Subgroups — C₃×#2977 #84841

Order 384.

422 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₃×#435	C₂	1	4	yes	maximal, complemented
3	C₃×C₈×D₄	C₂	1	4	yes	maximal, complemented
4	C₃×#433	C₂	1	2	yes	maximal, complemented
5	C₃×#408	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
7	#2977	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#41	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
11	C₂²×C₃×C₈	C₂²	1	4	yes	—
12	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
13	C₃×#123	C₄	1	2	yes	—
14	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
15	C₃×#121	C₄	1	2	yes	—
16	C₃×#120	C₄	1	2	yes	—
17	C₃×C₄×D₄	C₂²	1	4	yes	—
18	C₃×#117	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊C₈	C₂²	1	4	yes	complemented
20	C₃×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
24	C₃×#97	C₂²	1	2	yes	complemented
25	C₃×C₄×C₈	C₂²	1	2	yes	complemented
26	#525	C₂×C₃	1	1	yes	characteristic
27	#435	C₂×C₃	1	4	yes	complemented
28	C₈×D₄	C₂×C₃	1	4	yes	complemented
29	#433	C₂×C₃	1	2	yes	complemented
30	#408	C₂×C₃	1	2	yes	complemented
31	C₂×C₂²⋊C₈	C₂×C₃	1	2	yes	complemented
32	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
33	C₃×#41	—	4	2	no	—
34	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
36	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	—	4	2	no	—
39	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	C₂³	1	2	yes	—
44	C₃×M₁₆	—	4	2	no	—
45	C₂×C₃×C₈	—	4	4	no	—
46	C₂×C₃×C₈	—	4	2	no	—
47	C₂×C₃×C₈	C₂³	1	4	yes	—
48	C₂×C₃×C₈	C₂³	1	4	yes	—
49	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
51	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
53	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
54	C₃×C₄²	C₂³	1	2	yes	—
55	C₂×#41	C₂²×C₃	1	1	yes	characteristic
56	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×M₁₆	C₂²×C₃	1	2	yes	complemented
58	C₂²×C₈	C₂²×C₃	1	4	yes	—
59	C₂²×C₈	C₂²×C₃	1	2	yes	complemented
60	#123	C₃×C₄	1	2	yes	—
61	C₂²⋊Q₈	C₃×C₄	1	2	yes	—
62	#121	C₃×C₄	1	2	yes	—
63	#120	C₃×C₄	1	2	yes	—
64	C₄×D₄	C₂²×C₃	1	4	yes	—
65	#117	C₂²×C₃	1	1	yes	characteristic
66	C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
67	C₂²⋊C₈	C₂²×C₃	1	4	yes	—
68	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
69	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
70	C₂²⋊C₈	C₂²×C₃	1	4	yes	complemented
71	#97	C₂²×C₃	1	2	yes	complemented
72	C₄×C₈	C₂²×C₃	1	2	yes	complemented
73	C₂³×C₃	—	2	4	no	—
74	C₂³×C₃	—	4	1	no	—
75	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₃	C₂²×C₄	1	2	yes	—
77	C₂³×C₃	C₂²×C₄	1	2	yes	—
78	C₃×Q₈	—	4	1	no	—
79	C₃×D₄	—	2	8	no	—
80	C₃×D₄	—	4	1	no	—
81	C₃×D₄	—	4	2	no	—
82	C₂×C₃×C₄	—	2	4	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	4	1	no	—
87	C₂×C₃×C₄	—	4	2	no	—
88	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
89	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
90	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
91	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
93	C₃×C₈	—	2	4	no	—
94	C₃×C₈	—	2	4	no	—
95	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
96	#41	—	4	2	no	—
97	C₂×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
99	C₂×D₄	C₂×C₃×C₄	1	2	yes	—
100	C₂×D₄	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
103	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₃	1	2	yes	—
106	C₂²×C₄	C₂³×C₃	1	2	yes	—
107	M₁₆	—	4	2	no	—
108	C₂×C₈	—	4	4	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	C₂³×C₃	1	4	yes	—
111	C₂×C₈	C₂³×C₃	1	4	yes	—
112	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
113	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
114	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
115	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
116	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
117	C₄²	C₂³×C₃	1	2	yes	—
118	C₂²×C₃	—	2	2	no	—
119	C₂²×C₃	—	2	2	no	—
120	C₂²×C₃	—	2	1	no	—
121	C₂²×C₃	—	2	2	no	—
122	C₂²×C₃	—	2	4	no	—
123	C₂²×C₃	—	2	2	no	—
124	C₂²×C₃	—	4	4	no	—
125	C₂²×C₃	#131	1	4	yes	—
126	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
127	C₃×C₄	—	2	2	no	—
128	C₃×C₄	—	2	1	no	—
129	C₃×C₄	—	2	2	no	—
130	C₃×C₄	—	2	4	no	—
131	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
132	C₂³	—	2	4	no	—
133	C₂³	—	4	1	no	—
134	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂³	C₂²×C₃×C₄	1	2	yes	—
136	C₂³	C₂²×C₃×C₄	1	2	yes	—
137	Q₈	—	4	1	no	—
138	D₄	—	2	8	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
148	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
149	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
150	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
151	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	2	4	no	—
154	C₂×C₃	—	2	4	no	—
155	C₂×C₃	—	4	2	no	—
156	C₂×C₃	—	4	1	no	—
157	C₂×C₃	#518	1	1	yes	characteristic
158	C₂×C₃	C₂×#131	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	4	no	—
166	C₂²	C₃×#131	1	4	yes	—
167	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	2	no	—
172	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
173	C₃	#2977	1	1	yes	characteristic, direct product factor, Sylow
174	C₂	—	2	4	no	—
175	C₂	—	4	1	no	—
176	C₂	—	4	2	no	—
177	C₂	C₃×#518	1	1	yes	characteristic
178	C₂	C₂×C₃×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₄
prime radical	C₃×#525	C₂
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
2. C₃×#435	C₂	semidirect
3. C₃×C₈×D₄	C₂	semidirect
4. C₃×#433	C₂	semidirect
5. C₃×#408	C₂	semidirect
6. C₂×C₃×C₂²⋊C₈	C₂	semidirect
7. #2977	C₃	direct (N × H)
10. C₂×C₃×M₁₆	C₂²	semidirect
12. C₂²×C₃×C₈	C₂²	semidirect
19. C₃×C₄⋊C₈	C₂²	semidirect
23. C₃×C₂²⋊C₈	C₂²	semidirect
24. C₃×#97	C₂²	semidirect
25. C₃×C₄×C₈	C₂²	semidirect
27. #435	C₂×C₃	semidirect
28. C₈×D₄	C₂×C₃	semidirect
29. #433	C₂×C₃	semidirect
30. #408	C₂×C₃	semidirect
31. C₂×C₂²⋊C₈	C₂×C₃	semidirect
57. C₂×M₁₆	C₂²×C₃	semidirect
59. C₂²×C₈	C₂²×C₃	semidirect
66. C₄⋊C₈	C₂²×C₃	semidirect
70. C₂²⋊C₈	C₂²×C₃	semidirect
71. #97	C₂²×C₃	semidirect
72. C₄×C₈	C₂²×C₃	semidirect
173. C₃	#2977	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×#525	2
C₃×#435	2
C₃×C₈×D₄	2
C₃×#433	2
C₃×#408	2
C₂×C₃×C₂²⋊C₈	2
#2977	3

Subgroups — C3×#2977 #84841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2977 #84841