Finite Group Databasebeta

← C3×#24330 #10642937

Subgroups — C3×#24330 #10642937

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic, complemented
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C3×#2582C211yesmaximal, characteristic, complemented
5C3×#2058C212yesmaximal, complemented
6C3×#2024C212yesmaximal, complemented
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1984C211yesmaximal, characteristic, complemented
9C3×#1877C212yesmaximal
10C3×#1864C212yesmaximal
11C3×#1802C211yesmaximal, characteristic
12#24330C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8—21no—
14C22×C3×SD16C2211yescharacteristic
15C3×D4×Q8—24no—
16C3×D4×Q8—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—22no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#475—24no—
26C3×C4×SD16—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×C4×C8C2211yescharacteristic
35C3×#398—22no—
36C3×#391—21no—
37C3×#391C2212yes—
38C3×#389—21no—
39C3×#389C2212yes—
40C3×#384C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×D4×Q8C2×C311yescharacteristic
47C2×#475C2×C311yescharacteristic, complemented
48C2×C4×SD16C2×C311yescharacteristic
49#2582C2×C311yescharacteristic, complemented
50#2058C2×C312yescomplemented
51#2024C2×C312yescomplemented
52#1990C2×C311yescharacteristic, complemented
53#1984C2×C311yescharacteristic, complemented
54#1877C2×C312yes—
55#1864C2×C312yes—
56#1802C2×C311yescharacteristic
57C22×C3×Q8—28no—
58C22×C3×Q8—22no—
59C22×C3×Q8—21no—
60C22×C3×Q8—44no—
61C22×C3×Q8—44no—
62C22×C3×Q8D411yescharacteristic
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—22no—
66C23×C3×C4—21no—
67C2×C3×SD16—22no—
68C2×C3×SD16—22no—
69C2×C3×SD16Q814yescomplemented
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C3×C4⋊Q8—24no—
74C3×C4⋊Q8—24no—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8D414yes—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—216no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C3×C4×D4—22no—
86C3×C4×D4D414yes—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2312yes—
103C2×C3×C42C2311yescharacteristic
104C3×#106—22no—
105C3×#106—24no—
106C3×C4⋊C8—22no—
107C3×#103—22no—
108C3×#103—24no—
109C3×#102—22no—
110C3×#102—24no—
111C3×C4×C8—22no—
112C3×#95—22no—
113C3×#95—22no—
114C3×#95—22no—
115C23×Q8—21no—
116C22×SD16C22×C311yescharacteristic
117D4×Q8—24no—
118D4×Q8—24no—
119C2×C4⋊Q8C22×C312yes—
120C2×C4⋊Q8C22×C311yescharacteristic
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—22no—
123C2×C4×Q8C22×C311yescharacteristic
124C2×C4×Q8C22×C311yescharacteristic
125C2×C4×D4C22×C312yes—
126C2×C4×D4C22×C311yescharacteristic
127#475—24no—
128C4×SD16—24no—
129C2×#106C22×C312yes—
130C2×#106C22×C311yescharacteristic
131C2×C4⋊C8C22×C311yescharacteristic
132C2×#103C22×C312yes—
133C2×#103C22×C311yescharacteristic
134C2×#102C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×C4×C8C22×C311yescharacteristic
137#398—22no—
138#391—21no—
139#391C22×C312yes—
140#389—21no—
141#389C22×C312yes—
142#384C22×C311yescharacteristic
143#340C22×C312yes—
144#340C22×C312yes—
145#340C22×C312yes—
146#336C22×C312yes—
147#336C22×C312yes—
148C24×C3—21no—
149C2×C3×Q8—28no—
150C2×C3×Q8—22no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—44no—
159C2×C3×Q8—432no—
160C2×C3×Q8—44no—
161C2×C3×Q8—42no—
162C2×C3×Q8—42no—
163C2×C3×Q8SD1614yescomplemented
164C2×C3×Q8C2×Q812yescomplemented
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4C2×D412yes—
169C2×C3×D4C2×Q812yes—
170C2×C3×D4#4112yes—
171C22×C3×C4—28no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—44no—
179C22×C3×C4—44no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C3×SD16—28no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8C2×Q812yescomplemented
197C3×C4⋊C4—28no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—81no—
210C3×C4⋊C4—82no—
211C3×C4⋊C4C2×D414yes—
212C3×C4⋊C4C2×D412yes—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—24no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—42no—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220C22×Q8—28no—
221C22×Q8—22no—
222C22×Q8—21no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C3×D411yescharacteristic
226C22×Q8C23×C311yescharacteristic
227C22×D4C23×C311yescharacteristic
228C23×C4—22no—
229C23×C4—21no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×SD16C3×Q814yescomplemented
233C22×C8C23×C311yescharacteristic
234C22×C8C23×C312yes—
235C22×C8C23×C311yescharacteristic
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8C3×D414yes—
240C22⋊Q8—28no—
241C22⋊Q8—216no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4C3×D414yes—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C3×D411yescharacteristic
257C2×C4⋊C4C3×D412yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C312yes—
260C2×C4⋊C4C23×C312yes—
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C42—21no—
265C2×C42C23×C312yes—
266C2×C42C23×C311yescharacteristic
267#106—22no—
268#106—24no—
269C4⋊C8—22no—
270#103—22no—
271#103—24no—
272#102—22no—
273#102—24no—
274C4×C8—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3C22×D411yescharacteristic
286C3×Q8—28no—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—416no—
290C3×Q8—41no—
291C3×Q8—41no—
292C3×Q8—48no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4C22⋊Q814yes—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—416no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—48no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4#12012yes—
322C2×C3×C4#12012yes—
323C2×C3×C4C2×SD1614yes—
324C2×C3×C4C2×SD1612yes—
325C2×C3×C4AGL(1,ℤ/8)12yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×Q811yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C42−1+411yescharacteristic
331C2×C3×C42−1+411yescharacteristic
332C3×C8—22no—
333C3×C8—41no—
334C3×C8—42no—
335C24—21no—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—44no—
346C2×Q8—432no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8C3×SD1614yescomplemented
351C2×Q8C2×C3×Q812yescomplemented
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4C2×C3×D412yes—
356C2×D4C2×C3×Q812yes—
357C2×D4C3×#4112yes—
358C22×C4—28no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4C2×C3×D412yes—
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D412yes—
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C24×C311yescharacteristic
374SD16—28no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8C2×C3×Q812yescomplemented
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—81no—
397C4⋊C4—82no—
398C4⋊C4C2×C3×D414yes—
399C4⋊C4C2×C3×D412yes—
400C22⋊C4—28no—
401C22⋊C4—24no—
402C42—22no—
403C42—22no—
404C42—42no—
405C42C2×C3×D412yes—
406C42C2×C3×D412yes—
407C22×C3—24no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3C2×#12011yescharacteristic
415C22×C3#54811yescharacteristic
416C22×C3D4×Q811yescharacteristic
417C22×C3C22×SD1611yescharacteristic
418C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
419C22×C3#57711yescharacteristic
420C22×C3#57811yescharacteristic
421C3×C4—24no—
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—44no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—82no—
428C3×C4—81no—
429C3×C4#45014yes—
430C3×C4C2×C22⋊Q811yescharacteristic
431C3×C4#53611yescharacteristic
432C3×C4#54811yescharacteristic
433C3×C4D4×Q811yescharacteristic
434C23—22no—
435C23—22no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23C22×C3×D411yescharacteristic
442Q8—28no—
443Q8—22no—
444Q8—22no—
445Q8—416no—
446Q8—48no—
447Q8—41no—
448Q8—41no—
449D4—22no—
450D4—22no—
451D4—22no—
452D4C3×C22⋊Q814yes—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—416no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—48no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4C3×#12012yes—
478C2×C4C3×#12012yes—
479C2×C4C2×C3×SD1614yes—
480C2×C4C2×C3×SD1612yes—
481C2×C4C3×AGL(1,ℤ/8)12yes—
482C2×C4C22×C3×D411yescharacteristic
483C2×C4C22×C3×D411yescharacteristic
484C2×C4C22×C3×Q811yescharacteristic
485C2×C4C2×C3×#4111yescharacteristic
486C2×C4C3×2−1+411yescharacteristic
487C2×C4C3×2−1+411yescharacteristic
488C8—22no—
489C8—41no—
490C8—42no—
491C2×C3—44no—
492C2×C3#240911yescharacteristic
493C2×C3C2×#45011yescharacteristic
494C2×C3#301311yescharacteristic
495C2×C3#335911yescharacteristic
496C2×C3#337311yescharacteristic
497C2×C3Q8×SD1611yescharacteristic
498C2×C3#338611yescharacteristic
499C22—24no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22C2×C3×#12011yescharacteristic
507C22C3×#54811yescharacteristic
508C22C3×D4×Q811yescharacteristic
509C22C22×C3×SD1611yescharacteristic
510C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
511C22C3×#57711yescharacteristic
512C22C3×#57811yescharacteristic
513C4—24no—
514C4—22no—
515C4—42no—
516C4—44no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—82no—
521C4C3×#45014yes—
522C4C2×C3×C22⋊Q811yescharacteristic
523C4C3×#53611yescharacteristic
524C4C3×#54811yescharacteristic
525C4C3×D4×Q811yescharacteristic
526C3#2433011yescharacteristic, direct product factor, Sylow
527C2—44no—
528C2C3×#240911yescharacteristic
529C2C2×C3×#45011yescharacteristic
530C2C3×#301311yescharacteristic
531C2C3×#335911yescharacteristic
532C2C3×#337311yescharacteristic
533C2C3×Q8×SD1611yescharacteristic
534C2C3×#338611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
2. C2×C3×#475C2semidirect
4. C3×#2582C2semidirect
5. C3×#2058C2semidirect
6. C3×#2024C2semidirect
7. C3×#1990C2semidirect
8. C3×#1984C2semidirect
12. #24330C3direct (N × H)
47. C2×#475C2×C3semidirect
49. #2582C2×C3semidirect
50. #2058C2×C3semidirect
51. #2024C2×C3semidirect
52. #1990C2×C3semidirect
53. #1984C2×C3semidirect
69. C2×C3×SD16Q8semidirect
163. C2×C3×Q8SD16semidirect
164. C2×C3×Q8C2×Q8semidirect
196. C2×C3×C8C2×Q8semidirect
232. C2×SD16C3×Q8semidirect
350. C2×Q8C3×SD16semidirect
351. C2×Q8C2×C3×Q8semidirect
383. C2×C8C2×C3×Q8semidirect
526. C3#24330direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#4752
C2×C3×C4×SD162
C3×#25822
C3×#20582
C3×#20242
C3×#19902
C3×#19842
C3×#18772
C3×#18642
C3×#18022
#243303