Finite Group Databasebeta

← C3×#15129 #10633736

Subgroups — C3×#15129 #10633736

Order 768.

2354 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#394C212yesmaximal
3C2×C3×#389C211yesmaximal, characteristic, complemented
4C2×C3×#380C212yesmaximal, complemented
5C2×C3×#380C212yesmaximal, complemented
6C3×#1441C212yesmaximal, complemented
7C3×#1440C214yesmaximal, complemented
8C3×C4⋊#95C211yesmaximal, characteristic, complemented
9#15129C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4C2212yes—
12C2×C3×C4×D4—28no—
13C2×C3×C4×D4—24no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2214yes—
21C22×C3×C22⋊C4C2214yes—
22C22×C3×C42C2211yescharacteristic
23C3×#394—28no—
24C3×#389—24no—
25C3×#380—28no—
26C3×#380—28no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2214yes—
29C2×C3×#95C2214yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C22×C4×D4C2×C311yescharacteristic
33C2×#394C2×C312yes—
34C2×#389C2×C311yescharacteristic, complemented
35C2×#380C2×C312yescomplemented
36C2×#380C2×C312yescomplemented
37#1441C2×C312yescomplemented
38#1440C2×C314yescomplemented
39C4⋊#95C2×C311yescharacteristic, complemented
40C25×C3C2312yes—
41C22×C3×D4—24no—
42C22×C3×D4—24no—
43C22×C3×D4—22no—
44C22×C3×D4—22no—
45C22×C3×D4D414yes—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4D418yes—
53C23×C3×C4C2314yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C3×C4×D4—232no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4D414yescomplemented
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—28no—
78C2×C3×C22⋊C4—28no—
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—28no—
81C2×C3×C22⋊C4—28no—
82C2×C3×C22⋊C4—44no—
83C2×C3×C42—24no—
84C2×C3×C42D414yes—
85C3×#95—28no—
86C3×#95—28no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—42no—
90C23×D4C22×C311yescharacteristic
91C24×C4C22×C312yes—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C22×C4⋊C4C22×C312yes—
95C22×C4⋊C4C22×C311yescharacteristic
96C22×C4⋊C4C22×C312yes—
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C311yescharacteristic
99C22×C22⋊C4C22×C314yes—
100C22×C22⋊C4C22×C312yes—
101C22×C22⋊C4C22×C314yes—
102C22×C42C22×C311yescharacteristic
103#394—28no—
104#389—24no—
105#380—28no—
106#380—28no—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C314yes—
110C2×#95C22×C314yes—
111C2×#95C22×C312yes—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3C2×D414yes—
125C24×C3C2×D414yes—
126C24×C3#4114yes—
127C24×C3C2411yescharacteristic
128C2×C3×D4—28no—
129C2×C3×D4—28no—
130C2×C3×D4—216no—
131C2×C3×D4—216no—
132C2×C3×D4—28no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—216no—
152C22×C3×C4—216no—
153C22×C3×C4—216no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—24no—
166C22×C3×C4—48no—
167C22×C3×C4—48no—
168C22×C3×C4—48no—
169C22×C3×C4—48no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4114yes—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—44no—
184C3×C4⋊C4—48no—
185C3×C4⋊C4—48no—
186C3×C22⋊C4—216no—
187C3×C22⋊C4—416no—
188C3×C22⋊C4—416no—
189C3×C42—28no—
190C25C23×C312yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4C3×D414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4C3×D418yes—
203C23×C4C23×C312yes—
204C23×C4C23×C314yes—
205C23×C4C23×C312yes—
206C23×C4C23×C311yescharacteristic
207C23×C4C23×C312yes—
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C311yescharacteristic
210C4×D4—232no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4C3×D414yescomplemented
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C42—24no—
234C2×C42C3×D414yes—
235#95—28no—
236#95—28no—
237#95—24no—
238#95—24no—
239#95—42no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—24no—
243C23×C3—24no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3#12118yes—
275C23×C3C22×D412yes—
276C23×C3C22×D411yescharacteristic
277C23×C3C22×D411yescharacteristic
278C23×C3C22×D411yescharacteristic
279C23×C3C2×#4111yescharacteristic
280C23×C3C2×#4112yes—
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4112yes—
283C23×C3H(5,𝔽2)11yescharacteristic
284C23×C32−1+411yescharacteristic
285C23×C32−1+412yes—
286C3×D4—232no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—432no—
311C2×C3×C4—42no—
312C2×C3×C4—44no—
313C2×C3×C4—44no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—44no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4—44no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4#12014yes—
324C2×C3×C4#12114yes—
325C2×C3×C4#12114yes—
326C2×C3×C4#12118yes—
327C2×C3×C4#12318yes—
328C2×C3×C4#12414yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24C2×C3×D414yes—
342C24C2×C3×D414yes—
343C24C3×#4114yes—
344C24C24×C311yescharacteristic
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—216no—
348C2×D4—216no—
349C2×D4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4C2×C3×D412yes—
388C22×C4C2×C3×D414yes—
389C22×C4C2×C3×D414yes—
390C22×C4C2×C3×D412yes—
391C22×C4C2×C3×D412yes—
392C22×C4C2×C3×D414yes—
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×D412yes—
395C22×C4C3×#4112yes—
396C22×C4C3×#4112yes—
397C22×C4C3×#4114yes—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C22⋊C4—216no—
404C22⋊C4—416no—
405C22⋊C4—416no—
406C42—28no—
407C22×C3—28no—
408C22×C3—28no—
409C22×C3—28no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3C2×#12011yescharacteristic
423C22×C3C2×#12112yes—
424C22×C3C2×#12111yescharacteristic
425C22×C3C2×#12112yes—
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12312yes—
428C22×C3#52511yescharacteristic
429C22×C3#52512yes—
430C22×C3C2×#12411yescharacteristic
431C22×C3#53612yes—
432C22×C3#53611yescharacteristic
433C22×C3#53711yescharacteristic
434C22×C3#53712yes—
435C22×C3D4211yescharacteristic
436C22×C3#54711yescharacteristic
437C22×C3#54712yes—
438C22×C3#54711yescharacteristic
439C22×C3#54812yes—
440C22×C3#55212yes—
441C22×C3#55212yes—
442C22×C3#55312yes—
443C22×C3#55511yescharacteristic
444C22×C3#55612yes—
445C3×C4—24no—
446C3×C4—42no—
447C3×C4—44no—
448C3×C4—44no—
449C3×C4—82no—
450C3×C4—84no—
451C3×C4#39418yescomplemented
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C3×#12118yes—
487C23C22×C3×D412yes—
488C23C22×C3×D411yescharacteristic
489C23C22×C3×D411yescharacteristic
490C23C22×C3×D411yescharacteristic
491C23C2×C3×#4111yescharacteristic
492C23C2×C3×#4112yes—
493C23C2×C3×#4111yescharacteristic
494C23C2×C3×#4112yes—
495C23C3×H(5,𝔽2)11yescharacteristic
496C23C3×2−1+411yescharacteristic
497C23C3×2−1+412yes—
498D4—232no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—432no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4C3×#12014yes—
536C2×C4C3×#12114yes—
537C2×C4C3×#12114yes—
538C2×C4C3×#12118yes—
539C2×C4C3×#12318yes—
540C2×C4C3×#12414yes—
541C2×C3—48no—
542C2×C3C2×#39411yescharacteristic
543C2×C3#241512yes—
544C2×C3#241612yes—
545C2×C3#248512yes—
546C2×C3#248612yes—
547C2×C3#253611yescharacteristic
548C2×C3#253711yescharacteristic
549C2×C3#254711yescharacteristic
550C2×C3#256211yescharacteristic
551C2×C3#261611yescharacteristic
552C2×C3#284711yescharacteristic
553C22—28no—
554C22—28no—
555C22—28no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22C2×C3×#12011yescharacteristic
569C22C2×C3×#12112yes—
570C22C2×C3×#12111yescharacteristic
571C22C2×C3×#12112yes—
572C22C2×C3×#12111yescharacteristic
573C22C2×C3×#12312yes—
574C22C3×#52512yes—
575C22C3×#52511yescharacteristic
576C22C2×C3×#12411yescharacteristic
577C22C3×#53612yes—
578C22C3×#53611yescharacteristic
579C22C3×#53712yes—
580C22C3×#53711yescharacteristic
581C22C3×D4211yescharacteristic
582C22C3×#54711yescharacteristic
583C22C3×#54712yes—
584C22C3×#54711yescharacteristic
585C22C3×#54812yes—
586C22C3×#55212yes—
587C22C3×#55212yes—
588C22C3×#55312yes—
589C22C3×#55511yescharacteristic
590C22C3×#55612yes—
591C4—24no—
592C4—42no—
593C4—44no—
594C4—44no—
595C4—84no—
596C4—82no—
597C4C3×#39418yescomplemented
598C3#1512911yescharacteristic, direct product factor, Sylow
599C2—48no—
600C2C2×C3×#39411yescharacteristic
601C2C3×#241512yes—
602C2C3×#241612yes—
603C2C3×#248512yes—
604C2C3×#248612yes—
605C2C3×#253611yescharacteristic
606C2C3×#253711yescharacteristic
607C2C3×#254711yescharacteristic
608C2C3×#256211yescharacteristic
609C2C3×#261611yescharacteristic
610C2C3×#284711yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#389C2semidirect
4. C2×C3×#380C2semidirect
5. C2×C3×#380C2semidirect
6. C3×#1441C2semidirect
7. C3×#1440C2semidirect
8. C3×C4⋊#95C2semidirect
9. #15129C3direct (N × H)
34. C2×#389C2×C3semidirect
35. C2×#380C2×C3semidirect
36. C2×#380C2×C3semidirect
37. #1441C2×C3semidirect
38. #1440C2×C3semidirect
39. C4⋊#95C2×C3semidirect
73. C2×C3×C4⋊C4D4semidirect
223. C2×C4⋊C4C3×D4semidirect
451. C3×C4#394semidirect
597. C4C3×#394semidirect
598. C3#15129direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#3942
C2×C3×#3892
C2×C3×#3802
C3×#14412
C3×#14402
C3×C4⋊#952
#151293