Finite Group Databasebeta

← C3×#25602 #10644209

Subgroups — C3×#25602 #10644209

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C3×#2850C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1877C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#25602C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C3×#532—24no—
17C3×#532—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121—21no—
24C2×C3×#121C2212yes—
25C2×C3×#121C2211yescharacteristic
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2212yes—
28C22×C3×C42C2211yescharacteristic
29C3×#496—24no—
30C3×#489—24no—
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×#97C2211yescharacteristic
37C2×C3×#97C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#388—21no—
41C3×#388—21no—
42C3×#386—21no—
43C3×#386—21no—
44C3×#382C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#532C2×C311yescharacteristic, complemented
49C2×#496C2×C311yescharacteristic, complemented
50C2×#489C2×C311yescharacteristic, complemented
51#2850C2×C311yescharacteristic, complemented
52#2000C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1993C2×C311yescharacteristic, complemented
55#1961C2×C311yescharacteristic, complemented
56#1879C2×C312yescomplemented
57#1877C2×C312yescomplemented
58#1749C2×C311yescharacteristic, complemented
59C2×C3×#41—28no—
60C2×C3×#41—44no—
61C2×C3×#41—44no—
62C22×C3×Q8—21no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—22no—
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C22×C3×D4C2312yes—
70C23×C3×C4—22no—
71C23×C3×C4D411yescharacteristic
72C23×C3×C4D411yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C2×C3×SD16—24no—
75C2×C3×SD16—24no—
76C2×C3×SD16—42no—
77C2×C3×D8—24no—
78C2×C3×D8—24no—
79C2×C3×D8—42no—
80C22×C3×C8C2312yes—
81C22×C3×C8C2312yes—
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8D414yescomplemented
84C3×#127—22no—
85C3×#127D414yescomplemented
86C3×#124—28no—
87C3×#124—22no—
88C3×#124—22no—
89C3×C22⋊Q8—42no—
90C3×#121—28no—
91C3×#121—24no—
92C3×#121—24no—
93C3×#121—42no—
94C3×C4×D4—28no—
95C3×C4×D4—24no—
96C3×C4×D4—24no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—22no—
111C2×C3×C42—22no—
112C2×C3×C42D411yescharacteristic
113C2×C3×C42D411yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#103—24no—
117C3×#102—24no—
118C3×#102—42no—
119C3×#102—42no—
120C3×C22⋊C8—24no—
121C3×#97—22no—
122C3×#97—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C22×#41—21no—
128C22×SD16C22×C312yes—
129C22×D8C22×C312yes—
130#532—24no—
131#532—24no—
132C2×C4⋊Q8C22×C311yescharacteristic
133C2×#127C22×C311yescharacteristic
134C2×#124—21no—
135C2×#124C22×C311yescharacteristic
136C2×C22⋊Q8C22×C311yescharacteristic
137C2×#121—21no—
138C2×#121C22×C312yes—
139C2×#121C22×C311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C22×C312yes—
142C22×C42C22×C311yescharacteristic
143#496—24no—
144#489—24no—
145C2×#103C22×C312yes—
146C2×#102C22×C312yes—
147C2×#102C22×C312yes—
148C2×#102C22×C312yes—
149C2×C22⋊C8C22×C312yes—
150C2×#97C22×C311yescharacteristic
151C2×#97C22×C311yescharacteristic
152#391C22×C311yescharacteristic
153#390C22×C311yescharacteristic
154#388—21no—
155#388—21no—
156#386—21no—
157#386—21no—
158#382C22×C311yescharacteristic
159#340C22×C312yes—
160#340C22×C312yes—
161#336C22×C312yes—
162C24×C3—22no—
163C24×C3—21no—
164C24×C3C2×D411yescharacteristic
165C3×#41—432no—
166C2×C3×Q8—24no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—42no—
172C2×C3×Q8—42no—
173C2×C3×D4—28no—
174C2×C3×D4—24no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
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—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4C2×D414yes—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—28no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—41no—
209C22×C3×C4—41no—
210C22×C3×C4—44no—
211C22×C3×C4—44no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×SD16—48no—
219C3×D8—48no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8#4114yescomplemented
226C3×C4⋊C4—24no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4C2×D414yes—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—42no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42C2×D412yes—
245C3×C42C2×D412yes—
246C2×#41—28no—
247C2×#41—44no—
248C2×#41—44no—
249C22×Q8—21no—
250C22×Q8C23×C311yescharacteristic
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C23×C311yescharacteristic
256C22×D4C23×C312yes—
257C23×C4—22no—
258C23×C4C3×D411yescharacteristic
259C23×C4C3×D411yescharacteristic
260C23×C4C23×C311yescharacteristic
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—42no—
267C22×C8C23×C312yes—
268C22×C8C23×C312yes—
269C4⋊Q8—22no—
270C4⋊Q8C3×D414yescomplemented
271#127—22no—
272#127C3×D414yescomplemented
273#124—28no—
274#124—22no—
275#124—22no—
276C22⋊Q8—42no—
277#121—28no—
278#121—24no—
279#121—24no—
280#121—42no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C312yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C3×D411yescharacteristic
300C2×C42C3×D411yescharacteristic
301C2×C42C23×C311yescharacteristic
302C2×C42C23×C311yescharacteristic
303#103—24no—
304#102—24no—
305#102—42no—
306#102—42no—
307C22⋊C8—24no—
308#97—22no—
309#97—22no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—24no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3AGL(1,ℤ/8)12yes—
335C23×C3C22×D411yescharacteristic
336C3×Q8—48no—
337C3×Q8—42no—
338C3×Q8—42no—
339C3×D4—24no—
340C3×D4—24no—
341C3×D4—48no—
342C3×D4—48no—
343C3×D4—416no—
344C3×D4—42no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—84no—
349C2×C3×C4—28no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—24no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—44no—
364C2×C3×C4—416no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
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—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4#12012yes—
379C2×C3×C4#12012yes—
380C2×C3×C4AGL(1,ℤ/8)12yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4112yes—
384C2×C3×C4H(5,𝔽2)12yes—
385C3×C8—24no—
386C3×C8—42no—
387C24—22no—
388C24—21no—
389C24C2×C3×D411yescharacteristic
390#41—432no—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C3×D414yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D411yescharacteristic
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C24×C311yescharacteristic
443SD16—48no—
444D8—48no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8C3×#4114yescomplemented
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4C2×C3×D414yes—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42C2×C3×D412yes—
470C42C2×C3×D412yes—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3C2×#12011yescharacteristic
493C22×C3#53211yescharacteristic
494C22×C3#53511yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
497C22×C3#57511yescharacteristic
498C22×C3#57511yescharacteristic
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—44no—
504C3×C4—41no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4#54612yes—
509C3×C4#55012yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—24no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C3×AGL(1,ℤ/8)12yes—
531C23C22×C3×D411yescharacteristic
532Q8—48no—
533Q8—42no—
534Q8—42no—
535D4—24no—
536D4—24no—
537D4—48no—
538D4—48no—
539D4—416no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—84no—
545C2×C4—28no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—416no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4C3×#12012yes—
575C2×C4C3×#12012yes—
576C2×C4C3×AGL(1,ℤ/8)12yes—
577C2×C4C22×C3×D411yescharacteristic
578C2×C4C22×C3×D411yescharacteristic
579C2×C4C2×C3×#4112yes—
580C2×C4C3×H(5,𝔽2)12yes—
581C8—24no—
582C8—42no—
583C2×C3—42no—
584C2×C3—84no—
585C2×C3—161no—
586C2×C3#243111yescharacteristic
587C2×C3#300311yescharacteristic
588C2×C3#300511yescharacteristic
589C2×C3#317111yescharacteristic
590C2×C3#318011yescharacteristic
591C2×C3#321311yescharacteristic
592C2×C3#321411yescharacteristic
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C2×C3×#12011yescharacteristic
615C22C3×#53211yescharacteristic
616C22C3×#53511yescharacteristic
617C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
618C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
619C22C3×#57511yescharacteristic
620C22C3×#57511yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—44no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4C3×#54612yes—
631C4C3×#55012yes—
632C3#2560211yescharacteristic, direct product factor, Sylow
633C2—42no—
634C2—84no—
635C2—161no—
636C2C3×#243111yescharacteristic
637C2C3×#300311yescharacteristic
638C2C3×#300511yescharacteristic
639C2C3×#317111yescharacteristic
640C2C3×#318011yescharacteristic
641C2C3×#321311yescharacteristic
642C2C3×#321411yescharacteristic
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×#496C2semidirect
3. C2×C3×#489C2semidirect
4. C3×#2850C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1993C2semidirect
8. C3×#1961C2semidirect
9. C3×#1879C2semidirect
10. C3×#1877C2semidirect
11. C3×#1749C2semidirect
12. #25602C3direct (N × H)
48. C2×#532C2×C3semidirect
49. C2×#496C2×C3semidirect
50. C2×#489C2×C3semidirect
51. #2850C2×C3semidirect
52. #2000C2×C3semidirect
53. #1999C2×C3semidirect
54. #1993C2×C3semidirect
55. #1961C2×C3semidirect
56. #1879C2×C3semidirect
57. #1877C2×C3semidirect
58. #1749C2×C3semidirect
83. C3×C4⋊Q8D4semidirect
85. C3×#127D4semidirect
225. C2×C3×C8#41semidirect
270. C4⋊Q8C3×D4semidirect
272. #127C3×D4semidirect
450. C2×C8C3×#41semidirect
632. C3#25602direct (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×#4962
C2×C3×#4892
C3×#28502
C3×#20002
C3×#19992
C3×#19932
C3×#19612
C3×#18792
C3×#18772
C3×#17492
#256023