Finite Group Databasebeta

← C3×#73576 #11805751

Subgroups — C3×#73576 #11805751

Order 1152.

1305 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#67C211yesmaximal, characteristic
2C32×C4×C22⋊C4C211yesmaximal, characteristic, complemented
3C3×#5211C211yesmaximal, characteristic, complemented
4C3×#4956C212yesmaximal
5C3×#4944C212yesmaximal, complemented
6C3×C4×(C2×D3)⋊C4C212yesmaximal, complemented
7C3×#4933C212yesmaximal, complemented
8C3×#4667C211yesmaximal, characteristic, complemented
9C3×D3×#95C211yesmaximal, characteristic, complemented
10C3×#4651C211yesmaximal, characteristic, complemented
11C3×#4648C211yesmaximal, characteristic, complemented
12C3×#2353—31nomaximal
13#73576C311yesmaximal, characteristic, direct product factor
14C23×C32×C4C2211yescharacteristic
15C22×C3×#67C2211yescharacteristic
16C22×C3×C4×D3C2212yes—
17C22×C3×C4×D3C2211yescharacteristic
18C2×C32×C22⋊C4C2212yes—
19C2×C32×C42C2212yes—
20C2×C3×#217C2212yes—
21C2×C3×#217C2211yescharacteristic
22C3×C4×#67C418yescomplemented
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2211yescharacteristic
25C2×C3×Dic3⋊C4C2212yes—
26C2×C3×Dic3⋊C4C2211yescharacteristic
27C2×C3×C4×Dic3C2212yes—
28C2×C3×C4×Dic3C2211yescharacteristic
29C32×#95C2211yescharacteristic
30C32×#95C2211yescharacteristic
31C3×C3⋊#95C2211yescharacteristic
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2211yescharacteristic
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2212yes—
38C2×C3×C4×D4—31no—
39C2×C4×#67C2×C311yescharacteristic
40C3×#388—32no—
41C3×#387—32no—
42C3×#386—32no—
43C3×#383—31no—
44C3×#381—31no—
45C3×#380—31no—
46C3×C4×C22⋊C4—21no—
47C3×C4×C22⋊C4—32no—
48C3×C4×C22⋊C4D311yescharacteristic, complemented
49C3×C4×C22⋊C4C2×C311yescharacteristic, complemented
50C3×#376—31no—
51C2×C3×#95—31no—
52#5211C2×C311yescharacteristic, complemented
53#4956C2×C312yes—
54#4944C2×C312yescomplemented
55C4×(C2×D3)⋊C4C2×C312yescomplemented
56#4933C2×C312yescomplemented
57#4667C2×C311yescharacteristic, complemented
58D3×#95C2×C311yescharacteristic, complemented
59#4651C2×C311yescharacteristic, complemented
60#4648C2×C311yescharacteristic, complemented
61C24×C32C2311yescharacteristic
62C23×C3×D3C2311yescharacteristic
63C22×C32×C4—42no—
64C22×C32×C4C2×C414yescomplemented
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2312yes—
69C2×C3×#67—22no—
70C2×C3×#67—22no—
71C2×C3×#67C2×C414yes—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2311yescharacteristic
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2311yescharacteristic
77C2×C3×C4×D3—24no—
78C2×C3×C4×D3—24no—
79C2×C3×C4×D3—42no—
80C2×C3×C4×D3—42no—
81C2×C3×C4×D3C2×C414yes—
82C32×C22⋊C4—24no—
83C32×C42—42no—
84C3×#217—42no—
85C3×#217C2×C414yescomplemented
86C3×(C2×D3)⋊C4—24no—
87C3×(C2×D3)⋊C4C2×C414yes—
88C3×Dic3⋊C4—42no—
89C3×Dic3⋊C4C2×C414yescomplemented
90C3×C4×Dic3—24no—
91C3×C4×Dic3C2×C414yescomplemented
92#2353—31noSylow
93C22×C3×D4—31no—
94C23×C3×C4—21no—
95C23×C3×C4—31no—
96C23×C3×C4—32no—
97C23×C3×C4C2×D311yescharacteristic
98C23×C3×C4C22×C311yescharacteristic
99C22×#67C22×C311yescharacteristic
100C22×C4×D3C22×C312yes—
101C22×C4×D3C22×C311yescharacteristic
102C3×C4×D4—38no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—32no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—31no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—32no—
109C2×C3×C22⋊C4—32no—
110C2×C3×C22⋊C4C2×D312yes—
111C2×C3×C22⋊C4C22×C312yes—
112C2×C3×C42—22no—
113C2×C3×C42—31no—
114C2×C3×C42—32no—
115C2×C3×C42C2×D312yes—
116C2×C3×C42C22×C312yes—
117C2×#217C22×C312yes—
118C2×#217C22×C311yescharacteristic
119C4×#67C3×C418yescomplemented
120C2×(C2×D3)⋊C4C22×C312yes—
121C2×(C2×D3)⋊C4C22×C311yescharacteristic
122C2×Dic3⋊C4C22×C312yes—
123C2×Dic3⋊C4C22×C311yescharacteristic
124C2×C4×Dic3C22×C312yes—
125C2×C4×Dic3C22×C311yescharacteristic
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—31no—
129C3×#95—32no—
130C3×#95—31no—
131C3×#95—32no—
132C3×#95—32no—
133C3×#95—32no—
134C3×#95—32no—
135C3×#95C2×D311yescharacteristic
136C3×#95C2×D311yescharacteristic
137C3×#95C22×C311yescharacteristic
138C3×#95C22×C311yescharacteristic
139C3⋊#95C22×C311yescharacteristic
140C3⋊#95C22×C312yes—
141C3⋊#95C22×C311yescharacteristic
142C3⋊#95C22×C312yes—
143C3⋊#95C22×C312yes—
144C3⋊#95C22×C312yes—
145C3⋊#95C22×C312yes—
146C23×C32—22no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—22no—
150C23×C32C22×C412yes—
151C23×C32C2411yescharacteristic
152C22×C3×D3—22no—
153C22×C3×D3—22no—
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—22no—
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4—44no—
165C2×C32×C4C22×C412yes—
166C2×C32×C4C22×C412yes—
167C2×C32×C4#4114yes—
168C3×#67—28no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
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—22no—
179C2×C3×Dic3C22×C412yes—
180C2×C3×Dic3C22×C412yes—
181C2×C3×Dic3C22×C412yes—
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⋊C4C3×D311yescharacteristic, complemented
193#376—31no—
194C2×#95—31no—
195C24×C3—21no—
196C24×C3—31no—
197C24×C3C22×D311yescharacteristic
198C24×C3C23×C311yescharacteristic
199C23×D3C23×C311yescharacteristic
200C2×C3×D4—34no—
201C2×C3×D4—62no—
202C2×C3×D4—62no—
203C22×C3×C4—24no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—34no—
212C22×C3×C4—32no—
213C22×C3×C4—32no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4—44no—
217C22×C3×C4—64no—
218C22×C3×C4—64no—
219C22×C3×C4—122no—
220C22×C3×C4—122no—
221C22×C3×C4C4×D314yescomplemented
222C22×C3×C4C2×C3×C414yescomplemented
223C22×C3×C4C22×D312yes—
224C22×C3×C4C22×D311yescharacteristic
225C22×C3×C4C22×D311yescharacteristic
226C22×C3×C4C22×D312yes—
227C22×C3×C4C23×C312yes—
228C22×C3×C4C23×C311yescharacteristic
229C22×C3×C4C23×C311yescharacteristic
230C22×C3×C4C23×C312yes—
231C2×#67—22no—
232C2×#67—22no—
233C2×#67C2×C3×C414yes—
234C22×Dic3C23×C312yes—
235C22×Dic3C23×C312yes—
236C22×Dic3C23×C311yescharacteristic
237C22×Dic3C23×C311yescharacteristic
238C22×Dic3C23×C311yescharacteristic
239C2×C4×D3—24no—
240C2×C4×D3—24no—
241C2×C4×D3—42no—
242C2×C4×D3—42no—
243C2×C4×D3C2×C3×C414yes—
244C3×C4⋊C4—34no—
245C3×C4⋊C4—122no—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—34no—
250C3×C22⋊C4—34no—
251C3×C22⋊C4—64no—
252C3×C22⋊C4—122no—
253C3×C42—34no—
254C3×C42—42no—
255C3×C42—42no—
256C3×C42—44no—
257C3×C42—64no—
258#217—42no—
259#217C2×C3×C414yescomplemented
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4C2×C3×C414yes—
262Dic3⋊C4—42no—
263Dic3⋊C4C2×C3×C414yescomplemented
264C4×Dic3—24no—
265C4×Dic3C2×C3×C414yescomplemented
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—42no—
270C22×C32—42no—
271C22×C32C23×C411yescharacteristic
272C22×C32C2×#4112yes—
273C22×C32C2×#4111yescharacteristic
274C22×C32C2×#4111yescharacteristic
275C22×C32H(5,𝔽2)12yes—
276C2×C3×D3—22no—
277C2×C3×D3—22no—
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3#11714yes—
281C32×C4—22no—
282C32×C4—22no—
283C32×C4—24no—
284C32×C4—42no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—22no—
288C3×Dic3—42no—
289C3×Dic3—42no—
290C22×D4—31no—
291C23×C4—31no—
292C23×C4—32no—
293C23×C4C2×C3×D311yescharacteristic
294C4×D4—38no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—32no—
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—32no—
300C2×C22⋊C4—32no—
301C2×C22⋊C4C2×C3×D312yes—
302C2×C42—31no—
303C2×C42—32no—
304C2×C42C2×C3×D312yes—
305#95—31no—
306#95—32no—
307#95—31no—
308#95—32no—
309#95—32no—
310#95—32no—
311#95—32no—
312#95C2×C3×D311yescharacteristic
313#95C2×C3×D311yescharacteristic
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—22no—
325C23×C3—32no—
326C23×C3—32no—
327C23×C3—32no—
328C23×C3—42no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3C2×C4×D312yes—
334C23×C3C22×C3×C412yes—
335C23×C3C23×D311yescharacteristic
336C23×C3C24×C311yescharacteristic
337C22×D3—22no—
338C22×D3—22no—
339C22×D3C22×C3×C412yes—
340C22×D3C3×#4112yes—
341C22×D3C3×#4112yes—
342C3×D4—68no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
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—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—32no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—44no—
371C2×C3×C4—44no—
372C2×C3×C4—48no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—62no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—124no—
384C2×C3×C4—128no—
385C2×C3×C4C2×C4×D312yes—
386C2×C3×C4C2×C4×D312yes—
387C2×C3×C4#23514yes—
388C2×C3×C4C22×C3×C412yes—
389C2×C3×C4C22×C3×C412yes—
390C2×C3×C4C3×#4114yes—
391#67—28no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3C22×C3×C412yes—
403C2×Dic3C22×C3×C412yes—
404C2×Dic3C22×C3×C412yes—
405C4×D3—44no—
406C4×D3—48no—
407C2×C32—42no—
408C2×C32C2×#11711yescharacteristic
409C2×C32C4×#4111yescharacteristic
410C2×C32#51811yescharacteristic
411C2×C32#55112yes—
412C2×C32#55312yes—
413C3×D3—24no—
414C24—31no—
415C24C22×C3×D311yescharacteristic
416C2×D4—34no—
417C2×D4—62no—
418C2×D4—62no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—34no—
423C22×C4—32no—
424C22×C4—32no—
425C22×C4—42no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4C3×C4×D314yescomplemented
431C22×C4C22×C3×D312yes—
432C22×C4C22×C3×D311yescharacteristic
433C22×C4C22×C3×D311yescharacteristic
434C22×C4C22×C3×D312yes—
435C4⋊C4—34no—
436C4⋊C4—122no—
437C22⋊C4—24no—
438C22⋊C4—34no—
439C22⋊C4—34no—
440C22⋊C4—64no—
441C22⋊C4—122no—
442C42—34no—
443C42—42no—
444C42—64no—
445C22×C3—24no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—34no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—64no—
469C22×C3—64no—
470C22×C3C22×C4×D311yescharacteristic
471C22×C3C2×#23512yes—
472C22×C3#100412yes—
473C22×C3D3×#4111yescharacteristic
474C22×C3D3×#4111yescharacteristic
475C22×C3C23×C3×C411yescharacteristic
476C22×C3C2×C3×#4112yes—
477C22×C3C2×C3×#4111yescharacteristic
478C22×C3C2×C3×#4111yescharacteristic
479C22×C3C3×H(5,𝔽2)12yes—
480C2×D3—22no—
481C2×D3—22no—
482C2×D3—24no—
483C2×D3—24no—
484C2×D3C3×#11714yes—
485C3×C4—24no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—24no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—24no—
492C3×C4—42no—
493C3×C4—44no—
494C3×C4—44no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—62no—
500C3×C4—122no—
501C3×C4—122no—
502Dic3—22no—
503Dic3—22no—
504Dic3—22no—
505Dic3—42no—
506Dic3—42no—
507C32#235311yescharacteristic, complemented, Sylow
508C23—22no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—32no—
513C23—32no—
514C23—32no—
515C23—62no—
516C23—62no—
517C23C2×C3×C4×D312yes—
518C23C23×C3×D311yescharacteristic
519D4—68no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—44no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—124no—
542C2×C4—128no—
543C2×C4C2×C3×C4×D312yes—
544C2×C4C2×C3×C4×D312yes—
545C2×C4C3×#23514yes—
546C2×C3—21no—
547C2×C3—22no—
548C2×C3—22no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—44no—
552C2×C3—42no—
553C2×C3—42no—
554C2×C3—64no—
555C2×C3C4×#23511yescharacteristic
556C2×C3#548611yescharacteristic
557C2×C3D3×#11711yescharacteristic
558C2×C3#555612yes—
559C2×C3#555812yes—
560C2×C3C2×C3×#11711yescharacteristic
561C2×C3C3×C4×#4111yescharacteristic
562C2×C3C3×#51811yescharacteristic
563C2×C3C3×#55112yes—
564C2×C3C3×#55312yes—
565D3—24no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—34no—
570C22—42no—
571C22—42no—
572C22—64no—
573C22—64no—
574C22—62no—
575C22—62no—
576C22C22×C3×C4×D311yescharacteristic
577C22C2×C3×#23512yes—
578C22C3×#100412yes—
579C22C3×D3×#4111yescharacteristic
580C22C3×D3×#4111yescharacteristic
581C4—24no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—62no—
586C4—62no—
587C4—62no—
588C4—122no—
589C4—122no—
590C3—21no—
591C3#7357611yescharacteristic, direct product factor
592C3C3×#235311yescharacteristic, complemented
593C2—42no—
594C2—64no—
595C2C3×C4×#23511yescharacteristic
596C2C3×#548611yescharacteristic
597C2C3×D3×#11711yescharacteristic
598C2C3×#555612yes—
599C2C3×#555812yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×C4×C22⋊C4C2
socleC23×C32C24
prime radicalC22×C3×#67C22
cosocleC23×C3C24×C3
hypercenterC3×C4×C22⋊C4D3
hypocenterC3C3×#2353
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×C22⋊C4C2semidirect
3. C3×#5211C2semidirect
5. C3×#4944C2semidirect
6. C3×C4×(C2×D3)⋊C4C2semidirect
7. C3×#4933C2semidirect
8. C3×#4667C2semidirect
9. C3×D3×#95C2semidirect
10. C3×#4651C2semidirect
11. C3×#4648C2semidirect
13. #73576C3direct (N × H)
22. C3×C4×#67C4semidirect
48. C3×C4×C22⋊C4D3semidirect
49. C3×C4×C22⋊C4C2×C3semidirect
52. #5211C2×C3semidirect
54. #4944C2×C3semidirect
55. C4×(C2×D3)⋊C4C2×C3semidirect
56. #4933C2×C3semidirect
57. #4667C2×C3semidirect
58. D3×#95C2×C3semidirect
59. #4651C2×C3semidirect
60. #4648C2×C3semidirect
64. C22×C32×C4C2×C4semidirect
85. C3×#217C2×C4semidirect
89. C3×Dic3⋊C4C2×C4semidirect
91. C3×C4×Dic3C2×C4semidirect
119. C4×#67C3×C4semidirect
192. C4×C22⋊C4C3×D3semidirect
221. C22×C3×C4C4×D3semidirect
222. C22×C3×C4C2×C3×C4semidirect
259. #217C2×C3×C4semidirect
263. Dic3⋊C4C2×C3×C4semidirect
265. C4×Dic3C2×C3×C4semidirect
430. C22×C4C3×C4×D3semidirect
507. C32#2353semidirect
591. C3#73576direct (N × H)
592. 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×#672
C32×C4×C22⋊C42
C3×#52112
C3×#49562
C3×#49442
C3×C4×(C2×D3)⋊C42
C3×#49332
C3×#46672
C3×D3×#952
C3×#46512
C3×#46482
C3×#23533
#735763