Finite Group Databasebeta

← C3×#25617 #10644224

Subgroups — C3×#25617 #10644224

Order 768.

1106 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3512C211yesmaximal, characteristic, complemented
2C3×#3153C211yesmaximal, characteristic, complemented
3C3×#3040C211yesmaximal, characteristic, complemented
4C3×#2861C211yesmaximal, characteristic, complemented
5C3×#2007C212yesmaximal, complemented
6C3×#2006C212yesmaximal, complemented
7C3×#1998C211yesmaximal, characteristic, complemented
8C3×#1962C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1882C212yesmaximal, complemented
11C3×C4×#104C211yesmaximal, characteristic, complemented
12#25617C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C2×C3×#137C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
17C3×#553—22no—
18C3×#553—24no—
19C3×#552—22no—
20C3×#541—21no—
21C3×#540—21no—
22C3×#537C2212yes—
23C3×#532C2211yescharacteristic
24C3×#529—22no—
25C2×C3×#125C2211yescharacteristic
26C2×C3×#124C2211yescharacteristic
27C3×#520C2212yes—
28C3×C4×#41C2211yescharacteristic
29C3×#497—21no—
30C3×#496—21no—
31C3×#495—22no—
32C3×#464—22no—
33C3×#463—22no—
34C3×#423C2211yescharacteristic
35C2×C3×#104C2212yes—
36C2×C3×#104C2212yes—
37C2×C3×#104C2212yes—
38C3×#419C2211yescharacteristic
39C3×#418C2211yescharacteristic
40C3×C4×M16C2211yescharacteristic
41C3×#391C2212yes—
42C3×#390C2212yes—
43C3×#388—24no—
44C3×C43C2211yescharacteristic
45C3×#343C2212yes—
46C3×#343C2212yes—
47C3×#339C2212yes—
48#3512C2×C311yescharacteristic, complemented
49#3153C2×C311yescharacteristic, complemented
50#3040C2×C311yescharacteristic, complemented
51#2861C2×C311yescharacteristic, complemented
52#2007C2×C312yescomplemented
53#2006C2×C312yescomplemented
54#1998C2×C311yescharacteristic, complemented
55#1962C2×C311yescharacteristic, complemented
56#1883C2×C312yescomplemented
57#1882C2×C312yescomplemented
58C4×#104C2×C311yescharacteristic, complemented
59C2×C3×#41C2311yescharacteristic
60C2×C3×#41C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C3×#137—24no—
64C3×#137—42no—
65C3×#137—42no—
66C3×AGL(1,ℤ/8)—24no—
67C3×AGL(1,ℤ/8)—42no—
68C3×AGL(1,ℤ/8)—42no—
69C2×C3×Q16—21no—
70C2×C3×Q16—22no—
71C2×C3×SD16—21no—
72C2×C3×SD16—22no—
73C2×C3×SD16—21no—
74C2×C3×SD16—22no—
75C2×C3×D8—21no—
76C2×C3×D8—22no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2312yes—
80C3×C4⋊Q8D411yescharacteristic
81C3×#127D411yescharacteristic
82C3×#126—24no—
83C3×#126—24no—
84C3×#125—24no—
85C3×#125—21no—
86C3×#125—21no—
87C3×#125—22no—
88C3×#124—22no—
89C3×#124—21no—
90C3×#124—21no—
91C3×#124D412yes—
92C3×#123—24no—
93C3×#123—24no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—22no—
96C3×#121—24no—
97C3×#121—22no—
98C3×#121—22no—
99C3×#121—22no—
100C3×#121—24no—
101C3×C4×Q8—22no—
102C3×C4×Q8D412yes—
103C3×C4×D4—24no—
104C3×C4×D4—22no—
105C3×C4×D4—22no—
106C3×C4×D4—24no—
107C3×C4×D4—22no—
108C3×C4×D4D412yes—
109C3×#117—24no—
110C3×#117—22no—
111C3×#117—21no—
112C3×#117C2311yescharacteristic
113C2×C3×C4⋊C4—22no—
114C2×C3×C4⋊C4D414yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C22⋊C4—24no—
117C2×C3×C22⋊C4—22no—
118C2×C3×C42—21no—
119C2×C3×C42—22no—
120C2×C3×C42C2312yes—
121C2×C3×C42C2311yescharacteristic
122C3×C4⋊C8—22no—
123C3×#104—24no—
124C3×#104—42no—
125C3×#104—42no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#97—21no—
129C3×C4×C8—21no—
130C3×#95—24no—
131C2×#137C22×C312yes—
132C2×#137C22×C311yescharacteristic
133C2×AGL(1,ℤ/8)C22×C312yes—
134C2×AGL(1,ℤ/8)C22×C311yescharacteristic
135#553—22no—
136#553—24no—
137#552—22no—
138#541—21no—
139#540—21no—
140#537C22×C312yes—
141#532C22×C311yescharacteristic
142#529—22no—
143C2×#125C22×C311yescharacteristic
144C2×#124C22×C311yescharacteristic
145#520C22×C312yes—
146C4×#41C22×C311yescharacteristic
147#497—21no—
148#496—21no—
149#495—22no—
150#464—22no—
151#463—22no—
152#423C22×C311yescharacteristic
153C2×#104C22×C312yes—
154C2×#104C22×C312yes—
155C2×#104C22×C312yes—
156#419C22×C311yescharacteristic
157#418C22×C311yescharacteristic
158C4×M16C22×C311yescharacteristic
159#391C22×C312yes—
160#390C22×C312yes—
161#388—24no—
162C43C22×C311yescharacteristic
163#343C22×C312yes—
164#343C22×C312yes—
165#339C22×C312yes—
166C24×C3—21no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C3×#41—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8C2×D412yes—
178C2×C3×Q8C2×D411yescharacteristic
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—24no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4C2×D412yes—
188C2×C3×D4C2×D411yescharacteristic
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2411yescharacteristic
198C3×Q16—44no—
199C3×Q16—42no—
200C3×SD16—44no—
201C3×SD16—44no—
202C3×SD16—42no—
203C3×SD16—42no—
204C3×D8—44no—
205C3×D8—42no—
206C3×M16—22no—
207C3×M16—22no—
208C3×M16—22no—
209C3×M16#4114yes—
210C2×C3×C8—22no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4C2×D412yes—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—48no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C42—24no—
233C3×C42—22no—
234C3×C42—21no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—44no—
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D412yes—
241C2×#41C23×C311yescharacteristic
242C2×#41C23×C312yes—
243C22×Q8C23×C311yescharacteristic
244C22×D4C23×C311yescharacteristic
245#137—24no—
246#137—42no—
247#137—42no—
248AGL(1,ℤ/8)—24no—
249AGL(1,ℤ/8)—42no—
250AGL(1,ℤ/8)—42no—
251C2×Q16—21no—
252C2×Q16—22no—
253C2×SD16—21no—
254C2×SD16—22no—
255C2×SD16—21no—
256C2×SD16—22no—
257C2×D8—21no—
258C2×D8—22no—
259C2×M16C23×C311yescharacteristic
260C2×M16C23×C311yescharacteristic
261C2×M16C23×C312yes—
262C4⋊Q8C3×D411yescharacteristic
263#127C3×D411yescharacteristic
264#126—24no—
265#126—24no—
266#125—24no—
267#125—21no—
268#125—21no—
269#125—22no—
270#124—22no—
271#124—21no—
272#124—21no—
273#124C3×D412yes—
274#123—24no—
275#123—24no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278#121—24no—
279#121—22no—
280#121—22no—
281#121—22no—
282#121—24no—
283C4×Q8—22no—
284C4×Q8C3×D412yes—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4C3×D412yes—
291#117—24no—
292#117—22no—
293#117—21no—
294#117C23×C311yescharacteristic
295C2×C4⋊C4—22no—
296C2×C4⋊C4C3×D414yes—
297C2×C4⋊C4C23×C312yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42C23×C312yes—
303C2×C42C23×C311yescharacteristic
304C4⋊C8—22no—
305#104—24no—
306#104—42no—
307#104—42no—
308#103—22no—
309#102—22no—
310#97—21no—
311C4×C8—21no—
312#95—24no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—44no—
316C23×C3—41no—
317C23×C3—42no—
318C23×C3C22×D411yescharacteristic
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—42no—
326C3×D4—21no—
327C3×D4—21no—
328C3×D4—22no—
329C3×D4—22no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—42no—
336C3×D4—82no—
337C3×D4—84no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—24no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—44no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—82no—
353C2×C3×C4—84no—
354C2×C3×C4#12012yes—
355C2×C3×C4#12012yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C2×#4112yes—
359C2×C3×C4H(5,𝔽2)12yes—
360C3×C8—24no—
361C3×C8—41no—
362C3×C8—41no—
363C24—21no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—22no—
369#41—22no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C2×C3×D412yes—
375C2×Q8C2×C3×D411yescharacteristic
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4C2×C3×D412yes—
385C2×D4C2×C3×D411yescharacteristic
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4C2×C3×D414yes—
393C22×C4C2×C3×D412yes—
394C22×C4C24×C311yescharacteristic
395Q16—44no—
396Q16—42no—
397SD16—44no—
398SD16—44no—
399SD16—42no—
400SD16—42no—
401D8—44no—
402D8—42no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16C3×#4114yes—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—21no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4C2×C3×D412yes—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—21no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—48no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C42—24no—
430C42—22no—
431C42—21no—
432C42—22no—
433C42—22no—
434C42—44no—
435C42C2×C3×D411yescharacteristic
436C42C2×C3×D411yescharacteristic
437C42C2×C3×D412yes—
438C22×C3—22no—
439C22×C3—42no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—84no—
445C22×C3—81no—
446C22×C3—82no—
447C22×C3C2×#12011yescharacteristic
448C22×C3#55012yes—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—44no—
452C3×C4—42no—
453C3×C4—41no—
454C3×C4—44no—
455C3×C4—42no—
456C3×C4—81no—
457C3×C4#53211yescharacteristic
458C3×C4#53511yescharacteristic
459C3×C4#54612yes—
460C23—22no—
461C23—21no—
462C23—44no—
463C23—41no—
464C23—42no—
465C23C22×C3×D411yescharacteristic
466Q8—21no—
467Q8—21no—
468Q8—22no—
469Q8—22no—
470Q8—41no—
471Q8—41no—
472Q8—42no—
473D4—21no—
474D4—21no—
475D4—22no—
476D4—22no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—41no—
481D4—41no—
482D4—42no—
483D4—82no—
484D4—84no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4C3×#12012yes—
502C2×C4C3×#12012yes—
503C2×C4C22×C3×D411yescharacteristic
504C2×C4C22×C3×D411yescharacteristic
505C2×C4C2×C3×#4112yes—
506C2×C4C3×H(5,𝔽2)12yes—
507C8—24no—
508C8—41no—
509C8—41no—
510C2×C3—22no—
511C2×C3—82no—
512C2×C3—81no—
513C2×C3—161no—
514C2×C3#243111yescharacteristic
515C2×C3#301812yes—
516C22—22no—
517C22—42no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—84no—
523C22—81no—
524C22—82no—
525C22C2×C3×#12011yescharacteristic
526C22C3×#55012yes—
527C4—22no—
528C4—22no—
529C4—44no—
530C4—42no—
531C4—41no—
532C4—44no—
533C4—42no—
534C4—81no—
535C4C3×#53211yescharacteristic
536C4C3×#53511yescharacteristic
537C4C3×#54612yes—
538C3#2561711yescharacteristic, direct product factor, Sylow
539C2—22no—
540C2—82no—
541C2—81no—
542C2—161no—
543C2C3×#243111yescharacteristic
544C2C3×#301812yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3512C2semidirect
2. C3×#3153C2semidirect
3. C3×#3040C2semidirect
4. C3×#2861C2semidirect
5. C3×#2007C2semidirect
6. C3×#2006C2semidirect
7. C3×#1998C2semidirect
8. C3×#1962C2semidirect
9. C3×#1883C2semidirect
10. C3×#1882C2semidirect
11. C3×C4×#104C2semidirect
12. #25617C3direct (N × H)
48. #3512C2×C3semidirect
49. #3153C2×C3semidirect
50. #3040C2×C3semidirect
51. #2861C2×C3semidirect
52. #2007C2×C3semidirect
53. #2006C2×C3semidirect
54. #1998C2×C3semidirect
55. #1962C2×C3semidirect
56. #1883C2×C3semidirect
57. #1882C2×C3semidirect
58. C4×#104C2×C3semidirect
538. C3#25617direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35122
C3×#31532
C3×#30402
C3×#28612
C3×#20072
C3×#20062
C3×#19982
C3×#19622
C3×#18832
C3×#18822
C3×C4×#1042
#256173