Finite Group Databasebeta

← C3×#73641 #11805890

Subgroups — C3×#73641 #11805890

Order 1152.

1057 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#887C211yesmaximal, characteristic
2C32×C4×C4⋊C4C211yesmaximal, characteristic
3C3×#4975C211yesmaximal, characteristic
4C3×Dic3×C4⋊C4C211yesmaximal, characteristic
5C3×#4973C211yesmaximal, characteristic
6C3×#4933C212yesmaximal
7C3×C4×Dic3⋊C4C211yesmaximal, characteristic
8C3×C4×Dic3⋊C4C212yesmaximal
9C3×#4929C211yesmaximal, characteristic
10C3×#4652C212yesmaximal
11C3×#4650C212yesmaximal
12C3×#2355—31nomaximal
13#73641C311yesmaximal, characteristic, direct product factor
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C42C2211yescharacteristic
18C2×C32×C42C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4C2211yescharacteristic
21C2×C3×Dic3⋊C4C2212yes—
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×Dic3⋊C4C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×C4×Dic3C2212yes—
27C2×C3×C4×Dic3C2212yes—
28C2×C3×C4×Dic3C2211yescharacteristic
29C3×#887C418yescomplemented
30C32×#95C2212yes—
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2211yescharacteristic
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2211yescharacteristic
38C2×C3×C4×Q8—31no—
39C2×#887C2×C311yescharacteristic
40C3×#391—31no—
41C3×#391—31no—
42C3×#387—32no—
43C3×#387—32no—
44C3×#384—31no—
45C3×#382—32no—
46C3×C4×C4⋊C4—21no—
47C3×C4×C4⋊C4—31no—
48C3×C4×C4⋊C4—31no—
49C3×C4×C4⋊C4—32no—
50C3×C4×C4⋊C4D311yescharacteristic
51C3×C4×C4⋊C4C2×C311yescharacteristic
52#4975C2×C311yescharacteristic
53Dic3×C4⋊C4C2×C311yescharacteristic
54#4973C2×C311yescharacteristic
55#4933C2×C312yes—
56C4×Dic3⋊C4C2×C311yescharacteristic
57C4×Dic3⋊C4C2×C312yes—
58#4929C2×C311yescharacteristic
59#4652C2×C312yes—
60#4650C2×C312yes—
61C22×C32×C4C2312yes—
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2312yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C3×Dic3C2312yes—
67C22×C3×Dic3C2311yescharacteristic
68C22×C3×Dic3C2311yescharacteristic
69C22×C3×Dic3C2312yes—
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C2×C3×Dic6—22no—
73C2×C3×Dic6—22no—
74C2×C3×Dic6C2×C414yes—
75C32×C4⋊C4—22no—
76C32×C4⋊C4C2×C414yes—
77C32×C42—22no—
78C32×C42—42no—
79C3×C4⋊Dic3—22no—
80C3×Dic3⋊C4—22no—
81C3×Dic3⋊C4—24no—
82C3×Dic3⋊C4—42no—
83C3×Dic3⋊C4C2×C418yes—
84C3×Dic3⋊C4Q814yescomplemented
85C3×C4×Dic3—22no—
86C3×C4×Dic3—42no—
87C3×C4×Dic3C2×C414yes—
88C3×C4×Dic3C2×C418yes—
89#2355—31noSylow
90C22×C3×Q8—31no—
91C22×Dic6C22×C311yescharacteristic
92C3×C4×Q8—38no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C4⋊C4—32no—
97C2×C3×C4⋊C4—31no—
98C2×C3×C4⋊C4—32no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—32no—
101C2×C3×C4⋊C4C2×D311yescharacteristic
102C2×C3×C4⋊C4C2×D311yescharacteristic
103C2×C3×C4⋊C4C22×C311yescharacteristic
104C2×C3×C4⋊C4C22×C311yescharacteristic
105C2×C3×C42—21no—
106C2×C3×C42—22no—
107C2×C3×C42—32no—
108C2×C3×C42—31no—
109C2×C3×C42—31no—
110C2×C3×C42—32no—
111C2×C3×C42C2×D311yescharacteristic
112C2×C3×C42C2×D312yes—
113C2×C3×C42C22×C311yescharacteristic
114C2×C3×C42C22×C312yes—
115C2×C4⋊Dic3C22×C311yescharacteristic
116C2×Dic3⋊C4C22×C311yescharacteristic
117C2×Dic3⋊C4C22×C312yes—
118C2×Dic3⋊C4C22×C312yes—
119C2×Dic3⋊C4C22×C312yes—
120C2×Dic3⋊C4C22×C311yescharacteristic
121C2×C4×Dic3C22×C311yescharacteristic
122C2×C4×Dic3C22×C312yes—
123C2×C4×Dic3C22×C312yes—
124C2×C4×Dic3C22×C311yescharacteristic
125#887C3×C418yescomplemented
126C3×#95—22no—
127C3×#95—32no—
128C3×#95—31no—
129C3×#95—31no—
130C3×#95—32no—
131C3×#95—32no—
132C3×#95—32no—
133C3×#95—32no—
134C3×#95C2×D312yes—
135C3×#95C22×C312yes—
136C3⋊#95C22×C312yes—
137C3⋊#95C22×C312yes—
138C3⋊#95C22×C312yes—
139C3⋊#95C22×C311yescharacteristic
140C3⋊#95C22×C312yes—
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C311yescharacteristic
143C23×C32C2411yescharacteristic
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4C22×C414yes—
155C2×C32×C4C22×C412yes—
156C2×C32×C4C2×Q812yes—
157C2×C32×C4#4112yes—
158C2×C32×C4#4112yes—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3C22×C412yes—
172C2×C3×Dic3C22×C414yes—
173C2×C3×Dic3C22×C412yes—
174C2×C3×Dic3C2×Q812yes—
175C2×C3×Dic3C2×Q812yes—
176C2×C3×Dic3#4112yes—
177C3×Dic6—28no—
178C2×C4×Q8—31no—
179#391—31no—
180#391—31no—
181#387—32no—
182#387—32no—
183#384—31no—
184#382—32no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—32no—
188C4×C4⋊C4C3×D311yescharacteristic
189C2×C3×Q8—34no—
190C2×C3×Q8—62no—
191C2×C3×Q8—62no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—31no—
198C22×C3×C4—32no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—32no—
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C22×D312yes—
205C22×C3×C4C22×D312yes—
206C22×C3×C4C22×D311yescharacteristic
207C22×C3×C4C22×D311yescharacteristic
208C22×C3×C4C23×C311yescharacteristic
209C22×C3×C4C23×C312yes—
210C22×C3×C4C23×C312yes—
211C22×C3×C4C23×C311yescharacteristic
212C22×C3×C4C23×C311yescharacteristic
213C22×Dic3C23×C311yescharacteristic
214C22×Dic3C23×C312yes—
215C22×Dic3C23×C312yes—
216C22×Dic3C23×C311yescharacteristic
217C22×Dic3C23×C311yescharacteristic
218C22×Dic3C23×C311yescharacteristic
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6C2×C3×C414yes—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—38no—
227C3×C4⋊C4—34no—
228C3×C4⋊C4—62no—
229C3×C4⋊C4—64no—
230C3×C4⋊C4—62no—
231C3×C4⋊C4—122no—
232C3×C4⋊C4C4×D314yes—
233C3×C4⋊C4C2×C3×C414yes—
234C3×C42—24no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—34no—
238C3×C42—38no—
239C3×C42—42no—
240C3×C42—42no—
241C3×C42—44no—
242C3×C42—62no—
243C3×C42—122no—
244C4⋊Dic3—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—24no—
247Dic3⋊C4—42no—
248Dic3⋊C4C2×C3×C418yes—
249Dic3⋊C4C3×Q814yescomplemented
250C4×Dic3—22no—
251C4×Dic3—42no—
252C4×Dic3C2×C3×C414yes—
253C4×Dic3C2×C3×C418yes—
254C22×C32C23×C411yescharacteristic
255C22×C32C22×Q811yescharacteristic
256C22×C32C2×#4111yescharacteristic
257C22×C32C2×#4111yescharacteristic
258C22×C32C2×#4111yescharacteristic
259C22×C32H(5,𝔽2)11yescharacteristic
260C22×C322−1+411yescharacteristic
261C32×C4—22no—
262C32×C4—24no—
263C32×C4—41no—
264C32×C4—42no—
265C32×C4#11714yes—
266C3×Dic3—24no—
267C3×Dic3—22no—
268C3×Dic3—22no—
269C3×Dic3—41no—
270C3×Dic3—42no—
271C3×Dic3C4×Q814yes—
272C22×Q8—31no—
273C4×Q8—38no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4C2×C3×D311yescharacteristic
281C2×C4⋊C4C2×C3×D311yescharacteristic
282C2×C42—32no—
283C2×C42—31no—
284C2×C42—31no—
285C2×C42—32no—
286C2×C42C2×C3×D311yescharacteristic
287C2×C42C2×C3×D312yes—
288#95—32no—
289#95—31no—
290#95—31no—
291#95—32no—
292#95—32no—
293#95—32no—
294#95—32no—
295#95C2×C3×D312yes—
296C23×C3—21no—
297C23×C3C23×D311yescharacteristic
298C23×C3C24×C311yescharacteristic
299C3×Q8—68no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—34no—
334C2×C3×C4—32no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—61no—
353C2×C3×C4C2×C4×D314yes—
354C2×C3×C4C2×C4×D312yes—
355C2×C3×C4#23512yes—
356C2×C3×C4#23512yes—
357C2×C3×C4D3×Q812yes—
358C2×C3×C4C22×C3×C414yes—
359C2×C3×C4C22×C3×C412yes—
360C2×C3×C4C2×C3×Q812yes—
361C2×C3×C4C3×#4112yes—
362C2×C3×C4C3×#4112yes—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3C22×C3×C412yes—
376C2×Dic3C22×C3×C414yes—
377C2×Dic3C22×C3×C412yes—
378C2×Dic3C2×C3×Q812yes—
379C2×Dic3C2×C3×Q812yes—
380C2×Dic3C3×#4112yes—
381Dic6—28no—
382C2×C32C2×#11711yescharacteristic
383C2×C32C2×C4×Q811yescharacteristic
384C2×C32#52011yescharacteristic
385C2×C32#55411yescharacteristic
386C2×C32#55611yescharacteristic
387C2×C32#55711yescharacteristic
388C2×C32#55911yescharacteristic
389C2×Q8—34no—
390C2×Q8—62no—
391C2×Q8—62no—
392C22×C4—31no—
393C22×C4—32no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—32no—
398C22×C4C22×C3×D311yescharacteristic
399C22×C4C22×C3×D312yes—
400C22×C4C22×C3×D312yes—
401C22×C4C22×C3×D311yescharacteristic
402C22×C4C22×C3×D311yescharacteristic
403C4⋊C4—22no—
404C4⋊C4—38no—
405C4⋊C4—34no—
406C4⋊C4—62no—
407C4⋊C4—64no—
408C4⋊C4—62no—
409C4⋊C4—122no—
410C4⋊C4C3×C4×D314yes—
411C42—22no—
412C42—34no—
413C42—38no—
414C42—42no—
415C42—62no—
416C42—122no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3C22×C4×D311yescharacteristic
425C22×C3C2×#23511yescharacteristic
426C22×C3C2×#23511yescharacteristic
427C22×C3#100411yescharacteristic
428C22×C3C2×D3×Q811yescharacteristic
429C22×C3D3×#4111yescharacteristic
430C22×C3#101011yescharacteristic
431C22×C3C23×C3×C411yescharacteristic
432C22×C3C22×C3×Q811yescharacteristic
433C22×C3C2×C3×#4111yescharacteristic
434C22×C3C2×C3×#4111yescharacteristic
435C22×C3C2×C3×#4111yescharacteristic
436C22×C3C3×H(5,𝔽2)11yescharacteristic
437C22×C3C3×2−1+411yescharacteristic
438C3×C4—24no—
439C3×C4—22no—
440C3×C4—24no—
441C3×C4—24no—
442C3×C4—22no—
443C3×C4—24no—
444C3×C4—34no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—44no—
448C3×C4—42no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—64no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—121no—
456C3×C4—122no—
457C3×C4#87214yes—
458C3×C4C3×#11714yes—
459Dic3—24no—
460Dic3—22no—
461Dic3—22no—
462Dic3—42no—
463Dic3—41no—
464Dic3C3×C4×Q814yes—
465C32#235511yescharacteristic, complemented, Sylow
466C23C23×C3×D311yescharacteristic
467Q8—68no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—34no—
483C2×C4—32no—
484C2×C4—61no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—61no—
494C2×C4—62no—
495C2×C4—61no—
496C2×C4C2×C3×C4×D314yes—
497C2×C4C2×C3×C4×D312yes—
498C2×C4C3×#23512yes—
499C2×C4C3×#23512yes—
500C2×C4C3×D3×Q812yes—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3C2×#87211yescharacteristic
509C2×C3#551111yescharacteristic
510C2×C3#553311yescharacteristic
511C2×C3#554611yescharacteristic
512C2×C3#555911yescharacteristic
513C2×C3#556711yescharacteristic
514C2×C3C4×D3×Q811yescharacteristic
515C2×C3C2×C3×#11711yescharacteristic
516C2×C3C2×C3×C4×Q811yescharacteristic
517C2×C3C3×#52011yescharacteristic
518C2×C3C3×#55411yescharacteristic
519C2×C3C3×#55611yescharacteristic
520C2×C3C3×#55711yescharacteristic
521C2×C3C3×#55911yescharacteristic
522C22C22×C3×C4×D311yescharacteristic
523C22C2×C3×#23511yescharacteristic
524C22C2×C3×#23511yescharacteristic
525C22C3×#100411yescharacteristic
526C22C2×C3×D3×Q811yescharacteristic
527C22C3×D3×#4111yescharacteristic
528C22C3×#101011yescharacteristic
529C4—22no—
530C4—24no—
531C4—34no—
532C4—41no—
533C4—42no—
534C4—62no—
535C4—62no—
536C4—64no—
537C4—121no—
538C4—122no—
539C4C3×#87214yes—
540C3—21no—
541C3#7364111yescharacteristic, direct product factor
542C3C3×#235511yescharacteristic, complemented
543C2C2×C3×#87211yescharacteristic
544C2C3×#551111yescharacteristic
545C2C3×#553311yescharacteristic
546C2C3×#554611yescharacteristic
547C2C3×#555911yescharacteristic
548C2C3×#556711yescharacteristic
549C2C3×C4×D3×Q811yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×C4×C4⋊C4C2
socle, prime radicalC23×C32C24
cosocleC23×C3C24×C3
hypercenterC3×C4×C4⋊C4D3
hypocenterC3C3×#2355
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
13. #73641C3direct (N × H)
29. C3×#887C4semidirect
84. C3×Dic3⋊C4Q8semidirect
125. #887C3×C4semidirect
249. Dic3⋊C4C3×Q8semidirect
465. C32#2355semidirect
541. C3#73641direct (N × H)
542. C3C3×#2355semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8872
C32×C4×C4⋊C42
C3×#49752
C3×Dic3×C4⋊C42
C3×#49732
C3×#49332
C3×C4×Dic3⋊C42
C3×#49292
C3×#46522
C3×#46502
C3×#23553
#736413