Finite Group Databasebeta

← C3×#16668 #10635275

Subgroups — C3×#16668 #10635275

Order 768.

1398 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C211yesmaximal, characteristic
2C2×C3×#381C211yesmaximal, characteristic
3C3×#1445C212yesmaximal, complemented
4C3×#1442C212yesmaximal, complemented
5C3×#1441C212yesmaximal, complemented
6C3×#1441C212yesmaximal, complemented
7C3×#1439C211yesmaximal, characteristic
8C3×#1439C212yesmaximal
9C3×C22⋊#95C212yesmaximal
10#16668C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C3×#399—24no—
21C3×#381—24no—
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×#399C2×C311yescharacteristic
35C2×#381C2×C311yescharacteristic
36#1445C2×C312yescomplemented
37#1442C2×C312yescomplemented
38#1441C2×C312yescomplemented
39#1441C2×C312yescomplemented
40#1439C2×C312yes—
41#1439C2×C311yescharacteristic
42C22⋊#95C2×C312yes—
43C25×C3C2311yescharacteristic
44C23×C3×C4—22no—
45C23×C3×C4—22no—
46C23×C3×C4—41no—
47C23×C3×C4—41no—
48C23×C3×C4—41no—
49C23×C3×C4—41no—
50C23×C3×C4Q814yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—42no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4—41no—
67C2×C3×C4⋊C4—41no—
68C2×C3×C4⋊C4—41no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—41no—
76C2×C3×C22⋊C4—41no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4—42no—
79C2×C3×C22⋊C4—41no—
80C2×C3×C22⋊C4—41no—
81C2×C3×C22⋊C4D414yes—
82C3×#95—24no—
83C3×#95—24no—
84C3×#95—22no—
85C3×#95—24no—
86C3×#95—41no—
87C3×#95—42no—
88C3×#95—42no—
89C3×#95—42no—
90C3×#95—42no—
91C3×#95—42no—
92C3×#95—42no—
93C3×#95—42no—
94C24×C4C22×C311yescharacteristic
95C22×C4⋊C4C22×C312yes—
96C22×C4⋊C4C22×C312yes—
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C311yescharacteristic
99C22×C22⋊C4C22×C312yes—
100C22×C22⋊C4C22×C312yes—
101C22×C22⋊C4C22×C311yescharacteristic
102C22×C22⋊C4C22×C311yescharacteristic
103#399—24no—
104#381—24no—
105C2×#95C22×C311yescharacteristic
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C311yescharacteristic
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—22no—
125C24×C3C2×D412yes—
126C24×C3C2×Q812yes—
127C24×C3#4112yes—
128C24×C3C2411yescharacteristic
129C22×C3×C4—28no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
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—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
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—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
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—22no—
176C22×C3×C4—22no—
177C22×C3×C4—44no—
178C22×C3×C4—44no—
179C22×C3×C4—44no—
180C22×C3×C4—44no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×Q814yes—
186C22×C3×C4#4114yes—
187C22×C3×C4#4114yes—
188C3×C4⋊C4—48no—
189C3×C4⋊C4—48no—
190C3×C4⋊C4—44no—
191C3×C4⋊C4—44no—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—48no—
194C3×C22⋊C4—48no—
195C3×C22⋊C4—44no—
196C25C23×C311yescharacteristic
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4C3×Q814yes—
204C23×C4C23×C312yes—
205C23×C4C23×C312yes—
206C23×C4C23×C312yes—
207C23×C4C23×C311yescharacteristic
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C312yes—
210C23×C4C23×C312yes—
211C23×C4C23×C312yes—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4C3×D414yes—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—24no—
239#95—41no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—42no—
247C23×C3—24no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—41no—
269C23×C3C22⋊Q814yes—
270C23×C3C22×D411yescharacteristic
271C23×C3C22×Q811yescharacteristic
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4112yes—
274C23×C3C2×#4112yes—
275C23×C3H(5,𝔽2)11yescharacteristic
276C23×C3H(5,𝔽2)11yescharacteristic
277C23×C3H(5,𝔽2)11yescharacteristic
278C23×C3H(5,𝔽2)11yescharacteristic
279C23×C3H(5,𝔽2)12yes—
280C23×C32−1+412yes—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—416no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
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—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24C2×C3×D412yes—
343C24C2×C3×Q812yes—
344C24C3×#4112yes—
345C24C24×C311yescharacteristic
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D412yes—
402C22×C4C2×C3×Q814yes—
403C22×C4C3×#4114yes—
404C22×C4C3×#4114yes—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22×C3—24no—
414C22×C3—28no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C2×C22⋊Q811yescharacteristic
425C22×C3#53312yes—
426C22×C3#53411yescharacteristic
427C22×C3#53511yescharacteristic
428C22×C3#53811yescharacteristic
429C22×C3#53811yescharacteristic
430C22×C3#53912yes—
431C22×C3#53912yes—
432C22×C3#54012yes—
433C22×C3#54012yes—
434C22×C3#54311yescharacteristic
435C22×C3#54311yescharacteristic
436C22×C3#54412yes—
437C22×C3#54612yes—
438C22×C3#54712yes—
439C22×C3#55112yes—
440C22×C3#55112yes—
441C22×C3#55212yes—
442C22×C3#55911yescharacteristic
443C22×C3#55911yescharacteristic
444C22×C3#56011yescharacteristic
445C22×C3#56212yes—
446C22×C3#56311yescharacteristic
447C3×C4—44no—
448C3×C4—44no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—44no—
452C3×C4—82no—
453C3×C4—82no—
454C3×C4—82no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23C3×C22⋊Q814yes—
478C23C22×C3×D411yescharacteristic
479C23C22×C3×Q811yescharacteristic
480C23C2×C3×#4112yes—
481C23C2×C3×#4112yes—
482C23C2×C3×#4111yescharacteristic
483C23C3×H(5,𝔽2)11yescharacteristic
484C23C3×H(5,𝔽2)12yes—
485C23C3×H(5,𝔽2)11yescharacteristic
486C23C3×H(5,𝔽2)11yescharacteristic
487C23C3×H(5,𝔽2)11yescharacteristic
488C23C3×2−1+412yes—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—416no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
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—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C3—44no—
543C2×C3#243812yes—
544C2×C3#260711yescharacteristic
545C2×C3#262611yescharacteristic
546C2×C3#266712yes—
547C2×C3#270612yes—
548C2×C3#270911yescharacteristic
549C2×C3#272411yescharacteristic
550C2×C3#274312yes—
551C2×C3#275211yescharacteristic
552C2×C3#277111yescharacteristic
553C2×C3#284111yescharacteristic
554C22—24no—
555C22—28no—
556C22—44no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22C2×C3×C22⋊Q811yescharacteristic
566C22C3×#53312yes—
567C22C3×#53411yescharacteristic
568C22C3×#53511yescharacteristic
569C22C3×#53811yescharacteristic
570C22C3×#53811yescharacteristic
571C22C3×#53912yes—
572C22C3×#53912yes—
573C22C3×#54012yes—
574C22C3×#54012yes—
575C22C3×#54311yescharacteristic
576C22C3×#54311yescharacteristic
577C22C3×#54412yes—
578C22C3×#54612yes—
579C22C3×#54712yes—
580C22C3×#55112yes—
581C22C3×#55112yes—
582C22C3×#55212yes—
583C22C3×#55911yescharacteristic
584C22C3×#55911yescharacteristic
585C22C3×#56011yescharacteristic
586C22C3×#56212yes—
587C22C3×#56311yescharacteristic
588C4—44no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—44no—
593C4—82no—
594C4—82no—
595C4—82no—
596C3#1666811yescharacteristic, direct product factor, Sylow
597C2—44no—
598C2C3×#243812yes—
599C2C3×#260711yescharacteristic
600C2C3×#262611yescharacteristic
601C2C3×#266712yes—
602C2C3×#270612yes—
603C2C3×#270911yescharacteristic
604C2C3×#272411yescharacteristic
605C2C3×#274312yes—
606C2C3×#275211yescharacteristic
607C2C3×#277111yescharacteristic
608C2C3×#284111yescharacteristic
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
3. C3×#1445C2semidirect
4. C3×#1442C2semidirect
5. C3×#1441C2semidirect
6. C3×#1441C2semidirect
10. #16668C3direct (N × H)
36. #1445C2×C3semidirect
37. #1442C2×C3semidirect
38. #1441C2×C3semidirect
39. #1441C2×C3semidirect
596. C3#16668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3812
C3×#14452
C3×#14422
C3×#14412
C3×#14392
C3×C22⋊#952
#166683