Finite Group Databasebeta

← C3×#24208 #10642815

Subgroups — C3×#24208 #10642815

Order 768.

2202 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#127C211yesmaximal, characteristic, complemented
2C2×C3×#459C212yesmaximal, complemented
3C2×C3×#447C212yesmaximal, complemented
4C3×C4×#121C211yesmaximal, characteristic, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1982C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9C3×#1786C212yesmaximal, complemented
10#24208C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4—21no—
12C23×C3×D4—21no—
13C23×C3×D4C2212yes—
14C22×C3×D8C2214yescomplemented
15C2×C3×#127—28no—
16C2×C3×#127—21no—
17C2×C3×#127—21no—
18C2×C3×#127C2212yes—
19C2×C3×#127C2212yes—
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2211yescharacteristic, complemented
23C22×C3×C42C2211yescharacteristic
24C3×#459—28no—
25C3×#447—28no—
26C2×C3×C4⋊C8C2212yescomplemented
27C2×C3×#102C2214yescomplemented
28C2×C3×#102C2214yescomplemented
29C2×C3×#102C2214yescomplemented
30C2×C3×C22⋊C8C2212yes—
31C3×#390C2211yescharacteristic, complemented
32C3×#389C2211yescharacteristic, complemented
33C3×#388—21no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×C4×C4⋊C4C2211yescharacteristic, complemented
37C3×C4×C22⋊C4—21no—
38C3×#336C2212yescomplemented
39C22×#127C2×C311yescharacteristic, complemented
40C2×#459C2×C312yescomplemented
41C2×#447C2×C312yescomplemented
42C4×#121C2×C311yescharacteristic, complemented
43#2041C2×C312yescomplemented
44#2011C2×C312yescomplemented
45#1982C2×C312yescomplemented
46#1797C2×C311yescharacteristic, complemented
47#1786C2×C312yescomplemented
48C25×C3—22no—
49C22×C3×D4—28no—
50C22×C3×D4—28no—
51C22×C3×D4—21no—
52C22×C3×D4—28no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—24no—
59C22×C3×D4—24no—
60C22×C3×D4—44no—
61C22×C3×D4—44no—
62C22×C3×D4—44no—
63C22×C3×D4—44no—
64C22×C3×D4—44no—
65C22×C3×D4C2314yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C2×C3×D8—28no—
72C2×C3×D8—28no—
73C2×C3×D8—44no—
74C22×C3×C8C2314yes—
75C3×#127—28no—
76C3×#127—28no—
77C3×#127—28no—
78C3×#127—24no—
79C3×#127D418yescomplemented
80C3×#121—22no—
81C3×#121D414yescomplemented
82C3×C4×D4—24no—
83C3×C4×D4—44no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—24no—
94C2×C3×C42—21no—
95C2×C3×C42D411yescharacteristic
96C2×C3×C42D411yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C3×C4⋊C8—24no—
100C3×#102—28no—
101C3×#102—28no—
102C3×#102—44no—
103C3×C22⋊C8—24no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C23×D4—21no—
108C23×D4—21no—
109C23×D4C22×C312yes—
110C22×D8C22×C314yescomplemented
111C2×#127—28no—
112C2×#127—21no—
113C2×#127—21no—
114C2×#127C22×C312yes—
115C2×#127C22×C312yes—
116C2×#121C22×C311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4C22×C311yescharacteristic, complemented
119C22×C42C22×C311yescharacteristic
120#459—28no—
121#447—28no—
122C2×C4⋊C8C22×C312yescomplemented
123C2×#102C22×C314yescomplemented
124C2×#102C22×C314yescomplemented
125C2×#102C22×C314yescomplemented
126C2×C22⋊C8C22×C312yes—
127#390C22×C311yescharacteristic, complemented
128#389C22×C311yescharacteristic, complemented
129#388—21no—
130#386—21no—
131#380C22×C311yescharacteristic
132C4×C4⋊C4C22×C311yescharacteristic, complemented
133C4×C22⋊C4—21no—
134#336C22×C312yescomplemented
135C24×C3—28no—
136C24×C3—24no—
137C24×C3—21no—
138C24×C3—44no—
139C24×C3—44no—
140C24×C3—44no—
141C24×C3—44no—
142C24×C3—44no—
143C24×C3—44no—
144C24×C3C2×D411yescharacteristic
145C2×C3×D4—28no—
146C2×C3×D4—28no—
147C2×C3×D4—28no—
148C2×C3×D4—24no—
149C2×C3×D4—28no—
150C2×C3×D4—28no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—21no—
156C2×C3×D4—24no—
157C2×C3×D4—22no—
158C2×C3×D4—24no—
159C2×C3×D4—416no—
160C2×C3×D4—416no—
161C2×C3×D4—432no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4—432no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4C2×D418yes—
176C2×C3×D4C2×D412yescomplemented
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4D814yescomplemented
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C3×D8—416no—
200C2×C3×C8—24no—
201C2×C3×C8—24no—
202C2×C3×C8—24no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4C2×D412yescomplemented
208C3×C22⋊C4—24no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C22⋊C4—42no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42—42no—
217C3×C42C2×D414yes—
218C25—22no—
219C22×D4—28no—
220C22×D4—28no—
221C22×D4—21no—
222C22×D4—28no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4C23×C314yes—
236C22×D4C23×C311yescharacteristic
237C23×C4—21no—
238C23×C4C3×D411yescharacteristic
239C23×C4C3×D411yescharacteristic
240C23×C4C23×C311yescharacteristic
241C2×D8—28no—
242C2×D8—28no—
243C2×D8—44no—
244C22×C8C23×C314yes—
245#127—28no—
246#127—28no—
247#127—28no—
248#127—24no—
249#127C3×D418yescomplemented
250#121—22no—
251#121C3×D414yescomplemented
252C4×D4—24no—
253C4×D4—44no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42C3×D411yescharacteristic
266C2×C42C3×D411yescharacteristic
267C2×C42C23×C311yescharacteristic
268C2×C42C23×C311yescharacteristic
269C4⋊C8—24no—
270#102—28no—
271#102—28no—
272#102—44no—
273C22⋊C8—24no—
274#95—21no—
275#95—21no—
276#95—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—28no—
282C23×C3—22no—
283C23×C3—416no—
284C23×C3—416no—
285C23×C3—416no—
286C23×C3—41no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—44no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—41no—
298C23×C3—88no—
299C23×C3—88no—
300C23×C3—88no—
301C23×C3—88no—
302C23×C3#12112yes—
303C23×C3C2×D812yes—
304C23×C3AGL(1,ℤ/8)12yes—
305C23×C3C22×D411yescharacteristic
306C3×D4—24no—
307C3×D4—28no—
308C3×D4—28no—
309C3×D4—416no—
310C3×D4—416no—
311C3×D4—416no—
312C3×D4—416no—
313C3×D4—48no—
314C3×D4—44no—
315C3×D4—42no—
316C3×D4—44no—
317C3×D4—416no—
318C2×C3×C4—28no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—28no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—84no—
340C2×C3×C4#12012yes—
341C2×C3×C4#12012yes—
342C2×C3×C4#12112yes—
343C2×C3×C4C2×D812yes—
344C2×C3×C4C2×D812yes—
345C2×C3×C4AGL(1,ℤ/8)12yes—
346C2×C3×C4AGL(1,ℤ/8)12yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D412yes—
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)12yes—
351C3×C8—44no—
352C24—28no—
353C24—21no—
354C24—24no—
355C24—44no—
356C24—44no—
357C24—44no—
358C24—44no—
359C24—44no—
360C24—44no—
361C24C2×C3×D411yescharacteristic
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—24no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—432no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—432no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4C2×C3×D418yes—
393C2×D4C2×C3×D412yescomplemented
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4C3×D814yescomplemented
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D412yes—
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C3×#4112yes—
415C22×C4C24×C311yescharacteristic
416D8—416no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4C2×C3×D412yescomplemented
425C22⋊C4—24no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—42no—
434C42C2×C3×D414yes—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—48no—
442C22×C3—42no—
443C22×C3—48no—
444C22×C3—48no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—832no—
448C22×C3—81no—
449C22×C3—84no—
450C22×C3—81no—
451C22×C3—84no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—84no—
455C22×C3—84no—
456C22×C3#45914yescomplemented
457C22×C3C2×#12011yescharacteristic
458C22×C3C2×#12111yescharacteristic
459C22×C3#53511yescharacteristic
460C22×C3C22×D811yescharacteristic
461C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
462C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
463C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—42no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4#44714yescomplemented
472C3×C4D4212yes—
473C3×C4#54612yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—28no—
479C23—22no—
480C23—416no—
481C23—416no—
482C23—416no—
483C23—41no—
484C23—44no—
485C23—41no—
486C23—41no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—41no—
491C23—41no—
492C23—44no—
493C23—44no—
494C23—41no—
495C23—88no—
496C23—88no—
497C23—88no—
498C23—88no—
499C23C3×#12112yes—
500C23C2×C3×D812yes—
501C23C3×AGL(1,ℤ/8)12yes—
502C23C22×C3×D411yescharacteristic
503D4—24no—
504D4—28no—
505D4—28no—
506D4—416no—
507D4—416no—
508D4—416no—
509D4—416no—
510D4—48no—
511D4—416no—
512D4—44no—
513D4—44no—
514D4—42no—
515C2×C4—28no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—84no—
537C2×C4C3×#12012yes—
538C2×C4C3×#12012yes—
539C2×C4C3×#12112yes—
540C2×C4C2×C3×D812yes—
541C2×C4C2×C3×D812yes—
542C2×C4C3×AGL(1,ℤ/8)12yes—
543C2×C4C3×AGL(1,ℤ/8)12yes—
544C2×C4C22×C3×D411yescharacteristic
545C2×C4C22×C3×D412yes—
546C2×C4C2×C3×#4111yescharacteristic
547C2×C4C3×H(5,𝔽2)12yes—
548C8—44no—
549C2×C3—24no—
550C2×C3—82no—
551C2×C3—88no—
552C2×C3#240811yescharacteristic
553C2×C3C2×#44711yescharacteristic
554C2×C3#300311yescharacteristic
555C2×C3C2×#45911yescharacteristic
556C2×C3#303811yescharacteristic
557C2×C3#320511yescharacteristic
558C2×C3#321711yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—48no—
566C22—42no—
567C22—48no—
568C22—48no—
569C22—42no—
570C22—42no—
571C22—832no—
572C22—84no—
573C22—81no—
574C22—84no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—84no—
579C22—84no—
580C22C3×#45914yescomplemented
581C22C2×C3×#12011yescharacteristic
582C22C2×C3×#12111yescharacteristic
583C22C3×#53511yescharacteristic
584C22C22×C3×D811yescharacteristic
585C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
586C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
587C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4C3×#44714yescomplemented
596C4C3×D4212yes—
597C4C3×#54612yes—
598C3#2420811yescharacteristic, direct product factor, Sylow
599C2—24no—
600C2—88no—
601C2—82no—
602C2C3×#240811yescharacteristic
603C2C2×C3×#44711yescharacteristic
604C2C3×#300311yescharacteristic
605C2C2×C3×#45911yescharacteristic
606C2C3×#303811yescharacteristic
607C2C3×#320511yescharacteristic
608C2C3×#321711yescharacteristic
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. C22×C3×#127C2semidirect
2. C2×C3×#459C2semidirect
3. C2×C3×#447C2semidirect
4. C3×C4×#121C2semidirect
5. C3×#2041C2semidirect
6. C3×#2011C2semidirect
7. C3×#1982C2semidirect
8. C3×#1797C2semidirect
9. C3×#1786C2semidirect
10. #24208C3direct (N × H)
14. C22×C3×D8C22semidirect
22. C2×C3×C4×D4C22semidirect
26. C2×C3×C4⋊C8C22semidirect
27. C2×C3×#102C22semidirect
28. C2×C3×#102C22semidirect
29. C2×C3×#102C22semidirect
31. C3×#390C22semidirect
32. C3×#389C22semidirect
36. C3×C4×C4⋊C4C22semidirect
38. C3×#336C22semidirect
39. C22×#127C2×C3semidirect
40. C2×#459C2×C3semidirect
41. C2×#447C2×C3semidirect
42. C4×#121C2×C3semidirect
43. #2041C2×C3semidirect
44. #2011C2×C3semidirect
45. #1982C2×C3semidirect
46. #1797C2×C3semidirect
47. #1786C2×C3semidirect
79. C3×#127D4semidirect
81. C3×#121D4semidirect
110. C22×D8C22×C3semidirect
118. C2×C4×D4C22×C3semidirect
122. C2×C4⋊C8C22×C3semidirect
123. C2×#102C22×C3semidirect
124. C2×#102C22×C3semidirect
125. C2×#102C22×C3semidirect
127. #390C22×C3semidirect
128. #389C22×C3semidirect
132. C4×C4⋊C4C22×C3semidirect
134. #336C22×C3semidirect
176. C2×C3×D4C2×D4semidirect
190. C22×C3×C4D8semidirect
207. C3×C4⋊C4C2×D4semidirect
249. #127C3×D4semidirect
251. #121C3×D4semidirect
393. C2×D4C2×C3×D4semidirect
407. C22×C4C3×D8semidirect
424. C4⋊C4C2×C3×D4semidirect
456. C22×C3#459semidirect
471. C3×C4#447semidirect
580. C22C3×#459semidirect
595. C4C3×#447semidirect
598. C3#24208direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1272
C2×C3×#4592
C2×C3×#4472
C3×C4×#1212
C3×#20412
C3×#20112
C3×#19822
C3×#17972
C3×#17862
#242083