Finite Group Databasebeta

← C3×#73668 #11808652

Subgroups — C3×#73668 #11808652

Order 1152.

1153 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#872C211yesmaximal, characteristic
2C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
3C32×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#4994C212yesmaximal
5C3×#4981C212yesmaximal, complemented
6C3×C4×(C2×D3)⋊C4C212yesmaximal
7C3×C4×C4⋊Dic3C212yesmaximal, complemented
8C3×#4677C211yesmaximal, characteristic
9C3×#4673C211yesmaximal, characteristic
10C3×#4659C211yesmaximal, characteristic, complemented
11C3×#4656C211yesmaximal, characteristic, complemented
12C3×#2396—31nomaximal
13#73668C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×D3C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C32×C4⋊C4C2212yes—
17C2×C32×C42C2212yes—
18C2×C32×C42C2211yescharacteristic
19C2×C3×(C2×D3)⋊C4C2211yescharacteristic
20C2×C3×(C2×D3)⋊C4C2212yes—
21C2×C3×(C2×D3)⋊C4C2212yes—
22C2×C3×(C2×D3)⋊C4C2211yescharacteristic
23C2×C3×C4⋊Dic3C2212yes—
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×Dic3⋊C4C2211yescharacteristic
27C2×C3×Dic3⋊C4C2211yescharacteristic
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×C3×C4×Dic3C2212yes—
31C3×#872—24no—
32C3×C4×Dic6—24no—
33C32×#95C2211yescharacteristic
34C32×#95C2211yescharacteristic
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2212yes—
38C3×C3⋊#95C2212yes—
39C2×C3×C4×Q8—31no—
40C2×C3×#117—31no—
41C2×#872C2×C311yescharacteristic
42C2×C4×Dic6C2×C311yescharacteristic, complemented
43C3×#400—31no—
44C3×#399—31no—
45C3×#398—32no—
46C3×#396—32no—
47C3×#395—31no—
48C3×#393—31no—
49C3×C4×C4⋊C4—21no—
50C3×C4×C4⋊C4—32no—
51C3×C4×C4⋊C4D311yescharacteristic, complemented
52C3×C4×C4⋊C4C2×C311yescharacteristic, complemented
53C3×C4×C22⋊C4—32no—
54#4994C2×C312yes—
55#4981C2×C312yescomplemented
56C4×(C2×D3)⋊C4C2×C312yes—
57C4×C4⋊Dic3C2×C312yescomplemented
58#4677C2×C311yescharacteristic
59#4673C2×C311yescharacteristic
60#4659C2×C311yescharacteristic, complemented
61#4656C2×C311yescharacteristic, complemented
62C23×C3×D3C2311yescharacteristic
63C22×C32×C4C2312yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2312yes—
67C22×C32×C4C2311yescharacteristic
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C22×C3×Dic3C2312yes—
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3Q814yescomplemented
76C2×C3×Dic6—22no—
77C2×C3×Dic6—24no—
78C32×C4⋊C4—24no—
79C32×C42—24no—
80C32×C42D414yes—
81C3×(C2×D3)⋊C4—22no—
82C3×(C2×D3)⋊C4—22no—
83C3×(C2×D3)⋊C4—24no—
84C3×(C2×D3)⋊C4—24no—
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3—24no—
87C3×C4⋊Dic3—24no—
88C3×Dic3⋊C4—24no—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—22no—
91C3×C4×Dic3—24no—
92C3×C4×Dic3—22no—
93#2396—31noSylow
94C22×C3×Q8—31no—
95C23×C3×C4—31no—
96C22×C4×D3C22×C311yescharacteristic
97C22×Dic6C22×C311yescharacteristic
98C3×C4×Q8—64no—
99C3×#117—64no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—32no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—32no—
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C4⋊C4C2×D312yes—
108C2×C3×C4⋊C4C22×C312yes—
109C2×C3×C22⋊C4—32no—
110C2×C3×C22⋊C4—32no—
111C2×C3×C22⋊C4—31no—
112C2×C3×C22⋊C4—31no—
113C2×C3×C42—21no—
114C2×C3×C42—22no—
115C2×C3×C42—32no—
116C2×C3×C42—31no—
117C2×C3×C42C2×D312yes—
118C2×C3×C42C2×D311yescharacteristic
119C2×C3×C42C22×C312yes—
120C2×C3×C42C22×C311yescharacteristic
121C2×(C2×D3)⋊C4C22×C311yescharacteristic
122C2×(C2×D3)⋊C4C22×C312yes—
123C2×(C2×D3)⋊C4C22×C312yes—
124C2×(C2×D3)⋊C4C22×C311yescharacteristic
125C2×C4⋊Dic3C22×C312yes—
126C2×C4⋊Dic3C22×C311yescharacteristic
127C2×C4⋊Dic3C22×C312yes—
128C2×Dic3⋊C4C22×C311yescharacteristic
129C2×Dic3⋊C4C22×C312yes—
130C2×Dic3⋊C4C22×C311yescharacteristic
131C2×C4×Dic3C22×C311yescharacteristic
132C2×C4×Dic3C22×C312yes—
133#872—24no—
134C4×Dic6—24no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—32no—
138C3×#95—32no—
139C3×#95—32no—
140C3×#95—32no—
141C3×#95C2×D311yescharacteristic
142C3×#95C2×D311yescharacteristic
143C3×#95C22×C311yescharacteristic
144C3×#95C22×C311yescharacteristic
145C3⋊#95C22×C312yes—
146C3⋊#95C22×C312yes—
147C3⋊#95C22×C312yes—
148C3⋊#95C22×C312yes—
149C23×C32C2411yescharacteristic
150C22×C3×D3—21no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—21no—
154C22×C3×D3C2×Q812yes—
155C2×C32×C4—22no—
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×C4C2×D412yes—
163C2×C32×C4C2×D412yes—
164C2×C32×C4C2×D412yes—
165C2×C32×C4C2×Q812yes—
166C2×C32×C4#4114yes—
167C2×C32×C4#4112yes—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3C2×Q812yes—
180C2×C3×Dic3#4114yes—
181C3×C4×D3—28no—
182C3×Dic6—28no—
183C2×C4×Q8—31no—
184C2×#117—31no—
185#400—31no—
186#399—31no—
187#398—32no—
188#396—32no—
189#395—31no—
190#393—31no—
191C4×C4⋊C4—32no—
192C4×C4⋊C4C3×D311yescharacteristic, complemented
193C4×C22⋊C4—32no—
194C24×C3—31no—
195C23×D3C23×C311yescharacteristic
196C2×C3×Q8—64no—
197C2×C3×Q8—62no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—31no—
204C22×C3×C4—32no—
205C22×C3×C4—34no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—32no—
209C22×C3×C4—62no—
210C22×C3×C4—62no—
211C22×C3×C4C22×D312yes—
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D312yes—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C23×C312yes—
217C22×C3×C4C23×C311yescharacteristic
218C22×C3×C4C23×C311yescharacteristic
219C22×C3×C4C23×C312yes—
220C22×C3×C4C23×C311yescharacteristic
221C22×Dic3C23×C312yes—
222C22×Dic3C23×C311yescharacteristic
223C22×Dic3C23×C311yescharacteristic
224C22×Dic3C23×C311yescharacteristic
225C22×Dic3C23×C312yes—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3C3×Q814yescomplemented
229C2×Dic6—22no—
230C2×Dic6—24no—
231C3×C4⋊C4—28no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—64no—
235C3×C4⋊C4—62no—
236C3×C4⋊C4—62no—
237C3×C4⋊C4—64no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4—64no—
240C3×C22⋊C4—64no—
241C3×C22⋊C4—62no—
242C3×C22⋊C4—62no—
243C3×C22⋊C4—64no—
244C3×C42—24no—
245C3×C42—24no—
246C3×C42—24no—
247C3×C42—44no—
248C3×C42—62no—
249C3×C42—64no—
250C3×C42D1214yes—
251C3×C42C3×D414yes—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256C4⋊Dic3—22no—
257C4⋊Dic3—24no—
258C4⋊Dic3—24no—
259Dic3⋊C4—24no—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262C4×Dic3—24no—
263C4×Dic3—22no—
264C22×C32C22×D411yescharacteristic
265C22×C32C22×Q811yescharacteristic
266C22×C32C2×#4112yes—
267C22×C32C2×#4111yescharacteristic
268C22×C32C2×#4112yes—
269C2×C3×D3—22no—
270C2×C3×D3—24no—
271C2×C3×D3—41no—
272C2×C3×D3—42no—
273C2×C3×D3—41no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—24no—
277C32×C4—42no—
278C32×C4C22⋊Q814yes—
279C3×Dic3—22no—
280C3×Dic3—24no—
281C3×Dic3—41no—
282C3×Dic3—42no—
283C3×Dic3—41no—
284C22×Q8—31no—
285C23×C4—31no—
286C4×Q8—64no—
287#117—64no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—32no—
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×C42—31no—
300C2×C42—32no—
301C2×C42C2×C3×D312yes—
302C2×C42C2×C3×D311yescharacteristic
303#95—32no—
304#95—32no—
305#95—32no—
306#95—32no—
307#95C2×C3×D311yescharacteristic
308#95C2×C3×D311yescharacteristic
309C23×C3—21no—
310C23×C3—32no—
311C23×C3—61no—
312C23×C3—62no—
313C23×C3—61no—
314C23×C3—62no—
315C23×C3C23×D311yescharacteristic
316C23×C3C24×C311yescharacteristic
317C22×D3—21no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—21no—
321C22×D3C2×C3×Q812yes—
322C3×Q8—68no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
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—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—32no—
346C2×C3×C4—34no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—68no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4C2×D1212yes—
365C2×C3×C4C2×D1212yes—
366C2×C3×C4C2×D1212yes—
367C2×C3×C4#23514yes—
368C2×C3×C4#23712yes—
369C2×C3×C4D3×Q812yes—
370C2×C3×C4C2×C3×D412yes—
371C2×C3×C4C2×C3×D412yes—
372C2×C3×C4C2×C3×D412yes—
373C2×C3×C4C2×C3×Q812yes—
374C2×C3×C4C3×#4114yes—
375C2×C3×C4C3×#4112yes—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3C2×C3×Q812yes—
388C2×Dic3C3×#4114yes—
389C4×D3—28no—
390Dic6—28no—
391C2×C32C2×C22⋊Q811yescharacteristic
392C2×C32#52511yescharacteristic
393C2×C32#52912yes—
394C2×C32#53211yescharacteristic
395C2×C32#53312yes—
396C3×D3—42no—
397C24—31no—
398C2×Q8—64no—
399C2×Q8—62no—
400C22×C4—31no—
401C22×C4—32no—
402C22×C4—34no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—32no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4C22×C3×D312yes—
409C22×C4C22×C3×D311yescharacteristic
410C22×C4C22×C3×D311yescharacteristic
411C22×C4C22×C3×D312yes—
412C22×C4C22×C3×D311yescharacteristic
413C4⋊C4—24no—
414C4⋊C4—64no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—64no—
418C4⋊C4—62no—
419C4⋊C4—64no—
420C22⋊C4—64no—
421C22⋊C4—62no—
422C22⋊C4—62no—
423C22⋊C4—64no—
424C42—24no—
425C42—62no—
426C42—64no—
427C42C3×D1214yes—
428C22×C3—21no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—62no—
434C22×C3—64no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3—122no—
438C22×C3C22×D1211yescharacteristic
439C22×C3C2×#23512yes—
440C22×C3C2×#23711yescharacteristic
441C22×C3C2×D3×Q811yescharacteristic
442C22×C3D3×#4112yes—
443C22×C3C22×C3×D411yescharacteristic
444C22×C3C22×C3×Q811yescharacteristic
445C22×C3C2×C3×#4112yes—
446C22×C3C2×C3×#4111yescharacteristic
447C22×C3C2×C3×#4112yes—
448C2×D3—22no—
449C2×D3—24no—
450C2×D3—41no—
451C2×D3—42no—
452C2×D3—41no—
453C3×C4—24no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—24no—
459C3×C4—22no—
460C3×C4—24no—
461C3×C4—42no—
462C3×C4—44no—
463C3×C4—44no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—62no—
467C3×C4—64no—
468C3×C4—121no—
469C3×C4—122no—
470C3×C4—121no—
471C3×C4#89714yes—
472C3×C4C3×C22⋊Q814yes—
473Dic3—22no—
474Dic3—24no—
475Dic3—41no—
476Dic3—42no—
477Dic3—41no—
478C32#239611yescharacteristic, complemented, Sylow
479C23—32no—
480C23—61no—
481C23—62no—
482C23—61no—
483C23—62no—
484C23C23×C3×D311yescharacteristic
485Q8—68no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—32no—
494C2×C4—34no—
495C2×C4—62no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—62no—
499C2×C4—68no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4C2×C3×D1212yes—
508C2×C4C2×C3×D1212yes—
509C2×C4C2×C3×D1212yes—
510C2×C4C3×#23514yes—
511C2×C4C3×#23712yes—
512C2×C4C3×D3×Q812yes—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—22no—
516C2×C3—22no—
517C2×C3—21no—
518C2×C3—122no—
519C2×C3#547711yescharacteristic
520C2×C3C2×#89711yescharacteristic
521C2×C3#552411yescharacteristic
522C2×C3#553812yes—
523C2×C3#557812yes—
524C2×C3C2×C3×C22⋊Q811yescharacteristic
525C2×C3C3×#52511yescharacteristic
526C2×C3C3×#52912yes—
527C2×C3C3×#53211yescharacteristic
528C2×C3C3×#53312yes—
529D3—42no—
530C22—62no—
531C22—64no—
532C22—121no—
533C22—122no—
534C22—121no—
535C22C22×C3×D1211yescharacteristic
536C22C2×C3×#23512yes—
537C22C2×C3×#23711yescharacteristic
538C22C2×C3×D3×Q811yescharacteristic
539C22C3×D3×#4112yes—
540C4—22no—
541C4—24no—
542C4—22no—
543C4—42no—
544C4—62no—
545C4—64no—
546C4—121no—
547C4—121no—
548C4—122no—
549C4C3×#89714yes—
550C3—21no—
551C3#7366811yescharacteristic, direct product factor
552C3C3×#239611yescharacteristic, complemented
553C2—122no—
554C2C3×#547711yescharacteristic
555C2C2×C3×#89711yescharacteristic
556C2C3×#552411yescharacteristic
557C2C3×#553812yes—
558C2C3×#557812yes—
subgroupG/H
centerC22×C3×C4C22×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC32×C4×C4⋊C4C2
socleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC3×C4×C4⋊C4D3
hypocenterC3C3×#2396
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×Dic6C2semidirect
3. C32×C4×C4⋊C4C2semidirect
5. C3×#4981C2semidirect
7. C3×C4×C4⋊Dic3C2semidirect
10. C3×#4659C2semidirect
11. C3×#4656C2semidirect
13. #73668C3direct (N × H)
42. C2×C4×Dic6C2×C3semidirect
51. C3×C4×C4⋊C4D3semidirect
52. C3×C4×C4⋊C4C2×C3semidirect
55. #4981C2×C3semidirect
57. C4×C4⋊Dic3C2×C3semidirect
60. #4659C2×C3semidirect
61. #4656C2×C3semidirect
75. C2×C3×C4×D3Q8semidirect
192. C4×C4⋊C4C3×D3semidirect
228. C2×C4×D3C3×Q8semidirect
478. C32#2396semidirect
551. C3#73668direct (N × H)
552. C3C3×#2396semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8722
C2×C3×C4×Dic62
C32×C4×C4⋊C42
C3×#49942
C3×#49812
C3×C4×(C2×D3)⋊C42
C3×C4×C4⋊Dic32
C3×#46772
C3×#46732
C3×#46592
C3×#46562
C3×#23963
#736683