Finite Group Databasebeta

← C3×#25713 #10644320

Subgroups — C3×#25713 #10644320

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#553C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2563C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1854C212yesmaximal, complemented
10C3×#1852C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25713C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#553—24no—
14C3×#553—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#501—24no—
28C3×#474—24no—
29C2×C3×#107C2212yes—
30C2×C3×#107C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×#97C2211yescharacteristic
38C3×#393—21no—
39C3×#393—21no—
40C3×#390—21no—
41C3×#382—21no—
42C3×#380C2212yes—
43C3×#380C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#553C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#474C2×C311yescharacteristic, complemented
51#2563C2×C311yescharacteristic, complemented
52#2034C2×C312yescomplemented
53#2033C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic, complemented
56#1854C2×C312yescomplemented
57#1852C2×C312yescomplemented
58#1762C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4D411yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×#126—28no—
73C3×#125—22no—
74C3×#125D414yes—
75C3×#123—28no—
76C3×#121—28no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×C4×D4—28no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4D414yes—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42C2311yescharacteristic
111C3×#107—24no—
112C3×#107—24no—
113C3×#106—24no—
114C3×C4⋊C8—22no—
115C3×#102—24no—
116C3×#102—24no—
117C3×C22⋊C8—24no—
118C3×C22⋊C8—24no—
119C3×#97—22no—
120C3×#95—21no—
121C3×#95—22no—
122C3×#95—22no—
123#553—24no—
124#553—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#126—21no—
127C2×#125C22×C311yescharacteristic
128C2×#123—21no—
129C2×#121—21no—
130C2×#121C22×C311yescharacteristic
131C2×#121C22×C311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4C22×C311yescharacteristic
134C2×C4×D4C22×C311yescharacteristic
135C2×#117C22×C311yescharacteristic
136C22×C4⋊C4C22×C311yescharacteristic
137#501—24no—
138#474—24no—
139C2×#107C22×C312yes—
140C2×#107C22×C312yes—
141C2×#106C22×C312yes—
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#102C22×C312yes—
144C2×#102C22×C312yes—
145C2×C22⋊C8C22×C312yes—
146C2×C22⋊C8C22×C312yes—
147C2×#97C22×C311yescharacteristic
148#393—21no—
149#393—21no—
150#390—21no—
151#382—21no—
152#380C22×C312yes—
153#380C22×C312yes—
154C4×C4⋊C4C22×C311yescharacteristic
155#340C22×C312yes—
156#340C22×C312yes—
157#340C22×C312yes—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—42no—
162C2×C3×Q8—82no—
163C2×C3×D4—24no—
164C2×C3×D4—22no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—41no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4—82no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—24no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—82no—
191C22×C3×C4—82no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8#4114yes—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—82no—
225C3×C22⋊C4—82no—
226C3×C42—24no—
227C3×C42—42no—
228C3×C42C2×D412yes—
229C3×C42C2×D412yes—
230C22×Q8—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C3×D411yescharacteristic
234C22×D4C23×C311yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C23×C311yescharacteristic
238C23×C4C23×C311yescharacteristic
239C22×C8C23×C312yes—
240C22×C8C23×C312yes—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243#126—28no—
244#125—22no—
245#125C3×D414yes—
246#123—28no—
247#121—28no—
248#121—22no—
249#121—22no—
250#121—22no—
251#121—22no—
252C4×D4—28no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4C3×D414yes—
257#117—22no—
258#117—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C312yes—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C3×D411yescharacteristic
278C2×C22⋊C4C3×D411yescharacteristic
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C23×C311yescharacteristic
282#107—24no—
283#107—24no—
284#106—24no—
285C4⋊C8—22no—
286#102—24no—
287#102—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290#97—22no—
291#95—21no—
292#95—22no—
293#95—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3#13712yes—
309C23×C3C22×D411yescharacteristic
310C3×Q8—84no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—44no—
314C3×D4—48no—
315C3×D4—42no—
316C3×D4—84no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
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—84no—
344C2×C3×C4—84no—
345C2×C3×C4#12012yes—
346C2×C3×C4#12012yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C2×#4112yes—
350C2×C3×C42−1+412yes—
351C3×C8—24no—
352C3×C8—42no—
353C24—21no—
354C24C2×C3×D411yescharacteristic
355C24C2×C3×D411yescharacteristic
356C2×Q8—42no—
357C2×Q8—82no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—82no—
369C2×D4C2×C3×D412yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C24×C311yescharacteristic
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8C3×#4114yes—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4C2×C3×D412yes—
411C4⋊C4C2×C3×D412yes—
412C4⋊C4C2×C3×D412yes—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C42—24no—
422C42—42no—
423C42C2×C3×D412yes—
424C42C2×C3×D412yes—
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—41no—
433C22×C3—42no—
434C22×C3—41no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3C2×#12011yescharacteristic
441C22×C3#54712yes—
442C22×C3C2×#13711yescharacteristic
443C22×C3#57511yescharacteristic
444C22×C3#57612yes—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—82no—
453C3×C4—82no—
454C3×C4#53211yescharacteristic
455C3×C4#53611yescharacteristic
456C3×C4#54812yes—
457C23—21no—
458C23—22no—
459C23—21no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23C3×#13712yes—
472C23C22×C3×D411yescharacteristic
473Q8—84no—
474D4—22no—
475D4—22no—
476D4—44no—
477D4—48no—
478D4—42no—
479D4—84no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4C3×#12012yes—
509C2×C4C3×#12012yes—
510C2×C4C22×C3×D411yescharacteristic
511C2×C4C22×C3×D411yescharacteristic
512C2×C4C2×C3×#4112yes—
513C2×C4C3×2−1+412yes—
514C8—24no—
515C8—42no—
516C2×C3—42no—
517C2×C3—81no—
518C2×C3—82no—
519C2×C3#243211yescharacteristic
520C2×C3#300711yescharacteristic
521C2×C3#300811yescharacteristic
522C2×C3#334612yes—
523C2×C3#334712yes—
524C22—22no—
525C22—24no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22C2×C3×#12011yescharacteristic
540C22C3×#54712yes—
541C22C2×C3×#13711yescharacteristic
542C22C3×#57511yescharacteristic
543C22C3×#57612yes—
544C4—22no—
545C4—22no—
546C4—41no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—82no—
552C4—82no—
553C4C3×#53211yescharacteristic
554C4C3×#53611yescharacteristic
555C4C3×#54812yes—
556C3#2571311yescharacteristic, direct product factor, Sylow
557C2—42no—
558C2—81no—
559C2—82no—
560C2C3×#243211yescharacteristic
561C2C3×#300711yescharacteristic
562C2C3×#300811yescharacteristic
563C2C3×#334612yes—
564C2C3×#334712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2563C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1989C2semidirect
8. C3×#1955C2semidirect
9. C3×#1854C2semidirect
10. C3×#1852C2semidirect
11. C3×#1762C2semidirect
12. #25713C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#474C2×C3semidirect
51. #2563C2×C3semidirect
52. #2034C2×C3semidirect
53. #2033C2×C3semidirect
54. #1989C2×C3semidirect
55. #1955C2×C3semidirect
56. #1854C2×C3semidirect
57. #1852C2×C3semidirect
58. #1762C2×C3semidirect
556. C3#25713direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5532
C2×C3×#5012
C2×C3×#4742
C3×#25632
C3×#20342
C3×#20332
C3×#19892
C3×#19552
C3×#18542
C3×#18522
C3×#17622
#257133