Finite Group Databasebeta

← C3×#16673 #10635280

Subgroups — C3×#16673 #10635280

Order 768.

1534 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#126C211yesmaximal, characteristic
2C2×C3×#398C211yesmaximal, characteristic, complemented
3C2×C3×#397C211yesmaximal, characteristic
4C2×C3×#389C211yesmaximal, characteristic, complemented
5C3×#1444C212yesmaximal, complemented
6C3×#1442C212yesmaximal, complemented
7C3×#1441C211yesmaximal, characteristic, complemented
8C3×#1439C211yesmaximal, characteristic
9C3×#1439C212yesmaximal
10C3×#1439C212yesmaximal
11C3×#1434C211yesmaximal, characteristic, complemented
12#16673C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#126—24no—
14C2×C3×#126—24no—
15C2×C3×#126—24no—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2212yes—
22C22×C3×C4⋊C4C2212yes—
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C22⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4C2212yes—
26C22×C3×C22⋊C4C2211yescharacteristic
27C22×C3×C22⋊C4C2212yes—
28C22×C3×C42C2211yescharacteristic
29C3×#398—24no—
30C3×#397—24no—
31C3×#389—24no—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
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×#398C2×C311yescharacteristic, complemented
44C2×#397C2×C311yescharacteristic
45C2×#389C2×C311yescharacteristic, complemented
46#1444C2×C312yescomplemented
47#1442C2×C312yescomplemented
48#1441C2×C311yescharacteristic, complemented
49#1439C2×C312yes—
50#1439C2×C312yes—
51#1439C2×C311yescharacteristic
52#1434C2×C311yescharacteristic, complemented
53C25×C3C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C3×#126—232no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—42no—
81C2×C3×C4⋊C4—42no—
82C2×C3×C4⋊C4—42no—
83C2×C3×C4⋊C4—42no—
84C2×C3×C4⋊C4D414yes—
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—22no—
93C2×C3×C22⋊C4—24no—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4—24no—
96C2×C3×C22⋊C4—42no—
97C2×C3×C22⋊C4—42no—
98C2×C3×C22⋊C4—42no—
99C2×C3×C22⋊C4D414yes—
100C2×C3×C42—22no—
101C2×C3×C42—22no—
102C2×C3×C42D414yes—
103C3×#95—24no—
104C3×#95—22no—
105C3×#95—24no—
106C3×#95—22no—
107C3×#95—24no—
108C3×#95—24no—
109C3×#95—24no—
110C3×#95—24no—
111C3×#95—42no—
112C3×#95—42no—
113C2×#126—24no—
114C2×#126—24no—
115C2×#126—24no—
116C22×C4⋊C4C22×C311yescharacteristic
117C22×C4⋊C4C22×C311yescharacteristic
118C22×C4⋊C4C22×C311yescharacteristic
119C22×C4⋊C4C22×C311yescharacteristic
120C22×C4⋊C4C22×C311yescharacteristic
121C22×C4⋊C4C22×C312yes—
122C22×C4⋊C4C22×C312yes—
123C22×C22⋊C4C22×C311yescharacteristic
124C22×C22⋊C4C22×C311yescharacteristic
125C22×C22⋊C4C22×C312yes—
126C22×C22⋊C4C22×C311yescharacteristic
127C22×C22⋊C4C22×C312yes—
128C22×C42C22×C311yescharacteristic
129#398—24no—
130#397—24no—
131#389—24no—
132C2×#95C22×C311yescharacteristic
133C2×#95C22×C312yes—
134C2×#95C22×C311yescharacteristic
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—21no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3C2×D412yes—
153C24×C3C2411yescharacteristic
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
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×C4C2×D412yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4#4114yes—
215C22×C3×C4#4114yes—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—28no—
219C3×C4⋊C4—28no—
220C3×C4⋊C4—48no—
221C3×C4⋊C4—48no—
222C3×C4⋊C4—44no—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—28no—
226C3×C22⋊C4—48no—
227C3×C22⋊C4—48no—
228C3×C42—28no—
229C25C23×C311yescharacteristic
230C23×C4C23×C312yes—
231C23×C4C23×C312yes—
232C23×C4C23×C311yescharacteristic
233C23×C4C23×C311yescharacteristic
234C23×C4C23×C311yescharacteristic
235C23×C4C23×C311yescharacteristic
236C23×C4C23×C311yescharacteristic
237C23×C4C23×C312yes—
238C23×C4C23×C312yes—
239C23×C4C23×C311yescharacteristic
240#126—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C3×D414yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
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—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4C3×D414yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42C3×D414yes—
279#95—24no—
280#95—22no—
281#95—24no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—42no—
288#95—42no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3C22×D411yescharacteristic
313C23×C3C22×D411yescharacteristic
314C23×C3C22×D411yescharacteristic
315C23×C3C2×#4112yes—
316C23×C3C2×#4112yes—
317C23×C3H(5,𝔽2)11yescharacteristic
318C23×C3H(5,𝔽2)11yescharacteristic
319C23×C3H(5,𝔽2)11yescharacteristic
320C23×C32−1+412yes—
321C23×C32−1+412yes—
322C23×C32−1+411yescharacteristic
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
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—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
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#12014yes—
385C24—21no—
386C24—21no—
387C24—22no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24—21no—
392C24—21no—
393C24—22no—
394C24—22no—
395C24C2×C3×D412yes—
396C24C24×C311yescharacteristic
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
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×C4C2×C3×D412yes—
450C22×C4C2×C3×D412yes—
451C22×C4C2×C3×D412yes—
452C22×C4C2×C3×D412yes—
453C22×C4C2×C3×D412yes—
454C22×C4C2×C3×D412yes—
455C22×C4C2×C3×D412yes—
456C22×C4C2×C3×D412yes—
457C22×C4C3×#4114yes—
458C22×C4C3×#4114yes—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C42—28no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3C2×#12011yescharacteristic
484C22×C3#52511yescharacteristic
485C22×C3#53211yescharacteristic
486C22×C3#53511yescharacteristic
487C22×C3#53511yescharacteristic
488C22×C3#53511yescharacteristic
489C22×C3#53611yescharacteristic
490C22×C3#53611yescharacteristic
491C22×C3#53711yescharacteristic
492C22×C3#53711yescharacteristic
493C22×C3#53711yescharacteristic
494C22×C3#53912yes—
495C22×C3#54112yes—
496C22×C3#54112yes—
497C22×C3#54212yes—
498C22×C3#54712yes—
499C22×C3#54712yes—
500C22×C3#54712yes—
501C22×C3#54812yes—
502C22×C3#55112yes—
503C22×C3#55312yes—
504C22×C3#56312yes—
505C22×C3#56312yes—
506C3×C4—24no—
507C3×C4—44no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—44no—
514C3×C4—82no—
515C3×C4—82no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C22×C3×D411yescharacteristic
540C23C22×C3×D411yescharacteristic
541C23C22×C3×D411yescharacteristic
542C23C2×C3×#4112yes—
543C23C2×C3×#4112yes—
544C23C3×H(5,𝔽2)11yescharacteristic
545C23C3×H(5,𝔽2)11yescharacteristic
546C23C3×H(5,𝔽2)11yescharacteristic
547C23C3×2−1+412yes—
548C23C3×2−1+412yes—
549C23C3×2−1+411yescharacteristic
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
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×C4C3×#12014yes—
612C2×C3—82no—
613C2×C3#240611yescharacteristic
614C2×C3#243211yescharacteristic
615C2×C3#260311yescharacteristic
616C2×C3#261511yescharacteristic
617C2×C3#263711yescharacteristic
618C2×C3#266411yescharacteristic
619C2×C3#271812yes—
620C2×C3#272212yes—
621C2×C3#272312yes—
622C2×C3#272512yes—
623C2×C3#282111yescharacteristic
624C22—42no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22—82no—
633C22—81no—
634C22—81no—
635C22C2×C3×#12011yescharacteristic
636C22C3×#52511yescharacteristic
637C22C3×#53211yescharacteristic
638C22C3×#53511yescharacteristic
639C22C3×#53511yescharacteristic
640C22C3×#53511yescharacteristic
641C22C3×#53611yescharacteristic
642C22C3×#53611yescharacteristic
643C22C3×#53711yescharacteristic
644C22C3×#53711yescharacteristic
645C22C3×#53711yescharacteristic
646C22C3×#53912yes—
647C22C3×#54112yes—
648C22C3×#54112yes—
649C22C3×#54212yes—
650C22C3×#54712yes—
651C22C3×#54712yes—
652C22C3×#54712yes—
653C22C3×#54812yes—
654C22C3×#55112yes—
655C22C3×#55312yes—
656C22C3×#56312yes—
657C22C3×#56312yes—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C3#1667311yescharacteristic, direct product factor, Sylow
669C2—82no—
670C2C3×#240611yescharacteristic
671C2C3×#243211yescharacteristic
672C2C3×#260311yescharacteristic
673C2C3×#261511yescharacteristic
674C2C3×#263711yescharacteristic
675C2C3×#266411yescharacteristic
676C2C3×#271812yes—
677C2C3×#272212yes—
678C2C3×#272312yes—
679C2C3×#272512yes—
680C2C3×#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×#398C2semidirect
4. C2×C3×#389C2semidirect
5. C3×#1444C2semidirect
6. C3×#1442C2semidirect
7. C3×#1441C2semidirect
11. C3×#1434C2semidirect
12. #16673C3direct (N × H)
43. C2×#398C2×C3semidirect
45. C2×#389C2×C3semidirect
46. #1444C2×C3semidirect
47. #1442C2×C3semidirect
48. #1441C2×C3semidirect
52. #1434C2×C3semidirect
668. C3#16673direct (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×#3982
C2×C3×#3972
C2×C3×#3892
C3×#14442
C3×#14422
C3×#14412
C3×#14392
C3×#14342
#166733