Subgroups — 3₋¹⁺²×#472 #13145176

Order 1728.

432 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#100	C₂	1	2	yes	maximal, complemented
6	C₃²×#472	C₃	1	1	yes	maximal, characteristic
7	C₉×#472	C₃	1	3	yes	maximal, complemented
8	#41×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₂×D₄×3₋¹⁺²	C₂²	1	2	yes	—
10	SD₁₆×3₋¹⁺²	—	2	2	no	—
11	D₈×3₋¹⁺²	—	2	2	no	—
12	D₈×3₋¹⁺²	—	2	1	no	—
13	D₈×3₋¹⁺²	—	2	1	no	—
14	M₁₆×3₋¹⁺²	—	2	1	no	—
15	M₁₆×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	M₁₆×3₋¹⁺²	C₂²	1	2	yes	—
17	C₂×C₈×3₋¹⁺²	—	2	1	no	—
18	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
19	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	—
20	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic
21	C₃²×#131	C₂×C₃	1	1	yes	characteristic
22	C₃²×#108	C₂×C₃	1	1	yes	characteristic
23	C₃²×#100	C₂×C₃	1	2	yes	—
24	C₉×AGL(1,ℤ/8)	C₂×C₃	1	6	yes	complemented
25	C₂×C₉×D₈	C₂×C₃	1	3	yes	complemented
26	C₉×#131	C₂×C₃	1	3	yes	complemented
27	C₉×#108	C₂×C₃	1	3	yes	complemented
28	C₉×#100	C₂×C₃	1	6	yes	complemented
29	C₂³×3₋¹⁺²	—	2	2	no	—
30	Q₈×3₋¹⁺²	D₄	1	1	yes	characteristic
31	D₄×3₋¹⁺²	—	2	1	no	—
32	D₄×3₋¹⁺²	—	2	2	no	—
33	D₄×3₋¹⁺²	—	2	2	no	—
34	D₄×3₋¹⁺²	D₄	1	1	yes	characteristic
35	C₂×C₄×3₋¹⁺²	—	2	1	no	—
36	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
37	C₈×3₋¹⁺²	—	2	1	no	—
38	C₈×3₋¹⁺²	—	2	2	no	—
39	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
40	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
41	C₃×#472	—	3	1	no	—
42	C₃×#472	C₃²	1	1	yes	characteristic
43	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
45	C₃²×SD₁₆	—	2	2	no	—
46	C₃²×D₈	—	2	2	no	—
47	C₃²×D₈	—	2	1	no	—
48	C₃²×D₈	—	2	1	no	—
49	C₃²×M₁₆	—	2	1	no	—
50	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
51	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
52	C₂×C₃²×C₈	—	2	1	no	—
53	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₉×#41	C₂²×C₃	1	3	yes	—
55	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
56	C₉×SD₁₆	—	2	6	no	—
57	C₉×D₈	—	2	6	no	—
58	C₉×D₈	—	2	3	no	—
59	C₉×D₈	—	2	3	no	—
60	C₉×M₁₆	—	2	3	no	—
61	C₉×M₁₆	C₂²×C₃	1	3	yes	—
62	C₉×M₁₆	C₂²×C₃	1	6	yes	—
63	C₂×C₈×C₉	—	2	3	no	—
64	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
65	C₂²×3₋¹⁺²	—	2	1	no	—
66	C₂²×3₋¹⁺²	—	4	2	no	—
67	C₂²×3₋¹⁺²	—	8	2	no	—
68	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
69	C₄×3₋¹⁺²	—	2	1	no	—
70	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
71	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
72	C₃×AGL(1,ℤ/8)	—	3	2	no	—
73	C₃×AGL(1,ℤ/8)	C₂×C₃²	1	2	yes	—
74	C₂×C₃×D₈	—	3	1	no	—
75	C₂×C₃×D₈	C₂×C₃²	1	1	yes	characteristic
76	C₃×#131	—	3	1	no	—
77	C₃×#131	C₂×C₃²	1	1	yes	characteristic
78	C₃×#108	—	3	1	no	—
79	C₃×#108	C₂×C₃²	1	1	yes	characteristic
80	C₃×#100	—	3	2	no	—
81	C₃×#100	C₂×C₃²	1	2	yes	—
82	C₂³×C₃²	—	2	2	no	—
83	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
84	C₃²×D₄	—	2	1	no	—
85	C₃²×D₄	—	2	2	no	—
86	C₃²×D₄	—	2	2	no	—
87	C₃²×D₄	C₃×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	—	2	1	no	—
89	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂³×C₉	—	2	6	no	—
91	C₃²×C₈	—	2	1	no	—
92	C₃²×C₈	—	2	2	no	—
93	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
94	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
95	Q₈×C₉	C₃×D₄	1	3	yes	—
96	D₄×C₉	—	2	3	no	—
97	D₄×C₉	—	2	6	no	—
98	D₄×C₉	—	2	6	no	—
99	D₄×C₉	C₃×D₄	1	3	yes	—
100	C₂×C₄×C₉	—	2	3	no	—
101	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
102	C₈×C₉	—	2	3	no	—
103	C₈×C₉	—	2	6	no	—
104	C₈×C₉	C₃×D₄	1	3	yes	—
105	C₈×C₉	C₃×D₄	1	3	yes	—
106	#472	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×3₋¹⁺²	—	2	1	no	—
108	C₂×3₋¹⁺²	—	4	1	no	—
109	C₂×3₋¹⁺²	—	8	2	no	—
110	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
111	C₃×#41	—	3	1	no	—
112	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
113	C₂×C₃×D₄	—	3	2	no	—
114	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
115	C₃×SD₁₆	—	2	2	no	—
116	C₃×SD₁₆	—	6	2	no	—
117	C₃×D₈	—	2	2	no	—
118	C₃×D₈	—	2	1	no	—
119	C₃×D₈	—	2	1	no	—
120	C₃×D₈	—	6	1	no	—
121	C₃×D₈	—	6	1	no	—
122	C₃×D₈	—	6	2	no	—
123	C₃×M₁₆	—	2	1	no	—
124	C₃×M₁₆	—	3	1	no	—
125	C₃×M₁₆	—	3	2	no	—
126	C₃×M₁₆	—	6	1	no	—
127	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
128	C₃×M₁₆	C₂²×C₃²	1	2	yes	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	3	1	no	—
131	C₂×C₃×C₈	—	6	1	no	—
132	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
133	C₂²×C₃²	—	2	1	no	—
134	C₂²×C₃²	—	4	2	no	—
135	C₂²×C₃²	—	8	2	no	—
136	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
137	C₃²×C₄	—	2	1	no	—
138	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
139	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂²×C₉	—	2	3	no	—
141	C₂²×C₉	—	4	6	no	—
142	C₂²×C₉	—	8	6	no	—
143	C₂²×C₉	C₃×#41	1	3	yes	—
144	C₄×C₉	—	2	3	no	—
145	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
146	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
147	AGL(1,ℤ/8)	C₂×3₋¹⁺²	1	2	yes	complemented
148	C₂×D₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
149	#131	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
150	#108	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
151	#100	C₂×3₋¹⁺²	1	2	yes	complemented
152	3₋¹⁺²	#472	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	—	6	2	no	—
155	C₃×Q₈	—	3	1	no	—
156	C₃×Q₈	C₃²×D₄	1	1	yes	characteristic
157	C₃×D₄	—	2	1	no	—
158	C₃×D₄	—	2	2	no	—
159	C₃×D₄	—	2	2	no	—
160	C₃×D₄	—	3	1	no	—
161	C₃×D₄	—	6	2	no	—
162	C₃×D₄	—	6	2	no	—
163	C₃×D₄	—	6	1	no	—
164	C₃×D₄	C₃²×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	3	1	no	—
167	C₂×C₃×C₄	—	6	1	no	—
168	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	—	3	1	no	—
172	C₃×C₈	—	3	1	no	—
173	C₃×C₈	—	6	1	no	—
174	C₃×C₈	—	6	2	no	—
175	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
176	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
177	C₂×C₃²	—	2	1	no	—
178	C₂×C₃²	—	4	1	no	—
179	C₂×C₃²	—	8	2	no	—
180	C₂×C₃²	C₃×#121	1	1	yes	characteristic
181	C₂×C₉	—	2	3	no	—
182	C₂×C₉	—	4	3	no	—
183	C₂×C₉	—	8	6	no	—
184	C₂×C₉	C₃×#121	1	3	yes	—
185	#41	C₂²×3₋¹⁺²	1	1	yes	characteristic
186	C₂×D₄	C₂²×3₋¹⁺²	1	2	yes	—
187	SD₁₆	—	2	2	no	—
188	D₈	—	2	2	no	—
189	D₈	—	2	1	no	—
190	D₈	—	2	1	no	—
191	M₁₆	—	2	1	no	—
192	M₁₆	C₂²×3₋¹⁺²	1	1	yes	characteristic
193	M₁₆	C₂²×3₋¹⁺²	1	2	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	3	1	no	—
198	C₂²×C₃	—	4	2	no	—
199	C₂²×C₃	—	6	1	no	—
200	C₂²×C₃	—	8	2	no	—
201	C₂²×C₃	—	12	2	no	—
202	C₂²×C₃	—	24	2	no	—
203	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	3	1	no	—
206	C₃×C₄	—	3	1	no	—
207	C₃×C₄	—	6	1	no	—
208	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
209	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
210	C₃²	C₃×#472	1	1	yes	characteristic
211	C₉	C₃×#472	1	3	yes	complemented
212	C₂³	—	2	2	no	—
213	Q₈	D₄×3₋¹⁺²	1	1	yes	characteristic
214	D₄	—	2	1	no	—
215	D₄	—	2	2	no	—
216	D₄	—	2	2	no	—
217	D₄	D₄×3₋¹⁺²	1	1	yes	characteristic
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	2	no	—
222	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
223	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	3	1	no	—
226	C₂×C₃	—	4	1	no	—
227	C₂×C₃	—	6	1	no	—
228	C₂×C₃	—	8	2	no	—
229	C₂×C₃	—	12	1	no	—
230	C₂×C₃	—	24	2	no	—
231	C₂×C₃	C₃²×#121	1	1	yes	characteristic
232	C₂²	—	2	1	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	#41×3₋¹⁺²	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
238	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
239	C₃	—	3	1	no	—
240	C₃	C₃²×#472	1	1	yes	characteristic
241	C₂	—	2	1	no	—
242	C₂	—	4	1	no	—
243	C₂	—	8	2	no	—
244	C₂	3₋¹⁺²×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×#121
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₃²×#472	C₃
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×D₈×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²×#131	C₂	semidirect
4. 3₋¹⁺²×#108	C₂	semidirect
5. 3₋¹⁺²×#100	C₂	semidirect
7. C₉×#472	C₃	semidirect
24. C₉×AGL(1,ℤ/8)	C₂×C₃	semidirect
25. C₂×C₉×D₈	C₂×C₃	semidirect
26. C₉×#131	C₂×C₃	semidirect
27. C₉×#108	C₂×C₃	semidirect
28. C₉×#100	C₂×C₃	semidirect
106. #472	3₋¹⁺²	direct (N × H)
147. AGL(1,ℤ/8)	C₂×3₋¹⁺²	semidirect
148. C₂×D₈	C₂×3₋¹⁺²	semidirect
149. #131	C₂×3₋¹⁺²	semidirect
150. #108	C₂×3₋¹⁺²	semidirect
151. #100	C₂×3₋¹⁺²	semidirect
152. 3₋¹⁺²	#472	direct (N × H)
211. C₉	C₃×#472	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×AGL(1,ℤ/8)	2
C₂×D₈×3₋¹⁺²	2
3₋¹⁺²×#131	2
3₋¹⁺²×#108	2
3₋¹⁺²×#100	2
C₃²×#472	3
C₉×#472	3

Subgroups — 3−1+2×#472 #13145176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#472 #13145176