Finite Group Databasebeta

← C3×#24196 #10642803

Subgroups — C3×#24196 #10642803

Order 768.

1428 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic, complemented
3C2×C3×#442C211yesmaximal, characteristic, complemented
4C3×#2580C211yesmaximal, characteristic, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1864C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24196C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×D8C2211yescharacteristic
15C22×C3×D8C2212yes—
16C3×#550—24no—
17C3×#550—24no—
18C2×C3×#127C2212yes—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#121—22no—
21C2×C3×#121—21no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C3×#459—24no—
27C3×#442—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#102C2212yes—
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#399—21no—
38C3×#393—21no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#385—22no—
42C3×#382C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#550C2×C311yescharacteristic, complemented
47C2×#459C2×C311yescharacteristic, complemented
48C2×#442C2×C311yescharacteristic, complemented
49#2580C2×C311yescharacteristic, complemented
50#2041C2×C312yescomplemented
51#2011C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1961C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1864C2×C312yescomplemented
56#1804C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—22no—
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C2×C3×D8—22no—
71C2×C3×D8—22no—
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C2×C3×D8—22no—
75C2×C3×D8—42no—
76C22×C3×C8C2312yes—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#127—24no—
80C3×#127—24no—
81C3×#127—22no—
82C3×#127D414yes—
83C3×#121—28no—
84C3×#121—216no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×D4—24no—
88C3×C4×D4—24no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4D414yes—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C42C2312yes—
105C2×C3×C42C2311yescharacteristic
106C3×#106—22no—
107C3×C4⋊C8—22no—
108C3×#102—22no—
109C3×#102—24no—
110C3×#102—24no—
111C3×#102—22no—
112C3×#102—42no—
113C3×#102—42no—
114C3×#97—22no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—22no—
118C22×#41—21no—
119C22×D8C22×C311yescharacteristic
120C22×D8C22×C312yes—
121#550—24no—
122#550—24no—
123C2×#127C22×C312yes—
124C2×#127C22×C311yescharacteristic
125C2×#121—22no—
126C2×#121—21no—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×D4C22×C311yescharacteristic
129C2×C4×D4C22×C312yes—
130C2×C4×D4C22×C311yescharacteristic
131#459—24no—
132#442—24no—
133C2×#106C22×C311yescharacteristic
134C2×C4⋊C8C22×C311yescharacteristic
135C2×#102C22×C312yes—
136C2×#102C22×C312yes—
137C2×#102C22×C311yescharacteristic
138C2×#102C22×C311yescharacteristic
139C2×#102C22×C312yes—
140C2×#102C22×C312yes—
141C2×#97C22×C311yescharacteristic
142#399—21no—
143#393—21no—
144#390C22×C312yes—
145#389C22×C312yes—
146#385—22no—
147#382C22×C311yescharacteristic
148#340C22×C312yes—
149#336C22×C312yes—
150#336C22×C312yes—
151C24×C3—21no—
152C24×C3—22no—
153C24×C3—21no—
154C3×#41—432no—
155C2×C3×Q8—22no—
156C2×C3×Q8—24no—
157C2×C3×D4—28no—
158C2×C3×D4—28no—
159C2×C3×D4—24no—
160C2×C3×D4—24no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—44no—
177C2×C3×D4—44no—
178C2×C3×D4—44no—
179C2×C3×D4—82no—
180C2×C3×D4—82no—
181C2×C3×D4—82no—
182C2×C3×D4C2×D414yes—
183C2×C3×D4C2×D412yes—
184C2×C3×D4#4112yescomplemented
185C22×C3×C4—28no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—44no—
191C22×C3×C4—44no—
192C22×C3×C4—44no—
193C22×C3×C4—42no—
194C22×C3×C4—82no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2411yescharacteristic
200C3×D8—28no—
201C3×D8—48no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—21no—
208C2×C3×C8—22no—
209C2×C3×C8—21no—
210C2×C3×C8#4112yescomplemented
211C3×C4⋊C4—24no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—82no—
219C3×C4⋊C4C2×D412yes—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—82no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229C2×#41—28no—
230C2×#41—48no—
231C22×Q8C3×D411yescharacteristic
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4C23×C312yes—
237C22×D4C23×C311yescharacteristic
238C22×D4C23×C311yescharacteristic
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C2×D8—22no—
243C2×D8—22no—
244C2×D8—24no—
245C2×D8—24no—
246C2×D8—22no—
247C2×D8—42no—
248C22×C8C23×C312yes—
249C22×C8C23×C311yescharacteristic
250C22×C8C23×C311yescharacteristic
251#127—24no—
252#127—24no—
253#127—22no—
254#127C3×D414yes—
255#121—28no—
256#121—216no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4C3×D414yes—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C3×D412yes—
267C2×C4⋊C4C3×D411yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C42C23×C312yes—
277C2×C42C23×C311yescharacteristic
278#106—22no—
279C4⋊C8—22no—
280#102—22no—
281#102—24no—
282#102—24no—
283#102—22no—
284#102—42no—
285#102—42no—
286#97—22no—
287#95—21no—
288#95—21no—
289#95—22no—
290C23×C3—24no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—28no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—24no—
314C3×D4—24no—
315C3×D4—48no—
316C3×D4—416no—
317C3×D4—42no—
318C3×D4—41no—
319C3×D4—42no—
320C3×D4—41no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—84no—
324C3×D4—88no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—48no—
334C2×C3×C4—416no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—84no—
342C2×C3×C4#12012yes—
343C2×C3×C4#12012yes—
344C2×C3×C4AGL(1,ℤ/8)14yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4H(5,𝔽2)11yescharacteristic
350C2×C3×C42−1+411yescharacteristic
351C3×C8—22no—
352C3×C8—41no—
353C3×C8—42no—
354C24—21no—
355C24—22no—
356C24—21no—
357#41—432no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4C2×C3×D414yes—
386C2×D4C2×C3×D412yes—
387C2×D4C3×#4112yescomplemented
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4C2×C3×D412yes—
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D412yes—
402C22×C4C24×C311yescharacteristic
403D8—28no—
404D8—48no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8C3×#4112yescomplemented
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—82no—
422C4⋊C4C2×C3×D412yes—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—82no—
428C42—22no—
429C42—22no—
430C42C2×C3×D412yes—
431C42C2×C3×D412yes—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—44no—
436C22×C3—42no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3C2×#12011yescharacteristic
450C22×C3#54811yescharacteristic
451C22×C3#55011yescharacteristic
452C22×C3C2×AGL(1,ℤ/8)12yes—
453C22×C3#57612yes—
454C3×C4—22no—
455C3×C4—24no—
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—82no—
460C3×C4—81no—
461C3×C4#52511yescharacteristic
462C3×C4#53711yescharacteristic
463C3×C4#54611yescharacteristic
464C3×C4#54711yescharacteristic
465C23—24no—
466C23—22no—
467C23—22no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23C22×C3×D411yescharacteristic
483Q8—28no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4—22no—
488D4—24no—
489D4—24no—
490D4—48no—
491D4—416no—
492D4—42no—
493D4—41no—
494D4—42no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—84no—
499D4—88no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4C3×#12012yes—
518C2×C4C3×#12012yes—
519C2×C4C3×AGL(1,ℤ/8)14yes—
520C2×C4C22×C3×D411yescharacteristic
521C2×C4C22×C3×D411yescharacteristic
522C2×C4C2×C3×#4111yescharacteristic
523C2×C4C2×C3×#4111yescharacteristic
524C2×C4C3×H(5,𝔽2)11yescharacteristic
525C2×C4C3×2−1+411yescharacteristic
526C8—22no—
527C8—41no—
528C8—42no—
529C2×C3—82no—
530C2×C3—82no—
531C2×C3—84no—
532C2×C3#240711yescharacteristic
533C2×C3#300311yescharacteristic
534C2×C3#300811yescharacteristic
535C2×C3#336612yes—
536C2×C3#339912yes—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—44no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C2×C3×#12011yescharacteristic
555C22C3×#54811yescharacteristic
556C22C3×#55011yescharacteristic
557C22C2×C3×AGL(1,ℤ/8)12yes—
558C22C3×#57612yes—
559C4—22no—
560C4—24no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—81no—
566C4C3×#52511yescharacteristic
567C4C3×#53711yescharacteristic
568C4C3×#54611yescharacteristic
569C4C3×#54711yescharacteristic
570C3#2419611yescharacteristic, direct product factor, Sylow
571C2—82no—
572C2—84no—
573C2—82no—
574C2C3×#240711yescharacteristic
575C2C3×#300311yescharacteristic
576C2C3×#300811yescharacteristic
577C2C3×#336612yes—
578C2C3×#339912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#550C2semidirect
2. C2×C3×#459C2semidirect
3. C2×C3×#442C2semidirect
4. C3×#2580C2semidirect
5. C3×#2041C2semidirect
6. C3×#2011C2semidirect
7. C3×#1985C2semidirect
8. C3×#1961C2semidirect
9. C3×#1879C2semidirect
10. C3×#1864C2semidirect
11. C3×#1804C2semidirect
12. #24196C3direct (N × H)
46. C2×#550C2×C3semidirect
47. C2×#459C2×C3semidirect
48. C2×#442C2×C3semidirect
49. #2580C2×C3semidirect
50. #2041C2×C3semidirect
51. #2011C2×C3semidirect
52. #1985C2×C3semidirect
53. #1961C2×C3semidirect
54. #1879C2×C3semidirect
55. #1864C2×C3semidirect
56. #1804C2×C3semidirect
184. C2×C3×D4#41semidirect
210. C2×C3×C8#41semidirect
387. C2×D4C3×#41semidirect
413. C2×C8C3×#41semidirect
570. C3#24196direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4592
C2×C3×#4422
C3×#25802
C3×#20412
C3×#20112
C3×#19852
C3×#19612
C3×#18792
C3×#18642
C3×#18042
#241963