Finite Group Databasebeta

← C3×#16693 #10635300

Subgroups — C3×#16693 #10635300

Order 768.

1514 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2C2×C3×#383C211yesmaximal, characteristic, complemented
3C3×#1444C211yesmaximal, characteristic, complemented
4C3×#1442C211yesmaximal, characteristic, complemented
5C3×#1440C212yesmaximal, complemented
6C3×#1440C212yesmaximal, complemented
7C3×#1439C212yesmaximal, complemented
8C3×#1439C212yesmaximal, complemented
9C3×#1438C211yesmaximal, characteristic
10C3×#1438C211yesmaximal, characteristic
11C3×#1434C211yesmaximal, characteristic, complemented
12#16693C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C3×#390—24no—
24C3×#383—24no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×#390C2×C311yescharacteristic
39C2×#383C2×C311yescharacteristic, complemented
40#1444C2×C311yescharacteristic, complemented
41#1442C2×C311yescharacteristic, complemented
42#1440C2×C312yescomplemented
43#1440C2×C312yescomplemented
44#1439C2×C312yescomplemented
45#1439C2×C312yescomplemented
46#1438C2×C311yescharacteristic
47#1438C2×C311yescharacteristic
48#1434C2×C311yescharacteristic, complemented
49C25×C3C2312yes—
50C22×C3×D4—24no—
51C22×C3×D4—22no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—42no—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—42no—
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⋊C4—42no—
85C2×C3×C22⋊C4—42no—
86C2×C3×C22⋊C4—42no—
87C2×C3×C22⋊C4—42no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42D414yes—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—24no—
99C3×#95—22no—
100C3×#95—42no—
101C3×#95—42no—
102C3×#95—42no—
103C3×#95—42no—
104C23×D4C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C4⋊C4C22×C312yes—
107C22×C22⋊C4C22×C312yes—
108C22×C22⋊C4C22×C312yes—
109C22×C22⋊C4C22×C312yes—
110C22×C22⋊C4C22×C312yes—
111C22×C22⋊C4C22×C312yes—
112C22×C22⋊C4C22×C312yes—
113C22×C42C22×C311yescharacteristic
114#390—24no—
115#383—24no—
116C2×#95C22×C311yescharacteristic
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C312yes—
119C2×#95C22×C311yescharacteristic
120C2×#95C22×C311yescharacteristic
121C2×#95C22×C311yescharacteristic
122C2×#95C22×C311yescharacteristic
123C2×#95C22×C311yescharacteristic
124C2×#95C22×C311yescharacteristic
125C2×#95C22×C312yes—
126C2×#95C22×C312yes—
127C2×#95C22×C312yes—
128C2×#95C22×C312yes—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3#4114yes—
144C24×C3C2411yescharacteristic
145C2×C3×D4—28no—
146C2×C3×D4—48no—
147C2×C3×D4—48no—
148C2×C3×D4—44no—
149C2×C3×D4—44no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
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—21no—
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—22no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
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—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4#4114yes—
206C22×C3×C4#4112yes—
207C22×C3×C4#4112yes—
208C3×C4⋊C4—44no—
209C3×C4⋊C4—48no—
210C3×C22⋊C4—48no—
211C3×C22⋊C4—48no—
212C3×C22⋊C4—48no—
213C3×C22⋊C4—48no—
214C3×C22⋊C4—48no—
215C3×C22⋊C4—48no—
216C3×C42—28no—
217C25C23×C312yes—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—42no—
223C23×C4C23×C311yescharacteristic
224C23×C4C23×C312yes—
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C311yescharacteristic
227C23×C4C23×C311yescharacteristic
228C23×C4C23×C312yes—
229C23×C4C23×C311yescharacteristic
230C23×C4C23×C311yescharacteristic
231C23×C4C23×C312yes—
232C23×C4C23×C311yescharacteristic
233C2×C4⋊C4—22no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C3×D414yes—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—22no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—24no—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3C22×D411yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4112yes—
311C23×C3C2×#4112yes—
312C23×C3H(5,𝔽2)12yes—
313C23×C3H(5,𝔽2)12yes—
314C23×C3H(5,𝔽2)11yescharacteristic
315C23×C3H(5,𝔽2)11yescharacteristic
316C23×C3H(5,𝔽2)11yescharacteristic
317C23×C32−1+411yescharacteristic
318C3×D4—416no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
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—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
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—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
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—42no—
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—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#12314yes—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24C3×#4114yes—
392C24C24×C311yescharacteristic
393C2×D4—28no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—44no—
397C2×D4—44no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4C2×C3×D412yes—
451C22×C4C2×C3×D412yes—
452C22×C4C2×C3×D412yes—
453C22×C4C3×#4114yes—
454C22×C4C3×#4112yes—
455C22×C4C3×#4112yes—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C42—28no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3C2×#12311yescharacteristic
481C22×C3#52511yescharacteristic
482C22×C3#53211yescharacteristic
483C22×C3#53511yescharacteristic
484C22×C3#53511yescharacteristic
485C22×C3#53511yescharacteristic
486C22×C3#53711yescharacteristic
487C22×C3#53812yes—
488C22×C3#53812yes—
489C22×C3#53912yes—
490C22×C3#54011yescharacteristic
491C22×C3#54011yescharacteristic
492C22×C3#54211yescharacteristic
493C22×C3#54211yescharacteristic
494C22×C3#54212yes—
495C22×C3#55112yes—
496C22×C3#55112yes—
497C22×C3#55112yes—
498C22×C3#55312yes—
499C22×C3#55312yes—
500C22×C3#55512yes—
501C22×C3#56012yes—
502C22×C3#56012yes—
503C3×C4—24no—
504C3×C4—42no—
505C3×C4—44no—
506C3×C4—42no—
507C3×C4—82no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—82no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C22×C3×D411yescharacteristic
549C23C2×C3×#4111yescharacteristic
550C23C2×C3×#4111yescharacteristic
551C23C2×C3×#4112yes—
552C23C2×C3×#4112yes—
553C23C3×H(5,𝔽2)12yes—
554C23C3×H(5,𝔽2)12yes—
555C23C3×H(5,𝔽2)11yescharacteristic
556C23C3×H(5,𝔽2)11yescharacteristic
557C23C3×H(5,𝔽2)11yescharacteristic
558C23C3×2−1+411yescharacteristic
559D4—416no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
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×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
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×C4C3×#12314yes—
618C2×C3—84no—
619C2×C3#248211yescharacteristic
620C2×C3#250011yescharacteristic
621C2×C3#260311yescharacteristic
622C2×C3#262411yescharacteristic
623C2×C3#263611yescharacteristic
624C2×C3#266211yescharacteristic
625C2×C3#273712yes—
626C2×C3#274912yes—
627C2×C3#275612yes—
628C2×C3#275712yes—
629C2×C3#281811yescharacteristic
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22C2×C3×#12311yescharacteristic
646C22C3×#52511yescharacteristic
647C22C3×#53211yescharacteristic
648C22C3×#53511yescharacteristic
649C22C3×#53511yescharacteristic
650C22C3×#53511yescharacteristic
651C22C3×#53711yescharacteristic
652C22C3×#53812yes—
653C22C3×#53812yes—
654C22C3×#53912yes—
655C22C3×#54011yescharacteristic
656C22C3×#54011yescharacteristic
657C22C3×#54211yescharacteristic
658C22C3×#54211yescharacteristic
659C22C3×#54212yes—
660C22C3×#55112yes—
661C22C3×#55112yes—
662C22C3×#55112yes—
663C22C3×#55312yes—
664C22C3×#55312yes—
665C22C3×#55512yes—
666C22C3×#56012yes—
667C22C3×#56012yes—
668C4—24no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—81no—
678C3#1669311yescharacteristic, direct product factor, Sylow
679C2—84no—
680C2C3×#248211yescharacteristic
681C2C3×#250011yescharacteristic
682C2C3×#260311yescharacteristic
683C2C3×#262411yescharacteristic
684C2C3×#263611yescharacteristic
685C2C3×#266211yescharacteristic
686C2C3×#273712yes—
687C2C3×#274912yes—
688C2C3×#275612yes—
689C2C3×#275712yes—
690C2C3×#281811yescharacteristic
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
2. C2×C3×#383C2semidirect
3. C3×#1444C2semidirect
4. C3×#1442C2semidirect
5. C3×#1440C2semidirect
6. C3×#1440C2semidirect
7. C3×#1439C2semidirect
8. C3×#1439C2semidirect
11. C3×#1434C2semidirect
12. #16693C3direct (N × H)
39. C2×#383C2×C3semidirect
40. #1444C2×C3semidirect
41. #1442C2×C3semidirect
42. #1440C2×C3semidirect
43. #1440C2×C3semidirect
44. #1439C2×C3semidirect
45. #1439C2×C3semidirect
48. #1434C2×C3semidirect
678. C3#16693direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3902
C2×C3×#3832
C3×#14442
C3×#14422
C3×#14402
C3×#14392
C3×#14382
C3×#14342
#166933