Finite Group Databasebeta

← C3×#25714 #10644321

Subgroups — C3×#25714 #10644321

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 570 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×#476C211yesmaximal, characteristic, complemented
4C3×#2547C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1854C212yesmaximal, complemented
10C3×#1853C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25714C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#553—24no—
14C3×#553—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#123—21no—
20C2×C3×#121—21no—
21C2×C3×#121C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C3×#501—24no—
29C3×#476—24no—
30C2×C3×#107C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×#106C2212yes—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#397—21no—
40C3×#397—21no—
41C3×#391—21no—
42C3×#380C2212yes—
43C3×#380C2212yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#553C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#476C2×C311yescharacteristic, complemented
51#2547C2×C311yescharacteristic, complemented
52#2034C2×C312yescomplemented
53#2033C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic, complemented
56#1854C2×C312yescomplemented
57#1853C2×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×#124—44no—
76C3×#123—28no—
77C3×#121—28no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—22no—
82C3×C4×D4—28no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4D414yes—
87C3×#117—22no—
88C3×#117—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42C2311yescharacteristic
112C3×#107—24no—
113C3×#106—24no—
114C3×#106—24no—
115C3×C4⋊C8—22no—
116C3×#102—24no—
117C3×#102—24no—
118C3×C22⋊C8—24no—
119C3×C22⋊C8—24no—
120C3×#97—22no—
121C3×#95—21no—
122C3×#95—22no—
123C3×#95—22no—
124#553—24no—
125#553—24no—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#125C22×C311yescharacteristic
129C2×#124—21no—
130C2×#123—21no—
131C2×#121—21no—
132C2×#121C22×C311yescharacteristic
133C2×#121C22×C311yescharacteristic
134C2×C4×D4—21no—
135C2×C4×D4C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137C2×#117C22×C311yescharacteristic
138C22×C4⋊C4C22×C311yescharacteristic
139#501—24no—
140#476—24no—
141C2×#107C22×C312yes—
142C2×#106C22×C312yes—
143C2×#106C22×C312yes—
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148C2×C22⋊C8C22×C312yes—
149C2×#97C22×C311yescharacteristic
150#397—21no—
151#397—21no—
152#391—21no—
153#380C22×C312yes—
154#380C22×C312yes—
155C4×C4⋊C4C22×C311yescharacteristic
156#340C22×C312yes—
157#340C22×C312yes—
158#340C22×C312yes—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C24×C3C2×D411yescharacteristic
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C2×C3×Q8—42no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—24no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—41no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—82no—
176C2×C3×D4C2×D412yes—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—82no—
193C22×C3×C4—82no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8#4114yes—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4C2×D412yes—
218C3×C4⋊C4C2×D412yes—
219C3×C4⋊C4C2×D412yes—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—44no—
227C3×C22⋊C4—44no—
228C3×C42—24no—
229C3×C42—42no—
230C3×C42C2×D412yes—
231C3×C42C2×D412yes—
232C22×Q8—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C3×D411yescharacteristic
236C22×D4C23×C311yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C23×C311yescharacteristic
240C23×C4C23×C311yescharacteristic
241C22×C8C23×C312yes—
242C22×C8C23×C312yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245#126—28no—
246#125—22no—
247#125C3×D414yes—
248#124—44no—
249#123—28no—
250#121—28no—
251#121—22no—
252#121—22no—
253#121—22no—
254#121—22no—
255C4×D4—28no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4C3×D414yes—
260#117—22no—
261#117—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C3×D411yescharacteristic
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C312yes—
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C3×D411yescharacteristic
281C2×C22⋊C4C3×D411yescharacteristic
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C23×C311yescharacteristic
285#107—24no—
286#106—24no—
287#106—24no—
288C4⋊C8—22no—
289#102—24no—
290#102—24no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293#97—22no—
294#95—21no—
295#95—22no—
296#95—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—21no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—42no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—42no—
311C23×C3AGL(1,ℤ/8)12yes—
312C23×C3C22×D411yescharacteristic
313C3×Q8—84no—
314C3×D4—22no—
315C3×D4—22no—
316C3×D4—44no—
317C3×D4—48no—
318C3×D4—42no—
319C3×D4—84no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—84no—
347C2×C3×C4—84no—
348C2×C3×C4#12012yes—
349C2×C3×C4#12012yes—
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C2×#4112yes—
353C2×C3×C42−1+412yes—
354C3×C8—24no—
355C3×C8—42no—
356C24—21no—
357C24C2×C3×D411yescharacteristic
358C24C2×C3×D411yescharacteristic
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—82no—
373C2×D4C2×C3×D412yes—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C24×C311yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8C3×#4114yes—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4C2×C3×D412yes—
415C4⋊C4C2×C3×D412yes—
416C4⋊C4C2×C3×D412yes—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C42—24no—
426C42—42no—
427C42C2×C3×D412yes—
428C42C2×C3×D412yes—
429C22×C3—24no—
430C22×C3—22no—
431C22×C3—42no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—41no—
437C22×C3—42no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—41no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#54712yes—
446C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
447C22×C3#57511yescharacteristic
448C22×C3#57712yes—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—82no—
457C3×C4—82no—
458C3×C4#53211yescharacteristic
459C3×C4#53611yescharacteristic
460C3×C4#54812yes—
461C23—21no—
462C23—22no—
463C23—21no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23C3×AGL(1,ℤ/8)12yes—
476C23C22×C3×D411yescharacteristic
477Q8—84no—
478D4—22no—
479D4—22no—
480D4—44no—
481D4—48no—
482D4—42no—
483D4—84no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4C3×#12012yes—
513C2×C4C3×#12012yes—
514C2×C4C22×C3×D411yescharacteristic
515C2×C4C22×C3×D411yescharacteristic
516C2×C4C2×C3×#4112yes—
517C2×C4C3×2−1+412yes—
518C8—24no—
519C8—42no—
520C2×C3—42no—
521C2×C3—81no—
522C2×C3—82no—
523C2×C3#243211yescharacteristic
524C2×C3#300611yescharacteristic
525C2×C3#300911yescharacteristic
526C2×C3#334512yes—
527C2×C3#335012yes—
528C22—22no—
529C22—24no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—42no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—82no—
542C22—82no—
543C22C2×C3×#12011yescharacteristic
544C22C3×#54712yes—
545C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
546C22C3×#57511yescharacteristic
547C22C3×#57712yes—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4C3×#53211yescharacteristic
558C4C3×#53611yescharacteristic
559C4C3×#54812yes—
560C3#2571411yescharacteristic, direct product factor, Sylow
561C2—42no—
562C2—81no—
563C2—82no—
564C2C3×#243211yescharacteristic
565C2C3×#300611yescharacteristic
566C2C3×#300911yescharacteristic
567C2C3×#334512yes—
568C2C3×#335012yes—
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×#476C2semidirect
4. C3×#2547C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1989C2semidirect
8. C3×#1955C2semidirect
9. C3×#1854C2semidirect
10. C3×#1853C2semidirect
11. C3×#1762C2semidirect
12. #25714C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#476C2×C3semidirect
51. #2547C2×C3semidirect
52. #2034C2×C3semidirect
53. #2033C2×C3semidirect
54. #1989C2×C3semidirect
55. #1955C2×C3semidirect
56. #1854C2×C3semidirect
57. #1853C2×C3semidirect
58. #1762C2×C3semidirect
560. C3#25714direct (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×#4762
C3×#25472
C3×#20342
C3×#20332
C3×#19892
C3×#19552
C3×#18542
C3×#18532
C3×#17622
#257143