Finite Group Databasebeta

← C3×#24670 #10643277

Subgroups — C3×#24670 #10643277

Order 768.

838 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3038C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic, complemented
3C3×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4C3×#2925C211yesmaximal, characteristic, complemented
5C3×#2051C212yesmaximal, complemented
6C3×#2018C212yesmaximal, complemented
7C3×#1978C212yesmaximal, complemented
8C3×#1965C211yesmaximal, characteristic, complemented
9C3×#1950C211yesmaximal, characteristic, complemented
10C3×#1931C212yesmaximal, complemented
11C3×#1800C211yesmaximal, characteristic, complemented
12#24670C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#127C2211yescharacteristic
17C3×C4×#41C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#460—21no—
20C3×#460—21no—
21C3×#459—21no—
22C3×#459—21no—
23C3×#459—21no—
24C3×#459—21no—
25C3×#459—22no—
26C3×#442—21no—
27C3×#440—21no—
28C3×C4×SD16—21no—
29C3×C4×D8—21no—
30C3×#433—22no—
31C3×#432—21no—
32C3×#431—21no—
33C2×C3×#108C2212yes—
34C3×#428C2211yescharacteristic
35C3×#423C2211yescharacteristic
36C3×#423C2211yescharacteristic
37C2×C3×C4⋊C8C2211yescharacteristic
38C3×#418C2211yescharacteristic
39C3×#418C2212yes—
40C3×#417C2211yescharacteristic
41C2×C3×#102C2212yes—
42C2×C3×#100C2212yes—
43C2×C3×#100C2212yes—
44C2×C3×#100C2212yes—
45C3×#408C2212yes—
46C3×#405C2212yes—
47C3×C4×M16C2211yescharacteristic
48C3×#343C2212yes—
49C3×#341C2212yes—
50#3038C2×C311yescharacteristic, complemented
51C2×#459C2×C311yescharacteristic, complemented
52C4×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
53#2925C2×C311yescharacteristic, complemented
54#2051C2×C312yescomplemented
55#2018C2×C312yescomplemented
56#1978C2×C312yescomplemented
57#1965C2×C311yescharacteristic, complemented
58#1950C2×C311yescharacteristic, complemented
59#1931C2×C312yescomplemented
60#1800C2×C311yescharacteristic, complemented
61C2×C3×#41C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C3×AGL(1,ℤ/8)—22no—
68C3×AGL(1,ℤ/8)—22no—
69C3×AGL(1,ℤ/8)—42no—
70C3×AGL(1,ℤ/8)—42no—
71C2×C3×SD16—21no—
72C2×C3×SD16—22no—
73C2×C3×D8—21no—
74C2×C3×D8—22no—
75C2×C3×D8—24no—
76C2×C3×D8—22no—
77C2×C3×D8—22no—
78C2×C3×D8—42no—
79C2×C3×M16C2312yes—
80C2×C3×M16C2311yescharacteristic
81C2×C3×M16C2312yes—
82C2×C3×M16C2311yescharacteristic
83C22×C3×C8C2312yes—
84C3×#127—21no—
85C3×#127—21no—
86C3×#127—21no—
87C3×#127—21no—
88C3×C4×Q8D411yescharacteristic
89C3×C4×Q8D411yescharacteristic
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4D411yescharacteristic
97C3×C4×D4D411yescharacteristic
98C3×#117—21no—
99C3×#117C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C3×#108—42no—
105C3×#107—21no—
106C3×#106—21no—
107C3×C4⋊C8—21no—
108C3×C4⋊C8—21no—
109C3×C4⋊C8—21no—
110C3×C4⋊C8—21no—
111C3×C4⋊C8D411yescharacteristic
112C3×C4⋊C8D411yescharacteristic
113C3×C4⋊C8D412yes—
114C3×#103—21no—
115C3×#102—21no—
116C3×#102—22no—
117C3×#102—22no—
118C3×#102—21no—
119C3×#102—21no—
120C3×#102—22no—
121C3×#102—22no—
122C3×#100—24no—
123C3×#100—42no—
124C3×#100—42no—
125C3×C22⋊C8—22no—
126C3×C22⋊C8—22no—
127C3×#97—22no—
128C3×#97—21no—
129C3×C4×C8—22no—
130C3×C4×C8—21no—
131C2×AGL(1,ℤ/8)C22×C311yescharacteristic
132C2×AGL(1,ℤ/8)C22×C312yes—
133C22×D8C22×C312yes—
134C2×#127C22×C311yescharacteristic
135C4×#41C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137#460—21no—
138#460—21no—
139#459—21no—
140#459—21no—
141#459—21no—
142#459—21no—
143#459—22no—
144#442—21no—
145#440—21no—
146C4×SD16—21no—
147C4×D8—21no—
148#433—22no—
149#432—21no—
150#431—21no—
151C2×#108C22×C312yes—
152#428C22×C311yescharacteristic
153#423C22×C311yescharacteristic
154#423C22×C311yescharacteristic
155C2×C4⋊C8C22×C311yescharacteristic
156#418C22×C311yescharacteristic
157#418C22×C312yes—
158#417C22×C311yescharacteristic
159C2×#102C22×C312yes—
160C2×#100C22×C312yes—
161C2×#100C22×C312yes—
162C2×#100C22×C312yes—
163#408C22×C312yes—
164#405C22×C312yes—
165C4×M16C22×C311yescharacteristic
166#343C22×C312yes—
167#341C22×C312yes—
168C24×C3—21no—
169C24×C3—22no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C3×#41—21no—
174C2×C3×Q8C2×D411yescharacteristic
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—41no—
188C2×C3×D4—41no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4—82no—
192C2×C3×D4—82no—
193C2×C3×D4C2×D411yescharacteristic
194C2×C3×D4#4112yes—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—41no—
200C22×C3×C4—41no—
201C22×C3×C4—82no—
202C22×C3×C4#4111yescharacteristic
203C22×C3×C4#4111yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C3×SD16—24no—
206C3×SD16—44no—
207C3×D8—24no—
208C3×D8—44no—
209C3×D8—44no—
210C3×D8—44no—
211C3×M16—21no—
212C3×M16—22no—
213C3×M16—21no—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—21no—
217C3×M16—22no—
218C3×M16#4112yes—
219C2×C3×C8—21no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—21no—
224C2×C3×C8—22no—
225C2×C3×C8C2×D412yes—
226C2×C3×C8C2×D412yes—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C2×D411yescharacteristic
232C3×C4⋊C4C2×D411yescharacteristic
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—41no—
237C3×C22⋊C4—41no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C2×#41C23×C311yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C23×C312yes—
249C22×D4C23×C311yescharacteristic
250C23×C4—21no—
251AGL(1,ℤ/8)—22no—
252AGL(1,ℤ/8)—22no—
253AGL(1,ℤ/8)—42no—
254AGL(1,ℤ/8)—42no—
255C2×SD16—21no—
256C2×SD16—22no—
257C2×D8—21no—
258C2×D8—22no—
259C2×D8—24no—
260C2×D8—22no—
261C2×D8—22no—
262C2×D8—42no—
263C2×M16C23×C312yes—
264C2×M16C23×C312yes—
265C2×M16C23×C311yescharacteristic
266C2×M16C23×C311yescharacteristic
267C22×C8C23×C312yes—
268#127—21no—
269#127—21no—
270#127—21no—
271#127—21no—
272C4×Q8C3×D411yescharacteristic
273C4×Q8C3×D411yescharacteristic
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4C3×D411yescharacteristic
281C4×D4C3×D411yescharacteristic
282#117—21no—
283#117C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C42—21no—
287C2×C42C23×C311yescharacteristic
288#108—42no—
289#107—21no—
290#106—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8C3×D411yescharacteristic
296C4⋊C8C3×D411yescharacteristic
297C4⋊C8C3×D412yes—
298#103—21no—
299#102—21no—
300#102—22no—
301#102—22no—
302#102—21no—
303#102—21no—
304#102—22no—
305#102—22no—
306#100—24no—
307#100—42no—
308#100—42no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311#97—22no—
312#97—21no—
313C4×C8—22no—
314C4×C8—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—81no—
323C23×C3—81no—
324C23×C3—82no—
325C23×C3—82no—
326C23×C3C2×#4111yescharacteristic
327C3×Q8—21no—
328C3×Q8—21no—
329C3×D4—21no—
330C3×D4—22no—
331C3×D4—21no—
332C3×D4—22no—
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—42no—
336C3×D4—41no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—41no—
340C3×D4—42no—
341C3×D4—82no—
342C3×D4—82no—
343C3×D4—82no—
344C3×D4—82no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—81no—
355C2×C3×C4—82no—
356C2×C3×C4—82no—
357C2×C3×C4—81no—
358C2×C3×C4#12111yescharacteristic
359C2×C3×C4#12111yescharacteristic
360C2×C3×C4#12111yescharacteristic
361C2×C3×C4#12111yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C3×C8—22no—
369C3×C8—22no—
370C3×C8—22no—
371C3×C8—41no—
372C3×C8—42no—
373C3×C8—42no—
374C24—21no—
375C24—22no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—21no—
380C2×Q8C2×C3×D411yescharacteristic
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4C2×C3×D411yescharacteristic
400C2×D4C3×#4112yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—82no—
408C22×C4C3×#4111yescharacteristic
409C22×C4C3×#4111yescharacteristic
410C22×C4C24×C311yescharacteristic
411SD16—24no—
412SD16—44no—
413D8—24no—
414D8—44no—
415D8—44no—
416D8—44no—
417M16—21no—
418M16—22no—
419M16—21no—
420M16—22no—
421M16—22no—
422M16—21no—
423M16—22no—
424M16C3×#4112yes—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8C2×C3×D412yes—
432C2×C8C2×C3×D412yes—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4C2×C3×D411yescharacteristic
438C4⋊C4C2×C3×D411yescharacteristic
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42C2×C3×D411yescharacteristic
448C42C2×C3×D411yescharacteristic
449C42C2×C3×D411yescharacteristic
450C42C2×C3×D411yescharacteristic
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—41no—
454C22×C3—42no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3—162no—
465C22×C3—162no—
466C22×C3C2×#12111yescharacteristic
467C22×C3#55311yescharacteristic
468C22×C3#55511yescharacteristic
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—81no—
476C3×C4#47212yes—
477C3×C4#52511yescharacteristic
478C3×C4#53511yescharacteristic
479C3×C4#54611yescharacteristic
480C3×C4#55011yescharacteristic
481C23—21no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—81no—
489C23—81no—
490C23—82no—
491C23—82no—
492C23C2×C3×#4111yescharacteristic
493Q8—21no—
494Q8—21no—
495D4—21no—
496D4—22no—
497D4—21no—
498D4—22no—
499D4—41no—
500D4—41no—
501D4—42no—
502D4—41no—
503D4—42no—
504D4—42no—
505D4—41no—
506D4—42no—
507D4—82no—
508D4—82no—
509D4—82no—
510D4—82no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4C3×#12111yescharacteristic
525C2×C4C3×#12111yescharacteristic
526C2×C4C3×#12111yescharacteristic
527C2×C4C3×#12111yescharacteristic
528C2×C4C22×C3×D411yescharacteristic
529C2×C4C22×C3×D411yescharacteristic
530C2×C4C2×C3×#4111yescharacteristic
531C2×C4C2×C3×#4111yescharacteristic
532C2×C4C3×H(5,𝔽2)11yescharacteristic
533C2×C4C3×H(5,𝔽2)11yescharacteristic
534C8—22no—
535C8—22no—
536C8—22no—
537C8—42no—
538C8—41no—
539C8—42no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—81no—
543C2×C3—82no—
544C2×C3—162no—
545C2×C3#241411yescharacteristic
546C2×C3C2×#47211yescharacteristic
547C2×C3#306811yescharacteristic
548C22—21no—
549C22—21no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—82no—
555C22—82no—
556C22—81no—
557C22—81no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—162no—
562C22—162no—
563C22C2×C3×#12111yescharacteristic
564C22C3×#55311yescharacteristic
565C22C3×#55511yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4C3×#47212yes—
574C4C3×#52511yescharacteristic
575C4C3×#53511yescharacteristic
576C4C3×#54611yescharacteristic
577C4C3×#55011yescharacteristic
578C3#2467011yescharacteristic, direct product factor, Sylow
579C2—21no—
580C2—21no—
581C2—81no—
582C2—82no—
583C2—162no—
584C2C3×#241411yescharacteristic
585C2C2×C3×#47211yescharacteristic
586C2C3×#306811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
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×#3038C2semidirect
2. C2×C3×#459C2semidirect
3. C3×C4×AGL(1,ℤ/8)C2semidirect
4. C3×#2925C2semidirect
5. C3×#2051C2semidirect
6. C3×#2018C2semidirect
7. C3×#1978C2semidirect
8. C3×#1965C2semidirect
9. C3×#1950C2semidirect
10. C3×#1931C2semidirect
11. C3×#1800C2semidirect
12. #24670C3direct (N × H)
50. #3038C2×C3semidirect
51. C2×#459C2×C3semidirect
52. C4×AGL(1,ℤ/8)C2×C3semidirect
53. #2925C2×C3semidirect
54. #2051C2×C3semidirect
55. #2018C2×C3semidirect
56. #1978C2×C3semidirect
57. #1965C2×C3semidirect
58. #1950C2×C3semidirect
59. #1931C2×C3semidirect
60. #1800C2×C3semidirect
578. C3#24670direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30382
C2×C3×#4592
C3×C4×AGL(1,ℤ/8)2
C3×#29252
C3×#20512
C3×#20182
C3×#19782
C3×#19652
C3×#19502
C3×#19312
C3×#18002
#246703