Finite Group Databasebeta

← C3×#63002 #10681609

Subgroups — C3×#63002 #10681609

Order 768.

11160 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C211yesmaximal, characteristic, complemented
2C22×C3×2−1+4C211yesmaximal, characteristic, complemented
3C3×#3500C2110yesmaximal, complemented
4C3×#3497C215yesmaximal, complemented
5C3×#3489C2110yesmaximal, complemented
6C3×#3488C2110yesmaximal, complemented
7C3×#3487C215yesmaximal, complemented
8C3×#3486C215yesmaximal, complemented
9C3×D4×#41C2110yesmaximal, complemented
10C2×C3×#548C215yesmaximal, complemented
11C3×C4×2−1+4C211yesmaximal, characteristic, complemented
12#63002C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#585—216no—
14C2×C3×2−1+4—216no—
15C2×C3×2−1+4C2211yescharacteristic
16C2×C3×2−1+4C2215yescomplemented
17C2×C3×2−1+4C22110yescomplemented
18C2×C3×2−1+4C2215yescomplemented
19C2×C3×H(5,𝔽2)C22110yescomplemented
20C22×C3×#41C22110yescomplemented
21C22×C3×#41C2215yescomplemented
22C22×C3×#41C22110yescomplemented
23C23×C3×Q8C2215yescomplemented
24C3×#550C22110yescomplemented
25C3×#550C22120yescomplemented
26C3×D4×Q8C22110yescomplemented
27C3×D4×Q8C22120yescomplemented
28C3×#548C22110yescomplemented
29C3×#548C22120yescomplemented
30C3×#548C22120yescomplemented
31C3×#548C22120yescomplemented
32C3×#548C22110yescomplemented
33C3×#548C22120yescomplemented
34C3×#547C22130yescomplemented
35C3×#546C22110yescomplemented
36C3×#546C22130yescomplemented
37C3×#546C22160yescomplemented
38C3×D42C22130yescomplemented
39C3×#537C22130yescomplemented
40C3×#536C22130yescomplemented
41C3×#535C22130yescomplemented
42C3×#532C22130yescomplemented
43C2×C3×#124C22115yescomplemented
44C3×#525C22130yescomplemented
45C3×#525C22130yescomplemented
46C2×C3×C22⋊Q8C22115yescomplemented
47C2×C3×#121C22115yescomplemented
48C3×#520C22110yescomplemented
49C3×#519C2215yescomplemented
50C3×C4×#41C22110yescomplemented
51C2×C3×C4×Q8C2215yescomplemented
52C2×C3×C4×D4C22115yescomplemented
53C2×#585C2×C311yescharacteristic, complemented
54C22×2−1+4C2×C311yescharacteristic, complemented
55#3500C2×C3110yescomplemented
56#3497C2×C315yescomplemented
57#3489C2×C3110yescomplemented
58#3488C2×C3110yescomplemented
59#3487C2×C315yescomplemented
60#3486C2×C315yescomplemented
61D4×#41C2×C3110yescomplemented
62C2×#548C2×C315yescomplemented
63C4×2−1+4C2×C311yescharacteristic, complemented
64C3×2−1+4—280no—
65C3×2−1+4—240no—
66C3×2−1+4—240no—
67C3×2−1+4D4116yescomplemented
68C3×H(5,𝔽2)—280no—
69C2×C3×#41—280no—
70C2×C3×#41—280no—
71C2×C3×#41—240no—
72C2×C3×#41C23110yes—
73C2×C3×#41C23110yes—
74C2×C3×#41C23130yes—
75C2×C3×#41C23120yes—
76C2×C3×#41C23130yes—
77C2×C3×#41C23130yes—
78C2×C3×#41C23110yes—
79C2×C3×#41C23110yes—
80C22×C3×Q8—240no—
81C22×C3×Q8C23110yes—
82C22×C3×Q8C23115yes—
83C22×C3×Q8C23120yes—
84C22×C3×Q8C23110yes—
85C22×C3×Q8C2315yes—
86C22×C3×D4C23110yes—
87C22×C3×D4C23115yes—
88C22×C3×D4C23130yes—
89C22×C3×D4C23115yes—
90C23×C3×C4C23115yes—
91C23×C3×C4C23115yes—
92C3×C4⋊Q8C23130yes—
93C3×#127C23130yes—
94C3×#124C23130yes—
95C3×#124C23160yes—
96C3×#124C23110yes—
97C3×#123C23120yes—
98C3×#123C23160yes—
99C3×C22⋊Q8C23130yes—
100C3×C22⋊Q8C23160yes—
101C3×C22⋊Q8C23110yes—
102C3×C22⋊Q8C23130yes—
103C3×C22⋊Q8C23130yes—
104C3×#121C23130yes—
105C3×#121C23160yes—
106C3×#121C23110yes—
107C3×#121C23130yes—
108C3×#121C23130yes—
109C3×#121C23120yes—
110C3×#121C23160yes—
111C3×#120C23120yes—
112C3×#120C23160yes—
113C3×C4×Q8C23110yes—
114C3×C4×Q8C23120yes—
115C3×C4×Q8C23110yes—
116C3×C4×D4C23110yes—
117C3×C4×D4C23130yes—
118C3×C4×D4C23130yes—
119C3×C4×D4C23130yes—
120C3×C4×D4C23110yes—
121C3×C4×D4C23130yes—
122C3×C4×D4C23160yes—
123C3×#117C23130yes—
124C2×C3×C4⋊C4C23115yes—
125C2×C3×C4⋊C4C23115yes—
126C2×C3×C22⋊C4C2315yes—
127C2×C3×C22⋊C4C23115yes—
128C2×C3×C22⋊C4C23130yes—
129C2×C3×C42C23115yes—
130#585—216no—
131C2×2−1+4—216no—
132C2×2−1+4C22×C311yescharacteristic
133C2×2−1+4C22×C315yescomplemented
134C2×2−1+4C22×C3110yescomplemented
135C2×2−1+4C22×C315yescomplemented
136C2×H(5,𝔽2)C22×C3110yescomplemented
137C22×#41C22×C3110yescomplemented
138C22×#41C22×C315yescomplemented
139C22×#41C22×C3110yescomplemented
140C23×Q8C22×C315yescomplemented
141#550C22×C3110yescomplemented
142#550C22×C3120yescomplemented
143D4×Q8C22×C3110yescomplemented
144D4×Q8C22×C3120yescomplemented
145#548C22×C3110yescomplemented
146#548C22×C3120yescomplemented
147#548C22×C3120yescomplemented
148#548C22×C3120yescomplemented
149#548C22×C3110yescomplemented
150#548C22×C3120yescomplemented
151#547C22×C3130yescomplemented
152#546C22×C3110yescomplemented
153#546C22×C3130yescomplemented
154#546C22×C3160yescomplemented
155D42C22×C3130yescomplemented
156#537C22×C3130yescomplemented
157#536C22×C3130yescomplemented
158#535C22×C3130yescomplemented
159#532C22×C3130yescomplemented
160C2×#124C22×C3115yescomplemented
161#525C22×C3130yescomplemented
162#525C22×C3130yescomplemented
163C2×C22⋊Q8C22×C3115yescomplemented
164C2×#121C22×C3115yescomplemented
165#520C22×C3110yescomplemented
166#519C22×C315yescomplemented
167C4×#41C22×C3110yescomplemented
168C2×C4×Q8C22×C315yescomplemented
169C2×C4×D4C22×C3115yescomplemented
170C24×C3C2415yes—
171C24×C3C24115yes—
172C3×#41—240no—
173C3×#41—280no—
174C3×#41—240no—
175C3×#41—240no—
176C3×#41—2120no—
177C3×#41—2120no—
178C3×#41—2120no—
179C3×#41C2×D4180yescomplemented
180C2×C3×Q8—240no—
181C2×C3×Q8—280no—
182C2×C3×Q8—240no—
183C2×C3×Q8—260no—
184C2×C3×Q8C2×D4140yescomplemented
185C2×C3×Q8C24130yes—
186C2×C3×Q8C24110yes—
187C2×C3×Q8C24110yes—
188C2×C3×D4—240no—
189C2×C3×D4—2120no—
190C2×C3×D4—260no—
191C2×C3×D4—430no—
192C2×C3×D4C24110yes—
193C2×C3×D4C24120yes—
194C2×C3×D4C24120yes—
195C2×C3×D4C24130yes—
196C2×C3×D4C24110yes—
197C2×C3×D4C24120yes—
198C2×C3×D4C24130yes—
199C2×C3×D4C24110yes—
200C22×C3×C4—260no—
201C22×C3×C4—260no—
202C22×C3×C4C24115yes—
203C22×C3×C4C2415yes—
204C22×C3×C4C24110yes—
205C22×C3×C4C24130yes—
206C22×C3×C4C24110yes—
207C22×C3×C4C24130yes—
208C22×C3×C4C2415yes—
209C22×C3×C4C24115yes—
210C22×C3×C4C24115yes—
211C3×C4⋊C4C24110yes—
212C3×C4⋊C4C24130yes—
213C3×C4⋊C4C24130yes—
214C3×C4⋊C4C24110yes—
215C3×C4⋊C4C24120yes—
216C3×C22⋊C4C2411yescharacteristic
217C3×C22⋊C4C24110yes—
218C3×C22⋊C4C24160yes—
219C3×C22⋊C4C2415yes—
220C3×C22⋊C4C2415yes—
221C3×C22⋊C4C2415yes—
222C3×C22⋊C4C24130yes—
223C3×C22⋊C4C24120yes—
224C3×C22⋊C4C24120yes—
225C3×C42C24110yes—
226C3×C42C24130yes—
2272−1+4—280no—
2282−1+4—240no—
2292−1+4—240no—
2302−1+4C3×D4116yescomplemented
231H(5,𝔽2)—280no—
232C2×#41—280no—
233C2×#41—280no—
234C2×#41—240no—
235C2×#41C23×C3130yes—
236C2×#41C23×C3130yes—
237C2×#41C23×C3130yes—
238C2×#41C23×C3110yes—
239C2×#41C23×C3110yes—
240C2×#41C23×C3120yes—
241C2×#41C23×C3110yes—
242C2×#41C23×C3110yes—
243C22×Q8—240no—
244C22×Q8C23×C3115yes—
245C22×Q8C23×C3110yes—
246C22×Q8C23×C3120yes—
247C22×Q8C23×C3110yes—
248C22×Q8C23×C315yes—
249C22×D4C23×C3115yes—
250C22×D4C23×C3110yes—
251C22×D4C23×C3130yes—
252C22×D4C23×C3115yes—
253C23×C4C23×C3115yes—
254C23×C4C23×C3115yes—
255C4⋊Q8C23×C3130yes—
256#127C23×C3130yes—
257#124C23×C3130yes—
258#124C23×C3160yes—
259#124C23×C3110yes—
260#123C23×C3120yes—
261#123C23×C3160yes—
262C22⋊Q8C23×C3130yes—
263C22⋊Q8C23×C3160yes—
264C22⋊Q8C23×C3110yes—
265C22⋊Q8C23×C3130yes—
266C22⋊Q8C23×C3130yes—
267#121C23×C3130yes—
268#121C23×C3160yes—
269#121C23×C3110yes—
270#121C23×C3130yes—
271#121C23×C3130yes—
272#121C23×C3120yes—
273#121C23×C3160yes—
274#120C23×C3120yes—
275#120C23×C3160yes—
276C4×Q8C23×C3110yes—
277C4×Q8C23×C3120yes—
278C4×Q8C23×C3110yes—
279C4×D4C23×C3110yes—
280C4×D4C23×C3130yes—
281C4×D4C23×C3130yes—
282C4×D4C23×C3130yes—
283C4×D4C23×C3110yes—
284C4×D4C23×C3130yes—
285C4×D4C23×C3160yes—
286#117C23×C3130yes—
287C2×C4⋊C4C23×C3115yes—
288C2×C4⋊C4C23×C3115yes—
289C2×C22⋊C4C23×C315yes—
290C2×C22⋊C4C23×C3115yes—
291C2×C22⋊C4C23×C3130yes—
292C2×C42C23×C3115yes—
293C23×C3—210no—
294C23×C3—210no—
295C23×C3—230no—
296C23×C3—45no—
297C23×C3—415no—
298C23×C3—415no—
299C23×C3C25110yes—
300C23×C3C2515yes—
301C23×C3C2515yes—
302C23×C3C2511yescharacteristic
303C3×Q8—260no—
304C3×Q8—220no—
305C3×Q8C22×D4140yes—
306C3×D4—240no—
307C3×D4—240no—
308C3×D4—220no—
309C3×D4—260no—
310C3×D4—220no—
311C3×D4—430no—
312C3×D4—420no—
313C3×D4C22×D4140yes—
314C2×C3×C4—210no—
315C2×C3×C4—230no—
316C2×C3×C4—220no—
317C2×C3×C4—220no—
318C2×C3×C4—260no—
319C2×C3×C4—260no—
320C2×C3×C4—45no—
321C2×C3×C4—415no—
322C2×C3×C4C22×D4160yes—
323C2×C3×C4C2511yescharacteristic
324C2×C3×C4C2515yes—
325C2×C3×C4C25110yes—
326C2×C3×C4C25110yes—
327C2×C3×C4C2511yescharacteristic
328C2×C3×C4C2515yes—
329C2×C3×C4C25110yes—
330C24C24×C315yes—
331C24C24×C3115yes—
332#41—240no—
333#41—280no—
334#41—240no—
335#41—240no—
336#41—2120no—
337#41—2120no—
338#41—2120no—
339#41C2×C3×D4180yescomplemented
340C2×Q8—240no—
341C2×Q8—280no—
342C2×Q8—240no—
343C2×Q8—260no—
344C2×Q8C2×C3×D4140yescomplemented
345C2×Q8C24×C3130yes—
346C2×Q8C24×C3110yes—
347C2×Q8C24×C3110yes—
348C2×D4—240no—
349C2×D4—2120no—
350C2×D4—260no—
351C2×D4—430no—
352C2×D4C24×C3110yes—
353C2×D4C24×C3120yes—
354C2×D4C24×C3120yes—
355C2×D4C24×C3130yes—
356C2×D4C24×C3110yes—
357C2×D4C24×C3120yes—
358C2×D4C24×C3130yes—
359C2×D4C24×C3110yes—
360C22×C4—260no—
361C22×C4—260no—
362C22×C4C24×C3115yes—
363C22×C4C24×C315yes—
364C22×C4C24×C3110yes—
365C22×C4C24×C3130yes—
366C22×C4C24×C3110yes—
367C22×C4C24×C3130yes—
368C22×C4C24×C315yes—
369C22×C4C24×C3115yes—
370C22×C4C24×C3115yes—
371C4⋊C4C24×C3110yes—
372C4⋊C4C24×C3130yes—
373C4⋊C4C24×C3130yes—
374C4⋊C4C24×C3110yes—
375C4⋊C4C24×C3120yes—
376C22⋊C4C24×C311yescharacteristic
377C22⋊C4C24×C3110yes—
378C22⋊C4C24×C3160yes—
379C22⋊C4C24×C315yes—
380C22⋊C4C24×C315yes—
381C22⋊C4C24×C315yes—
382C22⋊C4C24×C3130yes—
383C22⋊C4C24×C3120yes—
384C22⋊C4C24×C3120yes—
385C42C24×C3110yes—
386C42C24×C3130yes—
387C22×C3—21no—
388C22×C3—25no—
389C22×C3—25no—
390C22×C3—21no—
391C22×C3—25no—
392C22×C3—210no—
393C22×C3—210no—
394C22×C3—25no—
395C22×C3—25no—
396C22×C3—210no—
397C22×C3—420no—
398C22×C3—830no—
399C22×C3C23×D4110yes—
400C22×C3#58512yescomplemented
401C22×C3C2611yescharacteristic
402C3×C4—21no—
403C3×C4—25no—
404C3×C4—210no—
405C3×C4—210no—
406C3×C4—21no—
407C3×C4—25no—
408C3×C4C23×D4120yes—
409C23—210no—
410C23—210no—
411C23—230no—
412C23—45no—
413C23—415no—
414C23—415no—
415C23C25×C3110yes—
416C23C25×C315yes—
417C23C25×C315yes—
418C23C25×C311yescharacteristic
419Q8—260no—
420Q8—220no—
421Q8C22×C3×D4140yes—
422D4—240no—
423D4—240no—
424D4—220no—
425D4—260no—
426D4—220no—
427D4—430no—
428D4—420no—
429D4C22×C3×D4140yes—
430C2×C4—210no—
431C2×C4—230no—
432C2×C4—220no—
433C2×C4—220no—
434C2×C4—260no—
435C2×C4—260no—
436C2×C4—415no—
437C2×C4—45no—
438C2×C4C22×C3×D4160yes—
439C2×C4C25×C311yescharacteristic
440C2×C4C25×C315yes—
441C2×C4C25×C3110yes—
442C2×C4C25×C3110yes—
443C2×C4C25×C311yescharacteristic
444C2×C4C25×C315yes—
445C2×C4C25×C3110yes—
446C2×C3—22no—
447C2×C3—210no—
448C2×C3—45no—
449C2×C3—410no—
450C2×C3C24×D411yescharacteristic
451C2×C3C2×#58511yescharacteristic
452C2×C32−1+611yescharacteristic
453C22—21no—
454C22—25no—
455C22—25no—
456C22—21no—
457C22—25no—
458C22—210no—
459C22—210no—
460C22—25no—
461C22—25no—
462C22—210no—
463C22—420no—
464C22—830no—
465C22C23×C3×D4110yes—
466C22C3×#58512yescomplemented
467C22C26×C311yescharacteristic
468C4—21no—
469C4—25no—
470C4—210no—
471C4—210no—
472C4—21no—
473C4—25no—
474C4C23×C3×D4120yes—
475C3#6300211yescharacteristic, direct product factor, Sylow
476C2—22no—
477C2—210no—
478C2—45no—
479C2—410no—
480C2C24×C3×D411yescharacteristic
481C2C2×C3×#58511yescharacteristic
482C2C3×2−1+611yescharacteristic
subgroupG/H
center, socleC22×C3C26
commutator, Frattini, cosocleC22C26×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#585C2semidirect
2. C22×C3×2−1+4C2semidirect
3. C3×#3500C2semidirect
4. C3×#3497C2semidirect
5. C3×#3489C2semidirect
6. C3×#3488C2semidirect
7. C3×#3487C2semidirect
8. C3×#3486C2semidirect
9. C3×D4×#41C2semidirect
10. C2×C3×#548C2semidirect
11. C3×C4×2−1+4C2semidirect
12. #63002C3direct (N × H)
16. C2×C3×2−1+4C22semidirect
17. C2×C3×2−1+4C22semidirect
18. C2×C3×2−1+4C22semidirect
19. C2×C3×H(5,𝔽2)C22semidirect
20. C22×C3×#41C22semidirect
21. C22×C3×#41C22semidirect
22. C22×C3×#41C22semidirect
23. C23×C3×Q8C22semidirect
24. C3×#550C22semidirect
25. C3×#550C22semidirect
26. C3×D4×Q8C22semidirect
27. C3×D4×Q8C22semidirect
28. C3×#548C22semidirect
29. C3×#548C22semidirect
30. C3×#548C22semidirect
31. C3×#548C22semidirect
32. C3×#548C22semidirect
33. C3×#548C22semidirect
34. C3×#547C22semidirect
35. C3×#546C22semidirect
36. C3×#546C22semidirect
37. C3×#546C22semidirect
38. C3×D42C22semidirect
39. C3×#537C22semidirect
40. C3×#536C22semidirect
41. C3×#535C22semidirect
42. C3×#532C22semidirect
43. C2×C3×#124C22semidirect
44. C3×#525C22semidirect
45. C3×#525C22semidirect
46. C2×C3×C22⋊Q8C22semidirect
47. C2×C3×#121C22semidirect
48. C3×#520C22semidirect
49. C3×#519C22semidirect
50. C3×C4×#41C22semidirect
51. C2×C3×C4×Q8C22semidirect
52. C2×C3×C4×D4C22semidirect
53. C2×#585C2×C3semidirect
54. C22×2−1+4C2×C3semidirect
55. #3500C2×C3semidirect
56. #3497C2×C3semidirect
57. #3489C2×C3semidirect
58. #3488C2×C3semidirect
59. #3487C2×C3semidirect
60. #3486C2×C3semidirect
61. D4×#41C2×C3semidirect
62. C2×#548C2×C3semidirect
63. C4×2−1+4C2×C3semidirect
67. C3×2−1+4D4semidirect
133. C2×2−1+4C22×C3semidirect
134. C2×2−1+4C22×C3semidirect
135. C2×2−1+4C22×C3semidirect
136. C2×H(5,𝔽2)C22×C3semidirect
137. C22×#41C22×C3semidirect
138. C22×#41C22×C3semidirect
139. C22×#41C22×C3semidirect
140. C23×Q8C22×C3semidirect
141. #550C22×C3semidirect
142. #550C22×C3semidirect
143. D4×Q8C22×C3semidirect
144. D4×Q8C22×C3semidirect
145. #548C22×C3semidirect
146. #548C22×C3semidirect
147. #548C22×C3semidirect
148. #548C22×C3semidirect
149. #548C22×C3semidirect
150. #548C22×C3semidirect
151. #547C22×C3semidirect
152. #546C22×C3semidirect
153. #546C22×C3semidirect
154. #546C22×C3semidirect
155. D42C22×C3semidirect
156. #537C22×C3semidirect
157. #536C22×C3semidirect
158. #535C22×C3semidirect
159. #532C22×C3semidirect
160. C2×#124C22×C3semidirect
161. #525C22×C3semidirect
162. #525C22×C3semidirect
163. C2×C22⋊Q8C22×C3semidirect
164. C2×#121C22×C3semidirect
165. #520C22×C3semidirect
166. #519C22×C3semidirect
167. C4×#41C22×C3semidirect
168. C2×C4×Q8C22×C3semidirect
169. C2×C4×D4C22×C3semidirect
179. C3×#41C2×D4semidirect
184. C2×C3×Q8C2×D4semidirect
230. 2−1+4C3×D4semidirect
339. #41C2×C3×D4semidirect
344. C2×Q8C2×C3×D4semidirect
400. C22×C3#585semidirect
466. C22C3×#585semidirect
475. C3#63002direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C22×C3×2−1+42
C3×#35002
C3×#34972
C3×#34892
C3×#34882
C3×#34872
C3×#34862
C3×D4×#412
C2×C3×#5482
C3×C4×2−1+42
#630023