×C3×D12C2×C414yes—
75C2×C3×C4×D3—24no—
76C2×C3×C4×D3—24no—
77C2×C3×C4×D3—42no—
78C2×C3×C4×D3—42no—
79C2×C3×C4×D3—42no—
80C2×C3×C4×D3C2×C414yes—
81C2×C3×C4×D3C2×C414yescomplemented
82C32×C42—24no—
83C32×C42C2×C414yescomplemented
84C3×(C2×D3)⋊C4—24no—
85C3×(C2×D3)⋊C4—24no—
86C3×(C2×D3)⋊C4—42no—
87C3×(C2×D3)⋊C4C2×C414yes—
88C3×(C2×D3)⋊C4C2×C414yescomplemented
89C3×C4⋊Dic3C2×C414yescomplemented
90C3×Dic3⋊C4—42no—
91C3×C4×Dic3—42no—
92#2353—31noSylow
93C22×C3×D4—31no—
94C23×C3×C4—31no—
95C23×C3×C4—31no—
96C23×C3×C4—32no—
97C22×D12C22×C311yescharacteristic
98C22×C4×D3C22×C312yes—
99C22×C4×D3C22×C311yescharacteristic
100C22×C4×D3C22×C311yescharacteristic
101C3×C4×D4—38no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C4⋊C4—32no—
104C2×C3×C22⋊C4—31no—
105C2×C3×C22⋊C4—31no—
106C2×C3×C22⋊C4—32no—
107C2×C3×C22⋊C4—32no—
108C2×C3×C22⋊C4—32no—
109C2×C3×C42—22no—
110C2×C3×C42—21no—
111C2×C3×C42—32no—
112C2×C3×C42C2×D312yes—
113C2×C3×C42C2×D311yescharacteristic
114C2×C3×C42C22×C312yes—
115C2×C3×C42C22×C311yescharacteristic
116C2×(C2×D3)⋊C4C22×C312yes—
117C2×(C2×D3)⋊C4C22×C312yes—
118C2×(C2×D3)⋊C4C22×C312yes—
119C2×(C2×D3)⋊C4C22×C311yescharacteristic
120C2×(C2×D3)⋊C4C22×C311yescharacteristic
121C2×C4⋊Dic3C22×C311yescharacteristic
122C2×Dic3⋊C4C22×C312yes—
123C2×C4×Dic3C22×C312yes—
124C4×D12C3×C418yescomplemented
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—22no—
128C3×#95—31no—
129C3×#95—32no—
130C3×#95—31no—
131C3×#95—32no—
132C3×#95—32no—
133C3×#95—32no—
134C3×#95C2×D311yescharacteristic
135C3×#95C2×D311yescharacteristic
136C3×#95C2×D312yes—
137C3×#95C22×C312yes—
138C3×#95C22×C311yescharacteristic
139C3×#95C22×C311yescharacteristic
140C3⋊#95C22×C312yes—
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C312yes—
143C3⋊#95C22×C311yescharacteristic
144C3⋊#95C22×C311yescharacteristic
145C3⋊#95C22×C312yes—
146C23×C32C2411yescharacteristic
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—21no—
151C22×C3×D3—21no—
152C22×C3×D3—22no—
153C22×C3×D3C22×C412yes—
154C22×C3×D3C22×C412yes—
155C22×C3×D3#4112yes—
156C22×C3×D3#4112yes—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4C22×C412yes—
167C2×C32×C4C22×C412yes—
168C2×C32×C4C22×C412yes—
169C2×C32×C4#4114yes—
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×Dic3C22×C412yes—
179C2×C3×Dic3C22×C412yes—
180C3×D12—28no—
181C3×C4×D3—44no—
182C3×C4×D3—44no—
183C3×C4×D3—48no—
184C2×C4×D4—31no—
185#388—32no—
186#387—32no—
187#386—32no—
188#383—31no—
189#381—31no—
190#380—31no—
191C4×C22⋊C4—32no—
192C4×C22⋊C4—31no—
193#376C3×D311yescharacteristic, complemented
194C2×#95—31no—
195C24×C3—31no—
196C24×C3—31no—
197C23×D3C23×C311yescharacteristic
198C23×D3C23×C311yescharacteristic
199C2×C3×D4—34no—
200C2×C3×D4—62no—
201C2×C3×D4—62no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—34no—
210C22×C3×C4—34no—
211C22×C3×C4—32no—
212C22×C3×C4—32no—
213C22×C3×C4—64no—
214C22×C3×C4—64no—
215C22×C3×C4—122no—
216C22×C3×C4—122no—
217C22×C3×C4—122no—
218C22×C3×C4C22×D312yes—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C22×D312yes—
223C22×C3×C4C23×C312yes—
224C22×C3×C4C23×C311yescharacteristic
225C22×C3×C4C23×C311yescharacteristic
226C22×C3×C4C23×C311yescharacteristic
227C22×C3×C4C23×C312yes—
228C22×Dic3C23×C312yes—
229C22×Dic3C23×C312yes—
230C22×Dic3C23×C311yescharacteristic
231C22×Dic3C23×C311yescharacteristic
232C2×D12—22no—
233C2×D12—22no—
234C2×D12C2×C3×C414yes—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3C2×C3×C414yes—
241C2×C4×D3C2×C3×C414yescomplemented
242C3×C4⋊C4—34no—
243C3×C4⋊C4—122no—
244C3×C22⋊C4—34no—
245C3×C22⋊C4—34no—
246C3×C22⋊C4—64no—
247C3×C22⋊C4—64no—
248C3×C22⋊C4—122no—
249C3×C42—28no—
250C3×C42—24no—
251C3×C42—24no—
252C3×C42—24no—
253C3×C42—122no—
254C3×C42C4×D314yescomplemented
255C3×C42C2×C3×C414yescomplemented
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4C2×C3×C414yes—
260(C2×D3)⋊C4C2×C3×C414yescomplemented
261C4⋊Dic3C2×C3×C414yescomplemented
262Dic3⋊C4—42no—
263C4×Dic3—42no—
264C22×C32C23×C411yescharacteristic
265C22×C32C2×#4112yes—
266C22×C32C2×#4111yescharacteristic
267C22×C32C2×#4111yescharacteristic
268C22×C32H(5,𝔽2)12yes—
269C2×C3×D3—22no—
270C2×C3×D3—22no—
271C2×C3×D3—24no—
272C2×C3×D3—24no—
273C2×C3×D3—22no—
274C2×C3×D3—22no—
275C2×C3×D3—22no—
276C2×C3×D3—42no—
277C2×C3×D3—42no—
278C2×C3×D3#11714yes—
279C32×C4—22no—
280C32×C4—22no—
281C32×C4—22no—
282C32×C4—24no—
283C32×C4—42no—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—42no—
287C3×Dic3—42no—
288C22×D4—31no—
289C23×C4—31no—
290C23×C4—31no—
291C23×C4—32no—
292C4×D4—38no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—32no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—32no—
298C2×C22⋊C4—32no—
299C2×C22⋊C4—32no—
300C2×C42—32no—
301C2×C42C2×C3×D312yes—
302C2×C42C2×C3×D311yescharacteristic
303#95—31no—
304#95—32no—
305#95—31no—
306#95—32no—
307#95—32no—
308#95—32no—
309#95C2×C3×D311yescharacteristic
310#95C2×C3×D311yescharacteristic
311#95C2×C3×D312yes—
312C23×C3—21no—
313C23×C3—32no—
314C23×C3—32no—
315C23×C3—32no—
316C23×C3—32no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—62no—
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—62no—
323C23×C3C23×D311yescharacteristic
324C23×C3C24×C311yescharacteristic
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—22no—
331C22×D3C22×C3×C412yes—
332C22×D3C22×C3×C412yes—
333C22×D3C3×#4112yes—
334C22×D3C3×#4112yes—
335C3×D4—68no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—32no—
361C2×C3×C4—32no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—124no—
378C2×C3×C4—124no—
379C2×C3×C4—128no—
380C2×C3×C4C2×C4×D312yes—
381C2×C3×C4C2×C4×D312yes—
382C2×C3×C4C2×C4×D312yes—
383C2×C3×C4#23514yes—
384C2×C3×C4C22×C3×C412yes—
385C2×C3×C4C22×C3×C412yes—
386C2×C3×C4C22×C3×C412yes—
387C2×C3×C4C3×#4114yes—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3C22×C3×C412yes—
397C2×Dic3C22×C3×C412yes—
398D12—28no—
399C4×D3—44no—
400C4×D3—44no—
401C4×D3—48no—
402C2×C32C2×#11711yescharacteristic
403C2×C32C4×#4111yescharacteristic
404C2×C32#51811yescharacteristic
405C2×C32#55112yes—
406C2×C32#55312yes—
407C3×D3—24no—
408C3×D3—42no—
409C24—31no—
410C24—31no—
411C2×D4—34no—
412C2×D4—62no—
413C2×D4—62no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—34no—
417C22×C4—34no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—64no—
421C22×C4—64no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4C22×C3×D312yes—
426C22×C4C22×C3×D311yescharacteristic
427C22×C4C22×C3×D311yescharacteristic
428C22×C4C22×C3×D311yescharacteristic
429C22×C4C22×C3×D312yes—
430C4⋊C4—34no—
431C4⋊C4—122no—
432C22⋊C4—34no—
433C22⋊C4—34no—
434C22⋊C4—64no—
435C22⋊C4—64no—
436C22⋊C4—122no—
437C42—24no—
438C42—122no—
439C42C3×C4×D314yescomplemented
440C22×C3—21no—
441C22×C3—22no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—22no—
445C22×C3—34no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—64no—
449C22×C3—64no—
450C22×C3—62no—
451C22×C3—62no—
452C22×C3—62no—
453C22×C3—122no—
454C22×C3—122no—
455C22×C3C22×C4×D311yescharacteristic
456C22×C3C2×#23512yes—
457C22×C3D3×#4111yescharacteristic
458C22×C3D3×#4111yescharacteristic
459C22×C3#100912yes—
460C22×C3C23×C3×C411yescharacteristic
461C22×C3C2×C3×#4112yes—
462C22×C3C2×C3×#4111yescharacteristic
463C22×C3C2×C3×#4111yescharacteristic
464C22×C3C3×H(5,𝔽2)12yes—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—24no—
468C2×D3—24no—
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—22no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3C3×#11714yes—
475C3×C4—24no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—22no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—24no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—44no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—62no—
491C3×C4—62no—
492C3×C4—122no—
493C3×C4—122no—
494Dic3—22no—
495Dic3—22no—
496Dic3—42no—
497Dic3—42no—
498C32#235311yescharacteristic, complemented, Sylow
499C23—32no—
500C23—32no—
501C23—32no—
502C23—32no—
503C23—62no—
504C23—62no—
505C23—62no—
506C23—61no—
507C23—61no—
508C23—62no—
509C23C23×C3×D311yescharacteristic
510D4—68no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—32no—
521C2×C4—32no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—124no—
531C2×C4—124no—
532C2×C4—128no—
533C2×C4C2×C3×C4×D312yes—
534C2×C4C2×C3×C4×D312yes—
535C2×C4C2×C3×C4×D312yes—
536C2×C4C3×#23514yes—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—64no—
543C2×C3—122no—
544C2×C3C4×#23511yescharacteristic
545C2×C3D3×#11711yescharacteristic
546C2×C3#552111yescharacteristic
547C2×C3#552712yes—
548C2×C3#553012yes—
549C2×C3C2×C3×#11711yescharacteristic
550C2×C3C3×C4×#4111yescharacteristic
551C2×C3C3×#51811yescharacteristic
552C2×C3C3×#55112yes—
553C2×C3C3×#55312yes—
554D3—24no—
555D3—42no—
556C22—34no—
557C22—62no—
558C22—64no—
559C22—64no—
560C22—62no—
561C22—62no—
562C22—62no—
563C22—62no—
564C22—122no—
565C22—122no—
566C22C22×C3×C4×D311yescharacteristic
567C22C2×C3×#23512yes—
568C22C3×D3×#4111yescharacteristic
569C22C3×D3×#4111yescharacteristic
570C22C3×#100912yes—
571C4—22no—
572C4—22no—
573C4—24no—
574C4—22no—
575C4—42no—
576C4—62no—
577C4—62no—
578C4—122no—
579C4—122no—
580C3—21no—
581C3#7341711yescharacteristic, direct product factor
582C3C3×#235311yescharacteristic, complemented
583C2—64no—
584C2—122no—
585C2C3×C4×#23511yescharacteristic
586C2C3×D3×#11711yescharacteristic
587C2C3×#552111yescharacteristic
588C2C3×#552712yes—
589C2C3×#553012yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#376C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×#376D3
hypocenterC3C3×#2353
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#376

Complements

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

NH ≅ G/Ntype
2. C32×#376C2semidirect
3. C3×#4940C2semidirect
4. C3×C4×(C2×D3)⋊C4C2semidirect
5. C3×#4937C2semidirect
6. C3×#4667C2semidirect
7. C3×#4666C2semidirect
9. C3×#4664C2semidirect
10. C3×D3×#95C2semidirect
11. C3×#4654C2semidirect
13. #73417C3direct (N × H)
28. C3×C4×D12C4semidirect
49. C3×#376D3semidirect
50. C3×#376C2×C3semidirect
52. #4940C2×C3semidirect
53. C4×(C2×D3)⋊C4C2×C3semidirect
54. #4937C2×C3semidirect
55. #4667C2×C3semidirect
56. #4666C2×C3semidirect
58. #4664C2×C3semidirect
59. D3×#95C2×C3semidirect
60. #4654C2×C3semidirect
81. C2×C3×C4×D3C2×C4semidirect
83. C32×C42C2×C4semidirect
88. C3×(C2×D3)⋊C4C2×C4semidirect
89. C3×C4⋊Dic3C2×C4semidirect
124. C4×D12C3×C4semidirect
193. #376C3×D3semidirect
241. C2×C4×D3C2×C3×C4semidirect
254. C3×C42C4×D3semidirect
255. C3×C42C2×C3×C4semidirect
260. (C2×D3)⋊C4C2×C3×C4semidirect
261. C4⋊Dic3C2×C3×C4semidirect
439. C42C3×C4×D3semidirect
498. C32#2353semidirect
581. C3#73417direct (N × H)
582. C3C3×#2353semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D122
C32×#3762
C3×#49402
C3×C4×(C2×D3)⋊C42
C3×#49372
C3×#46672
C3×#46662
C3×#46652
C3×#46642
C3×D3×#952
C3×#46542
C3×#23533
#734173