Finite Group Databasebeta

← C3×#25496 #10644103

Subgroups — C3×#25496 #10644103

Order 768.

1474 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×#532C211yesmaximal, characteristic, complemented
2C2×C3×#466C212yesmaximal, complemented
3C3×#2854C211yesmaximal, characteristic, complemented
4C3×#2057C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic, complemented
6C3×#2023C211yesmaximal, characteristic, complemented
7C3×#2011C211yesmaximal, characteristic, complemented
8C3×#1893C212yesmaximal, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C2×C3×#336C211yesmaximal, characteristic, complemented
12#25496C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×D8C2212yes—
15C23×C3×C8C2212yes—
16C3×#532—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124—21no—
20C2×C3×#123—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2211yescharacteristic
23C2×C3×#121C2212yes—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2212yes—
26C22×C3×C42C2211yescharacteristic
27C3×#466—28no—
28C2×C3×#107C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×#386—21no—
40C3×#380C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#532C2×C311yescharacteristic, complemented
47C2×#466C2×C312yescomplemented
48#2854C2×C311yescharacteristic, complemented
49#2057C2×C311yescharacteristic, complemented
50#2041C2×C311yescharacteristic, complemented
51#2023C2×C311yescharacteristic, complemented
52#2011C2×C311yescharacteristic, complemented
53#1893C2×C312yescomplemented
54#1879C2×C312yescomplemented
55#1862C2×C312yescomplemented
56C2×#336C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2311yescharacteristic
66C22×C3×D4C2312yes—
67C23×C3×C4—22no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C2×C3×D8—24no—
72C2×C3×D8—24no—
73C2×C3×D8—42no—
74C22×C3×C8—24no—
75C22×C3×C8—24no—
76C22×C3×C8—42no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C3×C4⋊Q8—24no—
80C3×#127—24no—
81C3×#124—28no—
82C3×#123—44no—
83C3×#121—28no—
84C3×#121—24no—
85C3×#121—24no—
86C3×#121D418yes—
87C3×C4×D4—28no—
88C3×C4×D4—24no—
89C3×C4×D4—24no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—24no—
106C2×C3×C42D411yescharacteristic
107C2×C3×C42D411yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#102—24no—
112C3×#102—24no—
113C3×#102—24no—
114C3×#102—42no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C22×#41—21no—
119C22×D8C22×C312yes—
120C23×C8C22×C312yes—
121#532—28no—
122C2×C4⋊Q8C22×C311yescharacteristic
123C2×#127C22×C311yescharacteristic
124C2×#124—21no—
125C2×#123—21no—
126C2×#121—21no—
127C2×#121C22×C311yescharacteristic
128C2×#121C22×C312yes—
129C2×C4×D4—21no—
130C2×C4×D4C22×C312yes—
131C22×C42C22×C311yescharacteristic
132#466—28no—
133C2×#107C22×C312yes—
134C2×#102C22×C312yes—
135C2×#102C22×C312yes—
136C2×#102C22×C312yes—
137C2×#102C22×C312yes—
138#390C22×C311yescharacteristic
139#390C22×C311yescharacteristic
140#389C22×C311yescharacteristic
141#389C22×C311yescharacteristic
142#388—21no—
143#388—21no—
144#386—21no—
145#380C22×C311yescharacteristic
146#340C22×C312yes—
147#340C22×C312yes—
148#336C22×C311yescharacteristic
149#336C22×C312yes—
150#336C22×C311yescharacteristic
151C24×C3—22no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C3×#41—432no—
155C2×C3×Q8—24no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×D4—24no—
159C2×C3×D4—24no—
160C2×C3×D4—28no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4C2×D414yes—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—28no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—41no—
193C22×C3×C4—44no—
194C22×C3×C4—44no—
195C22×C3×C4D814yescomplemented
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C3×D8—48no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—48no—
210C2×C3×C8#4114yescomplemented
211C3×C4⋊C4—24no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4C2×D414yes—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—42no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—22no—
232C2×#41—28no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C23×C311yescharacteristic
241C22×D4C23×C312yes—
242C23×C4—22no—
243C23×C4C3×D411yescharacteristic
244C23×C4C3×D411yescharacteristic
245C23×C4C23×C311yescharacteristic
246C2×D8—24no—
247C2×D8—24no—
248C2×D8—42no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—42no—
252C22×C8C23×C312yes—
253C22×C8C23×C312yes—
254C4⋊Q8—24no—
255#127—24no—
256#124—28no—
257#123—44no—
258#121—28no—
259#121—24no—
260#121—24no—
261#121C3×D418yes—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C312yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—24no—
281C2×C42C3×D411yescharacteristic
282C2×C42C3×D411yescharacteristic
283C2×C42C23×C311yescharacteristic
284C2×C42C23×C311yescharacteristic
285#107—24no—
286#102—24no—
287#102—24no—
288#102—24no—
289#102—42no—
290#95—21no—
291#95—21no—
292#95—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—24no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3#12012yes—
312C23×C3C2×D812yes—
313C23×C3AGL(1,ℤ/8)12yes—
314C23×C3C22×D411yescharacteristic
315C3×Q8—48no—
316C3×D4—24no—
317C3×D4—24no—
318C3×D4—48no—
319C3×D4—48no—
320C3×D4—416no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—84no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—28no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—21no—
338C2×C3×C4—24no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—44no—
342C2×C3×C4—416no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4#12012yes—
355C2×C3×C4C2×D812yes—
356C2×C3×C4C2×D812yes—
357C2×C3×C4#13512yes—
358C2×C3×C4C22×D412yes—
359C2×C3×C4C2×#4112yes—
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C42−1+411yescharacteristic
362C3×C8—24no—
363C3×C8—42no—
364C24—22no—
365C24—21no—
366C24C2×C3×D411yescharacteristic
367#41—432no—
368C2×Q8—24no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4C2×C3×D414yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4C3×D814yescomplemented
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D414yes—
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C24×C311yescharacteristic
416D8—48no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—48no—
423C2×C8C3×#4114yescomplemented
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4C2×C3×D414yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3#44714yes—
466C22×C3C2×#12011yescharacteristic
467C22×C3#53211yescharacteristic
468C22×C3#53711yescharacteristic
469C22×C3C22×D811yescharacteristic
470C22×C3C2×#13511yescharacteristic
471C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
472C22×C3#57511yescharacteristic
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—44no—
478C3×C4—41no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4#54612yes—
483C3×C4#54712yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C3×#12012yes—
503C23C2×C3×D812yes—
504C23C3×AGL(1,ℤ/8)12yes—
505C23C22×C3×D411yescharacteristic
506Q8—48no—
507D4—24no—
508D4—24no—
509D4—48no—
510D4—48no—
511D4—416no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—84no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—28no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—416no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4C3×#12012yes—
546C2×C4C2×C3×D812yes—
547C2×C4C2×C3×D812yes—
548C2×C4C3×#13512yes—
549C2×C4C22×C3×D412yes—
550C2×C4C2×C3×#4112yes—
551C2×C4C3×H(5,𝔽2)11yescharacteristic
552C2×C4C3×2−1+411yescharacteristic
553C8—24no—
554C8—42no—
555C2×C3—24no—
556C2×C3—84no—
557C2×C3—161no—
558C2×C3#243011yescharacteristic
559C2×C3C2×#44711yescharacteristic
560C2×C3#300211yescharacteristic
561C2×C3#316911yescharacteristic
562C2×C3#318011yescharacteristic
563C2×C3#324511yescharacteristic
564C2×C3#324711yescharacteristic
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C3×#44714yes—
586C22C2×C3×#12011yescharacteristic
587C22C3×#53211yescharacteristic
588C22C3×#53711yescharacteristic
589C22C22×C3×D811yescharacteristic
590C22C2×C3×#13511yescharacteristic
591C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
592C22C3×#57511yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—44no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4C3×#54612yes—
603C4C3×#54712yes—
604C3#2549611yescharacteristic, direct product factor, Sylow
605C2—24no—
606C2—84no—
607C2—161no—
608C2C3×#243011yescharacteristic
609C2C2×C3×#44711yescharacteristic
610C2C3×#300211yescharacteristic
611C2C3×#316911yescharacteristic
612C2C3×#318011yescharacteristic
613C2C3×#324511yescharacteristic
614C2C3×#324711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×C3×#466C2semidirect
3. C3×#2854C2semidirect
4. C3×#2057C2semidirect
5. C3×#2041C2semidirect
6. C3×#2023C2semidirect
7. C3×#2011C2semidirect
8. C3×#1893C2semidirect
9. C3×#1879C2semidirect
10. C3×#1862C2semidirect
11. C2×C3×#336C2semidirect
12. #25496C3direct (N × H)
46. C2×#532C2×C3semidirect
47. C2×#466C2×C3semidirect
48. #2854C2×C3semidirect
49. #2057C2×C3semidirect
50. #2041C2×C3semidirect
51. #2023C2×C3semidirect
52. #2011C2×C3semidirect
53. #1893C2×C3semidirect
54. #1879C2×C3semidirect
55. #1862C2×C3semidirect
56. C2×#336C2×C3semidirect
195. C22×C3×C4D8semidirect
210. C2×C3×C8#41semidirect
408. C22×C4C3×D8semidirect
423. C2×C8C3×#41semidirect
604. C3#25496direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4662
C3×#28542
C3×#20572
C3×#20412
C3×#20232
C3×#20112
C3×#18932
C3×#18792
C3×#18622
C2×C3×#3362
#254963