Finite Group Databasebeta

← C3×#74006 #11803030

Subgroups — C3×#74006 #11803030

Order 1152.

1253 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
1C2×C3×C4×#67C211yesmaximal, characteristic
2C32×#380C211yesmaximal, characteristic, complemented
3C3×#5211C211yesmaximal, characteristic, complemented
4C3×#4989C212yesmaximal, complemented
5C3×#4978C212yesmaximal, complemented
6C3×#4956C212yesmaximal
7C3×#4945C212yesmaximal, complemented
8C3×#4666C211yesmaximal, characteristic, complemented
9C3×D3×#95C211yesmaximal, characteristic, complemented
10C3×#4651C211yesmaximal, characteristic, complemented
11C3×#4650C211yesmaximal, characteristic, complemented
12C3×#2318—31nomaximal
13#74006C311yesmaximal, characteristic, direct product factor
14C23×C32×C4C2211yescharacteristic
15C22×C3×#67C2211yescharacteristic
16C22×C3×C4×D3C2212yes—
17C22×C3×C4×D3C2211yescharacteristic
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C22⋊C4C2212yes—
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×#385—32no—
44C3×#383—31no—
45C3×#382—31no—
46C3×#381—31no—
47C3×#380—21no—
48C3×#380D311yescharacteristic, complemented
49C3×#380C2×C311yescharacteristic, complemented
50C3×C4×C22⋊C4—31no—
51C2×C3×#95—31no—
52#5211C2×C311yescharacteristic, complemented
53#4989C2×C312yescomplemented
54#4978C2×C312yescomplemented
55#4956C2×C312yes—
56#4945C2×C312yescomplemented
57#4666C2×C311yescharacteristic, complemented
58D3×#95C2×C311yescharacteristic, complemented
59#4651C2×C311yescharacteristic, complemented
60#4650C2×C311yescharacteristic, complemented
61C24×C32C2311yescharacteristic
62C23×C3×D3C2311yescharacteristic
63C22×C32×C4—42no—
64C22×C32×C4C2×C414yescomplemented
65C22×C32×C4C2312yes—
66C22×C32×C4C2312yes—
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2311yescharacteristic
69C2×C3×#67—24no—
70C2×C3×#67C2×C414yes—
71C22×C3×Dic3C2312yes—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2311yescharacteristic
74C22×C3×Dic3C2311yescharacteristic
75C22×C3×Dic3C2311yescharacteristic
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—
81C32×C4⋊C4—42no—
82C32×C22⋊C4—42no—
83C3×#217—42no—
84C3×#217C2×C414yescomplemented
85C3×(C2×D3)⋊C4—24no—
86C3×(C2×D3)⋊C4C2×C414yes—
87C3×Dic3⋊C4—42no—
88C3×Dic3⋊C4C2×C414yescomplemented
89C3×C4×Dic3—42no—
90C3×C4×Dic3C2×C414yescomplemented
91#2318—31noSylow
92C22×C3×D4—31no—
93C23×C3×C4—21no—
94C23×C3×C4—31no—
95C23×C3×C4—32no—
96C23×C3×C4C2×D311yescharacteristic
97C23×C3×C4C22×C311yescharacteristic
98C22×#67C22×C311yescharacteristic
99C22×C4×D3C22×C312yes—
100C22×C4×D3C22×C311yescharacteristic
101C3×C4×D4—38no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—32no—
105C2×C3×C4⋊C4C2×D312yes—
106C2×C3×C4⋊C4C22×C312yes—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—31no—
109C2×C3×C22⋊C4—31no—
110C2×C3×C22⋊C4—32no—
111C2×C3×C22⋊C4—32no—
112C2×C3×C22⋊C4C2×D312yes—
113C2×C3×C22⋊C4C22×C312yes—
114C2×C3×C42—31no—
115C2×C3×C42—32no—
116C2×#217C22×C312yes—
117C2×#217C22×C311yescharacteristic
118C4×#67C3×C418yescomplemented
119C2×(C2×D3)⋊C4C22×C312yes—
120C2×(C2×D3)⋊C4C22×C311yescharacteristic
121C2×Dic3⋊C4C22×C312yes—
122C2×Dic3⋊C4C22×C311yescharacteristic
123C2×C4×Dic3C22×C312yes—
124C2×C4×Dic3C22×C311yescharacteristic
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—31no—
128C3×#95—32no—
129C3×#95—31no—
130C3×#95—32no—
131C3×#95—32no—
132C3×#95—32no—
133C3×#95—32no—
134C3×#95C2×D311yescharacteristic
135C3×#95C2×D311yescharacteristic
136C3×#95C22×C311yescharacteristic
137C3×#95C22×C311yescharacteristic
138C3⋊#95C22×C311yescharacteristic
139C3⋊#95C22×C312yes—
140C3⋊#95C22×C311yescharacteristic
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C312yes—
143C3⋊#95C22×C312yes—
144C3⋊#95C22×C312yes—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—22no—
148C23×C32—22no—
149C23×C32C22×C412yes—
150C23×C32C2411yescharacteristic
151C22×C3×D3—22no—
152C22×C3×D3—21no—
153C22×C3×D3—21no—
154C22×C3×D3C22×C412yes—
155C22×C3×D3#4114yes—
156C2×C32×C4—22no—
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—
167C3×#67—28no—
168C2×C3×Dic3—22no—
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×Dic3C22×C412yes—
179C2×C3×Dic3C22×C412yes—
180C2×C3×Dic3C22×C412yes—
181C3×C4×D3—44no—
182C3×C4×D3—48no—
183C2×C4×D4—31no—
184#388—32no—
185#387—32no—
186#386—32no—
187#385—32no—
188#383—31no—
189#382—31no—
190#381—31no—
191#380C3×D311yescharacteristic, complemented
192C4×C22⋊C4—31no—
193C2×#95—31no—
194C24×C3—21no—
195C24×C3—31no—
196C24×C3C22×D311yescharacteristic
197C24×C3C23×C311yescharacteristic
198C23×D3C23×C311yescharacteristic
199C2×C3×D4—34no—
200C2×C3×D4—64no—
201C22×C3×C4—24no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—34no—
210C22×C3×C4—32no—
211C22×C3×C4—32no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—44no—
215C22×C3×C4—64no—
216C22×C3×C4—122no—
217C22×C3×C4—122no—
218C22×C3×C4—122no—
219C22×C3×C4C4×D314yescomplemented
220C22×C3×C4C2×C3×C414yescomplemented
221C22×C3×C4C22×D312yes—
222C22×C3×C4C22×D312yes—
223C22×C3×C4C22×D311yescharacteristic
224C22×C3×C4C22×D311yescharacteristic
225C22×C3×C4C23×C312yes—
226C22×C3×C4C23×C312yes—
227C22×C3×C4C23×C311yescharacteristic
228C22×C3×C4C23×C311yescharacteristic
229C2×#67—24no—
230C2×#67C2×C3×C414yes—
231C22×Dic3C23×C312yes—
232C22×Dic3C23×C312yes—
233C22×Dic3C23×C311yescharacteristic
234C22×Dic3C23×C311yescharacteristic
235C22×Dic3C23×C311yescharacteristic
236C2×C4×D3—24no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3C2×C3×C414yes—
241C3×C4⋊C4—34no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—44no—
245C3×C4⋊C4—122no—
246C3×C22⋊C4—34no—
247C3×C22⋊C4—34no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—44no—
251C3×C22⋊C4—64no—
252C3×C22⋊C4—122no—
253C3×C42—34no—
254C3×C42—122no—
255#217—42no—
256#217C2×C3×C414yescomplemented
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4C2×C3×C414yes—
259Dic3⋊C4—42no—
260Dic3⋊C4C2×C3×C414yescomplemented
261C4×Dic3—42no—
262C4×Dic3C2×C3×C414yescomplemented
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32C23×C411yescharacteristic
269C22×C32C2×#4112yes—
270C22×C32H(5,𝔽2)12yes—
271C22×C32H(5,𝔽2)11yescharacteristic
272C22×C322−1+411yescharacteristic
273C2×C3×D3—22no—
274C2×C3×D3—22no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3#11714yes—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4—42no—
281C32×C4—42no—
282C3×Dic3—22no—
283C3×Dic3—22no—
284C3×Dic3—22no—
285C3×Dic3—42no—
286C3×Dic3—42no—
287C22×D4—31no—
288C23×C4—31no—
289C23×C4—32no—
290C23×C4C2×C3×D311yescharacteristic
291C4×D4—38no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—32no—
294C2×C4⋊C4C2×C3×D312yes—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—32no—
298C2×C22⋊C4—32no—
299C2×C22⋊C4C2×C3×D312yes—
300C2×C42—31no—
301C2×C42—32no—
302#95—31no—
303#95—32no—
304#95—31no—
305#95—32no—
306#95—32no—
307#95—32no—
308#95—32no—
309#95C2×C3×D311yescharacteristic
310#95C2×C3×D311yescharacteristic
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—32no—
324C23×C3—34no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—62no—
328C23×C3—61no—
329C23×C3—61no—
330C23×C3C2×C4×D312yes—
331C23×C3C22×C3×C412yes—
332C23×C3C23×D311yescharacteristic
333C23×C3C24×C311yescharacteristic
334C22×D3—22no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3C22×C3×C412yes—
338C22×D3C3×#4114yes—
339C3×D4—68no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
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—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—32no—
360C2×C3×C4—32no—
361C2×C3×C4—32no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—44no—
370C2×C3×C4—44no—
371C2×C3×C4—48no—
372C2×C3×C4—62no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—62no—
381C2×C3×C4—62no—
382C2×C3×C4—124no—
383C2×C3×C4—128no—
384C2×C3×C4C2×C4×D312yes—
385C2×C3×C4C2×C4×D312yes—
386C2×C3×C4C22×C3×C412yes—
387C2×C3×C4C22×C3×C412yes—
388#67—28no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3C22×C3×C412yes—
400C2×Dic3C22×C3×C412yes—
401C2×Dic3C22×C3×C412yes—
402C4×D3—44no—
403C4×D3—48no—
404C2×C32—42no—
405C2×C32C2×#11711yescharacteristic
406C2×C32#51811yescharacteristic
407C2×C32#52011yescharacteristic
408C2×C32#53812yes—
409C2×C32#53912yes—
410C3×D3—24no—
411C24—31no—
412C24C22×C3×D311yescharacteristic
413C2×D4—34no—
414C2×D4—64no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—34no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—42no—
422C22×C4—64no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4C3×C4×D314yescomplemented
427C22×C4C22×C3×D312yes—
428C22×C4C22×C3×D312yes—
429C22×C4C22×C3×D311yescharacteristic
430C22×C4C22×C3×D311yescharacteristic
431C4⋊C4—34no—
432C4⋊C4—42no—
433C4⋊C4—122no—
434C22⋊C4—34no—
435C22⋊C4—34no—
436C22⋊C4—42no—
437C22⋊C4—64no—
438C22⋊C4—122no—
439C42—34no—
440C42—122no—
441C22×C3—24no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—21no—
449C22×C3—22no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—34no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—64no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—64no—
466C22×C3C22×C4×D311yescharacteristic
467C22×C3#100412yes—
468C22×C3#100411yescharacteristic
469C22×C3#100711yescharacteristic
470C22×C3D3×#4112yes—
471C22×C3C23×C3×C411yescharacteristic
472C22×C3C2×C3×#4112yes—
473C22×C3C3×H(5,𝔽2)12yes—
474C22×C3C3×H(5,𝔽2)11yescharacteristic
475C22×C3C3×2−1+411yescharacteristic
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—24no—
479C2×D3—24no—
480C2×D3C3×#11714yes—
481C3×C4—24no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—62no—
492C3×C4—62no—
493C3×C4—62no—
494C3×C4—82no—
495C3×C4—82no—
496C3×C4—122no—
497C3×C4—122no—
498Dic3—22no—
499Dic3—22no—
500Dic3—22no—
501Dic3—42no—
502Dic3—42no—
503C32#231811yescharacteristic, complemented, Sylow
504C23—21no—
505C23—22no—
506C23—21no—
507C23—22no—
508C23—32no—
509C23—34no—
510C23—61no—
511C23—62no—
512C23—61no—
513C23C2×C3×C4×D312yes—
514C23C23×C3×D311yescharacteristic
515D4—68no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—32no—
525C2×C4—32no—
526C2×C4—32no—
527C2×C4—44no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—124no—
539C2×C4—128no—
540C2×C4C2×C3×C4×D312yes—
541C2×C4C2×C3×C4×D312yes—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—44no—
548C2×C3—42no—
549C2×C3—42no—
550C2×C3—64no—
551C2×C3#548611yescharacteristic
552C2×C3#550411yescharacteristic
553C2×C3D3×#11711yescharacteristic
554C2×C3#561312yes—
555C2×C3#563812yes—
556C2×C3C2×C3×#11711yescharacteristic
557C2×C3C3×#51811yescharacteristic
558C2×C3C3×#52011yescharacteristic
559C2×C3C3×#53812yes—
560C2×C3C3×#53912yes—
561D3—24no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—34no—
566C22—42no—
567C22—42no—
568C22—62no—
569C22—64no—
570C22—64no—
571C22—62no—
572C22C22×C3×C4×D311yescharacteristic
573C22C3×#100411yescharacteristic
574C22C3×#100412yes—
575C22C3×#100711yescharacteristic
576C22C3×D3×#4112yes—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—62no—
582C4—62no—
583C4—62no—
584C4—122no—
585C4—122no—
586C3—21no—
587C3#7400611yescharacteristic, direct product factor
588C3C3×#231811yescharacteristic, complemented
589C2—42no—
590C2—64no—
591C2C3×#548611yescharacteristic
592C2C3×#550411yescharacteristic
593C2C3×D3×#11711yescharacteristic
594C2C3×#561312yes—
595C2C3×#563812yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#380C2
socleC23×C32C24
prime radicalC22×C3×#67C22
cosocleC23×C3C24×C3
hypercenterC3×#380D3
hypocenterC3C3×#2318
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#380

Complements

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

NH ≅ G/Ntype
2. C32×#380C2semidirect
3. C3×#5211C2semidirect
4. C3×#4989C2semidirect
5. C3×#4978C2semidirect
7. C3×#4945C2semidirect
8. C3×#4666C2semidirect
9. C3×D3×#95C2semidirect
10. C3×#4651C2semidirect
11. C3×#4650C2semidirect
13. #74006C3direct (N × H)
22. C3×C4×#67C4semidirect
48. C3×#380D3semidirect
49. C3×#380C2×C3semidirect
52. #5211C2×C3semidirect
53. #4989C2×C3semidirect
54. #4978C2×C3semidirect
56. #4945C2×C3semidirect
57. #4666C2×C3semidirect
58. D3×#95C2×C3semidirect
59. #4651C2×C3semidirect
60. #4650C2×C3semidirect
64. C22×C32×C4C2×C4semidirect
84. C3×#217C2×C4semidirect
88. C3×Dic3⋊C4C2×C4semidirect
90. C3×C4×Dic3C2×C4semidirect
118. C4×#67C3×C4semidirect
191. #380C3×D3semidirect
219. C22×C3×C4C4×D3semidirect
220. C22×C3×C4C2×C3×C4semidirect
256. #217C2×C3×C4semidirect
260. Dic3⋊C4C2×C3×C4semidirect
262. C4×Dic3C2×C3×C4semidirect
426. C22×C4C3×C4×D3semidirect
503. C32#2318semidirect
587. C3#74006direct (N × H)
588. C3C3×#2318semidirect

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×#3802
C3×#52112
C3×#49892
C3×#49782
C3×#49562
C3×#49452
C3×#46662
C3×D3×#952
C3×#46512
C3×#46502
C3×#23183
#740063