Finite Group Databasebeta

← C3×#61843 #10680450

Subgroups — C3×#61843 #10680450

Order 768.

1984 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3522C211yesmaximal, characteristic, complemented
2C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
3C3×#3384C216yesmaximal, complemented
4C3×Q8×SD16C212yesmaximal, complemented
5C3×#3373C216yesmaximal, complemented
6C3×#3358C212yesmaximal, complemented
7C3×#3239C213yesmaximal, complemented
8C3×#3227C213yesmaximal, complemented
9C3×#3109C213yesmaximal, complemented
10C3×#3105C213yesmaximal, complemented
11C3×#2984C211yesmaximal, characteristic, complemented
12#61843C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8—21no—
14C3×#559C2213yescomplemented
15C3×Q82C2211yescharacteristic, complemented
16C3×#554C2213yescomplemented
17C3×#550C2211yescharacteristic, complemented
18C3×D4×Q8—21no—
19C3×D4×Q8—23no—
20C3×D4×Q8C2216yes—
21C3×D4×Q8C2212yes—
22C3×#548—21no—
23C3×#547—23no—
24C3×#542—23no—
25C3×#532C2213yes—
26C2×C3×C4⋊Q8C2213yes—
27C3×#529C2213yes—
28C2×C3×C22⋊Q8—23no—
29C3×#520—21no—
30C2×C3×C4×Q8C2211yescharacteristic
31C2×C3×C4×D4C2213yes—
32C3×#501C2216yescomplemented
33C3×#498C2216yescomplemented
34C3×#489C2216yescomplemented
35C3×#486C2216yescomplemented
36C3×#484C2213yes—
37C3×#481C2213yes—
38C3×#481C2216yes—
39C3×#478C2216yes—
40C3×#476C2216yes—
41C3×#476C22112yescomplemented
42C3×#475C2216yescomplemented
43C3×#474C2216yescomplemented
44C3×#463C2216yes—
45C3×#459C2216yes—
46C3×#450C2211yescharacteristic
47C3×#447C2213yes—
48C3×#446C2216yescomplemented
49C3×C8×Q8C2212yescomplemented
50C3×#442C22112yes—
51C3×C4×SD16C2216yes—
52C3×C4×D8C2216yes—
53C3×#432C2213yes—
54C3×#431C2213yes—
55#3522C2×C311yescharacteristic, complemented
56C2×D4×Q8C2×C311yescharacteristic, complemented
57#3384C2×C316yescomplemented
58Q8×SD16C2×C312yescomplemented
59#3373C2×C316yescomplemented
60#3358C2×C312yescomplemented
61#3239C2×C313yescomplemented
62#3227C2×C313yescomplemented
63#3109C2×C313yescomplemented
64#3105C2×C313yescomplemented
65#2984C2×C311yescharacteristic, complemented
66C2×C3×#41—21no—
67C2×C3×#41—23no—
68C22×C3×Q8—26no—
69C22×C3×Q8—22no—
70C22×C3×Q8—23no—
71C22×C3×Q8—48no—
72C22×C3×Q8D411yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—23no—
75C2×C3×SD16C2312yes—
76C2×C3×D8C2316yes—
77C3×C4⋊Q8—23no—
78C3×C4⋊Q8—23no—
79C3×C4⋊Q8—23no—
80C3×C4⋊Q8C2313yes—
81C3×C4⋊Q8C2316yes—
82C3×C4⋊Q8C2316yes—
83C3×#127C2313yes—
84C3×#126—23no—
85C3×#125C2313yes—
86C3×#124—23no—
87C3×#124—23no—
88C3×#124C2313yes—
89C3×#123—23no—
90C3×#123—26no—
91C3×C22⋊Q8—23no—
92C3×C22⋊Q8—26no—
93C3×C22⋊Q8—26no—
94C3×C22⋊Q8—212no—
95C3×C22⋊Q8—26no—
96C3×C22⋊Q8C2311yescharacteristic
97C3×#121—23no—
98C3×#121—23no—
99C3×#121C2313yes—
100C3×C4×Q8—21no—
101C3×C4×Q8D411yescharacteristic
102C3×C4×Q8D413yes—
103C3×C4×Q8C2313yes—
104C3×C4×Q8C2311yescharacteristic
105C3×C4×Q8C2313yes—
106C3×C4×D4—23no—
107C3×C4×D4—23no—
108C3×C4×D4—23no—
109C3×C4×D4—23no—
110C3×C4×D4—23no—
111C3×C4×D4C2316yes—
112C3×C4×D4C2313yes—
113C3×C4×D4C2316yes—
114C3×C4×D4C2316yes—
115C3×#117—23no—
116C2×C3×C4⋊C4—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×#106C2316yes—
124C3×#106C23112yes—
125C3×C4⋊C8C2316yes—
126C3×C4⋊C8C2316yes—
127C3×#103C2316yes—
128C3×#102C2312yes—
129C3×#102C2316yes—
130C3×#102C2316yes—
131C3×#102C23112yes—
132C3×C22⋊C8C2311yescharacteristic
133C3×C22⋊C8C2313yes—
134C3×#97C2316yes—
135C3×C4×C8C2316yes—
136C23×Q8—21no—
137#559C22×C313yescomplemented
138Q82C22×C311yescharacteristic, complemented
139#554C22×C313yescomplemented
140#550C22×C311yescharacteristic, complemented
141D4×Q8—21no—
142D4×Q8—23no—
143D4×Q8C22×C312yes—
144D4×Q8C22×C316yes—
145#548—21no—
146#547—23no—
147#542—23no—
148#532C22×C313yes—
149C2×C4⋊Q8C22×C313yes—
150#529C22×C313yes—
151C2×C22⋊Q8—23no—
152#520—21no—
153C2×C4×Q8C22×C311yescharacteristic
154C2×C4×D4C22×C313yes—
155#501C22×C316yescomplemented
156#498C22×C316yescomplemented
157#489C22×C316yescomplemented
158#486C22×C316yescomplemented
159#484C22×C313yes—
160#481C22×C313yes—
161#481C22×C316yes—
162#478C22×C316yes—
163#476C22×C316yes—
164#476C22×C3112yescomplemented
165#475C22×C316yescomplemented
166#474C22×C316yescomplemented
167#463C22×C316yes—
168#459C22×C316yes—
169#450C22×C311yescharacteristic
170#447C22×C313yes—
171#446C22×C316yescomplemented
172C8×Q8C22×C312yescomplemented
173#442C22×C3112yes—
174C4×SD16C22×C316yes—
175C4×D8C22×C316yes—
176#432C22×C313yes—
177#431C22×C313yes—
178C24×C3—21no—
179C3×#41—44no—
180C3×#41—86no—
181C2×C3×Q8—24no—
182C2×C3×Q8—26no—
183C2×C3×Q8—26no—
184C2×C3×Q8—23no—
185C2×C3×Q8—412no—
186C2×C3×Q8—424no—
187C2×C3×Q8—48no—
188C2×C3×Q8C2×D413yes—
189C2×C3×Q8C2×D411yescharacteristic
190C2×C3×Q8C2411yescharacteristic
191C2×C3×D4—24no—
192C2×C3×D4—23no—
193C2×C3×D4—21no—
194C2×C3×D4—23no—
195C2×C3×D4—23no—
196C2×C3×D4C2413yes—
197C2×C3×D4C2412yes—
198C22×C3×C4—23no—
199C22×C3×C4—23no—
200C22×C3×C4—26no—
201C22×C3×C4—26no—
202C22×C3×C4—26no—
203C22×C3×C4—412no—
204C22×C3×C4C2×D413yes—
205C22×C3×C4C2×D413yes—
206C22×C3×C4C2411yescharacteristic
207C3×SD16—24no—
208C3×D8—212no—
209C2×C3×C8C2412yes—
210C2×C3×C8C2416yes—
211C3×C4⋊C4—23no—
212C3×C4⋊C4—23no—
213C3×C4⋊C4—26no—
214C3×C4⋊C4—26no—
215C3×C4⋊C4—23no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—23no—
218C3×C4⋊C4—23no—
219C3×C4⋊C4C2×D413yes—
220C3×C4⋊C4C2×D413yes—
221C3×C4⋊C4C2×D416yes—
222C3×C4⋊C4C2416yes—
223C3×C4⋊C4C2413yes—
224C3×C4⋊C4C2411yescharacteristic
225C3×C22⋊C4—21no—
226C3×C22⋊C4—26no—
227C3×C22⋊C4—26no—
228C3×C22⋊C4—23no—
229C3×C22⋊C4—23no—
230C3×C22⋊C4—23no—
231C3×C22⋊C4—23no—
232C3×C22⋊C4—26no—
233C3×C42—23no—
234C3×C42C2×D413yes—
235C3×C42C2×D413yes—
236C3×C42C2413yes—
237C3×C42C2413yes—
238C2×#41—21no—
239C2×#41—23no—
240C22×Q8—26no—
241C22×Q8—22no—
242C22×Q8—23no—
243C22×Q8—48no—
244C22×Q8C3×D411yescharacteristic
245C22×D4C23×C311yescharacteristic
246C23×C4—23no—
247C2×SD16C23×C312yes—
248C2×D8C23×C316yes—
249C4⋊Q8—23no—
250C4⋊Q8—23no—
251C4⋊Q8—23no—
252C4⋊Q8C23×C316yes—
253C4⋊Q8C23×C316yes—
254C4⋊Q8C23×C313yes—
255#127C23×C313yes—
256#126—23no—
257#125C23×C313yes—
258#124—23no—
259#124—23no—
260#124C23×C313yes—
261#123—23no—
262#123—26no—
263C22⋊Q8—23no—
264C22⋊Q8—26no—
265C22⋊Q8—26no—
266C22⋊Q8—212no—
267C22⋊Q8—26no—
268C22⋊Q8C23×C311yescharacteristic
269#121—23no—
270#121—23no—
271#121C23×C313yes—
272C4×Q8—21no—
273C4×Q8C3×D411yescharacteristic
274C4×Q8C3×D413yes—
275C4×Q8C23×C313yes—
276C4×Q8C23×C311yescharacteristic
277C4×Q8C23×C313yes—
278C4×D4—23no—
279C4×D4—23no—
280C4×D4—23no—
281C4×D4—23no—
282C4×D4—23no—
283C4×D4C23×C316yes—
284C4×D4C23×C316yes—
285C4×D4C23×C313yes—
286C4×D4C23×C316yes—
287#117—23no—
288C2×C4⋊C4—23no—
289C2×C4⋊C4—23no—
290C2×C4⋊C4C3×D413yes—
291C2×C4⋊C4C23×C313yes—
292C2×C22⋊C4—23no—
293C2×C42C23×C313yes—
294#107C23×C316yes—
295#106C23×C316yes—
296#106C23×C3112yes—
297C4⋊C8C23×C316yes—
298C4⋊C8C23×C316yes—
299#103C23×C316yes—
300#102C23×C312yes—
301#102C23×C316yes—
302#102C23×C316yes—
303#102C23×C3112yes—
304C22⋊C8C23×C311yescharacteristic
305C22⋊C8C23×C313yes—
306#97C23×C316yes—
307C4×C8C23×C316yes—
308C23×C3—24no—
309C23×C3—22no—
310C23×C3—23no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3C22×D411yescharacteristic
314C3×Q8—26no—
315C3×Q8—22no—
316C3×Q8—412no—
317C3×Q8—412no—
318C3×Q8—83no—
319C3×Q8AGL(1,ℤ/8)14yes—
320C3×D4—24no—
321C3×D4—22no—
322C3×D4—26no—
323C3×D4—46no—
324C3×D4—46no—
325C3×D4—83no—
326C3×D42−1+414yes—
327C2×C3×C4—26no—
328C2×C3×C4—26no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—23no—
332C2×C3×C4—21no—
333C2×C3×C4—26no—
334C2×C3×C4—412no—
335C2×C3×C4—46no—
336C2×C3×C4—412no—
337C2×C3×C4—46no—
338C2×C3×C4—86no—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D413yes—
341C2×C3×C4C22×D413yes—
342C2×C3×C4C22×D413yes—
343C2×C3×C4C22×D413yes—
344C2×C3×C4C2511yescharacteristic
345C3×C8—26no—
346C3×C8—22no—
347C24—21no—
348#41—44no—
349#41—86no—
350C2×Q8—24no—
351C2×Q8—26no—
352C2×Q8—26no—
353C2×Q8—23no—
354C2×Q8—412no—
355C2×Q8—424no—
356C2×Q8—48no—
357C2×Q8C2×C3×D413yes—
358C2×Q8C2×C3×D411yescharacteristic
359C2×Q8C24×C311yescharacteristic
360C2×D4—24no—
361C2×D4—23no—
362C2×D4—21no—
363C2×D4—23no—
364C2×D4—23no—
365C2×D4C24×C313yes—
366C2×D4C24×C312yes—
367C22×C4—23no—
368C22×C4—23no—
369C22×C4—26no—
370C22×C4—26no—
371C22×C4—26no—
372C22×C4—412no—
373C22×C4C2×C3×D413yes—
374C22×C4C2×C3×D413yes—
375C22×C4C24×C311yescharacteristic
376SD16—24no—
377D8—212no—
378C2×C8C24×C312yes—
379C2×C8C24×C316yes—
380C4⋊C4—23no—
381C4⋊C4—23no—
382C4⋊C4—26no—
383C4⋊C4—26no—
384C4⋊C4—23no—
385C4⋊C4—21no—
386C4⋊C4—23no—
387C4⋊C4—23no—
388C4⋊C4C2×C3×D413yes—
389C4⋊C4C2×C3×D413yes—
390C4⋊C4C2×C3×D416yes—
391C4⋊C4C24×C316yes—
392C4⋊C4C24×C313yes—
393C4⋊C4C24×C311yescharacteristic
394C22⋊C4—21no—
395C22⋊C4—26no—
396C22⋊C4—26no—
397C22⋊C4—23no—
398C22⋊C4—23no—
399C22⋊C4—23no—
400C22⋊C4—23no—
401C22⋊C4—26no—
402C42—23no—
403C42C2×C3×D413yes—
404C42C2×C3×D413yes—
405C42C24×C313yes—
406C42C24×C313yes—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—24no—
410C22×C3—48no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—43no—
414C22×C3—43no—
415C22×C3—43no—
416C22×C3#57712yes—
417C22×C3C23×D411yescharacteristic
418C3×C4—21no—
419C3×C4—23no—
420C3×C4—23no—
421C3×C4—23no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—46no—
425C3×C4—43no—
426C3×C4C2×AGL(1,ℤ/8)16yes—
427C3×C4C2×2−1+412yes—
428C23—24no—
429C23—22no—
430C23—23no—
431C23—42no—
432C23—42no—
433C23C22×C3×D411yescharacteristic
434Q8—26no—
435Q8—22no—
436Q8—412no—
437Q8—412no—
438Q8—83no—
439Q8C3×AGL(1,ℤ/8)14yes—
440D4—24no—
441D4—22no—
442D4—26no—
443D4—46no—
444D4—46no—
445D4—83no—
446D4C3×2−1+414yes—
447C2×C4—26no—
448C2×C4—26no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—23no—
452C2×C4—21no—
453C2×C4—26no—
454C2×C4—412no—
455C2×C4—46no—
456C2×C4—412no—
457C2×C4—46no—
458C2×C4—86no—
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C22×C3×D413yes—
461C2×C4C22×C3×D413yes—
462C2×C4C22×C3×D413yes—
463C2×C4C22×C3×D413yes—
464C2×C4C25×C311yescharacteristic
465C8—26no—
466C8—22no—
467C2×C3—22no—
468C2×C3—44no—
469C2×C3—83no—
470C2×C3#348611yescharacteristic
471C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
472C2×C3C2×#57711yescharacteristic
473C22—21no—
474C22—21no—
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—43no—
480C22—43no—
481C22—43no—
482C22C3×#57712yes—
483C22C23×C3×D411yescharacteristic
484C4—21no—
485C4—23no—
486C4—23no—
487C4—23no—
488C4—41no—
489C4—46no—
490C4—41no—
491C4—43no—
492C4C2×C3×AGL(1,ℤ/8)16yes—
493C4C2×C3×2−1+412yes—
494C3#6184311yescharacteristic, direct product factor, Sylow
495C2—22no—
496C2—44no—
497C2—83no—
498C2C3×#348611yescharacteristic
499C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
500C2C2×C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
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×#3522C2semidirect
2. C2×C3×D4×Q8C2semidirect
3. C3×#3384C2semidirect
4. C3×Q8×SD16C2semidirect
5. C3×#3373C2semidirect
6. C3×#3358C2semidirect
7. C3×#3239C2semidirect
8. C3×#3227C2semidirect
9. C3×#3109C2semidirect
10. C3×#3105C2semidirect
11. C3×#2984C2semidirect
12. #61843C3direct (N × H)
14. C3×#559C22semidirect
15. C3×Q82C22semidirect
16. C3×#554C22semidirect
17. C3×#550C22semidirect
32. C3×#501C22semidirect
33. C3×#498C22semidirect
34. C3×#489C22semidirect
35. C3×#486C22semidirect
41. C3×#476C22semidirect
42. C3×#475C22semidirect
43. C3×#474C22semidirect
48. C3×#446C22semidirect
49. C3×C8×Q8C22semidirect
55. #3522C2×C3semidirect
56. C2×D4×Q8C2×C3semidirect
57. #3384C2×C3semidirect
58. Q8×SD16C2×C3semidirect
59. #3373C2×C3semidirect
60. #3358C2×C3semidirect
61. #3239C2×C3semidirect
62. #3227C2×C3semidirect
63. #3109C2×C3semidirect
64. #3105C2×C3semidirect
65. #2984C2×C3semidirect
137. #559C22×C3semidirect
138. Q82C22×C3semidirect
139. #554C22×C3semidirect
140. #550C22×C3semidirect
155. #501C22×C3semidirect
156. #498C22×C3semidirect
157. #489C22×C3semidirect
158. #486C22×C3semidirect
164. #476C22×C3semidirect
165. #475C22×C3semidirect
166. #474C22×C3semidirect
171. #446C22×C3semidirect
172. C8×Q8C22×C3semidirect
494. C3#61843direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35222
C2×C3×D4×Q82
C3×#33842
C3×Q8×SD162
C3×#33732
C3×#33582
C3×#32392
C3×#32272
C3×#31092
C3×#31052
C3×#29842
#618433