Finite Group Databasebeta

← C3×#34493 #10653100

Subgroups — C3×#34493 #10653100

Order 768.

3076 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D42C211yesmaximal, characteristic
2C3×#2375C214yesmaximal, complemented
3C3×#2366C211yesmaximal, characteristic, complemented
4C3×#2363C214yesmaximal, complemented
5C3×#2354C212yesmaximal, complemented
6C3×D4×C22⋊C4C214yesmaximal, complemented
7C3×#2324C214yesmaximal, complemented
8C3×#2321C214yesmaximal, complemented
9C3×#2309C214yesmaximal, complemented
10C3×C4×#127C211yesmaximal, characteristic, complemented
11C3×#2293C212yesmaximal, complemented
12#34493C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2214yes—
14C3×D42C4116yescomplemented
15C2×C3×#127C2211yescharacteristic
16C2×C3×#121C2214yes—
17C2×C3×#120C2214yes—
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2214yescomplemented
20C2×C3×C4×D4C2214yescomplemented
21C2×C3×C4×D4C2212yescomplemented
22C2×C3×#117C2214yescomplemented
23C2×C3×#117C2214yescomplemented
24C22×C3×C22⋊C4C2214yescomplemented
25C3×#390C2212yes—
26C3×#390C2212yes—
27C3×#390C2214yes—
28C3×#390C2214yescomplemented
29C3×#390C2218yescomplemented
30C3×#390C2212yescomplemented
31C3×#390C2214yescomplemented
32C3×#390C2218yescomplemented
33C3×#389C2212yescomplemented
34C3×#389C2212yescomplemented
35C3×#388C2218yescomplemented
36C3×#388C2214yescomplemented
37C3×#388C2218yescomplemented
38C3×#388C2214yescomplemented
39C3×#386C2218yescomplemented
40C3×#386C2214yescomplemented
41C3×#386C2218yescomplemented
42C3×#385C2214yescomplemented
43C3×#384C2211yescharacteristic, complemented
44C3×#380C2214yescomplemented
45C3×#379C2218yescomplemented
46C3×#379C2214yescomplemented
47C3×C4×C4⋊C4C2214yescomplemented
48C3×C4×C4⋊C4C2212yescomplemented
49C3×C4×C22⋊C4C2214yescomplemented
50C3×C4×C22⋊C4C2214yescomplemented
51C3×C43C2211yescharacteristic, complemented
52C2×D42C2×C311yescharacteristic
53#2375C2×C314yescomplemented
54#2366C2×C311yescharacteristic, complemented
55#2363C2×C314yescomplemented
56#2354C2×C312yescomplemented
57D4×C22⋊C4C2×C314yescomplemented
58#2324C2×C314yescomplemented
59#2321C2×C314yescomplemented
60#2309C2×C314yescomplemented
61C4×#127C2×C311yescharacteristic, complemented
62#2293C2×C312yescomplemented
63C25×C3C2314yes—
64C22×C3×D4—48no—
65C22×C3×D4—48no—
66C22×C3×D4C2×C4132yescomplemented
67C22×C3×D4C2312yes—
68C22×C3×D4C2314yes—
69C22×C3×D4C2318yes—
70C22×C3×D4C2312yes—
71C23×C3×C4C2314yes—
72C23×C3×C4C2314yes—
73C23×C3×C4C2314yes—
74C3×#127C2×C418yes—
75C3×#121C2×C4132yescomplemented
76C3×#120C2×C4132yescomplemented
77C3×C4×D4—48no—
78C3×C4×D4—48no—
79C3×C4×D4—44no—
80C3×C4×D4C2×C4116yescomplemented
81C3×#117—48no—
82C3×#117—48no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C22⋊C4—48no—
91C2×C3×C22⋊C4C2314yes—
92C2×C3×C22⋊C4C2314yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2314yes—
95C2×C3×C22⋊C4C2318yes—
96C2×C3×C22⋊C4C2314yes—
97C2×C3×C22⋊C4C2318yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C22⋊C4C2318yes—
100C2×C3×C22⋊C4C2318yes—
101C2×C3×C22⋊C4C2314yes—
102C2×C3×C22⋊C4C2314yes—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2312yes—
106C2×C3×C42C2314yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2318yes—
109C2×C3×C42C2311yescharacteristic
110C3×#95C2314yes—
111C3×#95C2314yes—
112C3×#95C2318yes—
113C3×#95C2314yes—
114C23×D4C22×C314yes—
115D42C3×C4116yescomplemented
116C2×#127C22×C311yescharacteristic
117C2×#121C22×C314yes—
118C2×#120C22×C314yes—
119C2×C4×D4C22×C312yes—
120C2×C4×D4C22×C312yescomplemented
121C2×C4×D4C22×C314yescomplemented
122C2×C4×D4C22×C314yescomplemented
123C2×#117C22×C314yescomplemented
124C2×#117C22×C314yescomplemented
125C22×C22⋊C4C22×C314yescomplemented
126#390C22×C312yes—
127#390C22×C314yes—
128#390C22×C312yes—
129#390C22×C314yescomplemented
130#390C22×C312yescomplemented
131#390C22×C318yescomplemented
132#390C22×C314yescomplemented
133#390C22×C318yescomplemented
134#389C22×C312yescomplemented
135#389C22×C312yescomplemented
136#388C22×C314yescomplemented
137#388C22×C318yescomplemented
138#388C22×C318yescomplemented
139#388C22×C314yescomplemented
140#386C22×C318yescomplemented
141#386C22×C314yescomplemented
142#386C22×C318yescomplemented
143#385C22×C314yescomplemented
144#384C22×C311yescharacteristic, complemented
145#380C22×C314yescomplemented
146#379C22×C314yescomplemented
147#379C22×C318yescomplemented
148C4×C4⋊C4C22×C314yescomplemented
149C4×C4⋊C4C22×C312yescomplemented
150C4×C22⋊C4C22×C314yescomplemented
151C4×C22⋊C4C22×C314yescomplemented
152C43C22×C311yescharacteristic, complemented
153C24×C3—44no—
154C24×C3—44no—
155C24×C3—48no—
156C24×C3—48no—
157C24×C3C22×C4116yes—
158C24×C3C2414yes—
159C24×C3C2414yes—
160C2×C3×D4—28no—
161C2×C3×D4—28no—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—464no—
165C2×C3×D4—42no—
166C2×C3×D4—48no—
167C2×C3×D4—48no—
168C2×C3×D4—42no—
169C2×C3×D4C22×C418yes—
170C2×C3×D4C22×C4116yes—
171C2×C3×D4C22×C418yes—
172C2×C3×D4C22×C4132yes—
173C22×C3×C4—28no—
174C22×C3×C4—28no—
175C22×C3×C4—28no—
176C22×C3×C4—44no—
177C22×C3×C4—48no—
178C22×C3×C4—48no—
179C22×C3×C4C22×C4116yes—
180C22×C3×C4C2414yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2414yes—
183C22×C3×C4C2414yes—
184C22×C3×C4C2411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×C3×C4C2414yes—
187C22×C3×C4C2411yescharacteristic
188C22×C3×C4C2412yes—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—44no—
193C3×C4⋊C4—44no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4C22×C418yes—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—48no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—44no—
200C3×C22⋊C4—44no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—48no—
203C3×C22⋊C4—48no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—48no—
206C3×C22⋊C4C22×C4116yes—
207C3×C22⋊C4C22×C4116yes—
208C3×C42—24no—
209C3×C42—28no—
210C3×C42—22no—
211C3×C42—24no—
212C3×C42—41no—
213C3×C42—48no—
214C3×C42C22×C414yes—
215C25C23×C314yes—
216C22×D4—48no—
217C22×D4—48no—
218C22×D4C2×C3×C4132yescomplemented
219C22×D4C23×C312yes—
220C22×D4C23×C314yes—
221C22×D4C23×C312yes—
222C22×D4C23×C318yes—
223C23×C4C23×C314yes—
224C23×C4C23×C314yes—
225C23×C4C23×C314yes—
226#127C2×C3×C418yes—
227#121C2×C3×C4132yescomplemented
228#120C2×C3×C4132yescomplemented
229C4×D4—48no—
230C4×D4—48no—
231C4×D4—44no—
232C4×D4C2×C3×C4116yescomplemented
233#117—48no—
234#117—48no—
235C2×C4⋊C4C23×C312yes—
236C2×C4⋊C4C23×C314yes—
237C2×C4⋊C4C23×C312yes—
238C2×C4⋊C4C23×C314yes—
239C2×C4⋊C4C23×C314yes—
240C2×C4⋊C4C23×C312yes—
241C2×C4⋊C4C23×C312yes—
242C2×C22⋊C4—48no—
243C2×C22⋊C4C23×C314yes—
244C2×C22⋊C4C23×C318yes—
245C2×C22⋊C4C23×C314yes—
246C2×C22⋊C4C23×C318yes—
247C2×C22⋊C4C23×C314yes—
248C2×C22⋊C4C23×C314yes—
249C2×C22⋊C4C23×C318yes—
250C2×C22⋊C4C23×C318yes—
251C2×C22⋊C4C23×C314yes—
252C2×C22⋊C4C23×C314yes—
253C2×C22⋊C4C23×C314yes—
254C2×C22⋊C4C23×C314yes—
255C2×C42C23×C312yes—
256C2×C42C23×C314yes—
257C2×C42C23×C312yes—
258C2×C42C23×C311yescharacteristic
259C2×C42C23×C318yes—
260C2×C42C23×C311yescharacteristic
261C2×C42C23×C311yescharacteristic
262#95C23×C314yes—
263#95C23×C314yes—
264#95C23×C318yes—
265#95C23×C314yes—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—24no—
269C23×C3—24no—
270C23×C3—28no—
271C23×C3—28no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—432no—
277C23×C3—416no—
278C23×C3—88no—
279C23×C3—816no—
280C23×C3—88no—
281C23×C3C23×C418yes—
282C23×C3C23×C418yes—
283C23×C3C2511yescharacteristic
284C3×D4—216no—
285C3×D4—232no—
286C3×D4—432no—
287C3×D4—48no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—21no—
293C2×C3×C4—232no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—28no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—88no—
308C2×C3×C4—88no—
309C2×C3×C4C23×C412yes—
310C2×C3×C4C23×C418yes—
311C2×C3×C4C23×C414yes—
312C2×C3×C4H(5,𝔽2)12yes—
313C2×C3×C4H(5,𝔽2)14yes—
314C2×C3×C4H(5,𝔽2)14yes—
315C24—44no—
316C24—48no—
317C24—48no—
318C24—44no—
319C24C22×C3×C4116yes—
320C24C24×C314yes—
321C24C24×C314yes—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—464no—
327C2×D4—42no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—42no—
331C2×D4C22×C3×C418yes—
332C2×D4C22×C3×C4116yes—
333C2×D4C22×C3×C418yes—
334C2×D4C22×C3×C4132yes—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—44no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4C22×C3×C4116yes—
342C22×C4C24×C314yes—
343C22×C4C24×C312yes—
344C22×C4C24×C314yes—
345C22×C4C24×C314yes—
346C22×C4C24×C311yescharacteristic
347C22×C4C24×C311yescharacteristic
348C22×C4C24×C314yes—
349C22×C4C24×C311yescharacteristic
350C22×C4C24×C312yes—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4C22×C3×C418yes—
358C22⋊C4—28no—
359C22⋊C4—48no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—44no—
365C22⋊C4—48no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4C22×C3×C4116yes—
369C22⋊C4C22×C3×C4116yes—
370C42—24no—
371C42—28no—
372C42—22no—
373C42—24no—
374C42—48no—
375C42—41no—
376C42C22×C3×C414yes—
377C22×C3—28no—
378C22×C3—28no—
379C22×C3—216no—
380C22×C3—28no—
381C22×C3—28no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—48no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3—44no—
388C22×C3—44no—
389C22×C3—832no—
390C22×C3C24×C411yescharacteristic
391C22×C3C2×H(5,𝔽2)12yes—
392C22×C3C2×H(5,𝔽2)12yes—
393C22×C3C2×H(5,𝔽2)11yescharacteristic
394C22×C3#58511yescharacteristic
395C3×C4—28no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—44no—
399C3×C4—42no—
400C3×C4—44no—
401C3×C4—41no—
402C3×C4—44no—
403C3×C4#51818yes—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—28no—
409C23—28no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—432no—
415C23—416no—
416C23—88no—
417C23—816no—
418C23—88no—
419C23C23×C3×C418yes—
420C23C23×C3×C418yes—
421C23C25×C311yescharacteristic
422D4—216no—
423D4—232no—
424D4—432no—
425D4—48no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—21no—
431C2×C4—232no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—88no—
446C2×C4—88no—
447C2×C4C23×C3×C412yes—
448C2×C4C23×C3×C418yes—
449C2×C4C23×C3×C414yes—
450C2×C4C3×H(5,𝔽2)12yes—
451C2×C4C3×H(5,𝔽2)14yes—
452C2×C4C3×H(5,𝔽2)14yes—
453C2×C3—48no—
454C2×C3—48no—
455C2×C3C2×#51812yes—
456C2×C3C4×H(5,𝔽2)11yescharacteristic
457C2×C3#354311yescharacteristic
458C2×C3#354612yes—
459C2×C3#354911yescharacteristic
460C22—28no—
461C22—28no—
462C22—216no—
463C22—28no—
464C22—28no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—48no—
470C22—44no—
471C22—44no—
472C22—832no—
473C22C24×C3×C411yescharacteristic
474C22C2×C3×H(5,𝔽2)12yes—
475C22C2×C3×H(5,𝔽2)12yes—
476C22C2×C3×H(5,𝔽2)11yescharacteristic
477C22C3×#58511yescharacteristic
478C4—28no—
479C4—22no—
480C4—22no—
481C4—44no—
482C4—42no—
483C4—44no—
484C4—44no—
485C4—41no—
486C4C3×#51818yes—
487C3#3449311yescharacteristic, direct product factor, Sylow
488C2—48no—
489C2—48no—
490C2C2×C3×#51812yes—
491C2C3×C4×H(5,𝔽2)11yescharacteristic
492C2C3×#354311yescharacteristic
493C2C3×#354612yes—
494C2C3×#354911yescharacteristic
subgroupG/H
center, socleC23×C3C25
commutatorC22C24×C3×C4
Frattini, cosocleC23C25×C3
prime radicalC2×C3×D42C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2375C2semidirect
3. C3×#2366C2semidirect
4. C3×#2363C2semidirect
5. C3×#2354C2semidirect
6. C3×D4×C22⋊C4C2semidirect
7. C3×#2324C2semidirect
8. C3×#2321C2semidirect
9. C3×#2309C2semidirect
10. C3×C4×#127C2semidirect
11. C3×#2293C2semidirect
12. #34493C3direct (N × H)
14. C3×D42C4semidirect
19. C2×C3×C4×D4C22semidirect
20. C2×C3×C4×D4C22semidirect
21. C2×C3×C4×D4C22semidirect
22. C2×C3×#117C22semidirect
23. C2×C3×#117C22semidirect
24. C22×C3×C22⋊C4C22semidirect
28. C3×#390C22semidirect
29. C3×#390C22semidirect
30. C3×#390C22semidirect
31. C3×#390C22semidirect
32. C3×#390C22semidirect
33. C3×#389C22semidirect
34. C3×#389C22semidirect
35. C3×#388C22semidirect
36. C3×#388C22semidirect
37. C3×#388C22semidirect
38. C3×#388C22semidirect
39. C3×#386C22semidirect
40. C3×#386C22semidirect
41. C3×#386C22semidirect
42. C3×#385C22semidirect
43. C3×#384C22semidirect
44. C3×#380C22semidirect
45. C3×#379C22semidirect
46. C3×#379C22semidirect
47. C3×C4×C4⋊C4C22semidirect
48. C3×C4×C4⋊C4C22semidirect
49. C3×C4×C22⋊C4C22semidirect
50. C3×C4×C22⋊C4C22semidirect
51. C3×C43C22semidirect
53. #2375C2×C3semidirect
54. #2366C2×C3semidirect
55. #2363C2×C3semidirect
56. #2354C2×C3semidirect
57. D4×C22⋊C4C2×C3semidirect
58. #2324C2×C3semidirect
59. #2321C2×C3semidirect
60. #2309C2×C3semidirect
61. C4×#127C2×C3semidirect
62. #2293C2×C3semidirect
66. C22×C3×D4C2×C4semidirect
75. C3×#121C2×C4semidirect
76. C3×#120C2×C4semidirect
80. C3×C4×D4C2×C4semidirect
115. D42C3×C4semidirect
120. C2×C4×D4C22×C3semidirect
121. C2×C4×D4C22×C3semidirect
122. C2×C4×D4C22×C3semidirect
123. C2×#117C22×C3semidirect
124. C2×#117C22×C3semidirect
125. C22×C22⋊C4C22×C3semidirect
129. #390C22×C3semidirect
130. #390C22×C3semidirect
131. #390C22×C3semidirect
132. #390C22×C3semidirect
133. #390C22×C3semidirect
134. #389C22×C3semidirect
135. #389C22×C3semidirect
136. #388C22×C3semidirect
137. #388C22×C3semidirect
138. #388C22×C3semidirect
139. #388C22×C3semidirect
140. #386C22×C3semidirect
141. #386C22×C3semidirect
142. #386C22×C3semidirect
143. #385C22×C3semidirect
144. #384C22×C3semidirect
145. #380C22×C3semidirect
146. #379C22×C3semidirect
147. #379C22×C3semidirect
148. C4×C4⋊C4C22×C3semidirect
149. C4×C4⋊C4C22×C3semidirect
150. C4×C22⋊C4C22×C3semidirect
151. C4×C22⋊C4C22×C3semidirect
152. C43C22×C3semidirect
218. C22×D4C2×C3×C4semidirect
227. #121C2×C3×C4semidirect
228. #120C2×C3×C4semidirect
232. C4×D4C2×C3×C4semidirect
487. C3#34493direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D422
C3×#23752
C3×#23662
C3×#23632
C3×#23542
C3×D4×C22⋊C42
C3×#23242
C3×#23212
C3×#23092
C3×C4×#1272
C3×#22932
#344933