Subgroups — H(3,𝔽₃)×#471 #13145172

Order 1728.

572 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	H(3,𝔽₃)×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	H(3,𝔽₃)×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,𝔽₃)×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	H(3,𝔽₃)×#101	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,𝔽₃)×#100	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#471	C₃	1	4	yes	maximal, complemented
9	#41×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₂×Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
11	C₂×D₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	Q₁₆×H(3,𝔽₃)	—	2	1	no	—
13	SD₁₆×H(3,𝔽₃)	—	2	1	no	—
14	SD₁₆×H(3,𝔽₃)	—	2	1	no	—
15	SD₁₆×H(3,𝔽₃)	—	2	1	no	—
16	SD₁₆×H(3,𝔽₃)	—	2	1	no	—
17	D₈×H(3,𝔽₃)	—	2	1	no	—
18	M₁₆×H(3,𝔽₃)	—	2	1	no	—
19	M₁₆×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
20	M₁₆×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
21	M₁₆×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
22	C₂×C₈×H(3,𝔽₃)	—	2	1	no	—
23	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
24	C₃²×#137	C₂×C₃	1	4	yes	complemented
25	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	4	yes	—
26	C₂×C₃²×SD₁₆	C₂×C₃	1	4	yes	complemented
27	C₃²×#131	C₂×C₃	1	4	yes	complemented
28	C₃²×#108	C₂×C₃	1	4	yes	complemented
29	C₃²×#101	C₂×C₃	1	4	yes	complemented
30	C₃²×#100	C₂×C₃	1	4	yes	complemented
31	C₂³×H(3,𝔽₃)	—	2	1	no	—
32	Q₈×H(3,𝔽₃)	—	2	1	no	—
33	Q₈×H(3,𝔽₃)	—	2	1	no	—
34	Q₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
35	D₄×H(3,𝔽₃)	—	2	1	no	—
36	D₄×H(3,𝔽₃)	—	2	1	no	—
37	D₄×H(3,𝔽₃)	—	2	1	no	—
38	D₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
39	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
40	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
41	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
42	C₈×H(3,𝔽₃)	—	2	1	no	—
43	C₈×H(3,𝔽₃)	—	2	1	no	—
44	C₈×H(3,𝔽₃)	—	2	1	no	—
45	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
46	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
47	C₃×#471	—	3	4	no	—
48	C₃×#471	C₃²	1	1	yes	characteristic
49	C₃²×#41	C₂²×C₃	1	4	yes	—
50	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
51	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
52	C₃²×Q₁₆	—	2	4	no	—
53	C₃²×SD₁₆	—	2	4	no	—
54	C₃²×SD₁₆	—	2	4	no	—
55	C₃²×SD₁₆	—	2	4	no	—
56	C₃²×SD₁₆	—	2	4	no	—
57	C₃²×D₈	—	2	4	no	—
58	C₃²×M₁₆	—	2	4	no	—
59	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
60	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
61	C₃²×M₁₆	C₂²×C₃	1	4	yes	—
62	C₂×C₃²×C₈	—	2	4	no	—
63	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
64	C₂²×H(3,𝔽₃)	—	2	1	no	—
65	C₂²×H(3,𝔽₃)	—	4	1	no	—
66	C₂²×H(3,𝔽₃)	—	8	1	no	—
67	C₂²×H(3,𝔽₃)	#41	1	1	yes	characteristic
68	C₄×H(3,𝔽₃)	—	2	1	no	—
69	C₄×H(3,𝔽₃)	—	4	1	no	—
70	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
71	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
72	C₃×#137	—	3	4	no	—
73	C₃×#137	C₂×C₃²	1	1	yes	characteristic
74	C₃×AGL(1,ℤ/8)	—	3	4	no	—
75	C₃×AGL(1,ℤ/8)	C₂×C₃²	1	1	yes	characteristic
76	C₂×C₃×SD₁₆	—	3	4	no	—
77	C₂×C₃×SD₁₆	C₂×C₃²	1	1	yes	characteristic
78	C₃×#131	—	3	4	no	—
79	C₃×#131	C₂×C₃²	1	1	yes	characteristic
80	C₃×#108	—	3	4	no	—
81	C₃×#108	C₂×C₃²	1	1	yes	characteristic
82	C₃×#101	—	3	4	no	—
83	C₃×#101	C₂×C₃²	1	1	yes	characteristic
84	C₃×#100	—	3	4	no	—
85	C₃×#100	C₂×C₃²	1	1	yes	characteristic
86	C₂³×C₃²	—	2	4	no	—
87	C₃²×Q₈	—	2	4	no	—
88	C₃²×Q₈	—	2	4	no	—
89	C₃²×Q₈	C₃×D₄	1	4	yes	—
90	C₃²×D₄	—	2	4	no	—
91	C₃²×D₄	—	2	4	no	—
92	C₃²×D₄	—	2	4	no	—
93	C₃²×D₄	C₃×D₄	1	4	yes	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	2	4	no	—
96	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
97	C₃²×C₈	—	2	4	no	—
98	C₃²×C₈	—	2	4	no	—
99	C₃²×C₈	—	2	4	no	—
100	C₃²×C₈	C₃×D₄	1	4	yes	—
101	C₃²×C₈	C₃×D₄	1	4	yes	—
102	#471	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×H(3,𝔽₃)	—	2	1	no	—
104	C₂×H(3,𝔽₃)	—	4	1	no	—
105	C₂×H(3,𝔽₃)	—	8	1	no	—
106	C₂×H(3,𝔽₃)	#121	1	1	yes	characteristic
107	C₃×#41	—	3	4	no	—
108	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
109	C₂×C₃×Q₈	—	3	4	no	—
110	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
111	C₂×C₃×D₄	—	3	4	no	—
112	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
113	C₃×Q₁₆	—	2	1	no	—
114	C₃×Q₁₆	—	6	4	no	—
115	C₃×SD₁₆	—	2	1	no	—
116	C₃×SD₁₆	—	2	1	no	—
117	C₃×SD₁₆	—	2	1	no	—
118	C₃×SD₁₆	—	2	1	no	—
119	C₃×SD₁₆	—	6	4	no	—
120	C₃×SD₁₆	—	6	4	no	—
121	C₃×SD₁₆	—	6	4	no	—
122	C₃×SD₁₆	—	6	4	no	—
123	C₃×D₈	—	2	1	no	—
124	C₃×D₈	—	6	4	no	—
125	C₃×M₁₆	—	2	1	no	—
126	C₃×M₁₆	—	3	4	no	—
127	C₃×M₁₆	—	3	4	no	—
128	C₃×M₁₆	—	3	4	no	—
129	C₃×M₁₆	—	6	4	no	—
130	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
131	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
132	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	3	4	no	—
135	C₂×C₃×C₈	—	6	4	no	—
136	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	4	4	no	—
139	C₂²×C₃²	—	8	4	no	—
140	C₂²×C₃²	C₃×#41	1	4	yes	—
141	C₃²×C₄	—	2	4	no	—
142	C₃²×C₄	—	4	4	no	—
143	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
144	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
145	#137	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
146	AGL(1,ℤ/8)	C₂×H(3,𝔽₃)	1	1	yes	characteristic
147	C₂×SD₁₆	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
148	#131	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
149	#108	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
150	#101	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
151	#100	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
152	H(3,𝔽₃)	#471	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×C₃	—	2	1	no	—
154	C₂³×C₃	—	6	4	no	—
155	C₃×Q₈	—	2	1	no	—
156	C₃×Q₈	—	2	1	no	—
157	C₃×Q₈	—	3	4	no	—
158	C₃×Q₈	—	6	4	no	—
159	C₃×Q₈	—	6	4	no	—
160	C₃×Q₈	C₃²×D₄	1	1	yes	characteristic
161	C₃×D₄	—	2	1	no	—
162	C₃×D₄	—	2	1	no	—
163	C₃×D₄	—	2	1	no	—
164	C₃×D₄	—	3	4	no	—
165	C₃×D₄	—	6	4	no	—
166	C₃×D₄	—	6	4	no	—
167	C₃×D₄	—	6	4	no	—
168	C₃×D₄	C₃²×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	3	4	no	—
172	C₂×C₃×C₄	—	6	4	no	—
173	C₂×C₃×C₄	—	6	4	no	—
174	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
175	C₃×C₈	—	2	1	no	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	3	4	no	—
179	C₃×C₈	—	3	4	no	—
180	C₃×C₈	—	6	4	no	—
181	C₃×C₈	—	6	4	no	—
182	C₃×C₈	—	6	4	no	—
183	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
184	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
185	C₂×C₃²	—	2	4	no	—
186	C₂×C₃²	—	4	4	no	—
187	C₂×C₃²	—	8	4	no	—
188	C₂×C₃²	C₃×#121	1	4	yes	—
189	#41	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
190	C₂×Q₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
191	C₂×D₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
192	Q₁₆	—	2	1	no	—
193	SD₁₆	—	2	1	no	—
194	SD₁₆	—	2	1	no	—
195	SD₁₆	—	2	1	no	—
196	SD₁₆	—	2	1	no	—
197	D₈	—	2	1	no	—
198	M₁₆	—	2	1	no	—
199	M₁₆	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
200	M₁₆	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
201	M₁₆	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	3	4	no	—
206	C₂²×C₃	—	4	1	no	—
207	C₂²×C₃	—	6	4	no	—
208	C₂²×C₃	—	8	1	no	—
209	C₂²×C₃	—	12	4	no	—
210	C₂²×C₃	—	24	4	no	—
211	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	3	4	no	—
214	C₃×C₄	—	3	4	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	—	6	4	no	—
217	C₃×C₄	—	12	4	no	—
218	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
219	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
220	C₃²	C₃×#471	1	4	yes	complemented
221	C₂³	—	2	1	no	—
222	Q₈	—	2	1	no	—
223	Q₈	—	2	1	no	—
224	Q₈	D₄×H(3,𝔽₃)	1	1	yes	characteristic
225	D₄	—	2	1	no	—
226	D₄	—	2	1	no	—
227	D₄	—	2	1	no	—
228	D₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	D₄×H(3,𝔽₃)	1	1	yes	characteristic
236	C₈	D₄×H(3,𝔽₃)	1	1	yes	characteristic
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	3	4	no	—
239	C₂×C₃	—	4	1	no	—
240	C₂×C₃	—	6	4	no	—
241	C₂×C₃	—	8	1	no	—
242	C₂×C₃	—	12	4	no	—
243	C₂×C₃	—	24	4	no	—
244	C₂×C₃	C₃²×#121	1	1	yes	characteristic
245	C₂²	—	2	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	8	1	no	—
248	C₂²	#41×H(3,𝔽₃)	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	4	1	no	—
251	C₄	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
252	C₄	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
253	C₃	—	3	4	no	—
254	C₃	C₃²×#471	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	4	1	no	—
257	C₂	—	8	1	no	—
258	C₂	H(3,𝔽₃)×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×#121
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	H(3,𝔽₃)×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)×#137	C₂	semidirect
3. C₂×SD₁₆×H(3,𝔽₃)	C₂	semidirect
4. H(3,𝔽₃)×#131	C₂	semidirect
5. H(3,𝔽₃)×#108	C₂	semidirect
6. H(3,𝔽₃)×#101	C₂	semidirect
7. H(3,𝔽₃)×#100	C₂	semidirect
8. C₃²×#471	C₃	semidirect
24. C₃²×#137	C₂×C₃	semidirect
26. C₂×C₃²×SD₁₆	C₂×C₃	semidirect
27. C₃²×#131	C₂×C₃	semidirect
28. C₃²×#108	C₂×C₃	semidirect
29. C₃²×#101	C₂×C₃	semidirect
30. C₃²×#100	C₂×C₃	semidirect
102. #471	H(3,𝔽₃)	direct (N × H)
145. #137	C₂×H(3,𝔽₃)	semidirect
147. C₂×SD₁₆	C₂×H(3,𝔽₃)	semidirect
148. #131	C₂×H(3,𝔽₃)	semidirect
149. #108	C₂×H(3,𝔽₃)	semidirect
150. #101	C₂×H(3,𝔽₃)	semidirect
151. #100	C₂×H(3,𝔽₃)	semidirect
152. H(3,𝔽₃)	#471	direct (N × H)
220. C₃²	C₃×#471	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)×#137	2
H(3,𝔽₃)×AGL(1,ℤ/8)	2
C₂×SD₁₆×H(3,𝔽₃)	2
H(3,𝔽₃)×#131	2
H(3,𝔽₃)×#108	2
H(3,𝔽₃)×#101	2
H(3,𝔽₃)×#100	2
C₃²×#471	3

Subgroups — H(3,𝔽3)×#471 #13145172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#471 #13145172