Subgroups — H(3,𝔽₃)×#464 #13145151

Order 1728.

616 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	C₂×Q₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	H(3,𝔽₃)×#124	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,𝔽₃)×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	H(3,𝔽₃)×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,𝔽₃)×#102	C₂	1	1	yes	maximal, characteristic
8	C₃²×#464	C₃	1	4	yes	maximal, complemented
9	C₂×Q₈×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	2	no	—
13	SD₁₆×H(3,𝔽₃)	—	2	2	no	—
14	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
15	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×H(3,𝔽₃)	—	2	1	no	—
17	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×H(3,𝔽₃)	—	2	1	no	—
19	C₂²⋊C₄×H(3,𝔽₃)	—	2	1	no	—
20	C₄²×H(3,𝔽₃)	—	2	1	no	—
21	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
22	C₂×C₃²×Q₁₆	C₂×C₃	1	4	yes	complemented
23	C₂×C₃²×SD₁₆	C₂×C₃	1	4	yes	—
24	C₃²×#124	C₂×C₃	1	4	yes	—
25	C₃²×C₄×Q₈	C₂×C₃	1	4	yes	complemented
26	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
27	C₃²×#103	C₂×C₃	1	4	yes	complemented
28	C₃²×#102	C₂×C₃	1	4	yes	—
29	C₂³×H(3,𝔽₃)	—	2	1	no	—
30	Q₈×H(3,𝔽₃)	—	2	1	no	—
31	Q₈×H(3,𝔽₃)	—	2	1	no	—
32	Q₈×H(3,𝔽₃)	—	2	1	no	—
33	Q₈×H(3,𝔽₃)	D₄	1	2	yes	—
34	D₄×H(3,𝔽₃)	—	2	1	no	—
35	D₄×H(3,𝔽₃)	—	2	1	no	—
36	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
37	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
38	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
39	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
40	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
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	C₃×#464	—	3	4	no	—
45	C₃×#464	C₃²	1	1	yes	characteristic
46	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
47	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
48	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
49	C₃²×Q₁₆	—	2	8	no	—
50	C₃²×SD₁₆	—	2	8	no	—
51	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
52	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
53	C₃²×C₄⋊C₄	—	2	4	no	—
54	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₃²×C₂²⋊C₄	—	2	4	no	—
56	C₃²×C₂²⋊C₄	—	2	4	no	—
57	C₃²×C₄²	—	2	4	no	—
58	C₃²×C₄²	C₂²×C₃	1	4	yes	—
59	C₂²×H(3,𝔽₃)	—	4	1	no	—
60	C₂²×H(3,𝔽₃)	—	4	1	no	—
61	C₂²×H(3,𝔽₃)	—	4	1	no	—
62	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
63	C₄×H(3,𝔽₃)	—	2	2	no	—
64	C₄×H(3,𝔽₃)	—	2	1	no	—
65	C₄×H(3,𝔽₃)	—	2	1	no	—
66	C₄×H(3,𝔽₃)	—	4	1	no	—
67	C₄×H(3,𝔽₃)	—	4	1	no	—
68	C₄×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
69	C₄×H(3,𝔽₃)	#41	1	1	yes	characteristic
70	C₂×C₃×Q₁₆	—	3	4	no	—
71	C₂×C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
72	C₂×C₃×SD₁₆	—	3	4	no	—
73	C₂×C₃×SD₁₆	C₂×C₃²	1	1	yes	characteristic
74	C₃×#124	—	3	4	no	—
75	C₃×#124	C₂×C₃²	1	1	yes	characteristic
76	C₃×C₄×Q₈	—	3	4	no	—
77	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
78	C₃×C₄⋊C₈	—	3	4	no	—
79	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
80	C₃×#103	—	3	4	no	—
81	C₃×#103	C₂×C₃²	1	1	yes	characteristic
82	C₃×#102	—	3	4	no	—
83	C₃×#102	C₂×C₃²	1	1	yes	characteristic
84	C₂³×C₃²	—	2	4	no	—
85	C₃²×Q₈	—	2	4	no	—
86	C₃²×Q₈	—	2	4	no	—
87	C₃²×Q₈	—	2	4	no	—
88	C₃²×Q₈	C₃×D₄	1	8	yes	—
89	C₃²×D₄	—	2	4	no	—
90	C₃²×D₄	—	2	4	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	2	4	no	—
94	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
95	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
96	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
97	C₃²×C₈	—	2	4	no	—
98	C₃²×C₈	—	2	4	no	—
99	#464	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×H(3,𝔽₃)	—	8	1	no	—
101	C₂×H(3,𝔽₃)	#121	1	1	yes	characteristic
102	C₂×H(3,𝔽₃)	#135	1	1	yes	characteristic
103	C₂×H(3,𝔽₃)	#137	1	1	yes	characteristic
104	C₂×C₃×Q₈	—	3	4	no	—
105	C₂×C₃×Q₈	—	3	4	no	—
106	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
107	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
108	C₂×C₃×D₄	—	3	4	no	—
109	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
110	C₃×Q₁₆	—	2	2	no	—
111	C₃×Q₁₆	—	6	8	no	—
112	C₃×SD₁₆	—	2	2	no	—
113	C₃×SD₁₆	—	6	8	no	—
114	C₂×C₃×C₈	—	3	4	no	—
115	C₂×C₃×C₈	—	3	4	no	—
116	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	2	1	no	—
119	C₃×C₄⋊C₄	—	3	4	no	—
120	C₃×C₄⋊C₄	—	6	4	no	—
121	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	—	2	1	no	—
123	C₃×C₂²⋊C₄	—	2	1	no	—
124	C₃×C₂²⋊C₄	—	6	4	no	—
125	C₃×C₂²⋊C₄	—	6	4	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	3	4	no	—
128	C₃×C₄²	—	6	4	no	—
129	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
130	C₂²×C₃²	—	4	4	no	—
131	C₂²×C₃²	—	4	4	no	—
132	C₂²×C₃²	—	4	4	no	—
133	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
134	C₃²×C₄	—	2	8	no	—
135	C₃²×C₄	—	2	4	no	—
136	C₃²×C₄	—	2	4	no	—
137	C₃²×C₄	—	4	4	no	—
138	C₃²×C₄	—	4	4	no	—
139	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
140	C₃²×C₄	C₃×#41	1	4	yes	—
141	C₂×Q₁₆	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
142	C₂×SD₁₆	C₂×H(3,𝔽₃)	1	1	yes	characteristic
143	#124	C₂×H(3,𝔽₃)	1	1	yes	characteristic
144	C₄×Q₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
145	C₄⋊C₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
146	#103	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
147	#102	C₂×H(3,𝔽₃)	1	1	yes	characteristic
148	H(3,𝔽₃)	#464	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	—	6	4	no	—
151	C₃×Q₈	—	2	1	no	—
152	C₃×Q₈	—	2	1	no	—
153	C₃×Q₈	—	2	1	no	—
154	C₃×Q₈	—	3	8	no	—
155	C₃×Q₈	—	6	4	no	—
156	C₃×Q₈	—	6	4	no	—
157	C₃×Q₈	—	6	4	no	—
158	C₃×Q₈	C₃²×D₄	1	2	yes	—
159	C₃×D₄	—	2	1	no	—
160	C₃×D₄	—	2	1	no	—
161	C₃×D₄	—	6	4	no	—
162	C₃×D₄	—	6	4	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	3	4	no	—
167	C₂×C₃×C₄	—	3	4	no	—
168	C₂×C₃×C₄	—	3	4	no	—
169	C₂×C₃×C₄	—	6	4	no	—
170	C₂×C₃×C₄	—	6	4	no	—
171	C₂×C₃×C₄	—	6	4	no	—
172	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
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	—	6	4	no	—
178	C₃×C₈	—	6	4	no	—
179	C₂×C₃²	—	8	4	no	—
180	C₂×C₃²	C₃×#121	1	4	yes	—
181	C₂×C₃²	C₃×#135	1	4	yes	—
182	C₂×C₃²	C₃×#137	1	4	yes	—
183	C₂×Q₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
184	C₂×Q₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
185	C₂×D₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
186	Q₁₆	—	2	2	no	—
187	SD₁₆	—	2	2	no	—
188	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
189	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
196	C₂²×C₃	—	3	4	no	—
197	C₂²×C₃	—	4	1	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	1	no	—
200	C₂²×C₃	—	12	4	no	—
201	C₂²×C₃	—	12	4	no	—
202	C₂²×C₃	—	12	4	no	—
203	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	3	4	no	—
208	C₃×C₄	—	3	4	no	—
209	C₃×C₄	—	4	1	no	—
210	C₃×C₄	—	4	1	no	—
211	C₃×C₄	—	6	4	no	—
212	C₃×C₄	—	6	4	no	—
213	C₃×C₄	—	6	8	no	—
214	C₃×C₄	—	12	4	no	—
215	C₃×C₄	—	12	4	no	—
216	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
217	C₃×C₄	C₃²×#41	1	1	yes	characteristic
218	C₃²	C₃×#464	1	4	yes	complemented
219	C₂³	—	2	1	no	—
220	Q₈	—	2	1	no	—
221	Q₈	—	2	1	no	—
222	Q₈	—	2	1	no	—
223	Q₈	D₄×H(3,𝔽₃)	1	2	yes	—
224	D₄	—	2	1	no	—
225	D₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
230	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
231	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₂×C₃	—	3	4	no	—
235	C₂×C₃	—	3	4	no	—
236	C₂×C₃	—	3	4	no	—
237	C₂×C₃	—	8	1	no	—
238	C₂×C₃	—	24	4	no	—
239	C₂×C₃	C₃²×#121	1	1	yes	characteristic
240	C₂×C₃	C₃²×#135	1	1	yes	characteristic
241	C₂×C₃	C₃²×#137	1	1	yes	characteristic
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
252	C₄	#41×H(3,𝔽₃)	1	1	yes	characteristic
253	C₃	—	3	4	no	—
254	C₃	C₃²×#464	1	1	yes	characteristic
255	C₂	—	8	1	no	—
256	C₂	H(3,𝔽₃)×#121	1	1	yes	characteristic
257	C₂	H(3,𝔽₃)×#135	1	1	yes	characteristic
258	C₂	H(3,𝔽₃)×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂×D₄×H(3,𝔽₃)	C₂²
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×H(3,𝔽₃)	C₂	semidirect
4. C₄×Q₈×H(3,𝔽₃)	C₂	semidirect
5. H(3,𝔽₃)×C₄⋊C₈	C₂	semidirect
6. H(3,𝔽₃)×#103	C₂	semidirect
8. C₃²×#464	C₃	semidirect
22. C₂×C₃²×Q₁₆	C₂×C₃	semidirect
25. C₃²×C₄×Q₈	C₂×C₃	semidirect
26. C₃²×C₄⋊C₈	C₂×C₃	semidirect
27. C₃²×#103	C₂×C₃	semidirect
99. #464	H(3,𝔽₃)	direct (N × H)
141. C₂×Q₁₆	C₂×H(3,𝔽₃)	semidirect
144. C₄×Q₈	C₂×H(3,𝔽₃)	semidirect
145. C₄⋊C₈	C₂×H(3,𝔽₃)	semidirect
146. #103	C₂×H(3,𝔽₃)	semidirect
148. H(3,𝔽₃)	#464	direct (N × H)
218. C₃²	C₃×#464	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×H(3,𝔽₃)	2
C₂×SD₁₆×H(3,𝔽₃)	2
H(3,𝔽₃)×#124	2
C₄×Q₈×H(3,𝔽₃)	2
H(3,𝔽₃)×C₄⋊C₈	2
H(3,𝔽₃)×#103	2
H(3,𝔽₃)×#102	2
C₃²×#464	3

Subgroups — H(3,𝔽3)×#464 #13145151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#464 #13145151