Finite Group Databasebeta

← C3×#25643 #10644250

Subgroups — C3×#25643 #10644250

Order 768.

1108 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C3×#2568C211yesmaximal, characteristic, complemented
5C3×#2016C212yesmaximal, complemented
6C3×#2015C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic
9C3×#1878C212yesmaximal, complemented
10C3×#1877C212yesmaximal
11C3×#1763C211yesmaximal, characteristic, complemented
12#25643C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#556—24no—
17C3×#556—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126—21no—
21C2×C3×#126—21no—
22C2×C3×#124C2211yescharacteristic
23C2×C3×#124C2211yescharacteristic
24C2×C3×C22⋊Q8—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117—21no—
30C3×#497—24no—
31C3×#462—24no—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#97C2211yescharacteristic
38C3×#400—21no—
39C3×#400—21no—
40C3×#391—21no—
41C3×#391C2212yes—
42C3×#391C2212yes—
43C3×#382—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#556C2×C311yescharacteristic
49C2×#497C2×C311yescharacteristic
50C2×#462C2×C311yescharacteristic, complemented
51#2568C2×C311yescharacteristic, complemented
52#2016C2×C312yescomplemented
53#2015C2×C312yes—
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic
56#1878C2×C312yescomplemented
57#1877C2×C312yes—
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C2×C3×Q16—24no—
66C2×C3×Q16—24no—
67C2×C3×Q16—24no—
68C2×C3×Q16—42no—
69C2×C3×Q16—42no—
70C2×C3×Q16—42no—
71C2×C3×SD16—24no—
72C2×C3×SD16—24no—
73C2×C3×SD16—42no—
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—22no—
80C3×#126—28no—
81C3×#126—28no—
82C3×#124—22no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124D414yes—
86C3×C22⋊Q8—28no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8D414yes—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×#117—28no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×C4⋊C8—22no—
119C3×#103—24no—
120C3×#103—24no—
121C3×#103—42no—
122C3×#102—42no—
123C3×#97—22no—
124C3×#95—21no—
125C3×#95—22no—
126C3×#95—22no—
127C3×#95—22no—
128C3×#95—22no—
129C22×Q16C22×C312yes—
130C22×Q16C22×C312yes—
131C22×SD16C22×C312yes—
132#556—24no—
133#556—24no—
134C2×C4⋊Q8C22×C311yescharacteristic
135C2×C4⋊Q8C22×C311yescharacteristic
136C2×#126—21no—
137C2×#126—21no—
138C2×#124C22×C311yescharacteristic
139C2×#124C22×C311yescharacteristic
140C2×C22⋊Q8—21no—
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×D4C22×C311yescharacteristic
145C2×#117—21no—
146#497—24no—
147#462—24no—
148C2×C4⋊C8C22×C311yescharacteristic
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#103C22×C312yes—
152C2×#102C22×C312yes—
153C2×#97C22×C311yescharacteristic
154#400—21no—
155#400—21no—
156#391—21no—
157#391C22×C312yes—
158#391C22×C312yes—
159#382—21no—
160C4×C4⋊C4C22×C311yescharacteristic
161#340C22×C312yes—
162#336C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—41no—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—24no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C3×Q16—48no—
202C3×Q16—48no—
203C3×SD16—48no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8#4114yes—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4C2×D412yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C42—24no—
229C3×C42—24no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—42no—
235C3×C42C2×D412yes—
236C3×C42C2×D412yes—
237C22×Q8C3×D411yescharacteristic
238C22×Q8C23×C312yes—
239C22×Q8C23×C311yescharacteristic
240C22×Q8C23×C311yescharacteristic
241C22×D4C23×C311yescharacteristic
242C23×C4—21no—
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—24no—
246C2×Q16—42no—
247C2×Q16—42no—
248C2×Q16—42no—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—42no—
252C22×C8C23×C312yes—
253C22×C8C23×C312yes—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258#126—28no—
259#126—28no—
260#124—22no—
261#124—22no—
262#124—22no—
263#124C3×D414yes—
264C22⋊Q8—28no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8C3×D414yes—
271C4×D4—22no—
272C4×D4—22no—
273#117—28no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C3×D411yescharacteristic
281C2×C4⋊C4C3×D411yescharacteristic
282C2×C4⋊C4C3×D411yescharacteristic
283C2×C4⋊C4C23×C312yes—
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296C4⋊C8—22no—
297#103—24no—
298#103—24no—
299#103—42no—
300#102—42no—
301#97—22no—
302#95—21no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307C23×C3—22no—
308C23×C3—42no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3C22×D411yescharacteristic
313C3×Q8—22no—
314C3×Q8—22no—
315C3×Q8—24no—
316C3×Q8—44no—
317C3×Q8—41no—
318C3×Q8—41no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×D4—24no—
325C3×D4—41no—
326C3×D4—41no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—48no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4#12012yes—
354C2×C3×C4#12012yes—
355C2×C3×C4#13712yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C2×#4112yes—
359C2×C3×C4H(5,𝔽2)12yes—
360C3×C8—24no—
361C3×C8—42no—
362C24—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—41no—
378C2×Q8C2×C3×D412yes—
379C2×Q8C2×C3×D412yes—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4C2×C3×D412yes—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C24×C311yescharacteristic
399Q16—48no—
400Q16—48no—
401SD16—48no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8C3×#4114yes—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4C2×C3×D412yes—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C42—24no—
427C42—24no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42C2×C3×D412yes—
434C42C2×C3×D412yes—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3C2×#12011yescharacteristic
441C22×C3#55012yes—
442C22×C3C2×#13711yescharacteristic
443C22×C3#57511yescharacteristic
444C22×C3#57812yes—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—42no—
450C3×C4—41no—
451C3×C4—42no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—82no—
455C3×C4—82no—
456C3×C4#53211yescharacteristic
457C3×C4#53511yescharacteristic
458C3×C4#54612yes—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23C22×C3×D411yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—24no—
468Q8—44no—
469Q8—41no—
470Q8—41no—
471Q8—42no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476D4—24no—
477D4—41no—
478D4—41no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—48no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4C3×#12012yes—
506C2×C4C3×#12012yes—
507C2×C4C3×#13712yes—
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C22×C3×D411yescharacteristic
510C2×C4C2×C3×#4112yes—
511C2×C4C3×H(5,𝔽2)12yes—
512C8—24no—
513C8—42no—
514C2×C3—82no—
515C2×C3#243111yescharacteristic
516C2×C3#300711yescharacteristic
517C2×C3#301111yescharacteristic
518C2×C3#339812yes—
519C2×C3#340212yes—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22C2×C3×#12011yescharacteristic
526C22C3×#55012yes—
527C22C2×C3×#13711yescharacteristic
528C22C3×#57511yescharacteristic
529C22C3×#57812yes—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—42no—
534C4—42no—
535C4—41no—
536C4—42no—
537C4—41no—
538C4—41no—
539C4—82no—
540C4—82no—
541C4C3×#53211yescharacteristic
542C4C3×#53511yescharacteristic
543C4C3×#54612yes—
544C3#2564311yescharacteristic, direct product factor, Sylow
545C2—82no—
546C2C3×#243111yescharacteristic
547C2C3×#300711yescharacteristic
548C2C3×#301111yescharacteristic
549C2C3×#339812yes—
550C2C3×#340212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#462C2semidirect
4. C3×#2568C2semidirect
5. C3×#2016C2semidirect
7. C3×#1985C2semidirect
9. C3×#1878C2semidirect
11. C3×#1763C2semidirect
12. #25643C3direct (N × H)
50. C2×#462C2×C3semidirect
51. #2568C2×C3semidirect
52. #2016C2×C3semidirect
54. #1985C2×C3semidirect
56. #1878C2×C3semidirect
58. #1763C2×C3semidirect
544. C3#25643direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4972
C2×C3×#4622
C3×#25682
C3×#20162
C3×#20152
C3×#19852
C3×#19832
C3×#18782
C3×#18772
C3×#17632
#256433