Finite Group Databasebeta

← C3×#15081 #10633688

Subgroups — C3×#15081 #10633688

Order 768.

1518 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#398C211yesmaximal, characteristic, complemented
2C2×C3×#397C211yesmaximal, characteristic
3C2×C3×#388C211yesmaximal, characteristic
4C2×C3×#388C212yesmaximal
5C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
6C3×#1441C212yesmaximal, complemented
7C3×#1441C211yesmaximal, characteristic, complemented
8C3×#1439C212yesmaximal
9C3×#1439C211yesmaximal, characteristic
10C3×C4⋊#95C211yesmaximal, characteristic, complemented
11C3×#1434C212yesmaximal, complemented
12#15081C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C42C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#398—24no—
26C3×#397—24no—
27C3×#388—28no—
28C3×#388—24no—
29C3×C4×C4⋊C4—24no—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2212yes—
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2212yes—
37C2×C3×#95C2211yescharacteristic
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×C3×#95C2212yes—
41C2×#398C2×C311yescharacteristic, complemented
42C2×#397C2×C311yescharacteristic
43C2×#388C2×C311yescharacteristic
44C2×#388C2×C312yes—
45C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
46#1441C2×C311yescharacteristic, complemented
47#1441C2×C312yescomplemented
48#1439C2×C311yescharacteristic
49#1439C2×C312yes—
50C4⋊#95C2×C311yescharacteristic, complemented
51#1434C2×C312yescomplemented
52C25×C3C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—42no—
75C2×C3×C4⋊C4—42no—
76C2×C3×C4⋊C4D414yes—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4—42no—
90C2×C3×C22⋊C4D414yes—
91C2×C3×C42—24no—
92C2×C3×C42—24no—
93C2×C3×C42—24no—
94C2×C3×C42—24no—
95C2×C3×C42D414yes—
96C3×#95—24no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—24no—
105C3×#95—24no—
106C3×#95—24no—
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C4⋊C4C22×C312yes—
110C22×C4⋊C4C22×C312yes—
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C22⋊C4C22×C311yescharacteristic
113C22×C22⋊C4C22×C312yes—
114C22×C22⋊C4C22×C311yescharacteristic
115C22×C22⋊C4C22×C312yes—
116C22×C22⋊C4C22×C311yescharacteristic
117C22×C42C22×C312yes—
118C22×C42C22×C311yescharacteristic
119#398—24no—
120#397—24no—
121#388—28no—
122#388—24no—
123C4×C4⋊C4—24no—
124C2×#95C22×C312yes—
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C312yes—
127C2×#95C22×C311yescharacteristic
128C2×#95C22×C311yescharacteristic
129C2×#95C22×C312yes—
130C2×#95C22×C312yes—
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C2×#95C22×C312yes—
134C2×#95C22×C311yescharacteristic
135C24×C3—21no—
136C24×C3—21no—
137C24×C3—21no—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—21no—
141C24×C3—22no—
142C24×C3—21no—
143C24×C3—22no—
144C24×C3—21no—
145C24×C3C2×D412yes—
146C24×C3C2411yescharacteristic
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4#4114yes—
203C22×C3×C4#4114yes—
204C22×C3×C4#4112yes—
205C22×C3×C4#4112yes—
206C22×C3×C4#4114yes—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—48no—
210C3×C4⋊C4—48no—
211C3×C4⋊C4—44no—
212C3×C22⋊C4—216no—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—48no—
215C3×C22⋊C4—44no—
216C3×C22⋊C4—44no—
217C3×C42—28no—
218C3×C42—48no—
219C25C23×C311yescharacteristic
220C23×C4C23×C312yes—
221C23×C4C23×C311yescharacteristic
222C23×C4C23×C311yescharacteristic
223C23×C4C23×C311yescharacteristic
224C23×C4C23×C312yes—
225C23×C4C23×C312yes—
226C23×C4C23×C312yes—
227C23×C4C23×C311yescharacteristic
228C23×C4C23×C311yescharacteristic
229C23×C4C23×C311yescharacteristic
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C3×D414yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4C3×D414yes—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42C3×D414yes—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—24no—
273#95—24no—
274C23×C3—22no—
275C23×C3—24no—
276C23×C3—22no—
277C23×C3—24no—
278C23×C3—22no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3C22×D411yescharacteristic
298C23×C3C22×D411yescharacteristic
299C23×C3C22×D411yescharacteristic
300C23×C3C2×#4111yescharacteristic
301C23×C3C2×#4111yescharacteristic
302C23×C3C2×#4112yes—
303C23×C3C2×#4112yes—
304C23×C3C2×#4112yes—
305C23×C3H(5,𝔽2)11yescharacteristic
306C23×C32−1+412yes—
307C23×C32−1+411yescharacteristic
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4#12114yes—
366C2×C3×C4#12114yes—
367C2×C3×C4#12314yes—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24—21no—
378C24C2×C3×D412yes—
379C24C24×C311yescharacteristic
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4C2×C3×D414yes—
429C22×C4C2×C3×D412yes—
430C22×C4C2×C3×D412yes—
431C22×C4C2×C3×D412yes—
432C22×C4C2×C3×D412yes—
433C22×C4C2×C3×D412yes—
434C22×C4C2×C3×D412yes—
435C22×C4C3×#4114yes—
436C22×C4C3×#4114yes—
437C22×C4C3×#4112yes—
438C22×C4C3×#4112yes—
439C22×C4C3×#4114yes—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C22⋊C4—216no—
446C22⋊C4—28no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C42—28no—
451C42—48no—
452C22×C3—42no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—81no—
463C22×C3C2×#12111yescharacteristic
464C22×C3C2×#12111yescharacteristic
465C22×C3C2×#12311yescharacteristic
466C22×C3#52511yescharacteristic
467C22×C3#52512yes—
468C22×C3#52912yes—
469C22×C3#52912yes—
470C22×C3#53212yes—
471C22×C3#53211yescharacteristic
472C22×C3#53611yescharacteristic
473C22×C3#53711yescharacteristic
474C22×C3#54112yes—
475C22×C3#54212yes—
476C22×C3D4211yescharacteristic
477C22×C3#54712yes—
478C22×C3#54711yescharacteristic
479C22×C3#54711yescharacteristic
480C22×C3#54812yes—
481C22×C3#55112yes—
482C22×C3#55212yes—
483C22×C3#55212yes—
484C22×C3#55511yescharacteristic
485C22×C3#55612yes—
486C3×C4—24no—
487C3×C4—24no—
488C3×C4—24no—
489C3×C4—44no—
490C3×C4—44no—
491C3×C4—42no—
492C3×C4—44no—
493C3×C4—81no—
494C3×C4—82no—
495C3×C4—81no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23C22×C3×D411yescharacteristic
520C23C22×C3×D411yescharacteristic
521C23C22×C3×D411yescharacteristic
522C23C2×C3×#4111yescharacteristic
523C23C2×C3×#4111yescharacteristic
524C23C2×C3×#4112yes—
525C23C2×C3×#4112yes—
526C23C2×C3×#4112yes—
527C23C3×H(5,𝔽2)11yescharacteristic
528C23C3×2−1+412yes—
529C23C3×2−1+411yescharacteristic
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4C3×#12114yes—
588C2×C4C3×#12114yes—
589C2×C4C3×#12314yes—
590C2×C3—82no—
591C2×C3#239211yescharacteristic
592C2×C3#246112yes—
593C2×C3#247812yes—
594C2×C3#248611yescharacteristic
595C2×C3#249811yescharacteristic
596C2×C3#251912yes—
597C2×C3#253212yes—
598C2×C3#253711yescharacteristic
599C2×C3#254511yescharacteristic
600C2×C3#263711yescharacteristic
601C2×C3#285411yescharacteristic
602C22—42no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—82no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22C2×C3×#12111yescharacteristic
614C22C2×C3×#12111yescharacteristic
615C22C2×C3×#12311yescharacteristic
616C22C3×#52511yescharacteristic
617C22C3×#52512yes—
618C22C3×#52912yes—
619C22C3×#52912yes—
620C22C3×#53211yescharacteristic
621C22C3×#53212yes—
622C22C3×#53611yescharacteristic
623C22C3×#53711yescharacteristic
624C22C3×#54112yes—
625C22C3×#54212yes—
626C22C3×D4211yescharacteristic
627C22C3×#54712yes—
628C22C3×#54711yescharacteristic
629C22C3×#54711yescharacteristic
630C22C3×#54812yes—
631C22C3×#55112yes—
632C22C3×#55212yes—
633C22C3×#55212yes—
634C22C3×#55511yescharacteristic
635C22C3×#55612yes—
636C4—24no—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4—81no—
645C4—81no—
646C3#1508111yescharacteristic, direct product factor, Sylow
647C2—82no—
648C2C3×#239211yescharacteristic
649C2C3×#246112yes—
650C2C3×#247812yes—
651C2C3×#248611yescharacteristic
652C2C3×#249811yescharacteristic
653C2C3×#251912yes—
654C2C3×#253212yes—
655C2C3×#253711yescharacteristic
656C2C3×#254511yescharacteristic
657C2C3×#263711yescharacteristic
658C2C3×#285411yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#398C2semidirect
5. C2×C3×C4×C4⋊C4C2semidirect
6. C3×#1441C2semidirect
7. C3×#1441C2semidirect
10. C3×C4⋊#95C2semidirect
11. C3×#1434C2semidirect
12. #15081C3direct (N × H)
41. C2×#398C2×C3semidirect
45. C2×C4×C4⋊C4C2×C3semidirect
46. #1441C2×C3semidirect
47. #1441C2×C3semidirect
50. C4⋊#95C2×C3semidirect
51. #1434C2×C3semidirect
646. C3#15081direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3982
C2×C3×#3972
C2×C3×#3882
C2×C3×C4×C4⋊C42
C3×#14412
C3×#14392
C3×C4⋊#952
C3×#14342
#150813