Subgroups — C₂×3₋¹⁺²×#108 #13145047

Order 1728.

424 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×3₋¹⁺²	C₂	1	2	yes	maximal
2	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#108	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×#108	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×#108	C₃	1	3	yes	maximal, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	M₁₆×3₋¹⁺²	—	2	2	no	—
8	M₁₆×3₋¹⁺²	C₂²	1	4	yes	—
9	C₂×C₈×3₋¹⁺²	—	2	2	no	—
10	C₂×C₈×3₋¹⁺²	C₄	1	2	yes	—
11	C₂×C₈×3₋¹⁺²	C₄	1	2	yes	—
12	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
13	C₂×C₃²×M₁₆	C₂×C₃	1	2	yes	—
14	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
15	C₃²×#108	C₂×C₃	1	4	yes	—
16	C₂×C₉×M₁₆	C₂×C₃	1	6	yes	—
17	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
18	C₉×#108	C₂×C₃	1	12	yes	complemented
19	C₂³×3₋¹⁺²	Q₈	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
23	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
24	C₂×C₄×3₋¹⁺²	Q₈	1	1	yes	characteristic
25	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
26	C₈×3₋¹⁺²	—	2	4	no	—
27	C₈×3₋¹⁺²	C₂×C₄	1	4	yes	—
28	C₂×C₃×#108	—	3	1	no	—
29	C₂×C₃×#108	C₃²	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
31	C₃²×M₁₆	—	2	2	no	—
32	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
35	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
36	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
37	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
38	C₉×M₁₆	—	2	6	no	—
39	C₉×M₁₆	C₂²×C₃	1	12	yes	—
40	C₂×C₈×C₉	—	2	6	no	—
41	C₂×C₈×C₉	C₃×C₄	1	6	yes	—
42	C₂×C₈×C₉	C₃×C₄	1	6	yes	—
43	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
44	C₂²×3₋¹⁺²	—	2	2	no	—
45	C₂²×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
48	C₄×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
49	C₄×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
50	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
51	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×M₁₆	—	3	2	no	—
53	C₂×C₃×M₁₆	C₂×C₃²	1	2	yes	—
54	C₂²×C₃×C₈	—	3	1	no	—
55	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
56	C₃×#108	—	3	4	no	—
57	C₃×#108	C₂×C₃²	1	4	yes	—
58	C₂³×C₃²	C₃×Q₈	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
65	C₂³×C₉	C₃×Q₈	1	3	yes	—
66	C₃²×C₈	—	2	4	no	—
67	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
68	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
69	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
70	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
71	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
72	C₂×C₄×C₉	C₃×Q₈	1	3	yes	—
73	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
74	C₈×C₉	—	2	12	no	—
75	C₈×C₉	C₂×C₃×C₄	1	12	yes	—
76	C₂×#108	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×3₋¹⁺²	—	2	1	no	—
78	C₂×3₋¹⁺²	—	2	1	no	—
79	C₂×3₋¹⁺²	#108	1	2	yes	direct product factor
80	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	3	1	no	—
82	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
83	C₃×M₁₆	—	2	2	no	—
84	C₃×M₁₆	—	3	4	no	—
85	C₃×M₁₆	—	6	2	no	—
86	C₃×M₁₆	C₂²×C₃²	1	4	yes	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	—	3	2	no	—
89	C₂×C₃×C₈	—	3	2	no	—
90	C₂×C₃×C₈	—	3	2	no	—
91	C₂×C₃×C₈	—	6	2	no	—
92	C₂×C₃×C₈	C₃²×C₄	1	2	yes	—
93	C₂×C₃×C₈	C₃²×C₄	1	2	yes	—
94	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
95	C₂²×C₃²	—	2	2	no	—
96	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
99	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
100	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
101	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
102	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
103	C₂²×C₉	—	2	6	no	—
104	C₂²×C₉	C₃×C₄⋊C₄	1	3	yes	—
105	C₂²×C₉	C₃×C₄⋊C₄	1	3	yes	—
106	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
107	C₄×C₉	C₃×C₄⋊C₄	1	3	yes	—
108	C₄×C₉	C₃×C₄⋊C₄	1	3	yes	—
109	C₄×C₉	C₂²×C₃×C₄	1	3	yes	—
110	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
111	C₂×M₁₆	C₂×3₋¹⁺²	1	2	yes	—
112	C₂²×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
113	#108	C₂×3₋¹⁺²	1	4	yes	direct product factor
114	3₋¹⁺²	C₂×#108	1	1	yes	characteristic, direct product factor, Sylow
115	C₂³×C₃	—	3	1	no	—
116	C₂³×C₃	C₃²×Q₈	1	1	yes	characteristic
117	C₂×C₃×C₄	—	3	1	no	—
118	C₂×C₃×C₄	—	3	1	no	—
119	C₂×C₃×C₄	—	3	1	no	—
120	C₂×C₃×C₄	—	3	1	no	—
121	C₂×C₃×C₄	—	3	1	no	—
122	C₂×C₃×C₄	—	3	1	no	—
123	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₃²×Q₈	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
129	C₃×C₈	—	2	4	no	—
130	C₃×C₈	—	3	4	no	—
131	C₃×C₈	—	6	4	no	—
132	C₃×C₈	C₂×C₃²×C₄	1	4	yes	—
133	C₂×C₃²	—	2	1	no	—
134	C₂×C₃²	—	2	1	no	—
135	C₂×C₃²	C₃×#108	1	2	yes	—
136	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₉	—	2	3	no	—
138	C₂×C₉	—	2	3	no	—
139	C₂×C₉	C₃×#108	1	6	yes	complemented
140	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
141	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	C₂²×3₋¹⁺²	1	4	yes	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	C₄×3₋¹⁺²	1	2	yes	—
146	C₂×C₈	C₄×3₋¹⁺²	1	2	yes	—
147	C₂×C₈	C₂²×3₋¹⁺²	1	2	yes	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	3	1	no	—
150	C₂²×C₃	—	3	1	no	—
151	C₂²×C₃	—	3	1	no	—
152	C₂²×C₃	—	6	2	no	—
153	C₂²×C₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
156	C₃×C₄	—	3	1	no	—
157	C₃×C₄	—	3	1	no	—
158	C₃×C₄	—	3	1	no	—
159	C₃×C₄	—	3	1	no	—
160	C₃×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
161	C₃×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
162	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
163	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
164	C₃²	C₂×C₃×#108	1	1	yes	characteristic
165	C₉	C₂×C₃×#108	1	3	yes	complemented
166	C₂³	Q₈×3₋¹⁺²	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
169	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
170	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
171	C₂×C₄	Q₈×3₋¹⁺²	1	1	yes	characteristic
172	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
173	C₈	—	2	4	no	—
174	C₈	C₂×C₄×3₋¹⁺²	1	4	yes	—
175	C₂×C₃	—	2	1	no	—
176	C₂×C₃	—	2	1	no	—
177	C₂×C₃	—	3	1	no	—
178	C₂×C₃	—	3	2	no	—
179	C₂×C₃	—	6	1	no	—
180	C₂×C₃	—	6	1	no	—
181	C₂×C₃	C₃²×#108	1	2	yes	—
182	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
185	C₂²	C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
186	C₂²	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
187	C₄	C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
188	C₄	C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
189	C₄	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
190	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
191	C₃	—	3	1	no	—
192	C₃	C₂×C₃²×#108	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	3₋¹⁺²×#108	1	2	yes	direct product factor
196	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₃²×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃	C₃²×C₄⋊C₄
prime radical	C₂³×C₃²	C₃×Q₈
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×#108	C₂	direct (N × H)
5. C₂×C₉×#108	C₃	semidirect
18. C₉×#108	C₂×C₃	semidirect
76. C₂×#108	3₋¹⁺²	direct (N × H)
79. C₂×3₋¹⁺²	#108	direct (N × H)
113. #108	C₂×3₋¹⁺²	direct (N × H)
114. 3₋¹⁺²	C₂×#108	direct (N × H)
139. C₂×C₉	C₃×#108	semidirect
165. C₉	C₂×C₃×#108	semidirect
195. C₂	3₋¹⁺²×#108	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×M₁₆×3₋¹⁺²	2
C₂²×C₈×3₋¹⁺²	2
3₋¹⁺²×#108	2
C₂×C₃²×#108	3
C₂×C₉×#108	3

Subgroups — C2×3−1+2×#108 #13145047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²×#108 #13145047