Finite Group Databasebeta

← C3×#16683 #10635290

Subgroups — C3×#16683 #10635290

Order 768.

1338 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic
2C2×C3×#400C211yesmaximal, characteristic
3C2×C3×#398C211yesmaximal, characteristic
4C2×C3×#387C211yesmaximal, characteristic
5C3×#1445C211yesmaximal, characteristic
6C3×#1445C211yesmaximal, characteristic
7C3×#1444C212yesmaximal
8C3×#1442C212yesmaximal
9C3×#1442C212yesmaximal
10C3×#1441C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#16683C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125—24no—
14C2×C3×#125—24no—
15C2×C3×#125—24no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C3×#400—24no—
25C3×#398—24no—
26C3×#387—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×C3×#95C2212yes—
41C2×C3×#95C2212yes—
42C22×#125C2×C311yescharacteristic
43C2×#400C2×C311yescharacteristic
44C2×#398C2×C311yescharacteristic
45C2×#387C2×C311yescharacteristic
46#1445C2×C311yescharacteristic
47#1445C2×C311yescharacteristic
48#1444C2×C312yes—
49#1442C2×C312yes—
50#1442C2×C312yes—
51#1441C2×C312yes—
52#1434C2×C311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C3×#125—232no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
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—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—42no—
84C2×C3×C4⋊C4D414yes—
85C2×C3×C4⋊C4D414yes—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42D414yes—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—22no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—24no—
101C3×#95—24no—
102C3×#95—42no—
103C3×#95—42no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125—24no—
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C4⋊C4C22×C312yes—
109C22×C4⋊C4C22×C312yes—
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C4⋊C4C22×C312yes—
112C22×C4⋊C4C22×C311yescharacteristic
113C22×C4⋊C4C22×C311yescharacteristic
114C22×C42C22×C311yescharacteristic
115#400—24no—
116#398—24no—
117#387—24no—
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C311yescharacteristic
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C311yescharacteristic
125C2×#95C22×C312yes—
126C2×#95C22×C311yescharacteristic
127C2×#95C22×C311yescharacteristic
128C2×#95C22×C312yes—
129C2×#95C22×C312yes—
130C2×#95C22×C311yescharacteristic
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C24×C3C2411yescharacteristic
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
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—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
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—22no—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D414yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4#4114yes—
199C22×C3×C4#4112yes—
200C22×C3×C4#4112yes—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—216no—
203C3×C4⋊C4—216no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—48no—
207C3×C4⋊C4—44no—
208C3×C42—28no—
209C23×C4C23×C311yescharacteristic
210C23×C4C23×C312yes—
211C23×C4C23×C311yescharacteristic
212C23×C4C23×C312yes—
213C23×C4C23×C312yes—
214C23×C4C23×C311yescharacteristic
215C23×C4C23×C311yescharacteristic
216C23×C4C23×C311yescharacteristic
217C23×C4C23×C311yescharacteristic
218C23×C4C23×C312yes—
219C23×C4C23×C311yescharacteristic
220#125—232no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C3×D414yes—
241C2×C4⋊C4C3×D414yes—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42C3×D414yes—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—42no—
259#95—42no—
260C23×C3C22×D411yescharacteristic
261C23×C3C22×D411yescharacteristic
262C23×C3C22×D411yescharacteristic
263C23×C3C2×#4111yescharacteristic
264C23×C3C2×#4111yescharacteristic
265C23×C3C2×#4112yes—
266C23×C3H(5,𝔽2)11yescharacteristic
267C23×C32−1+412yes—
268C23×C32−1+412yes—
269C23×C32−1+412yes—
270C23×C32−1+411yescharacteristic
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4#12014yes—
338C24C24×C311yescharacteristic
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D414yes—
398C22×C4C2×C3×D412yes—
399C22×C4C2×C3×D414yes—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D412yes—
402C22×C4C2×C3×D412yes—
403C22×C4C3×#4114yes—
404C22×C4C3×#4112yes—
405C22×C4C3×#4112yes—
406C4⋊C4—216no—
407C4⋊C4—216no—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C42—28no—
414C22×C3C2×#12011yescharacteristic
415C22×C3#52511yescharacteristic
416C22×C3#53211yescharacteristic
417C22×C3#53611yescharacteristic
418C22×C3#53611yescharacteristic
419C22×C3#53611yescharacteristic
420C22×C3#53612yes—
421C22×C3#53612yes—
422C22×C3#53711yescharacteristic
423C22×C3#53912yes—
424C22×C3#54112yes—
425C22×C3#54112yes—
426C22×C3#54112yes—
427C22×C3#54611yescharacteristic
428C22×C3#54711yescharacteristic
429C22×C3#54712yes—
430C22×C3#54811yescharacteristic
431C22×C3#54812yes—
432C22×C3#55011yescharacteristic
433C22×C3#55212yes—
434C22×C3#55212yes—
435C22×C3#56312yes—
436C22×C3#56412yes—
437C3×C4—24no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—44no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—44no—
444C3×C4—44no—
445C3×C4—81no—
446C3×C4—81no—
447C3×C4—82no—
448C23C22×C3×D411yescharacteristic
449C23C22×C3×D411yescharacteristic
450C23C22×C3×D411yescharacteristic
451C23C2×C3×#4111yescharacteristic
452C23C2×C3×#4111yescharacteristic
453C23C2×C3×#4112yes—
454C23C3×H(5,𝔽2)11yescharacteristic
455C23C3×2−1+412yes—
456C23C3×2−1+412yes—
457C23C3×2−1+411yescharacteristic
458C23C3×2−1+412yes—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4C3×#12014yes—
526C2×C3#240711yescharacteristic
527C2×C3#243211yescharacteristic
528C2×C3#261111yescharacteristic
529C2×C3#261411yescharacteristic
530C2×C3#263411yescharacteristic
531C2×C3#266411yescharacteristic
532C2×C3#269012yes—
533C2×C3#271912yes—
534C2×C3#271912yes—
535C2×C3#272112yes—
536C2×C3#282211yescharacteristic
537C22C2×C3×#12011yescharacteristic
538C22C3×#52511yescharacteristic
539C22C3×#53211yescharacteristic
540C22C3×#53611yescharacteristic
541C22C3×#53611yescharacteristic
542C22C3×#53612yes—
543C22C3×#53612yes—
544C22C3×#53611yescharacteristic
545C22C3×#53711yescharacteristic
546C22C3×#53912yes—
547C22C3×#54112yes—
548C22C3×#54112yes—
549C22C3×#54112yes—
550C22C3×#54611yescharacteristic
551C22C3×#54711yescharacteristic
552C22C3×#54712yes—
553C22C3×#54811yescharacteristic
554C22C3×#54812yes—
555C22C3×#55011yescharacteristic
556C22C3×#55212yes—
557C22C3×#55212yes—
558C22C3×#56312yes—
559C22C3×#56412yes—
560C4—24no—
561C4—42no—
562C4—44no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—44no—
567C4—42no—
568C4—81no—
569C4—82no—
570C4—81no—
571C3#1668311yescharacteristic, direct product factor, Sylow
572C2C3×#240711yescharacteristic
573C2C3×#243211yescharacteristic
574C2C3×#261111yescharacteristic
575C2C3×#261411yescharacteristic
576C2C3×#263411yescharacteristic
577C2C3×#266411yescharacteristic
578C2C3×#269012yes—
579C2C3×#271912yes—
580C2C3×#271912yes—
581C2C3×#272112yes—
582C2C3×#282211yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #16683C3direct (N × H)
571. C3#16683direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C3×#4002
C2×C3×#3982
C2×C3×#3872
C3×#14452
C3×#14442
C3×#14422
C3×#14412
C3×#14342
#166833