Finite Group Databasebeta

← C3×#14625 #10633232

Subgroups — C3×#14625 #10633232

Order 768.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#393C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic, complemented
3C2×C3×#388C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic, complemented
5C3×#1444C211yesmaximal, characteristic, complemented
6C3×#1442C212yesmaximal, complemented
7C3×#1442C212yesmaximal, complemented
8C3×#1440C212yesmaximal
9C3×#1439C212yesmaximal
10C3×#1439C211yesmaximal, characteristic
11C3×#1434C211yesmaximal, characteristic, complemented
12#14625C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C3×#393C418yescomplemented
24C3×#391—24no—
25C3×#388—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2211yescharacteristic
38C2×C3×#95C2212yes—
39C2×#393C2×C311yescharacteristic
40C2×#391C2×C311yescharacteristic, complemented
41C2×#388C2×C311yescharacteristic
42#1446C2×C311yescharacteristic, complemented
43#1444C2×C311yescharacteristic, complemented
44#1442C2×C312yescomplemented
45#1442C2×C312yescomplemented
46#1440C2×C312yes—
47#1439C2×C311yescharacteristic
48#1439C2×C312yes—
49#1434C2×C311yescharacteristic, complemented
50C25×C3C2311yescharacteristic
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—42no—
54C22×C3×Q8—42no—
55C22×C3×Q8C2×C414yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4—42no—
79C2×C3×C22⋊C4—42no—
80C2×C3×C22⋊C4—42no—
81C2×C3×C22⋊C4—42no—
82C2×C3×C22⋊C4—42no—
83C2×C3×C22⋊C4—42no—
84C2×C3×C22⋊C4C2×C414yes—
85C2×C3×C22⋊C4C2×C418yes—
86C2×C3×C42—42no—
87C2×C3×C42D414yes—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—22no—
93C3×#95—42no—
94C3×#95—42no—
95C3×#95—42no—
96C3×#95—42no—
97C3×#95—42no—
98C3×#95—42no—
99C3×#95C2×C418yes—
100C3×#95C2×C414yes—
101C23×Q8C22×C311yescharacteristic
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C4⋊C4C22×C312yes—
105C22×C22⋊C4C22×C311yescharacteristic
106C22×C22⋊C4C22×C311yescharacteristic
107C22×C22⋊C4C22×C311yescharacteristic
108C22×C22⋊C4C22×C312yes—
109C22×C22⋊C4C22×C312yes—
110C22×C42C22×C311yescharacteristic
111#393C3×C418yescomplemented
112#391—24no—
113#388—24no—
114C2×#95C22×C311yescharacteristic
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C312yes—
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C312yes—
127C24×C3—21no—
128C24×C3—21no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—21no—
133C24×C3—21no—
134C24×C3—21no—
135C24×C3—21no—
136C24×C3—22no—
137C24×C3C22×C412yes—
138C24×C3C2411yescharacteristic
139C2×C3×Q8—28no—
140C2×C3×Q8—48no—
141C2×C3×Q8—44no—
142C2×C3×Q8—44no—
143C2×C3×Q8—48no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
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—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4C22×C412yes—
195C22×C3×C4C22×C414yes—
196C22×C3×C4C22×C414yes—
197C22×C3×C4C22×C412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4#4112yes—
202C3×C4⋊C4—44no—
203C3×C4⋊C4—44no—
204C3×C4⋊C4—48no—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—48no—
207C3×C22⋊C4—44no—
208C3×C22⋊C4—44no—
209C3×C22⋊C4—48no—
210C3×C42—44no—
211C25C23×C311yescharacteristic
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8C2×C3×C414yes—
217C23×C4C23×C311yescharacteristic
218C23×C4C23×C312yes—
219C23×C4C23×C311yescharacteristic
220C23×C4C23×C312yes—
221C23×C4C23×C311yescharacteristic
222C23×C4C23×C312yes—
223C23×C4C23×C312yes—
224C23×C4C23×C311yescharacteristic
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C311yescharacteristic
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4C2×C3×C414yes—
246C2×C22⋊C4C2×C3×C418yes—
247C2×C42—42no—
248C2×C42C3×D414yes—
249#95—22no—
250#95—22no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95C2×C3×C418yes—
261#95C2×C3×C414yes—
262C23×C3—22no—
263C23×C3—24no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—24no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—42no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3C23×C411yescharacteristic
286C23×C3C22×D411yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3H(5,𝔽2)12yes—
289C23×C3H(5,𝔽2)12yes—
290C23×C3H(5,𝔽2)12yes—
291C23×C3H(5,𝔽2)11yescharacteristic
292C23×C32−1+411yescharacteristic
293C23×C32−1+411yescharacteristic
294C23×C32−1+411yescharacteristic
295C23×C32−1+412yes—
296C3×Q8—416no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4C4×D414yes—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24C22×C3×C412yes—
366C24C24×C311yescharacteristic
367C2×Q8—28no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4C22×C3×C412yes—
423C22×C4C22×C3×C414yes—
424C22×C4C22×C3×C414yes—
425C22×C4C22×C3×C412yes—
426C22×C4C2×C3×D412yes—
427C22×C4C2×C3×D412yes—
428C22×C4C2×C3×D412yes—
429C22×C4C3×#4112yes—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C22⋊C4—28no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C42—44no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—81no—
450C22×C3C2×C4×D411yescharacteristic
451C22×C3#51812yes—
452C22×C3#51911yescharacteristic
453C22×C3#52011yescharacteristic
454C22×C3#52012yes—
455C22×C3#53411yescharacteristic
456C22×C3#53511yescharacteristic
457C22×C3#53511yescharacteristic
458C22×C3#53611yescharacteristic
459C22×C3#53611yescharacteristic
460C22×C3#53711yescharacteristic
461C22×C3#53812yes—
462C22×C3#54111yescharacteristic
463C22×C3#54212yes—
464C22×C3#54211yescharacteristic
465C22×C3H(3,𝔽4)12yes—
466C22×C3#56212yes—
467C22×C3#56212yes—
468C22×C3#56212yes—
469C22×C3#56212yes—
470C22×C3#56312yes—
471C22×C3#56312yes—
472C22×C3#56312yes—
473C3×C4—24no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4—42no—
477C3×C4—44no—
478C3×C4—42no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—82no—
482C3×C4—82no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23C23×C3×C411yescharacteristic
507C23C22×C3×D411yescharacteristic
508C23C2×C3×#4111yescharacteristic
509C23C3×H(5,𝔽2)11yescharacteristic
510C23C3×H(5,𝔽2)12yes—
511C23C3×H(5,𝔽2)12yes—
512C23C3×H(5,𝔽2)12yes—
513C23C3×2−1+411yescharacteristic
514C23C3×2−1+411yescharacteristic
515C23C3×2−1+411yescharacteristic
516C23C3×2−1+412yes—
517Q8—416no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4C3×C4×D414yes—
576C2×C3—82no—
577C2×C3#232111yescharacteristic
578C2×C3#232311yescharacteristic
579C2×C3#232611yescharacteristic
580C2×C3#237812yes—
581C2×C3#237912yes—
582C2×C3#281411yescharacteristic
583C2×C3#281611yescharacteristic
584C2×C3#281911yescharacteristic
585C2×C3#282111yescharacteristic
586C2×C3#283112yes—
587C2×C3#283212yes—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—82no—
598C22—82no—
599C22C2×C3×C4×D411yescharacteristic
600C22C3×#51812yes—
601C22C3×#51911yescharacteristic
602C22C3×#52012yes—
603C22C3×#52011yescharacteristic
604C22C3×#53411yescharacteristic
605C22C3×#53511yescharacteristic
606C22C3×#53511yescharacteristic
607C22C3×#53611yescharacteristic
608C22C3×#53611yescharacteristic
609C22C3×#53711yescharacteristic
610C22C3×#53812yes—
611C22C3×#54111yescharacteristic
612C22C3×#54212yes—
613C22C3×#54211yescharacteristic
614C22C3×H(3,𝔽4)12yes—
615C22C3×#56212yes—
616C22C3×#56212yes—
617C22C3×#56212yes—
618C22C3×#56212yes—
619C22C3×#56312yes—
620C22C3×#56312yes—
621C22C3×#56312yes—
622C4—24no—
623C4—44no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—81no—
629C4—82no—
630C4—81no—
631C4—82no—
632C3#1462511yescharacteristic, direct product factor, Sylow
633C2—82no—
634C2C3×#232111yescharacteristic
635C2C3×#232311yescharacteristic
636C2C3×#232611yescharacteristic
637C2C3×#237812yes—
638C2C3×#237912yes—
639C2C3×#281411yescharacteristic
640C2C3×#281611yescharacteristic
641C2C3×#281911yescharacteristic
642C2C3×#282111yescharacteristic
643C2C3×#283112yes—
644C2C3×#283212yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#391C2semidirect
4. C3×#1446C2semidirect
5. C3×#1444C2semidirect
6. C3×#1442C2semidirect
7. C3×#1442C2semidirect
11. C3×#1434C2semidirect
12. #14625C3direct (N × H)
23. C3×#393C4semidirect
40. C2×#391C2×C3semidirect
42. #1446C2×C3semidirect
43. #1444C2×C3semidirect
44. #1442C2×C3semidirect
45. #1442C2×C3semidirect
49. #1434C2×C3semidirect
111. #393C3×C4semidirect
632. C3#14625direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3932
C2×C3×#3912
C2×C3×#3882
C3×#14462
C3×#14442
C3×#14422
C3×#14402
C3×#14392
C3×#14342
#146253