Finite Group Databasebeta

← C3×#16663 #10635270

Subgroups — C3×#16663 #10635270

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#395C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic
3C2×C3×#387C211yesmaximal, characteristic
4C2×C3×#383C211yesmaximal, characteristic
5C3×#1446C211yesmaximal, characteristic
6C3×#1445C212yesmaximal
7C3×#1443C212yesmaximal
8C3×#1442C212yesmaximal
9C3×#1442C212yesmaximal
10C3×#1442C211yesmaximal, characteristic
11C3×#1441C211yesmaximal, characteristic
12#16663C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C3×#395—24no—
20C3×#391—24no—
21C3×#387—24no—
22C3×#383—24no—
23C2×C3×#95C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2211yescharacteristic
40C2×#395C2×C311yescharacteristic
41C2×#391C2×C311yescharacteristic
42C2×#387C2×C311yescharacteristic
43C2×#383C2×C311yescharacteristic
44#1446C2×C311yescharacteristic
45#1445C2×C312yes—
46#1443C2×C312yes—
47#1442C2×C312yes—
48#1442C2×C312yes—
49#1442C2×C311yescharacteristic
50#1441C2×C311yescharacteristic
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—24no—
54C22×C3×Q8—22no—
55C22×C3×Q8—42no—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
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—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—42no—
77C2×C3×C4⋊C4—42no—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D414yes—
81C3×#95—24no—
82C3×#95—24no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—24no—
94C3×#95—42no—
95C3×#95—42no—
96C3×#95—42no—
97C3×#95D414yes—
98C23×Q8C22×C311yescharacteristic
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C311yescharacteristic
101C22×C4⋊C4C22×C312yes—
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C42C22×C311yescharacteristic
104#395—24no—
105#391—24no—
106#387—24no—
107#383—24no—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C311yescharacteristic
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C311yescharacteristic
120C2×#95C22×C311yescharacteristic
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C311yescharacteristic
125C24×C3C2411yescharacteristic
126C2×C3×Q8—28no—
127C2×C3×Q8—48no—
128C2×C3×Q8—48no—
129C2×C3×Q8—44no—
130C2×C3×Q8—44no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
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—21no—
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—21no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
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—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4#4112yes—
195C22×C3×C4#4112yes—
196C22×C3×C4#4114yes—
197C22×C3×C4#4112yes—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—48no—
200C3×C4⋊C4—44no—
201C3×C4⋊C4—44no—
202C3×C42—28no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—42no—
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C312yes—
210C23×C4C23×C312yes—
211C23×C4C23×C311yescharacteristic
212C23×C4C23×C312yes—
213C23×C4C23×C311yescharacteristic
214C23×C4C23×C311yescharacteristic
215C23×C4C23×C311yescharacteristic
216C23×C4C23×C312yes—
217C23×C4C23×C311yescharacteristic
218C23×C4C23×C311yescharacteristic
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C3×D414yes—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95C3×D414yes—
250C23×C3C22×D411yescharacteristic
251C23×C3C22×D411yescharacteristic
252C23×C3C2×#4111yescharacteristic
253C23×C3C2×#4112yes—
254C23×C3C2×#4111yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3H(5,𝔽2)12yes—
257C23×C32−1+412yes—
258C23×C32−1+412yes—
259C23×C32−1+411yescharacteristic
260C23×C32−1+411yescharacteristic
261C3×Q8—416no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
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#12114yes—
329C24C24×C311yescharacteristic
330C2×Q8—28no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—44no—
334C2×Q8—44no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×D414yes—
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D412yes—
398C22×C4C3×#4112yes—
399C22×C4C3×#4112yes—
400C22×C4C3×#4114yes—
401C22×C4C3×#4112yes—
402C4⋊C4—28no—
403C4⋊C4—48no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C42—28no—
407C22×C3C2×#12111yescharacteristic
408C22×C3#52511yescharacteristic
409C22×C3#52511yescharacteristic
410C22×C3#53511yescharacteristic
411C22×C3#53611yescharacteristic
412C22×C3#53611yescharacteristic
413C22×C3#53611yescharacteristic
414C22×C3#53612yes—
415C22×C3#54112yes—
416C22×C3#54112yes—
417C22×C3#54112yes—
418C22×C3#54212yes—
419C22×C3#54212yes—
420C22×C3#54612yes—
421C22×C3#54711yescharacteristic
422C22×C3#54811yescharacteristic
423C22×C3#55112yes—
424C22×C3#55212yes—
425C22×C3#55211yescharacteristic
426C22×C3#55612yes—
427C22×C3#55611yescharacteristic
428C22×C3#56312yes—
429C22×C3#56312yes—
430C3×C4—24no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—44no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—44no—
437C3×C4—81no—
438C3×C4—82no—
439C3×C4—81no—
440C3×C4—82no—
441C23C22×C3×D411yescharacteristic
442C23C22×C3×D411yescharacteristic
443C23C2×C3×#4111yescharacteristic
444C23C2×C3×#4112yes—
445C23C2×C3×#4111yescharacteristic
446C23C2×C3×#4111yescharacteristic
447C23C3×H(5,𝔽2)12yes—
448C23C3×2−1+412yes—
449C23C3×2−1+411yescharacteristic
450C23C3×2−1+411yescharacteristic
451C23C3×2−1+412yes—
452Q8—416no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
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×C4C3×#12114yes—
520C2×C3#241111yescharacteristic
521C2×C3#241511yescharacteristic
522C2×C3#259911yescharacteristic
523C2×C3#262111yescharacteristic
524C2×C3#263211yescharacteristic
525C2×C3#263411yescharacteristic
526C2×C3#269812yes—
527C2×C3#271612yes—
528C2×C3#275812yes—
529C2×C3#276212yes—
530C2×C3#282111yescharacteristic
531C22C2×C3×#12111yescharacteristic
532C22C3×#52511yescharacteristic
533C22C3×#52511yescharacteristic
534C22C3×#53511yescharacteristic
535C22C3×#53611yescharacteristic
536C22C3×#53612yes—
537C22C3×#53611yescharacteristic
538C22C3×#53611yescharacteristic
539C22C3×#54112yes—
540C22C3×#54112yes—
541C22C3×#54112yes—
542C22C3×#54212yes—
543C22C3×#54212yes—
544C22C3×#54612yes—
545C22C3×#54711yescharacteristic
546C22C3×#54811yescharacteristic
547C22C3×#55112yes—
548C22C3×#55211yescharacteristic
549C22C3×#55212yes—
550C22C3×#55611yescharacteristic
551C22C3×#55612yes—
552C22C3×#56312yes—
553C22C3×#56312yes—
554C4—24no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—42no—
561C4—82no—
562C4—81no—
563C4—82no—
564C4—81no—
565C3#1666311yescharacteristic, direct product factor, Sylow
566C2C3×#241111yescharacteristic
567C2C3×#241511yescharacteristic
568C2C3×#259911yescharacteristic
569C2C3×#262111yescharacteristic
570C2C3×#263211yescharacteristic
571C2C3×#263411yescharacteristic
572C2C3×#269812yes—
573C2C3×#271612yes—
574C2C3×#275812yes—
575C2C3×#276212yes—
576C2C3×#282111yescharacteristic
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. #16663C3direct (N × H)
565. C3#16663direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3952
C2×C3×#3912
C2×C3×#3872
C2×C3×#3832
C3×#14462
C3×#14452
C3×#14432
C3×#14422
C3×#14412
#166633