Subgroups — 3₋¹⁺²×#470 #13145170

Order 1728.

440 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	3₋¹⁺²×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#103	C₂	1	2	yes	maximal, complemented
6	C₃²×#470	C₃	1	1	yes	maximal, characteristic
7	C₉×#470	C₃	1	3	yes	maximal, complemented
8	C₂×Q₈×3₋¹⁺²	C₂²	1	2	yes	—
9	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	Q₁₆×3₋¹⁺²	—	2	2	no	—
11	M₁₆×3₋¹⁺²	—	2	2	no	—
12	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×3₋¹⁺²	—	2	2	no	—
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
16	C₂²⋊C₄×3₋¹⁺²	—	2	2	no	—
17	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
19	C₃²×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
20	C₃²×#106	C₂×C₃	1	1	yes	characteristic
21	C₃²×#103	C₂×C₃	1	2	yes	—
22	C₂×C₉×Q₁₆	C₂×C₃	1	3	yes	complemented
23	C₂×C₉×M₁₆	C₂×C₃	1	3	yes	—
24	C₉×C₂²⋊Q₈	C₂×C₃	1	6	yes	—
25	C₉×#106	C₂×C₃	1	3	yes	complemented
26	C₉×#103	C₂×C₃	1	6	yes	complemented
27	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
28	Q₈×3₋¹⁺²	—	2	2	no	—
29	Q₈×3₋¹⁺²	—	2	2	no	—
30	C₂×C₄×3₋¹⁺²	—	2	1	no	—
31	C₂×C₄×3₋¹⁺²	—	2	1	no	—
32	C₂×C₄×3₋¹⁺²	—	2	2	no	—
33	C₂×C₄×3₋¹⁺²	—	2	2	no	—
34	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
35	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
36	C₈×3₋¹⁺²	—	2	1	no	—
37	C₈×3₋¹⁺²	D₄	1	2	yes	—
38	C₃×#470	—	3	1	no	—
39	C₃×#470	C₃²	1	1	yes	characteristic
40	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
41	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₃²×Q₁₆	—	2	2	no	—
43	C₃²×M₁₆	—	2	2	no	—
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₄⋊C₄	—	2	2	no	—
47	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₃²×C₂²⋊C₄	—	2	2	no	—
49	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	—
50	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
51	C₉×Q₁₆	—	2	6	no	—
52	C₉×M₁₆	—	2	6	no	—
53	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
54	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
55	C₉×C₄⋊C₄	—	2	6	no	—
56	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
57	C₉×C₂²⋊C₄	—	2	6	no	—
58	C₂²×3₋¹⁺²	—	2	1	no	—
59	C₂²×3₋¹⁺²	—	2	2	no	—
60	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
61	C₄×3₋¹⁺²	—	2	1	no	—
62	C₄×3₋¹⁺²	—	4	2	no	—
63	C₄×3₋¹⁺²	—	4	2	no	—
64	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
65	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
66	C₂×C₃×Q₁₆	—	3	1	no	—
67	C₂×C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
68	C₂×C₃×M₁₆	—	3	1	no	—
69	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
70	C₃×C₂²⋊Q₈	—	3	2	no	—
71	C₃×C₂²⋊Q₈	C₂×C₃²	1	2	yes	—
72	C₃×#106	—	3	1	no	—
73	C₃×#106	C₂×C₃²	1	1	yes	characteristic
74	C₃×#103	—	3	2	no	—
75	C₃×#103	C₂×C₃²	1	2	yes	—
76	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
77	C₃²×Q₈	—	2	2	no	—
78	C₃²×Q₈	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	1	no	—
80	C₂×C₃²×C₄	—	2	1	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₂³×C₉	C₃×D₄	1	3	yes	—
86	C₃²×C₈	—	2	1	no	—
87	C₃²×C₈	C₃×D₄	1	2	yes	—
88	Q₈×C₉	—	2	6	no	—
89	Q₈×C₉	—	2	6	no	—
90	C₂×C₄×C₉	—	2	3	no	—
91	C₂×C₄×C₉	—	2	3	no	—
92	C₂×C₄×C₉	—	2	6	no	—
93	C₂×C₄×C₉	—	2	6	no	—
94	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
95	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
96	C₈×C₉	—	2	3	no	—
97	C₈×C₉	C₃×D₄	1	6	yes	—
98	#470	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×3₋¹⁺²	—	4	1	no	—
100	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
101	C₂×3₋¹⁺²	#137	1	2	yes	—
102	C₂×C₃×Q₈	—	3	2	no	—
103	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
104	C₂²×C₃×C₄	—	3	1	no	—
105	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
106	C₃×Q₁₆	—	2	2	no	—
107	C₃×Q₁₆	—	6	2	no	—
108	C₃×M₁₆	—	2	2	no	—
109	C₃×M₁₆	—	6	2	no	—
110	C₂×C₃×C₈	—	3	1	no	—
111	C₂×C₃×C₈	—	3	1	no	—
112	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	—	3	2	no	—
116	C₃×C₄⋊C₄	—	6	2	no	—
117	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
118	C₃×C₂²⋊C₄	—	2	2	no	—
119	C₃×C₂²⋊C₄	—	6	2	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	—	2	2	no	—
122	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	—	4	2	no	—
125	C₃²×C₄	—	4	2	no	—
126	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
127	C₃²×C₄	C₃×#41	1	1	yes	characteristic
128	C₂²×C₉	—	2	3	no	—
129	C₂²×C₉	—	2	6	no	—
130	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
131	C₄×C₉	—	2	3	no	—
132	C₄×C₉	—	4	6	no	—
133	C₄×C₉	—	4	6	no	—
134	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
135	C₄×C₉	C₃×#41	1	3	yes	—
136	C₂×Q₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
137	C₂×M₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
138	C₂²⋊Q₈	C₂×3₋¹⁺²	1	2	yes	—
139	#106	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
140	#103	C₂×3₋¹⁺²	1	2	yes	complemented
141	3₋¹⁺²	#470	1	1	yes	characteristic, direct product factor, Sylow
142	C₂³×C₃	—	3	1	no	—
143	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
144	C₃×Q₈	—	2	2	no	—
145	C₃×Q₈	—	2	2	no	—
146	C₃×Q₈	—	6	2	no	—
147	C₃×Q₈	—	6	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	3	1	no	—
153	C₂×C₃×C₄	—	3	1	no	—
154	C₂×C₃×C₄	—	6	1	no	—
155	C₂×C₃×C₄	—	6	1	no	—
156	C₂×C₃×C₄	—	6	2	no	—
157	C₂×C₃×C₄	—	6	2	no	—
158	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	3	2	no	—
162	C₃×C₈	—	6	1	no	—
163	C₃×C₈	C₃²×D₄	1	2	yes	—
164	C₂×C₃²	—	4	1	no	—
165	C₂×C₃²	C₃×#121	1	1	yes	characteristic
166	C₂×C₃²	C₃×#137	1	2	yes	—
167	C₂×C₉	—	4	3	no	—
168	C₂×C₉	C₃×#121	1	3	yes	—
169	C₂×C₉	C₃×#137	1	6	yes	—
170	C₂×Q₈	C₂²×3₋¹⁺²	1	2	yes	—
171	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
172	Q₁₆	—	2	2	no	—
173	M₁₆	—	2	2	no	—
174	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
175	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	C₂²×3₋¹⁺²	1	2	yes	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	3	1	no	—
182	C₂²×C₃	—	6	1	no	—
183	C₂²×C₃	—	6	2	no	—
184	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	3	1	no	—
187	C₃×C₄	—	3	1	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	—	4	2	no	—
190	C₃×C₄	—	6	1	no	—
191	C₃×C₄	—	12	2	no	—
192	C₃×C₄	—	12	2	no	—
193	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
194	C₃×C₄	C₃²×#41	1	1	yes	characteristic
195	C₃²	C₃×#470	1	1	yes	characteristic
196	C₉	C₃×#470	1	3	yes	complemented
197	C₂³	D₄×3₋¹⁺²	1	1	yes	characteristic
198	Q₈	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
205	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	D₄×3₋¹⁺²	1	2	yes	—
208	C₂×C₃	—	3	1	no	—
209	C₂×C₃	—	3	2	no	—
210	C₂×C₃	—	4	1	no	—
211	C₂×C₃	—	12	1	no	—
212	C₂×C₃	C₃²×#121	1	1	yes	characteristic
213	C₂×C₃	C₃²×#137	1	2	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	2	no	—
216	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
221	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
222	C₃	—	3	1	no	—
223	C₃	C₃²×#470	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	3₋¹⁺²×#121	1	1	yes	characteristic
226	C₂	3₋¹⁺²×#137	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×3₋¹⁺²	C₂	semidirect
4. 3₋¹⁺²×#106	C₂	semidirect
5. 3₋¹⁺²×#103	C₂	semidirect
7. C₉×#470	C₃	semidirect
22. C₂×C₉×Q₁₆	C₂×C₃	semidirect
25. C₉×#106	C₂×C₃	semidirect
26. C₉×#103	C₂×C₃	semidirect
98. #470	3₋¹⁺²	direct (N × H)
136. C₂×Q₁₆	C₂×3₋¹⁺²	semidirect
139. #106	C₂×3₋¹⁺²	semidirect
140. #103	C₂×3₋¹⁺²	semidirect
141. 3₋¹⁺²	#470	direct (N × H)
196. C₉	C₃×#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×3₋¹⁺²	2
C₂×M₁₆×3₋¹⁺²	2
3₋¹⁺²×C₂²⋊Q₈	2
3₋¹⁺²×#106	2
3₋¹⁺²×#103	2
C₃²×#470	3
C₉×#470	3

Subgroups — 3−1+2×#470 #13145170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#470 #13145170