Finite Group Databasebeta

← C3×#25642 #10644249

Subgroups — C3×#25642 #10644249

Order 768.

1276 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic, complemented
4C3×#2548C211yesmaximal, characteristic, complemented
5C3×#2012C212yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1879C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25642C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C22×C3×D8C2212yes—
16C3×#559—24no—
17C3×#559—24no—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124—21no—
21C2×C3×#124—21no—
22C2×C3×#124C2211yescharacteristic
23C2×C3×#124C2211yescharacteristic
24C2×C3×#123—21no—
25C2×C3×#123—21no—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4—21no—
29C2×C3×C4×D4C2211yescharacteristic
30C2×C3×C4×D4C2211yescharacteristic
31C3×#496—24no—
32C3×#459—24no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#394—21no—
40C3×#394—21no—
41C3×#390—21no—
42C3×#390C2212yes—
43C3×#390C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#559C2×C311yescharacteristic, complemented
49C2×#496C2×C311yescharacteristic, complemented
50C2×#459C2×C311yescharacteristic, complemented
51#2548C2×C311yescharacteristic, complemented
52#2012C2×C312yescomplemented
53#2011C2×C312yescomplemented
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic, complemented
56#1880C2×C312yescomplemented
57#1879C2×C312yescomplemented
58#1762C2×C311yescharacteristic, complemented
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C2×C3×D8—24no—
76C2×C3×D8—42no—
77C2×C3×D8—42no—
78C2×C3×D8—42no—
79C22×C3×C8C2312yes—
80C22×C3×C8C2312yes—
81C3×#127—22no—
82C3×#127—22no—
83C3×#127—22no—
84C3×#127—22no—
85C3×#124—28no—
86C3×#124—22no—
87C3×#124—22no—
88C3×#124—22no—
89C3×#124—44no—
90C3×#124D414yes—
91C3×#123—28no—
92C3×#123—28no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×D4—28no—
98C3×C4×D4—22no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4D414yes—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4—22no—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C3×C4⋊C8—22no—
124C3×#103—42no—
125C3×#102—24no—
126C3×#102—24no—
127C3×#102—42no—
128C3×#97—22no—
129C3×#95—21no—
130C22×SD16C22×C312yes—
131C22×D8C22×C312yes—
132C22×D8C22×C312yes—
133#559—24no—
134#559—24no—
135C2×#127C22×C311yescharacteristic
136C2×#127C22×C311yescharacteristic
137C2×#124—21no—
138C2×#124—21no—
139C2×#124C22×C311yescharacteristic
140C2×#124C22×C311yescharacteristic
141C2×#123—21no—
142C2×#123—21no—
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×Q8C22×C311yescharacteristic
145C2×C4×D4—21no—
146C2×C4×D4C22×C311yescharacteristic
147C2×C4×D4C22×C311yescharacteristic
148#496—24no—
149#459—24no—
150C2×C4⋊C8C22×C311yescharacteristic
151C2×#103C22×C312yes—
152C2×#102C22×C312yes—
153C2×#102C22×C312yes—
154C2×#102C22×C312yes—
155C2×#97C22×C311yescharacteristic
156#394—21no—
157#394—21no—
158#390—21no—
159#390C22×C312yes—
160#390C22×C312yes—
161C4×C4⋊C4C22×C311yescharacteristic
162#340C22×C312yes—
163#336C22×C312yes—
164#336C22×C312yes—
165C24×C3—21no—
166C24×C3—21no—
167C24×C3—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—41no—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4—42no—
195C2×C3×D4—82no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D412yes—
198C2×C3×D4C2×D412yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—22no—
204C22×C3×C4—82no—
205C22×C3×C4—82no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C3×SD16—48no—
214C3×D8—48no—
215C3×D8—48no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8#4114yes—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4C2×D412yes—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4—44no—
238C3×C22⋊C4—82no—
239C3×C22⋊C4—82no—
240C3×C42—24no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—42no—
246C3×C42C2×D412yes—
247C3×C42C2×D412yes—
248C22×Q8C3×D411yescharacteristic
249C22×Q8C23×C311yescharacteristic
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C23×C312yes—
255C22×D4C23×C311yescharacteristic
256C22×D4C23×C311yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×D8—24no—
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—42no—
266C2×D8—42no—
267C2×D8—42no—
268C22×C8C23×C312yes—
269C22×C8C23×C312yes—
270#127—22no—
271#127—22no—
272#127—22no—
273#127—22no—
274#124—28no—
275#124—22no—
276#124—22no—
277#124—22no—
278#124—44no—
279#124C3×D414yes—
280#123—28no—
281#123—28no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×D4—28no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4C3×D414yes—
291C2×C4⋊C4—21no—
292C2×C4⋊C4C3×D411yescharacteristic
293C2×C4⋊C4C3×D411yescharacteristic
294C2×C4⋊C4C3×D411yescharacteristic
295C2×C4⋊C4C23×C312yes—
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42C23×C311yescharacteristic
310C2×C42C23×C311yescharacteristic
311C2×C42C23×C311yescharacteristic
312C4⋊C8—22no—
313#103—42no—
314#102—24no—
315#102—24no—
316#102—42no—
317#97—22no—
318#95—21no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3C22×D411yescharacteristic
337C3×Q8—24no—
338C3×Q8—44no—
339C3×Q8—41no—
340C3×Q8—41no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—24no—
344C3×D4—48no—
345C3×D4—41no—
346C3×D4—42no—
347C3×D4—41no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—84no—
353C3×D4—84no—
354C3×D4—84no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—84no—
375C2×C3×C4—84no—
376C2×C3×C4#12012yes—
377C2×C3×C4#12012yes—
378C2×C3×C4AGL(1,ℤ/8)12yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4112yes—
382C2×C3×C4H(5,𝔽2)12yes—
383C3×C8—24no—
384C3×C8—42no—
385C24—21no—
386C24—21no—
387C24—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—41no—
394C2×Q8C2×C3×D412yes—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4C2×C3×D412yes—
418C2×D4C2×C3×D412yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D411yescharacteristic
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C24×C311yescharacteristic
433SD16—48no—
434D8—48no—
435D8—48no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8C3×#4114yes—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4C2×C3×D412yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C42—24no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42C2×C3×D412yes—
467C42C2×C3×D412yes—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—82no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3C2×#12011yescharacteristic
486C22×C3#55012yes—
487C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
488C22×C3#57511yescharacteristic
489C22×C3#57612yes—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—42no—
498C3×C4—82no—
499C3×C4#53211yescharacteristic
500C3×C4#53511yescharacteristic
501C3×C4#54612yes—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23C22×C3×D411yescharacteristic
520Q8—24no—
521Q8—44no—
522Q8—41no—
523Q8—41no—
524D4—22no—
525D4—22no—
526D4—24no—
527D4—48no—
528D4—41no—
529D4—42no—
530D4—41no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—84no—
536D4—84no—
537D4—84no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C3×#12012yes—
560C2×C4C3×#12012yes—
561C2×C4C3×AGL(1,ℤ/8)12yes—
562C2×C4C22×C3×D411yescharacteristic
563C2×C4C22×C3×D411yescharacteristic
564C2×C4C2×C3×#4112yes—
565C2×C4C3×H(5,𝔽2)12yes—
566C8—24no—
567C8—42no—
568C2×C3—82no—
569C2×C3—82no—
570C2×C3—162no—
571C2×C3#243111yescharacteristic
572C2×C3#300611yescharacteristic
573C2×C3#300811yescharacteristic
574C2×C3#339912yes—
575C2×C3#340312yes—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22C2×C3×#12011yescharacteristic
594C22C3×#55012yes—
595C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
596C22C3×#57511yescharacteristic
597C22C3×#57612yes—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—82no—
607C4C3×#53211yescharacteristic
608C4C3×#53511yescharacteristic
609C4C3×#54612yes—
610C3#2564211yescharacteristic, direct product factor, Sylow
611C2—82no—
612C2—82no—
613C2—162no—
614C2C3×#243111yescharacteristic
615C2C3×#300611yescharacteristic
616C2C3×#300811yescharacteristic
617C2C3×#339912yes—
618C2C3×#340312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#559C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#459C2semidirect
4. C3×#2548C2semidirect
5. C3×#2012C2semidirect
6. C3×#2011C2semidirect
7. C3×#1985C2semidirect
8. C3×#1983C2semidirect
9. C3×#1880C2semidirect
10. C3×#1879C2semidirect
11. C3×#1762C2semidirect
12. #25642C3direct (N × H)
48. C2×#559C2×C3semidirect
49. C2×#496C2×C3semidirect
50. C2×#459C2×C3semidirect
51. #2548C2×C3semidirect
52. #2012C2×C3semidirect
53. #2011C2×C3semidirect
54. #1985C2×C3semidirect
55. #1983C2×C3semidirect
56. #1880C2×C3semidirect
57. #1879C2×C3semidirect
58. #1762C2×C3semidirect
610. C3#25642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4962
C2×C3×#4592
C3×#25482
C3×#20122
C3×#20112
C3×#19852
C3×#19832
C3×#18802
C3×#18792
C3×#17622
#256423