Finite Group Databasebeta

← C3×#16656 #10635263

Subgroups — C3×#16656 #10635263

Order 768.

1598 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#126C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic, complemented
3C2×C3×#393C211yesmaximal, characteristic
4C2×C3×#391C211yesmaximal, characteristic, complemented
5C3×#1442C212yesmaximal, complemented
6C3×#1441C211yesmaximal, characteristic, complemented
7C3×#1441C212yesmaximal, complemented
8C3×#1440C212yesmaximal
9C3×#1439C211yesmaximal, characteristic
10C3×#1439C212yesmaximal
11C3×#1434C211yesmaximal, characteristic, complemented
12#16656C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×#126—24no—
15C2×C3×#126—24no—
16C2×C3×#126—24no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C22⋊C4C2212yes—
25C22×C3×C22⋊C4C2211yescharacteristic
26C22×C3×C22⋊C4C2212yes—
27C22×C3×C42C2211yescharacteristic
28C3×#395—24no—
29C3×#393—24no—
30C3×#391—24no—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×C3×#95C2212yes—
41C2×C3×#95C2212yes—
42C22×#126C2×C311yescharacteristic
43C2×#395C2×C311yescharacteristic, complemented
44C2×#393C2×C311yescharacteristic
45C2×#391C2×C311yescharacteristic, complemented
46#1442C2×C312yescomplemented
47#1441C2×C312yescomplemented
48#1441C2×C311yescharacteristic, complemented
49#1440C2×C312yes—
50#1439C2×C312yes—
51#1439C2×C311yescharacteristic
52#1434C2×C311yescharacteristic, complemented
53C25×C3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—24no—
58C22×C3×Q8—42no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C3×#126—232no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4—42no—
82C2×C3×C4⋊C4—42no—
83C2×C3×C4⋊C4D414yes—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—24no—
93C2×C3×C22⋊C4—24no—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4—42no—
96C2×C3×C22⋊C4—42no—
97C2×C3×C22⋊C4—42no—
98C2×C3×C22⋊C4D414yes—
99C2×C3×C42—22no—
100C2×C3×C42—22no—
101C2×C3×C42D414yes—
102C3×#95—24no—
103C3×#95—24no—
104C3×#95—24no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—24no—
108C3×#95—24no—
109C3×#95—24no—
110C3×#95—42no—
111C3×#95—42no—
112C3×#95—42no—
113C23×Q8C22×C311yescharacteristic
114C2×#126—24no—
115C2×#126—24no—
116C2×#126—24no—
117C22×C4⋊C4C22×C311yescharacteristic
118C22×C4⋊C4C22×C311yescharacteristic
119C22×C4⋊C4C22×C312yes—
120C22×C4⋊C4C22×C311yescharacteristic
121C22×C4⋊C4C22×C311yescharacteristic
122C22×C22⋊C4C22×C311yescharacteristic
123C22×C22⋊C4C22×C311yescharacteristic
124C22×C22⋊C4C22×C312yes—
125C22×C22⋊C4C22×C311yescharacteristic
126C22×C22⋊C4C22×C312yes—
127C22×C42C22×C311yescharacteristic
128#395—24no—
129#393—24no—
130#391—24no—
131C2×#95C22×C311yescharacteristic
132C2×#95C22×C312yes—
133C2×#95C22×C311yescharacteristic
134C2×#95C22×C312yes—
135C2×#95C22×C312yes—
136C2×#95C22×C312yes—
137C2×#95C22×C312yes—
138C2×#95C22×C312yes—
139C2×#95C22×C312yes—
140C2×#95C22×C312yes—
141C2×#95C22×C312yes—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—21no—
147C24×C3—22no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—21no—
151C24×C3—22no—
152C24×C3C2×D412yes—
153C24×C3C2411yescharacteristic
154C2×C3×Q8—28no—
155C2×C3×Q8—48no—
156C2×C3×Q8—48no—
157C2×C3×Q8—44no—
158C2×C3×Q8—44no—
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—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
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—21no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D412yes—
219C22×C3×C4#4114yes—
220C22×C3×C4#4114yes—
221C3×C4⋊C4—28no—
222C3×C4⋊C4—28no—
223C3×C4⋊C4—28no—
224C3×C4⋊C4—48no—
225C3×C4⋊C4—44no—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—28no—
229C3×C22⋊C4—48no—
230C3×C22⋊C4—48no—
231C3×C42—28no—
232C25C23×C311yescharacteristic
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8—42no—
238C23×C4C23×C312yes—
239C23×C4C23×C312yes—
240C23×C4C23×C311yescharacteristic
241C23×C4C23×C311yescharacteristic
242C23×C4C23×C311yescharacteristic
243C23×C4C23×C311yescharacteristic
244C23×C4C23×C311yescharacteristic
245C23×C4C23×C312yes—
246C23×C4C23×C312yes—
247C23×C4C23×C311yescharacteristic
248#126—232no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C3×D414yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4C3×D414yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42C3×D414yes—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—42no—
290#95—42no—
291#95—42no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3C22×D411yescharacteristic
316C23×C3C22×D411yescharacteristic
317C23×C3C22×D411yescharacteristic
318C23×C3C2×#4112yes—
319C23×C3C2×#4112yes—
320C23×C3H(5,𝔽2)12yes—
321C23×C3H(5,𝔽2)11yescharacteristic
322C23×C32−1+411yescharacteristic
323C23×C32−1+411yescharacteristic
324C23×C32−1+411yescharacteristic
325C23×C32−1+412yes—
326C3×Q8—416no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4#12014yes—
389C24—21no—
390C24—21no—
391C24—22no—
392C24—22no—
393C24—21no—
394C24—22no—
395C24—21no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24C2×C3×D412yes—
400C24C24×C311yescharacteristic
401C2×Q8—28no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—44no—
405C2×Q8—44no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4C2×C3×D412yes—
459C22×C4C2×C3×D412yes—
460C22×C4C2×C3×D412yes—
461C22×C4C2×C3×D412yes—
462C22×C4C2×C3×D412yes—
463C22×C4C2×C3×D412yes—
464C22×C4C2×C3×D412yes—
465C22×C4C2×C3×D412yes—
466C22×C4C3×#4114yes—
467C22×C4C3×#4114yes—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C42—28no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3C2×#12011yescharacteristic
491C22×C3#52511yescharacteristic
492C22×C3#53211yescharacteristic
493C22×C3#53511yescharacteristic
494C22×C3#53611yescharacteristic
495C22×C3#53611yescharacteristic
496C22×C3#53611yescharacteristic
497C22×C3#53611yescharacteristic
498C22×C3#53711yescharacteristic
499C22×C3#53711yescharacteristic
500C22×C3#53711yescharacteristic
501C22×C3#53812yes—
502C22×C3#54112yes—
503C22×C3#54212yes—
504C22×C3#54212yes—
505C22×C3#54612yes—
506C22×C3#54612yes—
507C22×C3#54712yes—
508C22×C3#54812yes—
509C22×C3#55212yes—
510C22×C3#55612yes—
511C22×C3#56312yes—
512C22×C3#56312yes—
513C3×C4—24no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—44no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—44no—
521C3×C4—82no—
522C3×C4—82no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C22×C3×D411yescharacteristic
547C23C22×C3×D411yescharacteristic
548C23C22×C3×D411yescharacteristic
549C23C2×C3×#4112yes—
550C23C2×C3×#4112yes—
551C23C3×H(5,𝔽2)12yes—
552C23C3×H(5,𝔽2)11yescharacteristic
553C23C3×2−1+411yescharacteristic
554C23C3×2−1+411yescharacteristic
555C23C3×2−1+412yes—
556C23C3×2−1+411yescharacteristic
557Q8—416no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4C3×#12014yes—
620C2×C3—82no—
621C2×C3#240411yescharacteristic
622C2×C3#243211yescharacteristic
623C2×C3#260411yescharacteristic
624C2×C3#261611yescharacteristic
625C2×C3#263711yescharacteristic
626C2×C3#266211yescharacteristic
627C2×C3#269812yes—
628C2×C3#270212yes—
629C2×C3#271512yes—
630C2×C3#271612yes—
631C2×C3#282111yescharacteristic
632C22—42no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22C2×C3×#12011yescharacteristic
644C22C3×#52511yescharacteristic
645C22C3×#53211yescharacteristic
646C22C3×#53511yescharacteristic
647C22C3×#53611yescharacteristic
648C22C3×#53611yescharacteristic
649C22C3×#53611yescharacteristic
650C22C3×#53611yescharacteristic
651C22C3×#53711yescharacteristic
652C22C3×#53711yescharacteristic
653C22C3×#53711yescharacteristic
654C22C3×#53812yes—
655C22C3×#54112yes—
656C22C3×#54212yes—
657C22C3×#54212yes—
658C22C3×#54612yes—
659C22C3×#54612yes—
660C22C3×#54712yes—
661C22C3×#54812yes—
662C22C3×#55212yes—
663C22C3×#55612yes—
664C22C3×#56312yes—
665C22C3×#56312yes—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—82no—
676C3#1665611yescharacteristic, direct product factor, Sylow
677C2—82no—
678C2C3×#240411yescharacteristic
679C2C3×#243211yescharacteristic
680C2C3×#260411yescharacteristic
681C2C3×#261611yescharacteristic
682C2C3×#263711yescharacteristic
683C2C3×#266211yescharacteristic
684C2C3×#269812yes—
685C2C3×#270212yes—
686C2C3×#271512yes—
687C2C3×#271612yes—
688C2C3×#282111yescharacteristic
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
2. C2×C3×#395C2semidirect
4. C2×C3×#391C2semidirect
5. C3×#1442C2semidirect
6. C3×#1441C2semidirect
7. C3×#1441C2semidirect
11. C3×#1434C2semidirect
12. #16656C3direct (N × H)
43. C2×#395C2×C3semidirect
45. C2×#391C2×C3semidirect
46. #1442C2×C3semidirect
47. #1441C2×C3semidirect
48. #1441C2×C3semidirect
52. #1434C2×C3semidirect
676. C3#16656direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1262
C2×C3×#3952
C2×C3×#3932
C2×C3×#3912
C3×#14422
C3×#14412
C3×#14402
C3×#14392
C3×#14342
#166563