Finite Group Databasebeta

← C3×#60625 #10679232

Subgroups — C3×#60625 #10679232

Order 768.

2066 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3503C211yesmaximal, characteristic, complemented
2C3×#3500C211yesmaximal, characteristic
3C3×#3141C213yesmaximal, complemented
4C3×#3140C213yesmaximal, complemented
5C3×#3137C216yesmaximal, complemented
6C3×#3136C216yesmaximal, complemented
7C3×#3135C213yesmaximal, complemented
8C2×C3×#489C213yesmaximal, complemented
9C3×#2898C212yesmaximal, complemented
10C3×#2897C212yesmaximal, complemented
11C3×#2875C211yesmaximal, characteristic, complemented
12#60625C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C2×C3×H(5,𝔽2)C2211yescharacteristic
15C2×C3×#131C2212yes—
16C3×Q82—21no—
17C3×#557—21no—
18C3×#554—23no—
19C3×D4×Q8—23no—
20C3×#548—21no—
21C3×#548—21no—
22C3×#546—23no—
23C3×#546—23no—
24C3×#544C2216yescomplemented
25C3×#536C2213yes—
26C3×#535C2213yes—
27C3×#533C2213yescomplemented
28C3×#532C2213yes—
29C2×C3×C4⋊Q8C2213yescomplemented
30C2×C3×#124C2213yes—
31C3×#520C2212yes—
32C3×C4×#41C2211yescharacteristic
33C3×#491C2216yescomplemented
34C3×#490C2216yescomplemented
35C3×#489C2213yescomplemented
36C3×#489C22112yescomplemented
37C3×#489C2216yescomplemented
38C3×#489C2211yescharacteristic, complemented
39C3×#489C2213yescomplemented
40C3×#489C2213yescomplemented
41C3×#488C22112yescomplemented
42C3×#487C2216yescomplemented
43C3×#486C2216yescomplemented
44C3×#419C2216yescomplemented
45C3×#418C2216yes—
46C3×#417C2216yes—
47C3×#417C2216yescomplemented
48C3×#416C2216yes—
49C3×#416C2216yescomplemented
50C2×C3×#103C2216yescomplemented
51C2×C3×#102C2216yes—
52C3×#405C2216yescomplemented
53C3×C4×M16C2213yescomplemented
54C2×C3×#97C2213yescomplemented
55#3503C2×C311yescharacteristic, complemented
56#3500C2×C311yescharacteristic
57#3141C2×C313yescomplemented
58#3140C2×C313yescomplemented
59#3137C2×C316yescomplemented
60#3136C2×C316yescomplemented
61#3135C2×C313yescomplemented
62C2×#489C2×C313yescomplemented
63#2898C2×C312yescomplemented
64#2897C2×C312yescomplemented
65#2875C2×C311yescharacteristic, complemented
66C3×2−1+4—28no—
67C3×H(5,𝔽2)—28no—
68C2×C3×#41—21no—
69C2×C3×#41—23no—
70C2×C3×#41C2313yes—
71C2×C3×#41C2313yes—
72C2×C3×#41C2311yescharacteristic
73C22×C3×Q8—21no—
74C22×C3×Q8—23no—
75C22×C3×Q8C2313yes—
76C22×C3×D4—23no—
77C22×C3×D4C2313yes—
78C3×#131—28no—
79C2×C3×M16C2316yes—
80C22×C3×C8C2316yes—
81C3×C4⋊Q8—26no—
82C3×C4⋊Q8—23no—
83C3×C4⋊Q8—23no—
84C3×C4⋊Q8C2313yes—
85C3×C4⋊Q8C2313yes—
86C3×C4⋊Q8C2316yes—
87C3×C4⋊Q8C2311yescharacteristic
88C3×#127C2313yes—
89C3×#125—26no—
90C3×#125C2316yes—
91C3×#124—23no—
92C3×#124—23no—
93C3×#124C2313yes—
94C3×#124C2316yes—
95C3×#124C2311yescharacteristic
96C3×#123—26no—
97C3×C22⋊Q8—23no—
98C3×C22⋊Q8—23no—
99C3×C22⋊Q8—23no—
100C3×C22⋊Q8—23no—
101C3×C22⋊Q8—212no—
102C3×#121—23no—
103C3×#121—23no—
104C3×#121—23no—
105C3×#121—23no—
106C3×#120—26no—
107C3×C4×Q8—22no—
108C3×C4×Q8—23no—
109C3×C4×Q8—23no—
110C3×C4×Q8D411yescharacteristic
111C3×C4×Q8D411yescharacteristic
112C3×C4×D4—26no—
113C3×C4×D4—23no—
114C3×C4×D4—23no—
115C3×C4×D4D413yes—
116C3×C4×D4D413yes—
117C3×#117C2316yes—
118C3×#117C2313yes—
119C2×C3×C4⋊C4—23no—
120C2×C3×C4⋊C4C2316yes—
121C2×C3×C22⋊C4—23no—
122C2×C3×C22⋊C4—23no—
123C2×C3×C42C2313yes—
124C3×#103C2316yes—
125C3×#103C23112yes—
126C3×#103C2312yes—
127C3×#103C2316yes—
128C3×#103C2316yes—
129C3×#102C2316yes—
130C3×#102C23112yes—
131C3×#102C2316yes—
132C3×#102C2312yes—
133C3×#102C2316yes—
134C3×#97C2311yescharacteristic
135C3×#97C2313yes—
136C3×#97C2316yes—
137C3×C4×C8C2316yes—
138C2×2−1+4C22×C311yescharacteristic
139C2×H(5,𝔽2)C22×C311yescharacteristic
140C2×#131C22×C312yes—
141Q82—21no—
142#557—21no—
143#554—23no—
144D4×Q8—23no—
145#548—21no—
146#548—21no—
147#546—23no—
148#546—23no—
149#544C22×C316yescomplemented
150#536C22×C313yes—
151#535C22×C313yes—
152#533C22×C313yescomplemented
153#532C22×C313yes—
154C2×C4⋊Q8C22×C313yescomplemented
155C2×#124C22×C313yes—
156#520C22×C312yes—
157C4×#41C22×C311yescharacteristic
158#491C22×C316yescomplemented
159#490C22×C316yescomplemented
160#489C22×C313yescomplemented
161#489C22×C3112yescomplemented
162#489C22×C316yescomplemented
163#489C22×C311yescharacteristic, complemented
164#489C22×C313yescomplemented
165#489C22×C313yescomplemented
166#488C22×C3112yescomplemented
167#487C22×C316yescomplemented
168#486C22×C316yescomplemented
169#419C22×C316yescomplemented
170#418C22×C316yes—
171#417C22×C316yes—
172#417C22×C316yescomplemented
173#416C22×C316yes—
174#416C22×C316yescomplemented
175C2×#103C22×C316yescomplemented
176C2×#102C22×C316yes—
177#405C22×C316yescomplemented
178C4×M16C22×C313yescomplemented
179C2×#97C22×C313yescomplemented
180C24×C3—23no—
181C3×#41—212no—
182C3×#41—212no—
183C3×#41—44no—
184C3×#41—412no—
185C3×#41#4118yes—
186C2×C3×Q8—21no—
187C2×C3×Q8—23no—
188C2×C3×Q8—23no—
189C2×C3×Q8—26no—
190C2×C3×Q8—212no—
191C2×C3×Q8—44no—
192C2×C3×Q8—86no—
193C2×C3×Q8C2×D411yescharacteristic
194C2×C3×Q8C2413yes—
195C2×C3×Q8C2411yescharacteristic
196C2×C3×D4—23no—
197C2×C3×D4—23no—
198C2×C3×D4—23no—
199C2×C3×D4—23no—
200C2×C3×D4—23no—
201C2×C3×D4—212no—
202C2×C3×D4—412no—
203C2×C3×D4C2×D413yes—
204C2×C3×D4C2413yes—
205C2×C3×D4C2411yescharacteristic
206C22×C3×C4—23no—
207C22×C3×C4—26no—
208C22×C3×C4—23no—
209C22×C3×C4—23no—
210C22×C3×C4C2×D413yes—
211C22×C3×C4C2×D413yes—
212C22×C3×C4C2413yes—
213C3×M16—212no—
214C2×C3×C8—212no—
215C2×C3×C8C2416yes—
216C2×C3×C8C2412yes—
217C3×C4⋊C4—23no—
218C3×C4⋊C4—26no—
219C3×C4⋊C4—26no—
220C3×C4⋊C4—23no—
221C3×C4⋊C4—23no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—26no—
224C3×C4⋊C4—26no—
225C3×C4⋊C4C2×D413yes—
226C3×C4⋊C4C2×D413yes—
227C3×C4⋊C4C2416yes—
228C3×C4⋊C4C2412yes—
229C3×C22⋊C4—26no—
230C3×C22⋊C4—23no—
231C3×C22⋊C4—23no—
232C3×C22⋊C4—26no—
233C3×C22⋊C4—26no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4C2×D416yes—
237C3×C42—26no—
238C3×C42C2×D413yes—
239C3×C42C2×D413yes—
240C3×C42C2413yes—
241C3×C42C2411yescharacteristic
2422−1+4—28no—
243H(5,𝔽2)—28no—
244C2×#41—21no—
245C2×#41—23no—
246C2×#41C23×C313yes—
247C2×#41C23×C313yes—
248C2×#41C23×C311yescharacteristic
249C22×Q8—21no—
250C22×Q8—23no—
251C22×Q8C23×C313yes—
252C22×D4—23no—
253C22×D4C23×C313yes—
254#131—28no—
255C2×M16C23×C316yes—
256C22×C8C23×C316yes—
257C4⋊Q8—26no—
258C4⋊Q8—23no—
259C4⋊Q8—23no—
260C4⋊Q8C23×C313yes—
261C4⋊Q8C23×C313yes—
262C4⋊Q8C23×C316yes—
263C4⋊Q8C23×C311yescharacteristic
264#127C23×C313yes—
265#125—26no—
266#125C23×C316yes—
267#124—23no—
268#124—23no—
269#124C23×C313yes—
270#124C23×C316yes—
271#124C23×C311yescharacteristic
272#123—26no—
273C22⋊Q8—23no—
274C22⋊Q8—23no—
275C22⋊Q8—23no—
276C22⋊Q8—23no—
277C22⋊Q8—212no—
278#121—23no—
279#121—23no—
280#121—23no—
281#121—23no—
282#120—26no—
283C4×Q8—22no—
284C4×Q8—23no—
285C4×Q8—23no—
286C4×Q8C3×D411yescharacteristic
287C4×Q8C3×D411yescharacteristic
288C4×D4—26no—
289C4×D4—23no—
290C4×D4—23no—
291C4×D4C3×D413yes—
292C4×D4C3×D413yes—
293#117C23×C316yes—
294#117C23×C313yes—
295C2×C4⋊C4—23no—
296C2×C4⋊C4C23×C316yes—
297C2×C22⋊C4—23no—
298C2×C22⋊C4—23no—
299C2×C42C23×C313yes—
300#103C23×C316yes—
301#103C23×C3112yes—
302#103C23×C312yes—
303#103C23×C316yes—
304#103C23×C316yes—
305#102C23×C316yes—
306#102C23×C3112yes—
307#102C23×C316yes—
308#102C23×C312yes—
309#102C23×C316yes—
310#97C23×C311yescharacteristic
311#97C23×C313yes—
312#97C23×C316yes—
313C4×C8C23×C316yes—
314C23×C3—23no—
315C23×C3—21no—
316C23×C3—46no—
317C23×C3—83no—
318C23×C3—83no—
319C23×C3C22×D413yes—
320C3×Q8—26no—
321C3×Q8—22no—
322C3×Q8—42no—
323C3×Q8—46no—
324C3×Q8—46no—
325C3×Q8—86no—
326C3×Q8#12414yes—
327C3×D4—26no—
328C3×D4—22no—
329C3×D4—46no—
330C3×D4—46no—
331C3×D4—46no—
332C3×D4—83no—
333C3×D4—83no—
334C3×D4#124112yes—
335C2×C3×C4—22no—
336C2×C3×C4—26no—
337C2×C3×C4—23no—
338C2×C3×C4—21no—
339C2×C3×C4—43no—
340C2×C3×C4—43no—
341C2×C3×C4—46no—
342C2×C3×C4—46no—
343C2×C3×C4—46no—
344C2×C3×C4—86no—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D413yes—
348C2×C3×C4C22×D413yes—
349C2×C3×C4C22×D413yes—
350C2×C3×C4C2×#41112yes—
351C2×C3×C4C2511yescharacteristic
352C3×C8—22no—
353C3×C8—26no—
354C24—23no—
355#41—212no—
356#41—212no—
357#41—44no—
358#41—412no—
359#41C3×#4118yes—
360C2×Q8—21no—
361C2×Q8—23no—
362C2×Q8—23no—
363C2×Q8—26no—
364C2×Q8—212no—
365C2×Q8—44no—
366C2×Q8—86no—
367C2×Q8C2×C3×D411yescharacteristic
368C2×Q8C24×C313yes—
369C2×Q8C24×C311yescharacteristic
370C2×D4—23no—
371C2×D4—23no—
372C2×D4—23no—
373C2×D4—23no—
374C2×D4—23no—
375C2×D4—212no—
376C2×D4—412no—
377C2×D4C2×C3×D413yes—
378C2×D4C24×C313yes—
379C2×D4C24×C311yescharacteristic
380C22×C4—23no—
381C22×C4—26no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4C2×C3×D413yes—
385C22×C4C2×C3×D413yes—
386C22×C4C24×C313yes—
387M16—212no—
388C2×C8—212no—
389C2×C8C24×C316yes—
390C2×C8C24×C312yes—
391C4⋊C4—23no—
392C4⋊C4—26no—
393C4⋊C4—26no—
394C4⋊C4—23no—
395C4⋊C4—23no—
396C4⋊C4—21no—
397C4⋊C4—26no—
398C4⋊C4—26no—
399C4⋊C4C2×C3×D413yes—
400C4⋊C4C2×C3×D413yes—
401C4⋊C4C24×C316yes—
402C4⋊C4C24×C312yes—
403C22⋊C4—26no—
404C22⋊C4—23no—
405C22⋊C4—23no—
406C22⋊C4—26no—
407C22⋊C4—26no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4C2×C3×D416yes—
411C42—26no—
412C42C2×C3×D413yes—
413C42C2×C3×D413yes—
414C42C24×C313yes—
415C42C24×C311yescharacteristic
416C22×C3—23no—
417C22×C3—23no—
418C22×C3—41no—
419C22×C3—43no—
420C22×C3—41no—
421C22×C3—43no—
422C22×C3—41no—
423C22×C3—43no—
424C22×C3—166no—
425C22×C3C2×#12416yes—
426C22×C3C23×D411yescharacteristic
427C3×C4—21no—
428C3×C4—23no—
429C3×C4—21no—
430C3×C4—23no—
431C3×C4—46no—
432C3×C4—42no—
433C3×C4—41no—
434C3×C4—43no—
435C3×C4C2×#12416yes—
436C3×C4C22×#4112yes—
437C23—23no—
438C23—21no—
439C23—46no—
440C23—83no—
441C23—83no—
442C23C22×C3×D413yes—
443Q8—26no—
444Q8—22no—
445Q8—46no—
446Q8—42no—
447Q8—46no—
448Q8—86no—
449Q8C3×#12414yes—
450D4—26no—
451D4—22no—
452D4—46no—
453D4—46no—
454D4—46no—
455D4—83no—
456D4—83no—
457D4C3×#124112yes—
458C2×C4—22no—
459C2×C4—26no—
460C2×C4—23no—
461C2×C4—21no—
462C2×C4—43no—
463C2×C4—43no—
464C2×C4—46no—
465C2×C4—46no—
466C2×C4—46no—
467C2×C4—86no—
468C2×C4C22×C3×D411yescharacteristic
469C2×C4C22×C3×D411yescharacteristic
470C2×C4C22×C3×D413yes—
471C2×C4C22×C3×D413yes—
472C2×C4C22×C3×D413yes—
473C2×C4C2×C3×#41112yes—
474C2×C4C25×C311yescharacteristic
475C8—22no—
476C8—26no—
477C2×C3—26no—
478C2×C3—81no—
479C2×C3—83no—
480C2×C3C22×#12411yescharacteristic
481C2×C3#358511yescharacteristic
482C2×C3#358611yescharacteristic
483C22—23no—
484C22—23no—
485C22—41no—
486C22—43no—
487C22—41no—
488C22—43no—
489C22—41no—
490C22—43no—
491C22—166no—
492C22C2×C3×#12416yes—
493C22C23×C3×D411yescharacteristic
494C4—21no—
495C4—23no—
496C4—21no—
497C4—23no—
498C4—41no—
499C4—43no—
500C4—46no—
501C4—42no—
502C4C2×C3×#12416yes—
503C4C22×C3×#4112yes—
504C3#6062511yescharacteristic, direct product factor, Sylow
505C2—26no—
506C2—83no—
507C2—81no—
508C2C22×C3×#12411yescharacteristic
509C2C3×#358511yescharacteristic
510C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3500C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3503C2semidirect
3. C3×#3141C2semidirect
4. C3×#3140C2semidirect
5. C3×#3137C2semidirect
6. C3×#3136C2semidirect
7. C3×#3135C2semidirect
8. C2×C3×#489C2semidirect
9. C3×#2898C2semidirect
10. C3×#2897C2semidirect
11. C3×#2875C2semidirect
12. #60625C3direct (N × H)
24. C3×#544C22semidirect
27. C3×#533C22semidirect
29. C2×C3×C4⋊Q8C22semidirect
33. C3×#491C22semidirect
34. C3×#490C22semidirect
35. C3×#489C22semidirect
36. C3×#489C22semidirect
37. C3×#489C22semidirect
38. C3×#489C22semidirect
39. C3×#489C22semidirect
40. C3×#489C22semidirect
41. C3×#488C22semidirect
42. C3×#487C22semidirect
43. C3×#486C22semidirect
44. C3×#419C22semidirect
47. C3×#417C22semidirect
49. C3×#416C22semidirect
50. C2×C3×#103C22semidirect
52. C3×#405C22semidirect
53. C3×C4×M16C22semidirect
54. C2×C3×#97C22semidirect
55. #3503C2×C3semidirect
57. #3141C2×C3semidirect
58. #3140C2×C3semidirect
59. #3137C2×C3semidirect
60. #3136C2×C3semidirect
61. #3135C2×C3semidirect
62. C2×#489C2×C3semidirect
63. #2898C2×C3semidirect
64. #2897C2×C3semidirect
65. #2875C2×C3semidirect
149. #544C22×C3semidirect
152. #533C22×C3semidirect
154. C2×C4⋊Q8C22×C3semidirect
158. #491C22×C3semidirect
159. #490C22×C3semidirect
160. #489C22×C3semidirect
161. #489C22×C3semidirect
162. #489C22×C3semidirect
163. #489C22×C3semidirect
164. #489C22×C3semidirect
165. #489C22×C3semidirect
166. #488C22×C3semidirect
167. #487C22×C3semidirect
168. #486C22×C3semidirect
169. #419C22×C3semidirect
172. #417C22×C3semidirect
174. #416C22×C3semidirect
175. C2×#103C22×C3semidirect
177. #405C22×C3semidirect
178. C4×M16C22×C3semidirect
179. C2×#97C22×C3semidirect
504. C3#60625direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×#35002
C3×#31412
C3×#31402
C3×#31372
C3×#31362
C3×#31352
C2×C3×#4892
C3×#28982
C3×#28972
C3×#28752
#606253