Subgroups — C₃×#546 #5876

Order 192.

334 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#123	C₂	1	2	yes	maximal, complemented
5	C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#121	C₂	1	2	yes	maximal, complemented
8	C₃×#120	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	#546	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂⁴×C₃	C₂²	1	2	yes	—
14	C₃×#41	—	2	4	no	—
15	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃×D₄	—	2	4	no	—
17	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×D₄	C₂²	1	2	yes	—
19	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×D₄	C₂²	1	2	yes	complemented
22	C₂×C₃×D₄	C₂²	1	2	yes	complemented
23	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂²	1	2	yes	complemented
25	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₃×C₄	C₂²	1	2	yes	complemented
27	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
33	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
34	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
35	C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
36	C₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
37	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
38	C₂×#41	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂²×D₄	C₂×C₃	1	1	yes	characteristic, complemented
40	#124	C₂×C₃	1	1	yes	characteristic, complemented
41	#123	C₂×C₃	1	2	yes	complemented
42	C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
43	#121	C₂×C₃	1	1	yes	characteristic, complemented
44	#121	C₂×C₃	1	2	yes	complemented
45	#120	C₂×C₃	1	2	yes	complemented
46	C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂×C₂²⋊C₄	C₂×C₃	1	2	yes	complemented
49	C₂³×C₃	—	2	4	no	—
50	C₂³×C₃	—	2	4	no	—
51	C₂³×C₃	—	4	2	no	—
52	C₂³×C₃	C₂³	1	2	yes	—
53	C₂³×C₃	C₂³	1	1	yes	characteristic
54	C₂³×C₃	C₂³	1	2	yes	—
55	C₂³×C₃	C₂³	1	1	yes	characteristic
56	C₃×Q₈	—	2	2	no	—
57	C₃×D₄	—	2	4	no	—
58	C₃×D₄	—	2	4	no	—
59	C₃×D₄	—	2	2	no	—
60	C₃×D₄	—	2	2	no	—
61	C₃×D₄	—	4	2	no	—
62	C₃×D₄	D₄	1	4	yes	complemented
63	C₂×C₃×C₄	—	2	4	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	—	2	2	no	—
66	C₂×C₃×C₄	—	4	2	no	—
67	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂³	1	2	yes	—
70	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂³	1	2	yes	—
72	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
74	C₂⁴	C₂²×C₃	1	2	yes	—
75	#41	—	2	4	no	—
76	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×D₄	C₂²×C₃	1	2	yes	—
80	C₂×D₄	C₂²×C₃	1	1	yes	characteristic
81	C₂×D₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×D₄	C₂²×C₃	1	2	yes	complemented
83	C₂×D₄	C₂²×C₃	1	2	yes	complemented
84	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₃	1	2	yes	complemented
86	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
87	C₂²×C₄	C₂²×C₃	1	2	yes	complemented
88	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
89	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
90	C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
91	C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
92	C₂²⋊C₄	C₂²×C₃	1	2	yes	—
93	C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
94	C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
95	C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
96	C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
97	C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
98	C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
99	C₂²×C₃	—	2	1	no	—
100	C₂²×C₃	—	2	2	no	—
101	C₂²×C₃	—	2	2	no	—
102	C₂²×C₃	—	2	1	no	—
103	C₂²×C₃	—	2	1	no	—
104	C₂²×C₃	—	2	2	no	—
105	C₂²×C₃	—	2	1	no	—
106	C₂²×C₃	—	2	2	no	—
107	C₂²×C₃	—	2	1	no	—
108	C₂²×C₃	—	2	2	no	—
109	C₂²×C₃	—	4	8	no	—
110	C₂²×C₃	C₂×D₄	1	4	yes	—
111	C₂²×C₃	#41	1	2	yes	complemented
112	C₂²×C₃	C₂⁴	1	1	yes	characteristic
113	C₃×C₄	—	2	1	no	—
114	C₃×C₄	—	2	1	no	—
115	C₃×C₄	—	2	2	no	—
116	C₃×C₄	—	2	1	no	—
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	C₂×D₄	1	2	yes	—
120	C₂³	—	2	4	no	—
121	C₂³	—	2	4	no	—
122	C₂³	—	4	2	no	—
123	C₂³	C₂³×C₃	1	2	yes	—
124	C₂³	C₂³×C₃	1	1	yes	characteristic
125	C₂³	C₂³×C₃	1	2	yes	—
126	C₂³	C₂³×C₃	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	D₄	—	2	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	2	2	no	—
131	D₄	—	2	2	no	—
132	D₄	—	4	2	no	—
133	D₄	C₃×D₄	1	4	yes	complemented
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₄	C₂³×C₃	1	2	yes	—
141	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₄	C₂³×C₃	1	2	yes	—
143	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂×C₃	—	2	2	no	—
146	C₂×C₃	—	2	4	no	—
147	C₂×C₃	—	4	1	no	—
148	C₂×C₃	—	4	2	no	—
149	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	4	8	no	—
163	C₂²	C₂×C₃×D₄	1	4	yes	—
164	C₂²	C₃×#41	1	2	yes	complemented
165	C₂²	C₂⁴×C₃	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	2	no	—
172	C₄	C₂×C₃×D₄	1	2	yes	—
173	C₃	#546	1	1	yes	characteristic, direct product factor, Sylow
174	C₂	—	2	2	no	—
175	C₂	—	2	4	no	—
176	C₂	—	4	2	no	—
177	C₂	—	4	1	no	—
178	C₂	C₂²×C₃×D₄	1	1	yes	characteristic
179	C₂	C₂×C₃×#41	1	1	yes	characteristic
180	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#41	C₂	semidirect
2. C₂²×C₃×D₄	C₂	semidirect
3. C₃×#124	C₂	semidirect
4. C₃×#123	C₂	semidirect
5. C₃×C₂²⋊Q₈	C₂	semidirect
6. C₃×#121	C₂	semidirect
7. C₃×#121	C₂	semidirect
8. C₃×#120	C₂	semidirect
9. C₃×C₄×D₄	C₂	semidirect
10. C₃×C₄×D₄	C₂	semidirect
11. C₂×C₃×C₂²⋊C₄	C₂	semidirect
12. #546	C₃	direct (N × H)
15. C₂×C₃×Q₈	C₂²	semidirect
21. C₂×C₃×D₄	C₂²	semidirect
22. C₂×C₃×D₄	C₂²	semidirect
24. C₂²×C₃×C₄	C₂²	semidirect
25. C₂²×C₃×C₄	C₂²	semidirect
26. C₂²×C₃×C₄	C₂²	semidirect
27. C₃×C₄⋊C₄	C₂²	semidirect
28. C₃×C₄⋊C₄	C₂²	semidirect
29. C₃×C₄⋊C₄	C₂²	semidirect
32. C₃×C₂²⋊C₄	C₂²	semidirect
33. C₃×C₂²⋊C₄	C₂²	semidirect
34. C₃×C₂²⋊C₄	C₂²	semidirect
35. C₃×C₂²⋊C₄	C₂²	semidirect
36. C₃×C₂²⋊C₄	C₂²	semidirect
37. C₃×C₄²	C₂²	semidirect
38. C₂×#41	C₂×C₃	semidirect
39. C₂²×D₄	C₂×C₃	semidirect
40. #124	C₂×C₃	semidirect
41. #123	C₂×C₃	semidirect
42. C₂²⋊Q₈	C₂×C₃	semidirect
43. #121	C₂×C₃	semidirect
44. #121	C₂×C₃	semidirect
45. #120	C₂×C₃	semidirect
46. C₄×D₄	C₂×C₃	semidirect
47. C₄×D₄	C₂×C₃	semidirect
48. C₂×C₂²⋊C₄	C₂×C₃	semidirect
62. C₃×D₄	D₄	semidirect
76. C₂×Q₈	C₂²×C₃	semidirect
82. C₂×D₄	C₂²×C₃	semidirect
83. C₂×D₄	C₂²×C₃	semidirect
85. C₂²×C₄	C₂²×C₃	semidirect
86. C₂²×C₄	C₂²×C₃	semidirect
87. C₂²×C₄	C₂²×C₃	semidirect
88. C₄⋊C₄	C₂²×C₃	semidirect
89. C₄⋊C₄	C₂²×C₃	semidirect
90. C₄⋊C₄	C₂²×C₃	semidirect
93. C₂²⋊C₄	C₂²×C₃	semidirect
94. C₂²⋊C₄	C₂²×C₃	semidirect
95. C₂²⋊C₄	C₂²×C₃	semidirect
96. C₂²⋊C₄	C₂²×C₃	semidirect
97. C₂²⋊C₄	C₂²×C₃	semidirect
98. C₄²	C₂²×C₃	semidirect
111. C₂²×C₃	#41	semidirect
133. D₄	C₃×D₄	semidirect
164. C₂²	C₃×#41	semidirect
173. C₃	#546	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×#41	2
C₂²×C₃×D₄	2
C₃×#124	2
C₃×#123	2
C₃×C₂²⋊Q₈	2
C₃×#121	2
C₃×#120	2
C₃×C₄×D₄	2
C₂×C₃×C₂²⋊C₄	2
#546	3

Subgroups — C3×#546 #5876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#546 #5876