Subgroups — C₃×#3455 #87339

Order 384.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#542	C₂	1	4	yes	maximal, complemented
2	C₃×#541	C₂	1	2	yes	maximal, complemented
3	C₃×#540	C₂	1	2	yes	maximal, complemented
4	C₃×#539	C₂	1	4	yes	maximal, complemented
5	C₃×#538	C₂	1	4	yes	maximal, complemented
6	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#529	C₂	1	8	yes	maximal, complemented
9	C₃×#525	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×#41	C₂	1	2	yes	maximal, complemented
11	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	#3455	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₃×#127	C₂²	1	1	yes	characteristic, complemented
18	C₃×#126	C₂²	1	8	yes	—
19	C₃×#126	C₂²	1	8	yes	complemented
20	C₃×#125	C₂²	1	8	yes	complemented
21	C₃×#125	C₂²	1	2	yes	complemented
22	C₃×#124	C₂²	1	8	yes	—
23	C₃×#124	C₂²	1	2	yes	complemented
24	C₃×#123	C₂²	1	4	yes	—
25	C₃×#123	C₂²	1	8	yes	—
26	C₃×#123	C₂²	1	8	yes	complemented
27	C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
28	C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
29	C₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
30	C₃×#121	C₂²	1	4	yes	—
31	C₃×#121	C₂²	1	8	yes	—
32	C₃×#121	C₂²	1	4	yes	—
33	C₃×#120	C₂²	1	4	yes	—
34	C₃×C₄×Q₈	C₂²	1	4	yes	complemented
35	C₃×C₄×Q₈	C₂²	1	4	yes	complemented
36	C₃×C₄×D₄	C₂²	1	8	yes	—
37	C₃×C₄×D₄	C₂²	1	8	yes	—
38	C₃×C₄×D₄	C₂²	1	4	yes	complemented
39	C₃×C₄×D₄	C₂²	1	4	yes	complemented
40	C₃×#117	C₂²	1	4	yes	—
41	C₃×#117	C₂²	1	2	yes	—
42	C₃×#117	C₂²	1	2	yes	—
43	C₃×#117	C₂²	1	2	yes	—
44	C₃×#117	C₂²	1	4	yes	complemented
45	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
46	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
47	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂²	1	4	yes	complemented
50	#542	C₂×C₃	1	4	yes	complemented
51	#541	C₂×C₃	1	2	yes	complemented
52	#540	C₂×C₃	1	2	yes	complemented
53	#539	C₂×C₃	1	4	yes	complemented
54	#538	C₂×C₃	1	4	yes	complemented
55	#533	C₂×C₃	1	1	yes	characteristic, complemented
56	#532	C₂×C₃	1	1	yes	characteristic, complemented
57	#529	C₂×C₃	1	8	yes	complemented
58	#525	C₂×C₃	1	2	yes	complemented
59	C₄×#41	C₂×C₃	1	2	yes	complemented
60	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂⁴×C₃	C₂³	1	1	yes	characteristic
62	C₃×#41	—	2	8	no	—
63	C₂×C₃×Q₈	C₂³	1	2	yes	—
64	C₂×C₃×Q₈	C₂³	1	4	yes	—
65	C₂×C₃×D₄	C₂³	1	4	yes	—
66	C₂×C₃×D₄	C₂³	1	2	yes	—
67	C₂×C₃×D₄	C₂³	1	8	yes	—
68	C₂×C₃×D₄	C₂³	1	4	yes	—
69	C₂²×C₃×C₄	—	2	4	no	—
70	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³	1	2	yes	—
72	C₂²×C₃×C₄	C₂³	1	4	yes	—
73	C₂²×C₃×C₄	C₂³	1	2	yes	—
74	C₂²×C₃×C₄	C₂³	1	2	yes	—
75	C₂²×C₃×C₄	C₂³	1	2	yes	—
76	C₂²×C₃×C₄	C₂³	1	8	yes	—
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₃×C₄⋊C₄	C₂³	1	4	yes	—
79	C₃×C₄⋊C₄	C₂³	1	4	yes	—
80	C₃×C₄⋊C₄	C₂³	1	4	yes	—
81	C₃×C₄⋊C₄	C₂³	1	8	yes	—
82	C₃×C₄⋊C₄	C₂³	1	8	yes	—
83	C₃×C₄⋊C₄	C₂³	1	4	yes	—
84	C₃×C₄⋊C₄	C₂³	1	8	yes	—
85	C₃×C₄⋊C₄	C₂³	1	4	yes	—
86	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
87	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
88	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
89	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
90	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
91	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
92	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
93	C₃×C₄²	C₂³	1	2	yes	—
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	C₃×C₄²	C₂³	1	1	yes	characteristic
96	C₃×C₄²	C₂³	1	8	yes	—
97	C₃×C₄²	C₂³	1	2	yes	—
98	C₃×C₄²	C₂³	1	2	yes	—
99	C₃×C₄²	C₂³	1	4	yes	—
100	C₂×#41	C₂²×C₃	1	2	yes	—
101	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
102	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
103	C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
104	#127	C₂²×C₃	1	1	yes	characteristic, complemented
105	#126	C₂²×C₃	1	8	yes	—
106	#126	C₂²×C₃	1	8	yes	complemented
107	#125	C₂²×C₃	1	8	yes	complemented
108	#125	C₂²×C₃	1	2	yes	complemented
109	#124	C₂²×C₃	1	8	yes	—
110	#124	C₂²×C₃	1	2	yes	complemented
111	#123	C₂²×C₃	1	4	yes	—
112	#123	C₂²×C₃	1	8	yes	—
113	#123	C₂²×C₃	1	8	yes	complemented
114	C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
115	C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
116	C₂²⋊Q₈	C₂²×C₃	1	8	yes	complemented
117	#121	C₂²×C₃	1	4	yes	—
118	#121	C₂²×C₃	1	8	yes	—
119	#121	C₂²×C₃	1	4	yes	—
120	#120	C₂²×C₃	1	4	yes	—
121	C₄×Q₈	C₂²×C₃	1	4	yes	complemented
122	C₄×Q₈	C₂²×C₃	1	4	yes	complemented
123	C₄×D₄	C₂²×C₃	1	8	yes	—
124	C₄×D₄	C₂²×C₃	1	8	yes	—
125	C₄×D₄	C₂²×C₃	1	4	yes	complemented
126	C₄×D₄	C₂²×C₃	1	4	yes	complemented
127	#117	C₂²×C₃	1	4	yes	—
128	#117	C₂²×C₃	1	2	yes	—
129	#117	C₂²×C₃	1	2	yes	—
130	#117	C₂²×C₃	1	2	yes	—
131	#117	C₂²×C₃	1	4	yes	complemented
132	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
134	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
136	C₂×C₄²	C₂²×C₃	1	4	yes	complemented
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	—	4	2	no	—
139	C₂³×C₃	C₂⁴	1	2	yes	—
140	C₂³×C₃	C₂⁴	1	4	yes	—
141	C₂³×C₃	C₂⁴	1	1	yes	characteristic
142	C₃×Q₈	—	2	4	no	—
143	C₃×Q₈	—	4	4	no	—
144	C₃×D₄	—	2	8	no	—
145	C₃×D₄	—	2	4	no	—
146	C₃×D₄	—	4	4	no	—
147	C₃×D₄	—	4	8	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	8	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	—	4	8	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	#41	1	4	yes	—
156	C₂×C₃×C₄	C₂⁴	1	4	yes	—
157	C₂×C₃×C₄	C₂⁴	1	2	yes	—
158	C₂×C₃×C₄	C₂⁴	1	4	yes	—
159	C₂×C₃×C₄	C₂⁴	1	8	yes	—
160	C₂×C₃×C₄	C₂⁴	1	2	yes	—
161	C₂×C₃×C₄	C₂⁴	1	2	yes	—
162	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
164	C₂⁴	C₂³×C₃	1	1	yes	characteristic
165	#41	—	2	8	no	—
166	C₂×Q₈	C₂³×C₃	1	2	yes	—
167	C₂×Q₈	C₂³×C₃	1	4	yes	—
168	C₂×D₄	C₂³×C₃	1	4	yes	—
169	C₂×D₄	C₂³×C₃	1	2	yes	—
170	C₂×D₄	C₂³×C₃	1	8	yes	—
171	C₂×D₄	C₂³×C₃	1	4	yes	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	C₂³×C₃	1	4	yes	—
174	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₃	1	8	yes	—
180	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₃	1	4	yes	—
182	C₄⋊C₄	C₂³×C₃	1	4	yes	—
183	C₄⋊C₄	C₂³×C₃	1	4	yes	—
184	C₄⋊C₄	C₂³×C₃	1	8	yes	—
185	C₄⋊C₄	C₂³×C₃	1	8	yes	—
186	C₄⋊C₄	C₂³×C₃	1	4	yes	—
187	C₄⋊C₄	C₂³×C₃	1	8	yes	—
188	C₄⋊C₄	C₂³×C₃	1	4	yes	—
189	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
190	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
191	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
192	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
193	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
194	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
195	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
196	C₄²	C₂³×C₃	1	2	yes	—
197	C₄²	C₂³×C₃	1	2	yes	—
198	C₄²	C₂³×C₃	1	2	yes	—
199	C₄²	C₂³×C₃	1	1	yes	characteristic
200	C₄²	C₂³×C₃	1	1	yes	characteristic
201	C₄²	C₂³×C₃	1	4	yes	—
202	C₄²	C₂³×C₃	1	8	yes	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	8	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	4	4	no	—
209	C₂²×C₃	C₂×#41	1	2	yes	—
210	C₂²×C₃	C₂⁵	1	1	yes	characteristic
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	2	8	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	C₂×#41	1	2	yes	—
218	C₃×C₄	C₂×#41	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂⁴×C₃	1	2	yes	—
222	C₂³	C₂⁴×C₃	1	4	yes	—
223	C₂³	C₂⁴×C₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	4	no	—
226	D₄	—	2	8	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₃×#41	1	4	yes	—
238	C₂×C₄	C₂⁴×C₃	1	4	yes	—
239	C₂×C₄	C₂⁴×C₃	1	2	yes	—
240	C₂×C₄	C₂⁴×C₃	1	4	yes	—
241	C₂×C₄	C₂⁴×C₃	1	8	yes	—
242	C₂×C₄	C₂⁴×C₃	1	2	yes	—
243	C₂×C₄	C₂⁴×C₃	1	2	yes	—
244	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
245	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	4	4	no	—
248	C₂×C₃	—	4	2	no	—
249	C₂×C₃	C₂²×#41	1	1	yes	characteristic
250	C₂×C₃	#585	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₂×C₃×#41	1	2	yes	—
258	C₂²	C₂⁵×C₃	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	8	no	—
264	C₄	—	2	2	no	—
265	C₄	C₂×C₃×#41	1	2	yes	—
266	C₄	C₂×C₃×#41	1	2	yes	—
267	C₃	#3455	1	1	yes	characteristic, direct product factor, Sylow
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	4	no	—
271	C₂	C₂²×C₃×#41	1	1	yes	characteristic
272	C₂	C₃×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂²×C₃	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#542	C₂	semidirect
2. C₃×#541	C₂	semidirect
3. C₃×#540	C₂	semidirect
4. C₃×#539	C₂	semidirect
5. C₃×#538	C₂	semidirect
6. C₃×#533	C₂	semidirect
7. C₃×#532	C₂	semidirect
8. C₃×#529	C₂	semidirect
9. C₃×#525	C₂	semidirect
10. C₃×C₄×#41	C₂	semidirect
11. C₂×C₃×#117	C₂	semidirect
12. #3455	C₃	direct (N × H)
15. C₃×C₄⋊Q₈	C₂²	semidirect
16. C₃×C₄⋊Q₈	C₂²	semidirect
17. C₃×#127	C₂²	semidirect
19. C₃×#126	C₂²	semidirect
20. C₃×#125	C₂²	semidirect
21. C₃×#125	C₂²	semidirect
23. C₃×#124	C₂²	semidirect
26. C₃×#123	C₂²	semidirect
29. C₃×C₂²⋊Q₈	C₂²	semidirect
34. C₃×C₄×Q₈	C₂²	semidirect
35. C₃×C₄×Q₈	C₂²	semidirect
38. C₃×C₄×D₄	C₂²	semidirect
39. C₃×C₄×D₄	C₂²	semidirect
44. C₃×#117	C₂²	semidirect
49. C₂×C₃×C₄²	C₂²	semidirect
50. #542	C₂×C₃	semidirect
51. #541	C₂×C₃	semidirect
52. #540	C₂×C₃	semidirect
53. #539	C₂×C₃	semidirect
54. #538	C₂×C₃	semidirect
55. #533	C₂×C₃	semidirect
56. #532	C₂×C₃	semidirect
57. #529	C₂×C₃	semidirect
58. #525	C₂×C₃	semidirect
59. C₄×#41	C₂×C₃	semidirect
60. C₂×#117	C₂×C₃	semidirect
102. C₄⋊Q₈	C₂²×C₃	semidirect
103. C₄⋊Q₈	C₂²×C₃	semidirect
104. #127	C₂²×C₃	semidirect
106. #126	C₂²×C₃	semidirect
107. #125	C₂²×C₃	semidirect
108. #125	C₂²×C₃	semidirect
110. #124	C₂²×C₃	semidirect
113. #123	C₂²×C₃	semidirect
116. C₂²⋊Q₈	C₂²×C₃	semidirect
121. C₄×Q₈	C₂²×C₃	semidirect
122. C₄×Q₈	C₂²×C₃	semidirect
125. C₄×D₄	C₂²×C₃	semidirect
126. C₄×D₄	C₂²×C₃	semidirect
131. #117	C₂²×C₃	semidirect
136. C₂×C₄²	C₂²×C₃	semidirect
267. C₃	#3455	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#542	2
C₃×#541	2
C₃×#540	2
C₃×#539	2
C₃×#538	2
C₃×#533	2
C₃×#532	2
C₃×#529	2
C₃×#525	2
C₃×C₄×#41	2
C₂×C₃×#117	2
#3455	3

Subgroups — C3×#3455 #87339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3455 #87339