Finite Group Databasebeta

← C3×#16652 #10635259

Subgroups — C3×#16652 #10635259

Order 768.

1654 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#394C211yesmaximal, characteristic
2C2×C3×#393C212yesmaximal, complemented
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C3×#1443C212yesmaximal, complemented
5C3×#1441C211yesmaximal, characteristic, complemented
6C3×#1440C212yesmaximal, complemented
7C3×#1439C212yesmaximal, complemented
8C3×#1438C212yesmaximal
9C3×#1437C212yesmaximal, complemented
10#16652C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C3×#394—24no—
23C3×#393—28no—
24C3×#391—24no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C2×#394C2×C311yescharacteristic
36C2×#393C2×C312yescomplemented
37C2×#391C2×C311yescharacteristic, complemented
38#1443C2×C312yescomplemented
39#1441C2×C311yescharacteristic, complemented
40#1440C2×C312yescomplemented
41#1439C2×C312yescomplemented
42#1438C2×C312yes—
43#1437C2×C312yescomplemented
44C25×C3C2312yes—
45C22×C3×Q8—22no—
46C22×C3×Q8—22no—
47C22×C3×Q8—41no—
48C22×C3×Q8—41no—
49C22×C3×Q8—41no—
50C22×C3×Q8—41no—
51C22×C3×Q8D414yescomplemented
52C22×C3×D4—24no—
53C22×C3×D4—22no—
54C22×C3×D4—42no—
55C22×C3×D4—41no—
56C22×C3×D4—41no—
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C2×C3×C4⋊C4—41no—
66C2×C3×C4⋊C4—41no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4—41no—
69C2×C3×C4⋊C4D414yes—
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—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
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—41no—
90C2×C3×C42—41no—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—24no—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—22no—
97C3×#95—24no—
98C3×#95—42no—
99C3×#95—41no—
100C3×#95—42no—
101C23×Q8C22×C311yescharacteristic
102C23×D4C22×C311yescharacteristic
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C22⋊C4C22×C312yes—
106C22×C22⋊C4C22×C312yes—
107C22×C22⋊C4C22×C312yes—
108C22×C22⋊C4C22×C312yes—
109C22×C22⋊C4C22×C312yes—
110C22×C22⋊C4C22×C312yes—
111C22×C42C22×C311yescharacteristic
112#394—24no—
113#393—28no—
114#391—24no—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C311yescharacteristic
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
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#4114yes—
140C24×C3C2411yescharacteristic
141C2×C3×Q8—28no—
142C2×C3×Q8—44no—
143C2×C3×Q8—44no—
144C2×C3×Q8—44no—
145C2×C3×Q8—44no—
146C2×C3×Q8—44no—
147C2×C3×Q8—44no—
148C2×C3×D4—48no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4—48no—
152C2×C3×D4—44no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
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—21no—
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—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
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—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×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4#4114yes—
204C22×C3×C4#4112yes—
205C3×C4⋊C4—44no—
206C3×C4⋊C4—44no—
207C3×C22⋊C4—48no—
208C3×C22⋊C4—48no—
209C3×C22⋊C4—48no—
210C3×C22⋊C4—48no—
211C3×C22⋊C4—48no—
212C3×C22⋊C4—48no—
213C3×C42—44no—
214C25C23×C312yes—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—41no—
218C22×Q8—41no—
219C22×Q8—41no—
220C22×Q8—41no—
221C22×Q8C3×D414yescomplemented
222C22×D4—24no—
223C22×D4—22no—
224C22×D4—42no—
225C22×D4—41no—
226C22×D4—41no—
227C23×C4C23×C312yes—
228C23×C4C23×C312yes—
229C23×C4C23×C312yes—
230C23×C4C23×C311yescharacteristic
231C23×C4C23×C312yes—
232C23×C4C23×C311yescharacteristic
233C23×C4C23×C312yes—
234C23×C4C23×C311yescharacteristic
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C3×D414yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
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×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C42—22no—
259C2×C42—41no—
260C2×C42—41no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—24no—
268#95—42no—
269#95—41no—
270#95—42no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—24no—
277C23×C3—24no—
278C23×C3—42no—
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×C3C22×D411yescharacteristic
307C23×C3C22×D411yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4112yes—
310C23×C3C2×#4112yes—
311C23×C3H(5,𝔽2)11yescharacteristic
312C23×C3H(5,𝔽2)11yescharacteristic
313C23×C3H(5,𝔽2)11yescharacteristic
314C23×C3H(5,𝔽2)11yescharacteristic
315C23×C3H(5,𝔽2)12yes—
316C23×C32−1+412yes—
317C3×Q8—416no—
318C3×D4—416no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
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#12114yes—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24C3×#4114yes—
386C24C24×C311yescharacteristic
387C2×Q8—28no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—48no—
398C2×D4—44no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
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—22no—
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—22no—
428C22×C4—22no—
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—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4C2×C3×D412yes—
445C22×C4C2×C3×D412yes—
446C22×C4C2×C3×D414yes—
447C22×C4C2×C3×D412yes—
448C22×C4C2×C3×D412yes—
449C22×C4C3×#4114yes—
450C22×C4C3×#4112yes—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C42—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—44no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3C2×#12111yescharacteristic
476C22×C3#52912yes—
477C22×C3#53411yescharacteristic
478C22×C3#53411yescharacteristic
479C22×C3#53411yescharacteristic
480C22×C3#53511yescharacteristic
481C22×C3#53811yescharacteristic
482C22×C3#53811yescharacteristic
483C22×C3#53812yes—
484C22×C3#53812yes—
485C22×C3#53912yes—
486C22×C3#54212yes—
487C22×C3#54212yes—
488C22×C3#54612yes—
489C22×C3#54812yes—
490C22×C3#54812yes—
491C22×C3#55012yes—
492C22×C3#55111yescharacteristic
493C22×C3#55112yes—
494C22×C3#55511yescharacteristic
495C22×C3H(3,𝔽4)11yescharacteristic
496C22×C3#56212yes—
497C22×C3#56311yescharacteristic
498C3×C4—24no—
499C3×C4—44no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—44no—
503C3×C4—82no—
504C3×C4—82no—
505C3×C4—82no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
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—
541C23C22×C3×D411yescharacteristic
542C23C22×C3×D411yescharacteristic
543C23C2×C3×#4111yescharacteristic
544C23C2×C3×#4112yes—
545C23C2×C3×#4112yes—
546C23C3×H(5,𝔽2)12yes—
547C23C3×H(5,𝔽2)11yescharacteristic
548C23C3×H(5,𝔽2)11yescharacteristic
549C23C3×H(5,𝔽2)11yescharacteristic
550C23C3×H(5,𝔽2)11yescharacteristic
551C23C3×2−1+412yes—
552Q8—416no—
553D4—416no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
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×C4C3×#12114yes—
606C2×C3—84no—
607C2×C3#242612yes—
608C2×C3#257211yescharacteristic
609C2×C3#261711yescharacteristic
610C2×C3#265312yes—
611C2×C3#268411yescharacteristic
612C2×C3#269612yes—
613C2×C3#272811yescharacteristic
614C2×C3#273012yes—
615C2×C3#273111yescharacteristic
616C2×C3#273311yescharacteristic
617C2×C3#283211yescharacteristic
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C2×C3×#12111yescharacteristic
634C22C3×#52912yes—
635C22C3×#53411yescharacteristic
636C22C3×#53411yescharacteristic
637C22C3×#53411yescharacteristic
638C22C3×#53511yescharacteristic
639C22C3×#53811yescharacteristic
640C22C3×#53811yescharacteristic
641C22C3×#53812yes—
642C22C3×#53812yes—
643C22C3×#53912yes—
644C22C3×#54212yes—
645C22C3×#54212yes—
646C22C3×#54612yes—
647C22C3×#54812yes—
648C22C3×#54812yes—
649C22C3×#55012yes—
650C22C3×#55111yescharacteristic
651C22C3×#55112yes—
652C22C3×#55511yescharacteristic
653C22C3×H(3,𝔽4)11yescharacteristic
654C22C3×#56212yes—
655C22C3×#56311yescharacteristic
656C4—24no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C3#1665211yescharacteristic, direct product factor, Sylow
665C2—84no—
666C2C3×#242612yes—
667C2C3×#257211yescharacteristic
668C2C3×#261711yescharacteristic
669C2C3×#265312yes—
670C2C3×#268411yescharacteristic
671C2C3×#269612yes—
672C2C3×#272811yescharacteristic
673C2C3×#273012yes—
674C2C3×#273111yescharacteristic
675C2C3×#273311yescharacteristic
676C2C3×#283211yescharacteristic
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×#393C2semidirect
3. C2×C3×#391C2semidirect
4. C3×#1443C2semidirect
5. C3×#1441C2semidirect
6. C3×#1440C2semidirect
7. C3×#1439C2semidirect
9. C3×#1437C2semidirect
10. #16652C3direct (N × H)
36. C2×#393C2×C3semidirect
37. C2×#391C2×C3semidirect
38. #1443C2×C3semidirect
39. #1441C2×C3semidirect
40. #1440C2×C3semidirect
41. #1439C2×C3semidirect
43. #1437C2×C3semidirect
51. C22×C3×Q8D4semidirect
221. C22×Q8C3×D4semidirect
664. C3#16652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3942
C2×C3×#3932
C2×C3×#3912
C3×#14432
C3×#14412
C3×#14402
C3×#14392
C3×#14382
C3×#14372
#166523