Finite Group Databasebeta

← C3×#61837 #10680444

Subgroups — C3×#61837 #10680444

Order 768.

1928 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3523C211yesmaximal, characteristic, complemented
2C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
3C3×#3386C216yesmaximal, complemented
4C3×Q8×D8C212yesmaximal, complemented
5C3×#3367C216yesmaximal, complemented
6C3×#3359C212yesmaximal, complemented
7C3×#3239C213yesmaximal, complemented
8C3×#3230C213yesmaximal, complemented
9C3×#3110C213yesmaximal, complemented
10C3×#3105C213yesmaximal
11C3×#2984C211yesmaximal, characteristic, complemented
12#61837C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8—21no—
14C3×#557C2211yescharacteristic, complemented
15C3×#557C2213yescomplemented
16C3×#556C2213yescomplemented
17C3×#552—23no—
18C3×D4×Q8—21no—
19C3×D4×Q8—21no—
20C3×D4×Q8—23no—
21C3×D4×Q8C2216yes—
22C3×D4×Q8C2212yes—
23C3×D4×Q8C2211yescharacteristic, complemented
24C3×#541—23no—
25C3×#533C2213yes—
26C2×C3×C4⋊Q8C2213yes—
27C3×#529C2213yes—
28C2×C3×C22⋊Q8—23no—
29C3×#519—21no—
30C2×C3×C4×Q8C2211yescharacteristic
31C2×C3×C4×D4C2213yes—
32C3×#501C2216yescomplemented
33C3×#500C2216yescomplemented
34C3×#489C2216yescomplemented
35C3×#487C2216yescomplemented
36C3×#484C2213yes—
37C3×#484C2216yes—
38C3×#481C2213yes—
39C3×#478C2216yes—
40C3×#477C22112yescomplemented
41C3×#475C2216yescomplemented
42C3×#474C2216yes—
43C3×#474C2216yescomplemented
44C3×#463C2216yes—
45C3×#461C2216yes—
46C3×#450C2213yes—
47C3×#447C2211yescharacteristic
48C3×#446C2216yescomplemented
49C3×C8×Q8C2212yescomplemented
50C3×#440C22112yes—
51C3×C4×SD16C2216yes—
52C3×C4×D8C2216yes—
53C3×#432C2213yes—
54C3×#431C2213yes—
55#3523C2×C311yescharacteristic, complemented
56C2×D4×Q8C2×C311yescharacteristic, complemented
57#3386C2×C316yescomplemented
58Q8×D8C2×C312yescomplemented
59#3367C2×C316yescomplemented
60#3359C2×C312yescomplemented
61#3239C2×C313yescomplemented
62#3230C2×C313yescomplemented
63#3110C2×C313yescomplemented
64#3105C2×C313yes—
65#2984C2×C311yescharacteristic, complemented
66C22×C3×Q8—26no—
67C22×C3×Q8—22no—
68C22×C3×Q8—21no—
69C22×C3×Q8—23no—
70C22×C3×Q8—48no—
71C22×C3×Q8D411yescharacteristic
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—23no—
74C2×C3×SD16C2316yes—
75C2×C3×D8C2312yes—
76C3×C4⋊Q8—23no—
77C3×C4⋊Q8—23no—
78C3×C4⋊Q8—23no—
79C3×C4⋊Q8C2313yes—
80C3×C4⋊Q8C2313yes—
81C3×C4⋊Q8C2316yes—
82C3×C4⋊Q8C2316yes—
83C3×#126—26no—
84C3×#126—23no—
85C3×#125—26no—
86C3×#125—23no—
87C3×#125—23no—
88C3×#125C2313yes—
89C3×#124C2313yes—
90C3×#123—23no—
91C3×C22⋊Q8—23no—
92C3×C22⋊Q8—23no—
93C3×C22⋊Q8—26no—
94C3×C22⋊Q8—26no—
95C3×C22⋊Q8—212no—
96C3×C22⋊Q8—23no—
97C3×C22⋊Q8C2313yes—
98C3×#121C2311yescharacteristic
99C3×C4×Q8—21no—
100C3×C4×Q8—23no—
101C3×C4×Q8—23no—
102C3×C4×Q8D411yescharacteristic
103C3×C4×Q8D413yes—
104C3×C4×Q8C2313yes—
105C3×C4×Q8C2316yes—
106C3×C4×Q8C2311yescharacteristic
107C3×C4×Q8C2313yes—
108C3×C4×D4—23no—
109C3×C4×D4—23no—
110C3×C4×D4—23no—
111C3×C4×D4C2313yes—
112C3×C4×D4C2316yes—
113C3×C4×D4C2316yes—
114C3×#117—23no—
115C3×#117—23no—
116C3×#117—23no—
117C2×C3×C4⋊C4—23no—
118C2×C3×C4⋊C4D413yes—
119C2×C3×C4⋊C4C2313yes—
120C2×C3×C22⋊C4—23no—
121C2×C3×C42C2313yes—
122C3×#107C2316yes—
123C3×#107C23112yes—
124C3×#106C2316yes—
125C3×C4⋊C8C2316yes—
126C3×C4⋊C8C2316yes—
127C3×#103C2316yes—
128C3×#103C23112yes—
129C3×#102C2316yes—
130C3×#102C2316yes—
131C3×#102C2312yes—
132C3×C22⋊C8C2311yescharacteristic
133C3×C22⋊C8C2313yes—
134C3×#97C2316yes—
135C3×C4×C8C2316yes—
136C23×Q8—21no—
137#557C22×C311yescharacteristic, complemented
138#557C22×C313yescomplemented
139#556C22×C313yescomplemented
140#552—23no—
141D4×Q8—21no—
142D4×Q8—21no—
143D4×Q8—23no—
144D4×Q8C22×C312yes—
145D4×Q8C22×C316yes—
146D4×Q8C22×C311yescharacteristic, complemented
147#541—23no—
148#533C22×C313yes—
149C2×C4⋊Q8C22×C313yes—
150#529C22×C313yes—
151C2×C22⋊Q8—23no—
152#519—21no—
153C2×C4×Q8C22×C311yescharacteristic
154C2×C4×D4C22×C313yes—
155#501C22×C316yescomplemented
156#500C22×C316yescomplemented
157#489C22×C316yescomplemented
158#487C22×C316yescomplemented
159#484C22×C313yes—
160#484C22×C316yes—
161#481C22×C313yes—
162#478C22×C316yes—
163#477C22×C3112yescomplemented
164#475C22×C316yescomplemented
165#474C22×C316yes—
166#474C22×C316yescomplemented
167#463C22×C316yes—
168#461C22×C316yes—
169#450C22×C313yes—
170#447C22×C311yescharacteristic
171#446C22×C316yescomplemented
172C8×Q8C22×C312yescomplemented
173#440C22×C3112yes—
174C4×SD16C22×C316yes—
175C4×D8C22×C316yes—
176#432C22×C313yes—
177#431C22×C313yes—
178C24×C3—21no—
179C2×C3×Q8—24no—
180C2×C3×Q8—23no—
181C2×C3×Q8—26no—
182C2×C3×Q8—26no—
183C2×C3×Q8—412no—
184C2×C3×Q8—424no—
185C2×C3×Q8—44no—
186C2×C3×Q8—48no—
187C2×C3×Q8C2×D413yes—
188C2×C3×Q8C2×D411yescharacteristic
189C2×C3×Q8C2413yes—
190C2×C3×D4—24no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4C2411yescharacteristic
194C2×C3×D4C2412yes—
195C22×C3×C4—23no—
196C22×C3×C4—23no—
197C22×C3×C4—26no—
198C22×C3×C4—26no—
199C22×C3×C4—412no—
200C22×C3×C4C2×D413yes—
201C22×C3×C4C2×D413yes—
202C22×C3×C4C2411yescharacteristic
203C3×SD16—212no—
204C3×D8—24no—
205C2×C3×C8C2412yes—
206C2×C3×C8C2416yes—
207C3×C4⋊C4—26no—
208C3×C4⋊C4—26no—
209C3×C4⋊C4—23no—
210C3×C4⋊C4—23no—
211C3×C4⋊C4—23no—
212C3×C4⋊C4—23no—
213C3×C4⋊C4—23no—
214C3×C4⋊C4—23no—
215C3×C4⋊C4—23no—
216C3×C4⋊C4—26no—
217C3×C4⋊C4—23no—
218C3×C4⋊C4C2×D413yes—
219C3×C4⋊C4C2×D413yes—
220C3×C4⋊C4C2×D416yes—
221C3×C4⋊C4C2411yescharacteristic
222C3×C4⋊C4C2413yes—
223C3×C4⋊C4C2416yes—
224C3×C22⋊C4—23no—
225C3×C22⋊C4—26no—
226C3×C22⋊C4—26no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—23no—
229C3×C22⋊C4—23no—
230C3×C22⋊C4—23no—
231C3×C42—26no—
232C3×C42—23no—
233C3×C42C2×D413yes—
234C3×C42C2×D413yes—
235C3×C42C2413yes—
236C3×C42C2413yes—
237C22×Q8—21no—
238C22×Q8—26no—
239C22×Q8—22no—
240C22×Q8—23no—
241C22×Q8—48no—
242C22×Q8C3×D411yescharacteristic
243C22×D4C23×C311yescharacteristic
244C23×C4—23no—
245C2×SD16C23×C316yes—
246C2×D8C23×C312yes—
247C4⋊Q8—23no—
248C4⋊Q8—23no—
249C4⋊Q8—23no—
250C4⋊Q8C23×C313yes—
251C4⋊Q8C23×C316yes—
252C4⋊Q8C23×C316yes—
253C4⋊Q8C23×C313yes—
254#126—26no—
255#126—23no—
256#125—26no—
257#125—23no—
258#125—23no—
259#125C23×C313yes—
260#124C23×C313yes—
261#123—23no—
262C22⋊Q8—23no—
263C22⋊Q8—23no—
264C22⋊Q8—26no—
265C22⋊Q8—26no—
266C22⋊Q8—212no—
267C22⋊Q8—23no—
268C22⋊Q8C23×C313yes—
269#121C23×C311yescharacteristic
270C4×Q8—21no—
271C4×Q8—23no—
272C4×Q8—23no—
273C4×Q8C3×D411yescharacteristic
274C4×Q8C3×D413yes—
275C4×Q8C23×C313yes—
276C4×Q8C23×C316yes—
277C4×Q8C23×C311yescharacteristic
278C4×Q8C23×C313yes—
279C4×D4—23no—
280C4×D4—23no—
281C4×D4—23no—
282C4×D4C23×C316yes—
283C4×D4C23×C316yes—
284C4×D4C23×C313yes—
285#117—23no—
286#117—23no—
287#117—23no—
288C2×C4⋊C4—23no—
289C2×C4⋊C4C3×D413yes—
290C2×C4⋊C4C23×C313yes—
291C2×C22⋊C4—23no—
292C2×C42C23×C313yes—
293#107C23×C316yes—
294#107C23×C3112yes—
295#106C23×C316yes—
296C4⋊C8C23×C316yes—
297C4⋊C8C23×C316yes—
298#103C23×C316yes—
299#103C23×C3112yes—
300#102C23×C316yes—
301#102C23×C316yes—
302#102C23×C312yes—
303C22⋊C8C23×C311yescharacteristic
304C22⋊C8C23×C313yes—
305#97C23×C316yes—
306C4×C8C23×C316yes—
307C23×C3—24no—
308C23×C3—22no—
309C23×C3—21no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3C22×D411yescharacteristic
313C3×Q8—26no—
314C3×Q8—26no—
315C3×Q8—412no—
316C3×Q8—46no—
317C3×Q8—412no—
318C3×Q8AGL(1,ℤ/8)14yes—
319C3×D4—24no—
320C3×D4—22no—
321C3×D4—22no—
322C3×D4—42no—
323C3×D42−1+414yes—
324C2×C3×C4—26no—
325C2×C3×C4—26no—
326C2×C3×C4—22no—
327C2×C3×C4—23no—
328C2×C3×C4—21no—
329C2×C3×C4—23no—
330C2×C3×C4—26no—
331C2×C3×C4—412no—
332C2×C3×C4—46no—
333C2×C3×C4—412no—
334C2×C3×C4—46no—
335C2×C3×C4C22×D413yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D413yes—
338C2×C3×C4C22×D413yes—
339C2×C3×C4C22×D413yes—
340C2×C3×C4C2511yescharacteristic
341C3×C8—22no—
342C3×C8—26no—
343C24—21no—
344C2×Q8—24no—
345C2×Q8—23no—
346C2×Q8—26no—
347C2×Q8—26no—
348C2×Q8—412no—
349C2×Q8—424no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8C2×C3×D413yes—
353C2×Q8C2×C3×D411yescharacteristic
354C2×Q8C24×C313yes—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4C24×C311yescharacteristic
359C2×D4C24×C312yes—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—26no—
363C22×C4—26no—
364C22×C4—412no—
365C22×C4C2×C3×D413yes—
366C22×C4C2×C3×D413yes—
367C22×C4C24×C311yescharacteristic
368SD16—212no—
369D8—24no—
370C2×C8C24×C312yes—
371C2×C8C24×C316yes—
372C4⋊C4—26no—
373C4⋊C4—26no—
374C4⋊C4—23no—
375C4⋊C4—23no—
376C4⋊C4—23no—
377C4⋊C4—23no—
378C4⋊C4—23no—
379C4⋊C4—23no—
380C4⋊C4—23no—
381C4⋊C4—26no—
382C4⋊C4—23no—
383C4⋊C4C2×C3×D413yes—
384C4⋊C4C2×C3×D413yes—
385C4⋊C4C2×C3×D416yes—
386C4⋊C4C24×C311yescharacteristic
387C4⋊C4C24×C313yes—
388C4⋊C4C24×C316yes—
389C22⋊C4—23no—
390C22⋊C4—26no—
391C22⋊C4—26no—
392C22⋊C4—21no—
393C22⋊C4—23no—
394C22⋊C4—23no—
395C22⋊C4—23no—
396C42—26no—
397C42—23no—
398C42C2×C3×D413yes—
399C42C2×C3×D413yes—
400C42C24×C313yes—
401C42C24×C313yes—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—24no—
405C22×C3—48no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3#57812yes—
412C22×C3C23×D411yescharacteristic
413C3×C4—21no—
414C3×C4—23no—
415C3×C4—23no—
416C3×C4—23no—
417C3×C4—43no—
418C3×C4—46no—
419C3×C4—43no—
420C3×C4—41no—
421C3×C4C2×AGL(1,ℤ/8)16yes—
422C3×C4C2×2−1+412yes—
423C23—24no—
424C23—22no—
425C23—21no—
426C23—42no—
427C23—42no—
428C23C22×C3×D411yescharacteristic
429Q8—26no—
430Q8—26no—
431Q8—412no—
432Q8—412no—
433Q8—46no—
434Q8C3×AGL(1,ℤ/8)14yes—
435D4—24no—
436D4—22no—
437D4—22no—
438D4—42no—
439D4C3×2−1+414yes—
440C2×C4—26no—
441C2×C4—26no—
442C2×C4—22no—
443C2×C4—23no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—26no—
447C2×C4—412no—
448C2×C4—46no—
449C2×C4—412no—
450C2×C4—46no—
451C2×C4C22×C3×D413yes—
452C2×C4C22×C3×D411yescharacteristic
453C2×C4C22×C3×D413yes—
454C2×C4C22×C3×D413yes—
455C2×C4C22×C3×D413yes—
456C2×C4C25×C311yescharacteristic
457C8—22no—
458C8—26no—
459C2×C3—22no—
460C2×C3—44no—
461C2×C3—81no—
462C2×C3#348611yescharacteristic
463C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
464C2×C3C2×#57811yescharacteristic
465C22—21no—
466C22—21no—
467C22—24no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22C3×#57812yes—
475C22C23×C3×D411yescharacteristic
476C4—21no—
477C4—23no—
478C4—23no—
479C4—23no—
480C4—43no—
481C4—46no—
482C4—43no—
483C4—41no—
484C4C2×C3×AGL(1,ℤ/8)16yes—
485C4C2×C3×2−1+412yes—
486C3#6183711yescharacteristic, direct product factor, Sylow
487C2—22no—
488C2—44no—
489C2—81no—
490C2C3×#348611yescharacteristic
491C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
492C2C2×C3×#57811yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#447C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3523C2semidirect
2. C2×C3×D4×Q8C2semidirect
3. C3×#3386C2semidirect
4. C3×Q8×D8C2semidirect
5. C3×#3367C2semidirect
6. C3×#3359C2semidirect
7. C3×#3239C2semidirect
8. C3×#3230C2semidirect
9. C3×#3110C2semidirect
11. C3×#2984C2semidirect
12. #61837C3direct (N × H)
14. C3×#557C22semidirect
15. C3×#557C22semidirect
16. C3×#556C22semidirect
23. C3×D4×Q8C22semidirect
32. C3×#501C22semidirect
33. C3×#500C22semidirect
34. C3×#489C22semidirect
35. C3×#487C22semidirect
40. C3×#477C22semidirect
41. C3×#475C22semidirect
43. C3×#474C22semidirect
48. C3×#446C22semidirect
49. C3×C8×Q8C22semidirect
55. #3523C2×C3semidirect
56. C2×D4×Q8C2×C3semidirect
57. #3386C2×C3semidirect
58. Q8×D8C2×C3semidirect
59. #3367C2×C3semidirect
60. #3359C2×C3semidirect
61. #3239C2×C3semidirect
62. #3230C2×C3semidirect
63. #3110C2×C3semidirect
65. #2984C2×C3semidirect
137. #557C22×C3semidirect
138. #557C22×C3semidirect
139. #556C22×C3semidirect
146. D4×Q8C22×C3semidirect
155. #501C22×C3semidirect
156. #500C22×C3semidirect
157. #489C22×C3semidirect
158. #487C22×C3semidirect
163. #477C22×C3semidirect
164. #475C22×C3semidirect
166. #474C22×C3semidirect
171. #446C22×C3semidirect
172. C8×Q8C22×C3semidirect
486. C3#61837direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35232
C2×C3×D4×Q82
C3×#33862
C3×Q8×D82
C3×#33672
C3×#33592
C3×#32392
C3×#32302
C3×#31102
C3×#31052
C3×#29842
#618373