Subgroups — C₃×#2897 #84761

Order 384.

778 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#418	C₂	1	3	yes	maximal, complemented
5	C₃×#417	C₂	1	3	yes	maximal, complemented
6	C₃×#416	C₂	1	3	yes	maximal, complemented
7	C₂×C₃×#102	C₂	1	3	yes	maximal, complemented
8	#2897	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×H(5,𝔽₂)	—	2	4	no	—
10	C₃×H(5,𝔽₂)	C₄	1	8	yes	complemented
11	C₂×C₃×#41	—	2	1	no	—
12	C₂×C₃×#41	C₂²	1	3	yes	—
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	—	2	3	no	—
15	C₂²×C₃×D₄	C₂²	1	3	yes	—
16	C₃×#131	—	2	4	no	—
17	C₂×C₃×M₁₆	C₂²	1	3	yes	complemented
18	C₂²×C₃×C₈	C₂²	1	3	yes	complemented
19	C₃×C₄×Q₈	—	2	1	no	—
20	C₃×C₄×D₄	—	2	3	no	—
21	C₃×#117	C₂²	1	3	yes	complemented
22	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	complemented
23	C₃×#103	C₂²	1	1	yes	characteristic, complemented
24	C₃×#103	C₂²	1	3	yes	complemented
25	C₃×#102	C₂²	1	3	yes	complemented
26	C₃×#102	C₂²	1	6	yes	complemented
27	C₃×#102	C₂²	1	3	yes	complemented
28	C₂×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic
29	C₂×#131	C₂×C₃	1	1	yes	characteristic, complemented
30	#520	C₂×C₃	1	1	yes	characteristic, complemented
31	#418	C₂×C₃	1	3	yes	complemented
32	#417	C₂×C₃	1	3	yes	complemented
33	#416	C₂×C₃	1	3	yes	complemented
34	C₂×#102	C₂×C₃	1	3	yes	complemented
35	C₂⁴×C₃	—	2	3	no	—
36	C₃×#41	—	2	8	no	—
37	C₃×#41	—	2	6	no	—
38	C₃×#41	C₂×C₄	1	4	yes	—
39	C₃×#41	C₂×C₄	1	12	yes	complemented
40	C₃×#41	D₄	1	4	yes	—
41	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
42	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
43	C₂×C₃×D₄	—	2	3	no	—
44	C₂×C₃×D₄	—	2	24	no	—
45	C₂×C₃×D₄	—	2	3	no	—
46	C₂×C₃×D₄	—	2	6	no	—
47	C₂×C₃×D₄	C₂×C₄	1	12	yes	complemented
48	C₂×C₃×D₄	D₄	1	3	yes	—
49	C₂×C₃×D₄	C₂³	1	3	yes	—
50	C₂²×C₃×C₄	—	2	3	no	—
51	C₂²×C₃×C₄	—	2	3	no	—
52	C₂²×C₃×C₄	C₂³	1	3	yes	—
53	C₃×M₁₆	—	2	6	no	—
54	C₂×C₃×C₈	—	2	6	no	—
55	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂³	1	3	yes	—
57	C₃×C₄⋊C₄	—	2	3	no	—
58	C₃×C₄⋊C₄	C₂³	1	3	yes	—
59	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₃×C₂²⋊C₄	—	2	3	no	—
61	C₃×C₄²	—	2	3	no	—
62	H(5,𝔽₂)	—	2	4	no	—
63	H(5,𝔽₂)	C₃×C₄	1	8	yes	complemented
64	C₂×#41	—	2	1	no	—
65	C₂×#41	C₂²×C₃	1	3	yes	—
66	C₂×#41	C₂²×C₃	1	1	yes	characteristic
67	C₂²×D₄	—	2	3	no	—
68	C₂²×D₄	C₂²×C₃	1	3	yes	—
69	#131	—	2	4	no	—
70	C₂×M₁₆	C₂²×C₃	1	3	yes	complemented
71	C₂²×C₈	C₂²×C₃	1	3	yes	complemented
72	C₄×Q₈	—	2	1	no	—
73	C₄×D₄	—	2	3	no	—
74	#117	C₂²×C₃	1	3	yes	complemented
75	C₂×C₄⋊C₄	C₂²×C₃	1	3	yes	complemented
76	#103	C₂²×C₃	1	1	yes	characteristic, complemented
77	#103	C₂²×C₃	1	3	yes	complemented
78	#102	C₂²×C₃	1	3	yes	complemented
79	#102	C₂²×C₃	1	6	yes	complemented
80	#102	C₂²×C₃	1	3	yes	complemented
81	C₂³×C₃	—	2	3	no	—
82	C₂³×C₃	—	2	12	no	—
83	C₂³×C₃	—	8	3	no	—
84	C₂³×C₃	—	8	3	no	—
85	C₂³×C₃	C₂×D₄	1	3	yes	—
86	C₃×Q₈	—	2	1	no	—
87	C₃×Q₈	C₂²⋊C₄	1	4	yes	—
88	C₃×Q₈	C₂²×C₄	1	2	yes	—
89	C₃×D₄	—	2	12	no	—
90	C₃×D₄	—	2	12	no	—
91	C₃×D₄	—	2	3	no	—
92	C₃×D₄	C₂²⋊C₄	1	12	yes	—
93	C₃×D₄	C₂²×C₄	1	6	yes	—
94	C₂×C₃×C₄	—	2	1	no	—
95	C₂×C₃×C₄	—	2	12	no	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	2	3	no	—
98	C₂×C₃×C₄	—	8	3	no	—
99	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
100	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
101	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
102	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
103	C₃×C₈	—	2	1	no	—
104	C₃×C₈	—	2	3	no	—
105	C₂⁴	—	2	3	no	—
106	#41	—	2	8	no	—
107	#41	—	2	6	no	—
108	#41	C₂×C₃×C₄	1	4	yes	—
109	#41	C₂×C₃×C₄	1	12	yes	complemented
110	#41	C₃×D₄	1	4	yes	—
111	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
112	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
113	C₂×D₄	—	2	3	no	—
114	C₂×D₄	—	2	24	no	—
115	C₂×D₄	—	2	3	no	—
116	C₂×D₄	—	2	6	no	—
117	C₂×D₄	C₂×C₃×C₄	1	12	yes	complemented
118	C₂×D₄	C₃×D₄	1	3	yes	—
119	C₂×D₄	C₂³×C₃	1	3	yes	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	—	2	3	no	—
122	C₂²×C₄	C₂³×C₃	1	3	yes	—
123	M₁₆	—	2	6	no	—
124	C₂×C₈	—	2	6	no	—
125	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₃	1	3	yes	—
127	C₄⋊C₄	—	2	3	no	—
128	C₄⋊C₄	C₂³×C₃	1	3	yes	—
129	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂²⋊C₄	—	2	3	no	—
131	C₄²	—	2	3	no	—
132	C₂²×C₃	—	2	3	no	—
133	C₂²×C₃	—	2	3	no	—
134	C₂²×C₃	—	2	6	no	—
135	C₂²×C₃	—	4	3	no	—
136	C₂²×C₃	—	4	3	no	—
137	C₂²×C₃	—	8	12	no	—
138	C₂²×C₃	C₂×C₂²⋊C₄	1	6	yes	—
139	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
140	C₃×C₄	—	2	2	no	—
141	C₃×C₄	—	4	3	no	—
142	C₃×C₄	—	4	1	no	—
143	C₃×C₄	C₂×C₂²⋊C₄	1	6	yes	—
144	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
145	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₂³	—	2	3	no	—
147	C₂³	—	2	12	no	—
148	C₂³	—	8	3	no	—
149	C₂³	—	8	3	no	—
150	C₂³	C₂×C₃×D₄	1	3	yes	—
151	Q₈	—	2	1	no	—
152	Q₈	C₃×C₂²⋊C₄	1	4	yes	—
153	Q₈	C₂²×C₃×C₄	1	2	yes	—
154	D₄	—	2	12	no	—
155	D₄	—	2	12	no	—
156	D₄	—	2	3	no	—
157	D₄	C₃×C₂²⋊C₄	1	12	yes	—
158	D₄	C₂²×C₃×C₄	1	6	yes	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	12	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	3	no	—
163	C₂×C₄	—	8	3	no	—
164	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
165	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
166	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
167	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	3	no	—
170	C₂×C₃	—	2	6	no	—
171	C₂×C₃	—	4	6	no	—
172	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₃	#576	1	1	yes	characteristic
174	C₂×C₃	#577	1	1	yes	characteristic
175	C₂²	—	2	3	no	—
176	C₂²	—	2	3	no	—
177	C₂²	—	2	6	no	—
178	C₂²	—	4	3	no	—
179	C₂²	—	4	3	no	—
180	C₂²	—	8	12	no	—
181	C₂²	C₂×C₃×C₂²⋊C₄	1	6	yes	—
182	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	3	no	—
185	C₄	—	4	1	no	—
186	C₄	C₂×C₃×C₂²⋊C₄	1	6	yes	—
187	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
188	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₃	#2897	1	1	yes	characteristic, direct product factor, Sylow
190	C₂	—	2	6	no	—
191	C₂	—	4	6	no	—
192	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₂	C₃×#576	1	1	yes	characteristic
194	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#131	C₂	semidirect
3. C₃×#520	C₂	semidirect
4. C₃×#418	C₂	semidirect
5. C₃×#417	C₂	semidirect
6. C₃×#416	C₂	semidirect
7. C₂×C₃×#102	C₂	semidirect
8. #2897	C₃	direct (N × H)
10. C₃×H(5,𝔽₂)	C₄	semidirect
17. C₂×C₃×M₁₆	C₂²	semidirect
18. C₂²×C₃×C₈	C₂²	semidirect
21. C₃×#117	C₂²	semidirect
22. C₂×C₃×C₄⋊C₄	C₂²	semidirect
23. C₃×#103	C₂²	semidirect
24. C₃×#103	C₂²	semidirect
25. C₃×#102	C₂²	semidirect
26. C₃×#102	C₂²	semidirect
27. C₃×#102	C₂²	semidirect
29. C₂×#131	C₂×C₃	semidirect
30. #520	C₂×C₃	semidirect
31. #418	C₂×C₃	semidirect
32. #417	C₂×C₃	semidirect
33. #416	C₂×C₃	semidirect
34. C₂×#102	C₂×C₃	semidirect
39. C₃×#41	C₂×C₄	semidirect
47. C₂×C₃×D₄	C₂×C₄	semidirect
63. H(5,𝔽₂)	C₃×C₄	semidirect
70. C₂×M₁₆	C₂²×C₃	semidirect
71. C₂²×C₈	C₂²×C₃	semidirect
74. #117	C₂²×C₃	semidirect
75. C₂×C₄⋊C₄	C₂²×C₃	semidirect
76. #103	C₂²×C₃	semidirect
77. #103	C₂²×C₃	semidirect
78. #102	C₂²×C₃	semidirect
79. #102	C₂²×C₃	semidirect
80. #102	C₂²×C₃	semidirect
109. #41	C₂×C₃×C₄	semidirect
117. C₂×D₄	C₂×C₃×C₄	semidirect
189. C₃	#2897	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×H(5,𝔽₂)	2
C₂×C₃×#131	2
C₃×#520	2
C₃×#418	2
C₃×#417	2
C₃×#416	2
C₂×C₃×#102	2
#2897	3

Subgroups — C3×#2897 #84761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2897 #84761