Finite Group Databasebeta

← C3×#75969 #11803108

Subgroups — C3×#75969 #11803108

Order 1152.

1165 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#893C211yesmaximal, characteristic
2C32×#389C211yesmaximal, characteristic, complemented
3C3×#4990C212yesmaximal, complemented
4C3×#4988C211yesmaximal, characteristic
5C3×#4987C212yesmaximal, complemented
6C3×#4984C211yesmaximal, characteristic, complemented
7C3×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
8C3×#4939C211yesmaximal, characteristic
9C3×#4932C211yesmaximal, characteristic, complemented
10C3×#4670C212yesmaximal, complemented
11C3×#4664C212yesmaximal, complemented
12C3×#2319—31nomaximal
13#75969C311yesmaximal, characteristic, direct product factor
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3C2212yes—
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C4⋊C4C2212yes—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C42C2211yescharacteristic
20C2×C3×(C2×D3)⋊C4C2212yes—
21C2×C3×(C2×D3)⋊C4C2212yes—
22C2×C3×(C2×D3)⋊C4C2212yes—
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×C4⋊Dic3C2212yes—
27C2×C3×C4×Dic3C2211yescharacteristic
28C2×C3×C4×Dic3C2212yes—
29C2×C3×C4×Dic3C2211yescharacteristic
30C3×#893C418yescomplemented
31C32×#95C2212yes—
32C3×C3⋊#95C2211yescharacteristic
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2211yescharacteristic
36C3×C3⋊#95C2211yescharacteristic
37C3×C3⋊#95C2211yescharacteristic
38C2×C3×C4×D4—31no—
39C2×#893C2×C311yescharacteristic
40C3×#390—31no—
41C3×#390—31no—
42C3×#389—21no—
43C3×#389—31no—
44C3×#389D311yescharacteristic, complemented
45C3×#389C2×C311yescharacteristic, complemented
46C3×#388—32no—
47C3×#388—32no—
48C3×#381—32no—
49C3×#380—32no—
50C3×C4×C4⋊C4—31no—
51C3×#376—31no—
52#4990C2×C312yescomplemented
53#4988C2×C311yescharacteristic
54#4987C2×C312yescomplemented
55#4984C2×C311yescharacteristic, complemented
56Dic3×C4⋊C4C2×C311yescharacteristic, complemented
57#4939C2×C311yescharacteristic
58#4932C2×C311yescharacteristic, complemented
59#4670C2×C312yescomplemented
60#4664C2×C312yescomplemented
61C23×C3×D3C2312yes—
62C22×C32×C4C2312yes—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C22×C3×Dic3C2311yescharacteristic
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C2×C3×D12—22no—
73C2×C3×D12—22no—
74C2×C3×D12C2×C414yes—
75C2×C3×C4×D3—42no—
76C2×C3×C4×D3—42no—
77C2×C3×C4×D3C2×C418yes—
78C32×C4⋊C4—22no—
79C32×C4⋊C4—42no—
80C32×C4⋊C4C2×C414yes—
81C32×C42—22no—
82C3×(C2×D3)⋊C4—42no—
83C3×(C2×D3)⋊C4—42no—
84C3×(C2×D3)⋊C4—42no—
85C3×(C2×D3)⋊C4—42no—
86C3×(C2×D3)⋊C4C2×C418yes—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3—42no—
89C3×C4×Dic3—22no—
90C3×C4×Dic3—42no—
91C3×C4×Dic3C2×C414yes—
92#2319—31noSylow
93C22×C3×D4—31no—
94C23×C3×C4—32no—
95C22×D12C22×C311yescharacteristic
96C22×C4×D3C22×C312yes—
97C3×C4×D4—38no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—32no—
103C2×C3×C4⋊C4C2×D311yescharacteristic
104C2×C3×C4⋊C4C2×D312yes—
105C2×C3×C4⋊C4C2×D311yescharacteristic
106C2×C3×C4⋊C4C22×C311yescharacteristic
107C2×C3×C4⋊C4C22×C312yes—
108C2×C3×C4⋊C4C22×C311yescharacteristic
109C2×C3×C22⋊C4—32no—
110C2×C3×C22⋊C4—32no—
111C2×C3×C22⋊C4—32no—
112C2×C3×C22⋊C4—32no—
113C2×C3×C22⋊C4—32no—
114C2×C3×C42—21no—
115C2×C3×C42—31no—
116C2×C3×C42—31no—
117C2×C3×C42—32no—
118C2×C3×C42C2×D311yescharacteristic
119C2×C3×C42C22×C311yescharacteristic
120C2×(C2×D3)⋊C4C22×C312yes—
121C2×(C2×D3)⋊C4C22×C312yes—
122C2×(C2×D3)⋊C4C22×C312yes—
123C2×(C2×D3)⋊C4C22×C312yes—
124C2×(C2×D3)⋊C4C22×C312yes—
125C2×C4⋊Dic3C22×C311yescharacteristic
126C2×C4⋊Dic3C22×C312yes—
127C2×C4×Dic3C22×C311yescharacteristic
128C2×C4×Dic3C22×C312yes—
129C2×C4×Dic3C22×C311yescharacteristic
130#893C3×C418yescomplemented
131C3×#95—22no—
132C3×#95—31no—
133C3×#95—31no—
134C3×#95—31no—
135C3×#95—31no—
136C3×#95—32no—
137C3×#95—32no—
138C3×#95C2×D312yes—
139C3×#95C22×C312yes—
140C3⋊#95C22×C311yescharacteristic
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C312yes—
143C3⋊#95C22×C311yescharacteristic
144C3⋊#95C22×C311yescharacteristic
145C3⋊#95C22×C311yescharacteristic
146C23×C32C2411yescharacteristic
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3C22×C414yes—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4C22×C414yes—
166C2×C32×C4C22×C412yes—
167C2×C32×C4#4112yes—
168C2×C32×C4#4112yes—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3C22×C412yes—
183C2×C3×Dic3C22×C412yes—
184C3×D12—28no—
185C3×C4×D3—48no—
186C2×C4×D4—31no—
187#390—31no—
188#390—31no—
189#389—31no—
190#389C3×D311yescharacteristic, complemented
191#388—32no—
192#388—32no—
193#381—32no—
194#380—32no—
195C4×C4⋊C4—31no—
196#376—31no—
197C24×C3—32no—
198C23×D3C23×C312yes—
199C2×C3×D4—34no—
200C2×C3×D4—62no—
201C2×C3×D4—62no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—38no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—32no—
213C22×C3×C4—122no—
214C22×C3×C4—122no—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D312yes—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C23×C311yescharacteristic
221C22×C3×C4C23×C311yescharacteristic
222C22×C3×C4C23×C312yes—
223C22×C3×C4C23×C312yes—
224C22×C3×C4C23×C311yescharacteristic
225C22×Dic3C23×C311yescharacteristic
226C22×Dic3C23×C311yescharacteristic
227C22×Dic3C23×C312yes—
228C22×Dic3C23×C311yescharacteristic
229C22×Dic3C23×C311yescharacteristic
230C2×D12—22no—
231C2×D12—22no—
232C2×D12C2×C3×C414yes—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3C2×C3×C418yes—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—44no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—122no—
245C3×C4⋊C4C4×D314yes—
246C3×C4⋊C4C2×C3×C414yes—
247C3×C22⋊C4—38no—
248C3×C22⋊C4—122no—
249C3×C22⋊C4—122no—
250C3×C22⋊C4—122no—
251C3×C22⋊C4—122no—
252C3×C42—22no—
253C3×C42—22no—
254C3×C42—34no—
255C3×C42—42no—
256C3×C42—62no—
257C3×C42—122no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4C2×C3×C418yes—
263C4⋊Dic3—22no—
264C4⋊Dic3—42no—
265C4×Dic3—22no—
266C4×Dic3—42no—
267C4×Dic3C2×C3×C414yes—
268C22×C32C23×C411yescharacteristic
269C22×C32C2×#4111yescharacteristic
270C22×C32C2×#4111yescharacteristic
271C22×C32H(5,𝔽2)11yescharacteristic
272C22×C32H(5,𝔽2)11yescharacteristic
273C22×C32H(5,𝔽2)11yescharacteristic
274C22×C322−1+411yescharacteristic
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—24no—
278C2×C3×D3—42no—
279C2×C3×D3—42no—
280C2×C3×D3—42no—
281C2×C3×D3—42no—
282C32×C4—24no—
283C32×C4—41no—
284C32×C4—41no—
285C32×C4—42no—
286C32×C4#11714yes—
287C3×Dic3—22no—
288C3×Dic3—22no—
289C3×Dic3—41no—
290C3×Dic3—42no—
291C3×Dic3—41no—
292C22×D4—31no—
293C23×C4—32no—
294C4×D4—38no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4C2×C3×D311yescharacteristic
298C2×C4⋊C4C2×C3×D312yes—
299C2×C4⋊C4C2×C3×D311yescharacteristic
300C2×C22⋊C4—32no—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—32no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—32no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—32no—
308C2×C42C2×C3×D311yescharacteristic
309#95—31no—
310#95—31no—
311#95—31no—
312#95—31no—
313#95—32no—
314#95—32no—
315#95C2×C3×D312yes—
316C23×C3—21no—
317C23×C3—34no—
318C23×C3—62no—
319C23×C3—62no—
320C23×C3—62no—
321C23×C3—62no—
322C23×C3—62no—
323C23×C3—62no—
324C23×C3C23×D311yescharacteristic
325C23×C3C24×C311yescharacteristic
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3C22×C3×C414yes—
333C3×D4—68no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—62no—
381C2×C3×C4—61no—
382C2×C3×C4—62no—
383C2×C3×C4—61no—
384C2×C3×C4—62no—
385C2×C3×C4—61no—
386C2×C3×C4—128no—
387C2×C3×C4C2×C4×D314yes—
388C2×C3×C4C2×C4×D312yes—
389C2×C3×C4#23712yes—
390C2×C3×C4#23912yes—
391C2×C3×C4C22×C3×C414yes—
392C2×C3×C4C22×C3×C412yes—
393C2×C3×C4C3×#4112yes—
394C2×C3×C4C3×#4112yes—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3C22×C3×C412yes—
409C2×Dic3C22×C3×C412yes—
410D12—28no—
411C4×D3—48no—
412C2×C32C2×#11711yescharacteristic
413C2×C32#51811yescharacteristic
414C2×C32#52011yescharacteristic
415C2×C32#54011yescharacteristic
416C2×C32#54011yescharacteristic
417C2×C32#54211yescharacteristic
418C2×C32#54211yescharacteristic
419C3×D3—44no—
420C24—32no—
421C2×D4—34no—
422C2×D4—62no—
423C2×D4—62no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—38no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4C22×C3×D311yescharacteristic
433C22×C4C22×C3×D311yescharacteristic
434C22×C4C22×C3×D312yes—
435C22×C4C22×C3×D312yes—
436C22×C4C22×C3×D311yescharacteristic
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—62no—
440C4⋊C4—122no—
441C4⋊C4C3×C4×D314yes—
442C22⋊C4—38no—
443C22⋊C4—122no—
444C22⋊C4—122no—
445C22⋊C4—122no—
446C22⋊C4—122no—
447C42—22no—
448C42—34no—
449C42—62no—
450C42—122no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—64no—
459C22×C3—64no—
460C22×C3—64no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—122no—
464C22×C3—122no—
465C22×C3C22×C4×D311yescharacteristic
466C22×C3C2×#23711yescharacteristic
467C22×C3#100411yescharacteristic
468C22×C3C2×#23911yescharacteristic
469C22×C3#100711yescharacteristic
470C22×C3#100911yescharacteristic
471C22×C3#100911yescharacteristic
472C22×C3C23×C3×C411yescharacteristic
473C22×C3C2×C3×#4111yescharacteristic
474C22×C3C2×C3×#4111yescharacteristic
475C22×C3C3×H(5,𝔽2)11yescharacteristic
476C22×C3C3×H(5,𝔽2)11yescharacteristic
477C22×C3C3×H(5,𝔽2)11yescharacteristic
478C22×C3C3×2−1+411yescharacteristic
479C2×D3—24no—
480C2×D3—24no—
481C2×D3—24no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C3×C4—24no—
487C3×C4—24no—
488C3×C4—24no—
489C3×C4—44no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—42no—
498C3×C4—62no—
499C3×C4—62no—
500C3×C4—82no—
501C3×C4—121no—
502C3×C4—122no—
503C3×C4—121no—
504C3×C4#89214yes—
505C3×C4C3×#11714yes—
506Dic3—22no—
507Dic3—22no—
508Dic3—41no—
509Dic3—42no—
510Dic3—41no—
511C32#231911yescharacteristic, complemented, Sylow
512C23—34no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—62no—
517C23—62no—
518C23—62no—
519C23C23×C3×D311yescharacteristic
520D4—68no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—128no—
548C2×C4C2×C3×C4×D314yes—
549C2×C4C2×C3×C4×D312yes—
550C2×C4C3×#23712yes—
551C2×C4C3×#23912yes—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—124no—
560C2×C3C2×#89211yescharacteristic
561C2×C3#554511yescharacteristic
562C2×C3#557411yescharacteristic
563C2×C3#561911yescharacteristic
564C2×C3#564311yescharacteristic
565C2×C3#569711yescharacteristic
566C2×C3#573311yescharacteristic
567C2×C3C2×C3×#11711yescharacteristic
568C2×C3C3×#51811yescharacteristic
569C2×C3C3×#52011yescharacteristic
570C2×C3C3×#54011yescharacteristic
571C2×C3C3×#54011yescharacteristic
572C2×C3C3×#54211yescharacteristic
573C2×C3C3×#54211yescharacteristic
574D3—44no—
575C22—64no—
576C22—64no—
577C22—64no—
578C22—122no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22C22×C3×C4×D311yescharacteristic
583C22C2×C3×#23711yescharacteristic
584C22C3×#100411yescharacteristic
585C22C2×C3×#23911yescharacteristic
586C22C3×#100711yescharacteristic
587C22C3×#100911yescharacteristic
588C22C3×#100911yescharacteristic
589C4—24no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—62no—
594C4—62no—
595C4—121no—
596C4—122no—
597C4—121no—
598C4C3×#89214yes—
599C3—21no—
600C3#7596911yescharacteristic, direct product factor
601C3C3×#231911yescharacteristic, complemented
602C2—124no—
603C2C2×C3×#89211yescharacteristic
604C2C3×#554511yescharacteristic
605C2C3×#557411yescharacteristic
606C2C3×#561911yescharacteristic
607C2C3×#564311yescharacteristic
608C2C3×#569711yescharacteristic
609C2C3×#573311yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#389C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×#389D3
hypocenterC3C3×#2319
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#389

Complements

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

NH ≅ G/Ntype
2. C32×#389C2semidirect
3. C3×#4990C2semidirect
5. C3×#4987C2semidirect
6. C3×#4984C2semidirect
7. C3×Dic3×C4⋊C4C2semidirect
9. C3×#4932C2semidirect
10. C3×#4670C2semidirect
11. C3×#4664C2semidirect
13. #75969C3direct (N × H)
30. C3×#893C4semidirect
44. C3×#389D3semidirect
45. C3×#389C2×C3semidirect
52. #4990C2×C3semidirect
54. #4987C2×C3semidirect
55. #4984C2×C3semidirect
56. Dic3×C4⋊C4C2×C3semidirect
58. #4932C2×C3semidirect
59. #4670C2×C3semidirect
60. #4664C2×C3semidirect
130. #893C3×C4semidirect
190. #389C3×D3semidirect
511. C32#2319semidirect
600. C3#75969direct (N × H)
601. C3C3×#2319semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8932
C32×#3892
C3×#49902
C3×#49882
C3×#49872
C3×#49842
C3×Dic3×C4⋊C42
C3×#49392
C3×#49322
C3×#46702
C3×#46642
C3×#23193
#759693