Finite Group Databasebeta

← C3×#25583 #10644190

Subgroups — C3×#25583 #10644190

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#552C211yesmaximal, characteristic, complemented
2C2×C3×#490C211yesmaximal, characteristic, complemented
3C2×C3×#460C211yesmaximal, characteristic, complemented
4C3×#2541C211yesmaximal, characteristic
5C3×#2000C212yesmaximal, complemented
6C3×#2000C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1875C212yesmaximal, complemented
10C3×#1874C212yesmaximal, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25583C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C3×#552—24no—
15C3×#552—24no—
16C2×C3×#127—21no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#126—21no—
19C2×C3×#125—21no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×#123—21no—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×#121C2212yes—
25C2×C3×#121C2212yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117—21no—
29C2×C3×#117C2211yescharacteristic
30C22×C3×C4⋊C4C2211yescharacteristic
31C3×#490—24no—
32C3×#460—24no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C2×C3×#97C2211yescharacteristic
41C3×#392—21no—
42C3×#392—21no—
43C3×#390—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#552C2×C311yescharacteristic, complemented
49C2×#490C2×C311yescharacteristic, complemented
50C2×#460C2×C311yescharacteristic, complemented
51#2541C2×C311yescharacteristic
52#2000C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic, complemented
56#1875C2×C312yescomplemented
57#1874C2×C312yescomplemented
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4D411yescharacteristic
65C22×C3×D4C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C2×C3×SD16—24no—
70C2×C3×SD16—42no—
71C2×C3×SD16—42no—
72C22×C3×C8C2312yes—
73C22×C3×C8C2312yes—
74C3×#127—22no—
75C3×#127—22no—
76C3×#127—44no—
77C3×#126—28no—
78C3×#125—28no—
79C3×#125—22no—
80C3×#125D414yescomplemented
81C3×#123—28no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#121—42no—
89C3×#121—42no—
90C3×C4×Q8—22no—
91C3×C4×Q8D414yescomplemented
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×#117—28no—
95C3×#117—22no—
96C3×#117—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C3×C4⋊C8—22no—
122C3×#103—24no—
123C3×#102—24no—
124C3×#102—24no—
125C3×#102—24no—
126C3×C22⋊C8—42no—
127C3×C22⋊C8—42no—
128C3×#97—22no—
129C3×#95—21no—
130C22×SD16C22×C312yes—
131#552—24no—
132#552—24no—
133C2×#127—21no—
134C2×#127C22×C311yescharacteristic
135C2×#126—21no—
136C2×#125—21no—
137C2×#125C22×C311yescharacteristic
138C2×#123—21no—
139C2×C22⋊Q8C22×C311yescharacteristic
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#121C22×C312yes—
142C2×#121C22×C312yes—
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×D4C22×C311yescharacteristic
145C2×#117—21no—
146C2×#117C22×C311yescharacteristic
147C22×C4⋊C4C22×C311yescharacteristic
148#490—24no—
149#460—24no—
150C2×C4⋊C8C22×C311yescharacteristic
151C2×#103C22×C312yes—
152C2×#102C22×C312yes—
153C2×#102C22×C312yes—
154C2×#102C22×C312yes—
155C2×C22⋊C8C22×C312yes—
156C2×C22⋊C8C22×C312yes—
157C2×#97C22×C311yescharacteristic
158#392—21no—
159#392—21no—
160#390—21no—
161C4×C4⋊C4C22×C311yescharacteristic
162#340C22×C312yes—
163#340C22×C312yes—
164#340C22×C312yes—
165C24×C3—22no—
166C24×C3C2×D411yescharacteristic
167C24×C3C2×D411yescharacteristic
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8C2×D412yes—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—41no—
181C2×C3×D4—44no—
182C2×C3×D4—44no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—82no—
190C2×C3×D4—82no—
191C2×C3×D4—82no—
192C22×C3×C4—24no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C3×SD16—48no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C4⋊C4C2×D412yes—
234C3×C4⋊C4#4114yes—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—44no—
241C3×C22⋊C4—44no—
242C3×C42—24no—
243C3×C42—24no—
244C3×C42—42no—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247C22×Q8C23×C311yescharacteristic
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C3×D411yescharacteristic
253C22×D4C23×C312yes—
254C23×C4—21no—
255C23×C4C23×C311yescharacteristic
256C23×C4C23×C311yescharacteristic
257C2×SD16—24no—
258C2×SD16—42no—
259C2×SD16—42no—
260C22×C8C23×C312yes—
261C22×C8C23×C312yes—
262#127—22no—
263#127—22no—
264#127—44no—
265#126—28no—
266#125—28no—
267#125—22no—
268#125C3×D414yescomplemented
269#123—28no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274#121—24no—
275#121—24no—
276#121—42no—
277#121—42no—
278C4×Q8—22no—
279C4×Q8C3×D414yescomplemented
280C4×D4—22no—
281C4×D4—22no—
282#117—28no—
283#117—22no—
284#117—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4C3×D411yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C312yes—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4C3×D411yescharacteristic
304C2×C22⋊C4C3×D411yescharacteristic
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42C23×C311yescharacteristic
309C4⋊C8—22no—
310#103—24no—
311#102—24no—
312#102—24no—
313#102—24no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316#97—22no—
317#95—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—22no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3AGL(1,ℤ/8)12yes—
335C23×C3C22×D411yescharacteristic
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—42no—
339C3×D4—44no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—84no—
345C3×D4—84no—
346C3×D4—88no—
347C3×D4—88no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—48no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4#12012yes—
376C2×C3×C4#12012yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4112yes—
380C2×C3×C4H(5,𝔽2)12yes—
381C3×C8—42no—
382C3×C8—42no—
383C24—22no—
384C24C2×C3×D411yescharacteristic
385C24C2×C3×D411yescharacteristic
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8C2×C3×D412yes—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—41no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C24×C311yescharacteristic
431SD16—48no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4C2×C3×D412yes—
450C4⋊C4C2×C3×D412yes—
451C4⋊C4C2×C3×D412yes—
452C4⋊C4C3×#4114yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C42—24no—
461C42—24no—
462C42—42no—
463C42C2×C3×D412yes—
464C42C2×C3×D412yes—
465C22×C3—22no—
466C22×C3—24no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—82no—
481C22×C3—82no—
482C22×C3C2×#12011yescharacteristic
483C22×C3#54612yes—
484C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
485C22×C3#57511yescharacteristic
486C22×C3#57712yes—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4#53211yescharacteristic
497C3×C4#53511yescharacteristic
498C3×C4#55012yes—
499C23—21no—
500C23—22no—
501C23—21no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—21no—
507C23—22no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C3×AGL(1,ℤ/8)12yes—
516C23C22×C3×D411yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—42no—
520D4—44no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—84no—
526D4—84no—
527D4—88no—
528D4—88no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4C3×#12012yes—
557C2×C4C3×#12012yes—
558C2×C4C22×C3×D411yescharacteristic
559C2×C4C22×C3×D411yescharacteristic
560C2×C4C2×C3×#4112yes—
561C2×C4C3×H(5,𝔽2)12yes—
562C8—42no—
563C8—42no—
564C2×C3—42no—
565C2×C3—81no—
566C2×C3—162no—
567C2×C3#243111yescharacteristic
568C2×C3#300611yescharacteristic
569C2×C3#301011yescharacteristic
570C2×C3#331012yes—
571C2×C3#331712yes—
572C22—22no—
573C22—24no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22C2×C3×#12011yescharacteristic
590C22C3×#54612yes—
591C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
592C22C3×#57511yescharacteristic
593C22C3×#57712yes—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—44no—
603C4C3×#53211yescharacteristic
604C4C3×#53511yescharacteristic
605C4C3×#55012yes—
606C3#2558311yescharacteristic, direct product factor, Sylow
607C2—42no—
608C2—81no—
609C2—162no—
610C2C3×#243111yescharacteristic
611C2C3×#300611yescharacteristic
612C2C3×#301011yescharacteristic
613C2C3×#331012yes—
614C2C3×#331712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#552C2semidirect
2. C2×C3×#490C2semidirect
3. C2×C3×#460C2semidirect
5. C3×#2000C2semidirect
6. C3×#2000C2semidirect
7. C3×#1989C2semidirect
8. C3×#1955C2semidirect
9. C3×#1875C2semidirect
10. C3×#1874C2semidirect
11. C3×#1763C2semidirect
12. #25583C3direct (N × H)
48. C2×#552C2×C3semidirect
49. C2×#490C2×C3semidirect
50. C2×#460C2×C3semidirect
52. #2000C2×C3semidirect
53. #2000C2×C3semidirect
54. #1989C2×C3semidirect
55. #1955C2×C3semidirect
56. #1875C2×C3semidirect
57. #1874C2×C3semidirect
58. #1763C2×C3semidirect
80. C3×#125D4semidirect
91. C3×C4×Q8D4semidirect
268. #125C3×D4semidirect
279. C4×Q8C3×D4semidirect
606. C3#25583direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×C3×#4902
C2×C3×#4602
C3×#25412
C3×#20002
C3×#19892
C3×#19552
C3×#18752
C3×#18742
C3×#17632
#255833