Finite Group Databasebeta

← C3×#71564 #11773224

Subgroups — C3×#71564 #11773224

Order 1152.

1433 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4⋊C4C211yesmaximal, characteristic
2C22×C3×C4×Dic3C211yesmaximal, characteristic
3C2×C3×C3⋊#95C211yesmaximal, characteristic
4C2×C3×C3⋊#95C212yesmaximal
5C2×C3×C3⋊#95C212yesmaximal
6C3×#1436—31nomaximal
7#71564C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C23×C32×C4C2211yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C23×C3×Dic3C2212yes—
12C23×C3×Dic3C2212yes—
13C2×C32×C4⋊C4—22no—
14C2×C32×C4⋊C4—22no—
15C2×C32×C4⋊C4C414yescomplemented
16C2×C3×C4×Dic3—22no—
17C2×C3×C4×Dic3—24no—
18C3×C3⋊#95—22no—
19C3×C3⋊#95—24no—
20C3×C3⋊#95C418yescomplemented
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4D311yescharacteristic
23C22×C3×C4⋊C4C2×C311yescharacteristic
24C22×C3×C42—31no—
25C22×C4×Dic3C2×C311yescharacteristic
26C2×C3×#95—32no—
27C2×C3×#95—32no—
28C2×C3×#95—31no—
29C2×C3⋊#95C2×C311yescharacteristic
30C2×C3⋊#95C2×C312yes—
31C2×C3⋊#95C2×C312yes—
32C24×C32C2311yescharacteristic
33C22×C32×C4—21no—
34C22×C32×C4—21no—
35C22×C32×C4—22no—
36C22×C32×C4—22no—
37C22×C32×C4—22no—
38C22×C32×C4—21no—
39C22×C32×C4—21no—
40C22×C32×C4—22no—
41C22×C32×C4—22no—
42C22×C32×C4C2×C412yes—
43C22×C32×C4C2×C414yes—
44C22×C32×C4C2×C412yes—
45C22×C32×C4C2×C412yes—
46C22×C32×C4D412yes—
47C22×C32×C4D412yes—
48C22×C3×Dic3—22no—
49C22×C3×Dic3—22no—
50C22×C3×Dic3—22no—
51C22×C3×Dic3—22no—
52C22×C3×Dic3—22no—
53C22×C3×Dic3—22no—
54C22×C3×Dic3—22no—
55C22×C3×Dic3—22no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3—22no—
58C22×C3×Dic3C2×C414yes—
59C22×C3×Dic3C2×C414yes—
60C22×C3×Dic3D414yes—
61C22×C3×Dic3Q814yes—
62C32×C4⋊C4—28no—
63C3×C4×Dic3—28no—
64#1436—31noSylow
65C23×C3×C4—21no—
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4—32no—
69C23×C3×C4—32no—
70C23×C3×C4C2×D311yescharacteristic
71C23×C3×C4C2×D311yescharacteristic
72C23×C3×C4C2×D311yescharacteristic
73C23×C3×C4C22×C311yescharacteristic
74C23×C3×C4C22×C311yescharacteristic
75C23×C3×C4C22×C311yescharacteristic
76C23×Dic3C22×C312yes—
77C23×Dic3C22×C312yes—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4Dic314yescomplemented
86C2×C3×C4⋊C4C3×C414yescomplemented
87C2×C3×C42—64no—
88C2×C3×C42—62no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—24no—
91C3×#95—38no—
92C3×#95—64no—
93C3×#95—62no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95C3×C418yescomplemented
97C23×C32C22×C412yes—
98C23×C32C22×C411yescharacteristic
99C23×C32C2×D412yes—
100C23×C32C2×D411yescharacteristic
101C23×C32C2×D411yescharacteristic
102C23×C32C2×Q812yes—
103C23×C32#4112yes—
104C23×C32#4112yes—
105C23×C32#4112yes—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—22no—
111C2×C32×C4—24no—
112C2×C32×C4—22no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—24no—
116C2×C32×C4—41no—
117C2×C32×C4—42no—
118C2×C32×C4—41no—
119C2×C32×C4C4214yescomplemented
120C2×C32×C4C22⋊C414yes—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—24no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—42no—
131C2×C3×Dic3—42no—
132C2×C3×Dic3—42no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3C4⋊C418yescomplemented
135C22×C4⋊C4C3×D311yescharacteristic
136C22×C42—31no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—31no—
140C24×C3—21no—
141C24×C3C22×D311yescharacteristic
142C24×C3C23×C311yescharacteristic
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—34no—
173C22×C3×C4—34no—
174C22×C3×C4—34no—
175C22×C3×C4—34no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—62no—
181C22×C3×C4—62no—
182C22×C3×C4—62no—
183C22×C3×C4—62no—
184C22×C3×C4—62no—
185C22×C3×C4—62no—
186C22×C3×C4—62no—
187C22×C3×C4—62no—
188C22×C3×C4—62no—
189C22×C3×C4—62no—
190C22×C3×C4C4×D314yes—
191C22×C3×C4C2×Dic312yes—
192C22×C3×C4C2×Dic312yes—
193C22×C3×C4C2×Dic312yes—
194C22×C3×C4#6712yes—
195C22×C3×C4#6712yes—
196C22×C3×C4C2×C3×C414yes—
197C22×C3×C4C2×C3×C412yes—
198C22×C3×C4C2×C3×C412yes—
199C22×C3×C4C2×C3×C412yes—
200C22×C3×C4C3×D412yes—
201C22×C3×C4C3×D412yes—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3C2×C3×C414yes—
213C22×Dic3C2×C3×C414yes—
214C22×Dic3C3×D414yes—
215C22×Dic3C3×Q814yes—
216C3×C4⋊C4—216no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—28no—
219C3×C42—68no—
220C4×Dic3—28no—
221C22×C32C2×C4211yescharacteristic
222C22×C32C2×C22⋊C411yescharacteristic
223C22×C32C2×C4⋊C412yes—
224C22×C32#11712yes—
225C22×C32#11711yescharacteristic
226C22×C32C4×D412yes—
227C22×C32C4×D412yes—
228C22×C32C4×D412yes—
229C22×C32C4×D412yes—
230C22×C32C4×Q812yes—
231C22×C32C4×Q812yes—
232C22×C32#12011yescharacteristic
233C22×C32#12112yes—
234C22×C32C22⋊Q812yes—
235C22×C32C22⋊Q812yes—
236C22×C32#12312yes—
237C22×C32#12311yescharacteristic
238C22×C32#12411yescharacteristic
239C22×C32#12512yes—
240C22×C32#12612yes—
241C22×C32C4⋊Q811yescharacteristic
242C32×C4—24no—
243C32×C4—24no—
244C32×C4—42no—
245C3×Dic3—28no—
246C3×Dic3—44no—
247C23×C4—32no—
248C23×C4—32no—
249C23×C4C2×C3×D311yescharacteristic
250C23×C4C2×C3×D311yescharacteristic
251C23×C4C2×C3×D311yescharacteristic
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C3×Dic314yescomplemented
255C2×C42—64no—
256C2×C42—62no—
257#95—38no—
258#95—64no—
259#95—62no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3C2×C4×D312yes—
270C23×C3#23512yes—
271C23×C3D3×D412yes—
272C23×C3#23712yes—
273C23×C3D3×Q812yes—
274C23×C3#23912yes—
275C23×C3C22×Dic311yescharacteristic
276C23×C3C2×#6711yescharacteristic
277C23×C3C2×#6711yescharacteristic
278C23×C3C22×C3×C412yes—
279C23×C3C22×C3×C411yescharacteristic
280C23×C3C2×C3×D412yes—
281C23×C3C2×C3×D411yescharacteristic
282C23×C3C2×C3×D411yescharacteristic
283C23×C3C2×C3×Q812yes—
284C23×C3C3×#4112yes—
285C23×C3C3×#4112yes—
286C23×C3C3×#4112yes—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—28no—
290C2×C3×C4—28no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—38no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—44no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—64no—
330C2×C3×C4—64no—
331C2×C3×C4—64no—
332C2×C3×C4—64no—
333C2×C3×C4—64no—
334C2×C3×C4—64no—
335C2×C3×C4—64no—
336C2×C3×C4—64no—
337C2×C3×C4—64no—
338C2×C3×C4—122no—
339C2×C3×C4—122no—
340C2×C3×C4—122no—
341C2×C3×C4—122no—
342C2×C3×C4C4×Dic314yescomplemented
343C2×C3×C4#21714yes—
344C2×C3×C4C3×C4214yescomplemented
345C2×C3×C4C3×C22⋊C414yes—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3C3×C4⋊C418yescomplemented
360C2×C32C4×C22⋊C411yescharacteristic
361C2×C32C4×C4⋊C412yes—
362C2×C32#38512yes—
363C2×C32#38611yescharacteristic
364C2×C32#38712yes—
365C2×C32#38712yes—
366C2×C32#38812yes—
367C2×C32#38912yes—
368C2×C32#39111yescharacteristic
369C24C22×C3×D311yescharacteristic
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—34no—
380C22×C4—34no—
381C22×C4—34no—
382C22×C4—34no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4C3×C4×D314yes—
394C22×C4C2×C3×Dic312yes—
395C22×C4C2×C3×Dic312yes—
396C22×C4C2×C3×Dic312yes—
397C22×C4C3×#6712yes—
398C22×C4C3×#6712yes—
399C4⋊C4—28no—
400C42—68no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3#88712yes—
423C22×C3#88912yes—
424C22×C3D3×C4⋊C412yes—
425C22×C3#89212yes—
426C22×C3#89312yes—
427C22×C3#89412yes—
428C22×C3#89612yes—
429C22×C3#89812yes—
430C22×C3C2×C4×Dic311yescharacteristic
431C22×C3#92611yescharacteristic
432C22×C3C4×#6712yes—
433C22×C3C4×#6712yes—
434C22×C3D4×Dic312yes—
435C22×C3#93511yescharacteristic
436C22×C3#93711yescharacteristic
437C22×C3#93912yes—
438C22×C3#94411yescharacteristic
439C22×C3Q8×Dic312yes—
440C22×C3#94612yes—
441C22×C3#94711yescharacteristic
442C22×C3C2×#21711yescharacteristic
443C22×C3C2×C3×C4211yescharacteristic
444C22×C3C2×C3×C22⋊C411yescharacteristic
445C22×C3C2×C3×C4⋊C412yes—
446C22×C3C3×#11712yes—
447C22×C3C3×#11711yescharacteristic
448C22×C3C3×C4×D412yes—
449C22×C3C3×C4×D412yes—
450C22×C3C3×C4×D412yes—
451C22×C3C3×C4×D412yes—
452C22×C3C3×C4×Q812yes—
453C22×C3C3×C4×Q812yes—
454C22×C3C3×#12011yescharacteristic
455C22×C3C3×#12112yes—
456C22×C3C3×C22⋊Q812yes—
457C22×C3C3×C22⋊Q812yes—
458C22×C3C3×#12312yes—
459C22×C3C3×#12311yescharacteristic
460C22×C3C3×#12411yescharacteristic
461C22×C3C3×#12512yes—
462C22×C3C3×#12612yes—
463C22×C3C3×C4⋊Q811yescharacteristic
464C3×C4—28no—
465C3×C4—24no—
466C3×C4—24no—
467C3×C4—24no—
468C3×C4—24no—
469C3×C4—44no—
470C3×C4—44no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—68no—
474C3×C4—124no—
475Dic3—28no—
476Dic3—44no—
477C32#143611yescharacteristic, complemented, Sylow
478C23C2×C3×C4×D312yes—
479C23C3×#23512yes—
480C23C3×D3×D412yes—
481C23C3×#23712yes—
482C23C3×D3×Q812yes—
483C23C3×#23912yes—
484C23C22×C3×Dic311yescharacteristic
485C23C2×C3×#6711yescharacteristic
486C23C2×C3×#6711yescharacteristic
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—38no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4C3×C4×Dic314yescomplemented
515C2×C4C3×#21714yes—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—22no—
525C2×C3Dic3×C4⋊C412yes—
526C2×C3#497612yes—
527C2×C3#497812yes—
528C2×C3#498012yes—
529C2×C3#498912yes—
530C2×C3#499012yes—
531C2×C3C4×#21711yescharacteristic
532C2×C3#522011yescharacteristic
533C2×C3#522911yescharacteristic
534C2×C3C3×C4×C22⋊C411yescharacteristic
535C2×C3C3×C4×C4⋊C412yes—
536C2×C3C3×#38512yes—
537C2×C3C3×#38611yescharacteristic
538C2×C3C3×#38712yes—
539C2×C3C3×#38712yes—
540C2×C3C3×#38812yes—
541C2×C3C3×#38912yes—
542C2×C3C3×#39111yescharacteristic
543C22C3×#88712yes—
544C22C3×#88912yes—
545C22C3×D3×C4⋊C412yes—
546C22C3×#89212yes—
547C22C3×#89312yes—
548C22C3×#89412yes—
549C22C3×#89612yes—
550C22C3×#89812yes—
551C22C2×C3×C4×Dic311yescharacteristic
552C22C3×#92611yescharacteristic
553C22C3×C4×#6712yes—
554C22C3×C4×#6712yes—
555C22C3×D4×Dic312yes—
556C22C3×#93511yescharacteristic
557C22C3×#93711yescharacteristic
558C22C3×#93912yes—
559C22C3×#94411yescharacteristic
560C22C3×Q8×Dic312yes—
561C22C3×#94612yes—
562C22C3×#94711yescharacteristic
563C22C2×C3×#21711yescharacteristic
564C4—24no—
565C4—24no—
566C4—42no—
567C4—68no—
568C4—124no—
569C3—21no—
570C3#7156411yescharacteristic, direct product factor
571C3C3×#143611yescharacteristic, complemented
572C2C3×Dic3×C4⋊C412yes—
573C2C3×#497612yes—
574C2C3×#497812yes—
575C2C3×#498012yes—
576C2C3×#498912yes—
577C2C3×#499012yes—
578C2C3×C4×#21711yescharacteristic
579C2C3×#522011yescharacteristic
580C2C3×#522911yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutatorC22×C3C2×C3×C42
FrattiniC24C22×C3×D3
FittingC22×C32×C4⋊C4C2
socle, prime radicalC24×C32C23
cosocleC24×C3C23×C3
hypercenterC22×C3×C4⋊C4D3
hypocenterC3C3×#1436
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #71564C3direct (N × H)
15. C2×C32×C4⋊C4C4semidirect
20. C3×C3⋊#95C4semidirect
85. C2×C3×C4⋊C4Dic3semidirect
86. C2×C3×C4⋊C4C3×C4semidirect
96. C3⋊#95C3×C4semidirect
119. C2×C32×C4C42semidirect
134. C2×C3×Dic3C4⋊C4semidirect
254. C2×C4⋊C4C3×Dic3semidirect
342. C2×C3×C4C4×Dic3semidirect
344. C2×C3×C4C3×C42semidirect
359. C2×Dic3C3×C4⋊C4semidirect
477. C32#1436semidirect
514. C2×C4C3×C4×Dic3semidirect
570. C3#71564direct (N × H)
571. C3C3×#1436semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C4⋊C42
C22×C3×C4×Dic32
C2×C3×C3⋊#952
C3×#14363
#715643