Finite Group Databasebeta

← C3×#5770 #10595867

Subgroups — C3×#5770 #10595867

Order 576.

547 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1010C211yesmaximal, characteristic, complemented
2C3×#1009C211yesmaximal, characteristic, complemented
3C32×#135C211yesmaximal, characteristic, complemented
4C3×#951C211yesmaximal, characteristic, complemented
5C3×#949C211yesmaximal, characteristic, complemented
6C3×#918C212yesmaximal, complemented
7C3×#916C212yesmaximal, complemented
8C3×D3×SD16C212yesmaximal, complemented
9C3×#911C212yesmaximal, complemented
10C2×C3×#204C211yesmaximal, characteristic, complemented
11C3×#901C211yesmaximal, characteristic, complemented
12C3×#577—31nomaximal
13#5770C311yesmaximal, characteristic, direct product factor
14C32×#41C2211yescharacteristic
15C32×#41C2211yescharacteristic, complemented
16C3×#239C2212yes—
17C3×D3×Q8C2212yescomplemented
18C3×#237—22no—
19C3×#237C2212yescomplemented
20C3×D3×D4—22no—
21C3×D3×D4C2212yes—
22C3×#235—21no—
23C3×#235—21no—
24C3×#235C2211yescharacteristic
25C2×C3×D12—21no—
26C2×C3×D12C2211yescharacteristic
27C2×C3×Dic6—21no—
28C2×C3×Dic6C2211yescharacteristic, complemented
29C32×Q16C2211yescharacteristic, complemented
30C32×SD16C2211yescharacteristic, complemented
31C32×SD16C2211yescharacteristic, complemented
32C32×D8C2211yescharacteristic, complemented
33C2×C32×C8C2211yescharacteristic, complemented
34C3×#216C2212yescomplemented
35C3×#215C2212yescomplemented
36C3×#214C2212yescomplemented
37C3×#213C2212yescomplemented
38C3×#208C2211yescharacteristic, complemented
39C3×#204C2212yescomplemented
40C3×#204C2212yescomplemented
41C3×ΓL(1,𝔽25)C2212yescomplemented
42C3×D3×C8C2212yescomplemented
43C3×2−1+4—31no—
44C3×H(5,𝔽2)—31no—
45#1010C2×C311yescharacteristic, complemented
46#1009C2×C311yescharacteristic, complemented
47C3×#137—31no—
48C3×#137—32no—
49C3×AGL(1,ℤ/8)—31no—
50C3×AGL(1,ℤ/8)—32no—
51C3×#135—21no—
52C3×#135—32no—
53C3×#135D311yescharacteristic, complemented
54C3×#135C2×C311yescharacteristic, complemented
55C2×C3×SD16—32no—
56C2×C3×SD16—31no—
57C3×#131—31no—
58#951C2×C311yescharacteristic, complemented
59#949C2×C311yescharacteristic, complemented
60#918C2×C312yescomplemented
61#916C2×C312yescomplemented
62D3×SD16C2×C312yescomplemented
63#911C2×C312yescomplemented
64C2×#204C2×C311yescharacteristic, complemented
65#901C2×C311yescharacteristic, complemented
66C22×C3×D3—21no—
67C22×C3×D3—22no—
68C32×Q8C2311yescharacteristic
69C32×Q8C2311yescharacteristic
70C32×D4—21no—
71C32×D4—21no—
72C32×D4C2311yescharacteristic
73C32×D4C2311yescharacteristic
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4C2311yescharacteristic
77C3×#67—22no—
78C3×#67—22no—
79C3×#67D412yes—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—22no—
82C3×D12—22no—
83C3×D12—21no—
84C3×D12D411yescharacteristic
85C3×D12C2312yes—
86C3×C4×D3—22no—
87C3×C4×D3—22no—
88C3×C4×D3C2312yes—
89C3×Dic6—21no—
90C3×Dic6—22no—
91C3×Dic6D411yescharacteristic
92C3×Dic6C2312yes—
93C32×C8C2311yescharacteristic
94C32×C8C2311yescharacteristic
95C3×C3⋊C8C2312yes—
96#577—31noSylow
97C3×#41—21no—
98C3×#41—21no—
99C3×#41—31no—
100C3×#41—32no—
101C3×#41—32no—
102C3×#41—61no—
103C3×#41—61no—
104C3×#41—62no—
105C3×#41C2×D311yescharacteristic
106C3×#41C2×D311yescharacteristic, complemented
107C3×#41C22×C311yescharacteristic
108C3×#41C22×C311yescharacteristic, complemented
109C2×C3×Q8—32no—
110C2×C3×Q8—31no—
111C2×C3×Q8—61no—
112C2×C3×D4—32no—
113C2×C3×D4—31no—
114C2×C3×D4—61no—
115C2×C3×D4—62no—
116#239C22×C312yes—
117D3×Q8C22×C312yescomplemented
118#237—22no—
119#237C22×C312yescomplemented
120D3×D4—22no—
121D3×D4C22×C312yes—
122#235—21no—
123#235—21no—
124#235C22×C311yescharacteristic
125C2×D12—21no—
126C2×D12C22×C311yescharacteristic
127C2×Dic6—21no—
128C2×Dic6C22×C311yescharacteristic, complemented
129C3×Q16—21no—
130C3×Q16—32no—
131C3×Q16C2×D311yescharacteristic, complemented
132C3×Q16C22×C311yescharacteristic, complemented
133C3×SD16—21no—
134C3×SD16—21no—
135C3×SD16—32no—
136C3×SD16—32no—
137C3×SD16—32no—
138C3×SD16—32no—
139C3×SD16C2×D311yescharacteristic, complemented
140C3×SD16C2×D311yescharacteristic, complemented
141C3×SD16C22×C311yescharacteristic, complemented
142C3×SD16C22×C311yescharacteristic, complemented
143C3×D8—21no—
144C3×D8—32no—
145C3×D8C2×D311yescharacteristic, complemented
146C3×D8C22×C311yescharacteristic, complemented
147C3×M16—31no—
148C3×M16—32no—
149C2×C3×C8—21no—
150C2×C3×C8—32no—
151C2×C3×C8C2×D311yescharacteristic, complemented
152C2×C3×C8C22×C311yescharacteristic, complemented
153#216C22×C312yescomplemented
154#215C22×C312yescomplemented
155#214C22×C312yescomplemented
156#213C22×C312yescomplemented
157#208C22×C311yescharacteristic, complemented
158#204C22×C312yescomplemented
159#204C22×C312yescomplemented
160ΓL(1,𝔽25)C22×C312yescomplemented
161D3×C8C22×C312yescomplemented
162C22×C32—21no—
163C22×C32—21no—
164C22×C32C2×D411yescharacteristic
165C2×C3×D3—22no—
166C2×C3×D3—82no—
167C2×C3×D3—81no—
168C2×C3×D3C2×D412yes—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4C2×D411yescharacteristic
172C32×C4C2411yescharacteristic
173C3×Dic3—22no—
174C3×Dic3C2×D412yes—
1752−1+4—31no—
176H(5,𝔽2)—31no—
177#137—31no—
178#137—32no—
179AGL(1,ℤ/8)—31no—
180AGL(1,ℤ/8)—32no—
181#135—32no—
182#135C3×D311yescharacteristic, complemented
183C2×SD16—32no—
184C2×SD16—31no—
185#131—31no—
186C23×C3—62no—
187C23×C3—61no—
188C22×D3—21no—
189C22×D3—22no—
190C3×Q8—21no—
191C3×Q8—21no—
192C3×Q8—31no—
193C3×Q8—32no—
194C3×Q8—62no—
195C3×Q8—61no—
196C3×Q8C22×D311yescharacteristic
197C3×Q8C22×D311yescharacteristic
198C3×Q8C23×C311yescharacteristic
199C3×Q8C23×C311yescharacteristic
200C3×D4—21no—
201C3×D4—21no—
202C3×D4—21no—
203C3×D4—21no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4—32no—
207C3×D4—31no—
208C3×D4—32no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—62no—
212C3×D4—62no—
213C3×D4—62no—
214C3×D4—61no—
215C3×D4C22×D311yescharacteristic
216C3×D4C22×D311yescharacteristic
217C3×D4C23×C311yescharacteristic
218C3×D4C23×C311yescharacteristic
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—32no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—61no—
231C2×C3×C4C22×D311yescharacteristic
232C2×C3×C4C23×C311yescharacteristic
233#67—22no—
234#67—22no—
235#67C3×D412yes—
236C2×Dic3—21no—
237C2×Dic3—22no—
238D12—22no—
239D12—21no—
240D12C3×D411yescharacteristic
241D12C23×C312yes—
242C4×D3—22no—
243C4×D3—22no—
244C4×D3C23×C312yes—
245Dic6—21no—
246Dic6—22no—
247Dic6C3×D411yescharacteristic
248Dic6C23×C312yes—
249C3×C8—21no—
250C3×C8—21no—
251C3×C8—32no—
252C3×C8C22×D311yescharacteristic
253C3×C8C22×D311yescharacteristic
254C3×C8C23×C311yescharacteristic
255C3×C8C23×C311yescharacteristic
256C3⋊C8C23×C312yes—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32C22×D411yescharacteristic
261C3×D3—22no—
262C3×D3—42no—
263#41—31no—
264#41—32no—
265#41—32no—
266#41—61no—
267#41—61no—
268#41—62no—
269#41C2×C3×D311yescharacteristic
270#41C2×C3×D311yescharacteristic, complemented
271C2×Q8—32no—
272C2×Q8—31no—
273C2×Q8—61no—
274C2×D4—32no—
275C2×D4—31no—
276C2×D4—61no—
277C2×D4—62no—
278Q16—32no—
279Q16C2×C3×D311yescharacteristic, complemented
280SD16—32no—
281SD16—32no—
282SD16—32no—
283SD16—32no—
284SD16C2×C3×D311yescharacteristic, complemented
285SD16C2×C3×D311yescharacteristic, complemented
286D8—32no—
287D8C2×C3×D311yescharacteristic, complemented
288M16—31no—
289M16—32no—
290C2×C8—32no—
291C2×C8C2×C3×D311yescharacteristic, complemented
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—32no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—62no—
301C22×C3—241no—
302C22×C3—242no—
303C22×C3D3×D411yescharacteristic
304C22×C3C2×C3×D411yescharacteristic
305C2×D3—22no—
306C2×D3—82no—
307C2×D3—81no—
308C2×D3C2×C3×D412yes—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—32no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—62no—
319C3×C4D3×D411yescharacteristic
320C3×C4C2×C3×D411yescharacteristic
321C3×C4C23×D311yescharacteristic
322C3×C4C24×C311yescharacteristic
323Dic3—22no—
324Dic3C2×C3×D412yes—
325C32#57711yescharacteristic, complemented, Sylow
326C23—62no—
327C23—61no—
328Q8—31no—
329Q8—32no—
330Q8—62no—
331Q8—61no—
332Q8C22×C3×D311yescharacteristic
333Q8C22×C3×D311yescharacteristic
334D4—21no—
335D4—21no—
336D4—32no—
337D4—32no—
338D4—31no—
339D4—62no—
340D4—62no—
341D4—62no—
342D4—61no—
343D4C22×C3×D311yescharacteristic
344D4C22×C3×D311yescharacteristic
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—32no—
348C2×C4—62no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—61no—
352C2×C4C22×C3×D311yescharacteristic
353C8—32no—
354C8C22×C3×D311yescharacteristic
355C8C22×C3×D311yescharacteristic
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—62no—
365C2×C3—81no—
366C2×C3—81no—
367C2×C3—122no—
368C2×C3C2×D3×D411yescharacteristic
369C2×C3C22×C3×D411yescharacteristic
370D3—22no—
371D3—42no—
372C22—21no—
373C22—21no—
374C22—32no—
375C22—62no—
376C22—241no—
377C22—242no—
378C22C3×D3×D411yescharacteristic
379C4—21no—
380C4—21no—
381C4—32no—
382C4—62no—
383C4C3×D3×D411yescharacteristic
384C4C23×C3×D311yescharacteristic
385C3—21no—
386C3#577011yescharacteristic, direct product factor
387C3C3×#57711yescharacteristic, complemented
388C2—21no—
389C2—41no—
390C2—41no—
391C2—62no—
392C2—122no—
393C2C2×C3×D3×D411yescharacteristic
subgroupG/H
centerC2×C3C2×D3×D4
commutator, cosocleC3×C4C24×C3
FrattiniC4C23×C3×D3
FittingC32×#135C2
socleC2×C32C22×D4
hypercenterC3×#135D3
hypocenterC3C3×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#135

Complements

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

NH ≅ G/Ntype
1. C3×#1010C2semidirect
2. C3×#1009C2semidirect
3. C32×#135C2semidirect
4. C3×#951C2semidirect
5. C3×#949C2semidirect
6. C3×#918C2semidirect
7. C3×#916C2semidirect
8. C3×D3×SD16C2semidirect
9. C3×#911C2semidirect
10. C2×C3×#204C2semidirect
11. C3×#901C2semidirect
13. #5770C3direct (N × H)
15. C32×#41C22semidirect
17. C3×D3×Q8C22semidirect
19. C3×#237C22semidirect
28. C2×C3×Dic6C22semidirect
29. C32×Q16C22semidirect
30. C32×SD16C22semidirect
31. C32×SD16C22semidirect
32. C32×D8C22semidirect
33. C2×C32×C8C22semidirect
34. C3×#216C22semidirect
35. C3×#215C22semidirect
36. C3×#214C22semidirect
37. C3×#213C22semidirect
38. C3×#208C22semidirect
39. C3×#204C22semidirect
40. C3×#204C22semidirect
41. C3×ΓL(1,𝔽25)C22semidirect
42. C3×D3×C8C22semidirect
45. #1010C2×C3semidirect
46. #1009C2×C3semidirect
53. C3×#135D3semidirect
54. C3×#135C2×C3semidirect
58. #951C2×C3semidirect
59. #949C2×C3semidirect
60. #918C2×C3semidirect
61. #916C2×C3semidirect
62. D3×SD16C2×C3semidirect
63. #911C2×C3semidirect
64. C2×#204C2×C3semidirect
65. #901C2×C3semidirect
106. C3×#41C2×D3semidirect
108. C3×#41C22×C3semidirect
117. D3×Q8C22×C3semidirect
119. #237C22×C3semidirect
128. C2×Dic6C22×C3semidirect
131. C3×Q16C2×D3semidirect
132. C3×Q16C22×C3semidirect
139. C3×SD16C2×D3semidirect
140. C3×SD16C2×D3semidirect
141. C3×SD16C22×C3semidirect
142. C3×SD16C22×C3semidirect
145. C3×D8C2×D3semidirect
146. C3×D8C22×C3semidirect
151. C2×C3×C8C2×D3semidirect
152. C2×C3×C8C22×C3semidirect
153. #216C22×C3semidirect
154. #215C22×C3semidirect
155. #214C22×C3semidirect
156. #213C22×C3semidirect
157. #208C22×C3semidirect
158. #204C22×C3semidirect
159. #204C22×C3semidirect
160. ΓL(1,𝔽25)C22×C3semidirect
161. D3×C8C22×C3semidirect
182. #135C3×D3semidirect
270. #41C2×C3×D3semidirect
279. Q16C2×C3×D3semidirect
284. SD16C2×C3×D3semidirect
285. SD16C2×C3×D3semidirect
287. D8C2×C3×D3semidirect
291. C2×C8C2×C3×D3semidirect
325. C32#577semidirect
386. C3#5770direct (N × H)
387. C3C3×#577semidirect

Maximal subgroups

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

subgroupindex
C3×#10102
C3×#10092
C32×#1352
C3×#9512
C3×#9492
C3×#9182
C3×#9162
C3×D3×SD162
C3×#9112
C2×C3×#2042
C3×#9012
C3×#5773
#57703