Finite Group Databasebeta

← C3×#73870 #11803117

Subgroups — C3×#73870 #11803117

Order 1152.

1197 conjugacy classes of subgroups.

The subgroup lattice has 593 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×#4990C212yesmaximal, complemented
5C3×#4976C212yesmaximal, complemented
6C3×#4957C212yesmaximal
7C3×#4944C212yesmaximal, complemented
8C3×#4667C211yesmaximal, characteristic, complemented
9C3×#4664C211yesmaximal, characteristic, complemented
10C3×#4649C211yesmaximal, characteristic, complemented
11C3×#4648C211yesmaximal, characteristic, complemented
12C3×#2319—31nomaximal
13#73870C311yesmaximal, characteristic, direct product factor
14C23×C32×C4C2211yescharacteristic
15C22×C3×#67C2211yescharacteristic
16C22×C3×C4×D3C2211yescharacteristic
17C2×C32×C4⋊C4C2212yes—
18C2×C32×C22⋊C4C2212yes—
19C2×C3×#217C2212yes—
20C2×C3×#217C2211yescharacteristic
21C3×C4×#67C418yescomplemented
22C2×C3×(C2×D3)⋊C4C2212yes—
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2211yescharacteristic
25C2×C3×Dic3⋊C4C2212yes—
26C2×C3×Dic3⋊C4C2212yes—
27C2×C3×Dic3⋊C4C2211yescharacteristic
28C2×C3×C4×Dic3C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C2×C3×C4×Dic3C2211yescharacteristic
31C32×#95C2211yescharacteristic
32C32×#95C2211yescharacteristic
33C3×C3⋊#95C2211yescharacteristic
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2211yescharacteristic
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2212yes—
38C2×C3×C4×D4—31no—
39C2×C4×#67C2×C311yescharacteristic
40C3×#390—32no—
41C3×#389—32no—
42C3×#388—32no—
43C3×#388—32no—
44C3×#381—31no—
45C3×#381—31no—
46C3×#380—21no—
47C3×#380—31no—
48C3×#380D311yescharacteristic, complemented
49C3×#380C2×C311yescharacteristic, complemented
50C3×C4×C4⋊C4—31no—
51C3×#376—31no—
52#5211C2×C311yescharacteristic, complemented
53#4990C2×C312yescomplemented
54#4976C2×C312yescomplemented
55#4957C2×C312yes—
56#4944C2×C312yescomplemented
57#4667C2×C311yescharacteristic, complemented
58#4664C2×C311yescharacteristic, complemented
59#4649C2×C311yescharacteristic, complemented
60#4648C2×C311yescharacteristic, complemented
61C24×C32C2311yescharacteristic
62C23×C3×D3C2311yescharacteristic
63C22×C32×C4—42no—
64C22×C32×C4C2×C414yes—
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—42no—
77C2×C3×C4×D3C2×C414yes—
78C32×C4⋊C4—42no—
79C32×C22⋊C4—42no—
80C3×#217—42no—
81C3×#217C2×C414yes—
82C3×(C2×D3)⋊C4—42no—
83C3×(C2×D3)⋊C4—42no—
84C3×(C2×D3)⋊C4C2×C414yes—
85C3×Dic3⋊C4—24no—
86C3×Dic3⋊C4—42no—
87C3×Dic3⋊C4C2×C414yes—
88C3×C4×Dic3—24no—
89C3×C4×Dic3—42no—
90C3×C4×Dic3C2×C414yes—
91#2319—31noSylow
92C22×C3×D4—31no—
93C23×C3×C4—21no—
94C23×C3×C4—31no—
95C23×C3×C4C2×D311yescharacteristic
96C23×C3×C4C22×C311yescharacteristic
97C22×#67C22×C311yescharacteristic
98C22×C4×D3C22×C311yescharacteristic
99C3×C4×D4—38no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—32no—
103C2×C3×C4⋊C4—32no—
104C2×C3×C4⋊C4C2×D312yes—
105C2×C3×C4⋊C4C22×C312yes—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—31no—
109C2×C3×C22⋊C4—32no—
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×C3×C42—32no—
117C2×#217C22×C312yes—
118C2×#217C22×C311yescharacteristic
119C4×#67C3×C418yescomplemented
120C2×(C2×D3)⋊C4C22×C312yes—
121C2×(C2×D3)⋊C4C22×C312yes—
122C2×(C2×D3)⋊C4C22×C311yescharacteristic
123C2×Dic3⋊C4C22×C312yes—
124C2×Dic3⋊C4C22×C312yes—
125C2×Dic3⋊C4C22×C311yescharacteristic
126C2×C4×Dic3C22×C312yes—
127C2×C4×Dic3C22×C312yes—
128C2×C4×Dic3C22×C311yescharacteristic
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—31no—
132C3×#95—32no—
133C3×#95—31no—
134C3×#95—32no—
135C3×#95—32no—
136C3×#95C2×D311yescharacteristic
137C3×#95C2×D311yescharacteristic
138C3×#95C22×C311yescharacteristic
139C3×#95C22×C311yescharacteristic
140C3⋊#95C22×C311yescharacteristic
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C311yescharacteristic
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—22no—
154C22×C3×D3—21no—
155C22×C3×D3C22×C412yes—
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—21no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—21no—
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×Dic3C22×C412yes—
178C2×C3×Dic3C22×C412yes—
179C2×C3×Dic3C22×C412yes—
180C2×C3×Dic3#4114yes—
181C3×C4×D3—44no—
182C2×C4×D4—31no—
183#390—32no—
184#389—32no—
185#388—32no—
186#388—32no—
187#381—31no—
188#381—31no—
189#380—31no—
190#380C3×D311yescharacteristic, complemented
191C4×C4⋊C4—31no—
192#376—31no—
193C24×C3—21no—
194C24×C3—31no—
195C24×C3C22×D311yescharacteristic
196C24×C3C23×C311yescharacteristic
197C23×D3C23×C311yescharacteristic
198C2×C3×D4—34no—
199C2×C3×D4—64no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—34no—
209C22×C3×C4—32no—
210C22×C3×C4—32no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4—44no—
214C22×C3×C4—122no—
215C22×C3×C4C4×D314yes—
216C22×C3×C4C2×C3×C414yes—
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D312yes—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C23×C312yes—
222C22×C3×C4C23×C312yes—
223C22×C3×C4C23×C311yescharacteristic
224C22×C3×C4C23×C311yescharacteristic
225C2×#67—24no—
226C2×#67C2×C3×C414yes—
227C22×Dic3C23×C312yes—
228C22×Dic3C23×C312yes—
229C22×Dic3C23×C311yescharacteristic
230C22×Dic3C23×C311yescharacteristic
231C22×Dic3C23×C311yescharacteristic
232C2×C4×D3—42no—
233C2×C4×D3C2×C3×C414yes—
234C3×C4⋊C4—34no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—44no—
238C3×C4⋊C4—64no—
239C3×C4⋊C4—122no—
240C3×C22⋊C4—34no—
241C3×C22⋊C4—34no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—44no—
245C3×C22⋊C4—122no—
246C3×C22⋊C4—122no—
247C3×C22⋊C4—122no—
248C3×C42—34no—
249C3×C42—64no—
250C3×C42—122no—
251#217—42no—
252#217C2×C3×C414yes—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4C2×C3×C414yes—
256Dic3⋊C4—24no—
257Dic3⋊C4—42no—
258Dic3⋊C4C2×C3×C414yes—
259C4×Dic3—24no—
260C4×Dic3—42no—
261C4×Dic3C2×C3×C414yes—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32C23×C411yescharacteristic
268C22×C32C2×#4112yes—
269C22×C32H(5,𝔽2)12yes—
270C22×C32H(5,𝔽2)11yescharacteristic
271C22×C322−1+411yescharacteristic
272C2×C3×D3—22no—
273C2×C3×D3—22no—
274C2×C3×D3—22no—
275C2×C3×D3—42no—
276C2×C3×D3—42no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—42no—
280C32×C4—42no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—42no—
284C3×Dic3—42no—
285C3×Dic3#11714yes—
286C22×D4—31no—
287C23×C4—31no—
288C23×C4C2×C3×D311yescharacteristic
289C4×D4—38no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4C2×C3×D312yes—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—32no—
297C2×C22⋊C4—32no—
298C2×C22⋊C4—32no—
299C2×C22⋊C4C2×C3×D312yes—
300C2×C42—31no—
301C2×C42—32no—
302C2×C42—32no—
303#95—31no—
304#95—32no—
305#95—31no—
306#95—32no—
307#95—32no—
308#95C2×C3×D311yescharacteristic
309#95C2×C3×D311yescharacteristic
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—32no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—61no—
326C23×C3—62no—
327C23×C3—61no—
328C23×C3—62no—
329C23×C3C2×C4×D312yes—
330C23×C3C22×C3×C412yes—
331C23×C3C23×D311yescharacteristic
332C23×C3C24×C311yescharacteristic
333C22×D3—22no—
334C22×D3—21no—
335C22×D3—22no—
336C22×D3—21no—
337C22×D3C22×C3×C412yes—
338C3×D4—68no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
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—32no—
359C2×C3×C4—34no—
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—61no—
373C2×C3×C4—62no—
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—124no—
382C2×C3×C4C2×C4×D312yes—
383C2×C3×C4C2×C4×D312yes—
384C2×C3×C4C22×C3×C412yes—
385C2×C3×C4C22×C3×C412yes—
386#67—28no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
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—
398C2×Dic3C22×C3×C412yes—
399C2×Dic3C3×#4114yes—
400C4×D3—44no—
401C2×C32—42no—
402C2×C32C2×#11711yescharacteristic
403C2×C32#51811yescharacteristic
404C2×C32#52011yescharacteristic
405C2×C32#54012yes—
406C2×C32#54212yes—
407C3×D3—42no—
408C24—31no—
409C24C22×C3×D311yescharacteristic
410C2×D4—34no—
411C2×D4—64no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—34no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—42no—
419C22×C4—122no—
420C22×C4C3×C4×D314yes—
421C22×C4C22×C3×D312yes—
422C22×C4C22×C3×D312yes—
423C22×C4C22×C3×D311yescharacteristic
424C22×C4C22×C3×D311yescharacteristic
425C4⋊C4—34no—
426C4⋊C4—42no—
427C4⋊C4—64no—
428C4⋊C4—122no—
429C22⋊C4—34no—
430C22⋊C4—34no—
431C22⋊C4—42no—
432C22⋊C4—122no—
433C22⋊C4—122no—
434C22⋊C4—122no—
435C42—34no—
436C42—64no—
437C42—122no—
438C22×C3—24no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—22no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—62no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3C22×C4×D311yescharacteristic
464C22×C3#100412yes—
465C22×C3#100411yescharacteristic
466C22×C3#100711yescharacteristic
467C22×C3D3×#4112yes—
468C22×C3C23×C3×C411yescharacteristic
469C22×C3C2×C3×#4112yes—
470C22×C3C3×H(5,𝔽2)12yes—
471C22×C3C3×H(5,𝔽2)11yescharacteristic
472C22×C3C3×2−1+411yescharacteristic
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—22no—
476C2×D3—42no—
477C2×D3—42no—
478C3×C4—24no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—34no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—82no—
492C3×C4—82no—
493C3×C4—122no—
494C3×C4—122no—
495Dic3—22no—
496Dic3—22no—
497Dic3—42no—
498Dic3—42no—
499Dic3C3×#11714yes—
500C32#231911yescharacteristic, complemented, Sylow
501C23—21no—
502C23—22no—
503C23—21no—
504C23—22no—
505C23—32no—
506C23—61no—
507C23—62no—
508C23—61no—
509C23—62no—
510C23C2×C3×C4×D312yes—
511C23C23×C3×D311yescharacteristic
512D4—68no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—32no—
522C2×C4—34no—
523C2×C4—32no—
524C2×C4—32no—
525C2×C4—44no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—124no—
536C2×C4C2×C3×C4×D312yes—
537C2×C4C2×C3×C4×D312yes—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—44no—
544C2×C3—42no—
545C2×C3—42no—
546C2×C3—122no—
547C2×C3#548611yescharacteristic
548C2×C3#550411yescharacteristic
549C2×C3D3×#11711yescharacteristic
550C2×C3#561512yes—
551C2×C3#564012yes—
552C2×C3C2×C3×#11711yescharacteristic
553C2×C3C3×#51811yescharacteristic
554C2×C3C3×#52011yescharacteristic
555C2×C3C3×#54012yes—
556C2×C3C3×#54212yes—
557D3—42no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—42no—
562C22—42no—
563C22—62no—
564C22—62no—
565C22—62no—
566C22—122no—
567C22—122no—
568C22C22×C3×C4×D311yescharacteristic
569C22C3×#100411yescharacteristic
570C22C3×#100412yes—
571C22C3×#100711yescharacteristic
572C22C3×D3×#4112yes—
573C4—22no—
574C4—22no—
575C4—34no—
576C4—42no—
577C4—42no—
578C4—62no—
579C4—62no—
580C4—122no—
581C4—122no—
582C3—21no—
583C3#7387011yescharacteristic, direct product factor
584C3C3×#231911yescharacteristic, complemented
585C2—42no—
586C2—122no—
587C2C3×#548611yescharacteristic
588C2C3×#550411yescharacteristic
589C2C3×D3×#11711yescharacteristic
590C2C3×#561512yes—
591C2C3×#564012yes—
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×#2319
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×#4990C2semidirect
5. C3×#4976C2semidirect
7. C3×#4944C2semidirect
8. C3×#4667C2semidirect
9. C3×#4664C2semidirect
10. C3×#4649C2semidirect
11. C3×#4648C2semidirect
13. #73870C3direct (N × H)
21. C3×C4×#67C4semidirect
48. C3×#380D3semidirect
49. C3×#380C2×C3semidirect
52. #5211C2×C3semidirect
53. #4990C2×C3semidirect
54. #4976C2×C3semidirect
56. #4944C2×C3semidirect
57. #4667C2×C3semidirect
58. #4664C2×C3semidirect
59. #4649C2×C3semidirect
60. #4648C2×C3semidirect
119. C4×#67C3×C4semidirect
190. #380C3×D3semidirect
500. C32#2319semidirect
583. C3#73870direct (N × H)
584. C3C3×#2319semidirect

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×#49902
C3×#49762
C3×#49572
C3×#49442
C3×#46672
C3×#46642
C3×#46492
C3×#46482
C3×#23193
#738703