Subgroups — H(3,𝔽₃)×#478 #13145193

Order 1728.

561 conjugacy classes of subgroups.

The subgroup lattice has 235 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	H(3,𝔽₃)×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	H(3,𝔽₃)×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,𝔽₃)×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	H(3,𝔽₃)×#102	C₂	1	1	yes	maximal, characteristic
7	H(3,𝔽₃)×#102	C₂	1	1	yes	maximal, characteristic
8	C₃²×#478	C₃	1	4	yes	maximal, complemented
9	C₂×D₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₂²×C₄×H(3,𝔽₃)	—	2	1	no	—
11	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×H(3,𝔽₃)	—	2	1	no	—
14	C₄⋊C₄×H(3,𝔽₃)	—	2	1	no	—
15	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×H(3,𝔽₃)	—	2	1	no	—
19	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
20	C₃²×#125	C₂×C₃	1	4	yes	complemented
21	C₃²×C₄×D₄	C₂×C₃	1	4	yes	—
22	C₃²×#107	C₂×C₃	1	4	yes	complemented
23	C₃²×#106	C₂×C₃	1	4	yes	complemented
24	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
25	C₃²×#102	C₂×C₃	1	4	yes	—
26	C₃²×#102	C₂×C₃	1	4	yes	—
27	C₂³×H(3,𝔽₃)	—	2	1	no	—
28	D₄×H(3,𝔽₃)	—	2	1	no	—
29	D₄×H(3,𝔽₃)	Q₈	1	2	yes	—
30	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
31	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
32	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
33	C₂×C₄×H(3,𝔽₃)	—	4	2	no	—
34	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
35	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
36	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
37	C₈×H(3,𝔽₃)	—	2	1	no	—
38	C₈×H(3,𝔽₃)	—	2	1	no	—
39	C₃×#478	—	3	4	no	—
40	C₃×#478	C₃²	1	1	yes	characteristic
41	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
42	C₂²×C₃²×C₄	—	2	4	no	—
43	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
44	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
45	C₃²×C₄⋊C₄	—	2	4	no	—
46	C₃²×C₄⋊C₄	—	2	4	no	—
47	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
48	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	C₃²×C₂²⋊C₄	—	2	4	no	—
51	C₃²×C₄²	C₂²×C₃	1	4	yes	—
52	C₂²×H(3,𝔽₃)	—	2	2	no	—
53	C₂²×H(3,𝔽₃)	—	4	1	no	—
54	C₂²×H(3,𝔽₃)	—	4	1	no	—
55	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
56	C₄×H(3,𝔽₃)	—	2	1	no	—
57	C₄×H(3,𝔽₃)	—	2	1	no	—
58	C₄×H(3,𝔽₃)	—	4	1	no	—
59	C₄×H(3,𝔽₃)	—	4	1	no	—
60	C₄×H(3,𝔽₃)	—	4	1	no	—
61	C₄×H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic
62	C₄×H(3,𝔽₃)	#41	1	1	yes	characteristic
63	C₃×#125	—	3	4	no	—
64	C₃×#125	C₂×C₃²	1	1	yes	characteristic
65	C₃×C₄×D₄	—	3	4	no	—
66	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
67	C₃×#107	—	3	4	no	—
68	C₃×#107	C₂×C₃²	1	1	yes	characteristic
69	C₃×#106	—	3	4	no	—
70	C₃×#106	C₂×C₃²	1	1	yes	characteristic
71	C₃×C₄⋊C₈	—	3	4	no	—
72	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
73	C₃×#102	—	3	4	no	—
74	C₃×#102	—	3	4	no	—
75	C₃×#102	C₂×C₃²	1	1	yes	characteristic
76	C₃×#102	C₂×C₃²	1	1	yes	characteristic
77	C₂³×C₃²	—	2	4	no	—
78	C₃²×D₄	—	2	4	no	—
79	C₃²×D₄	C₃×Q₈	1	8	yes	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	4	8	no	—
84	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
85	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
87	C₃²×C₈	—	2	4	no	—
88	C₃²×C₈	—	2	4	no	—
89	#478	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×H(3,𝔽₃)	—	4	2	no	—
91	C₂×H(3,𝔽₃)	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×H(3,𝔽₃)	#135	1	1	yes	characteristic
93	C₂×H(3,𝔽₃)	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₃×D₄	—	3	4	no	—
95	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	6	4	no	—
98	C₂×C₃×C₈	—	3	4	no	—
99	C₂×C₃×C₈	—	3	4	no	—
100	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
102	C₃×C₄⋊C₄	—	2	1	no	—
103	C₃×C₄⋊C₄	—	2	1	no	—
104	C₃×C₄⋊C₄	—	3	4	no	—
105	C₃×C₄⋊C₄	—	3	4	no	—
106	C₃×C₄⋊C₄	—	3	4	no	—
107	C₃×C₄⋊C₄	—	6	4	no	—
108	C₃×C₄⋊C₄	—	6	4	no	—
109	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
110	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
111	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
112	C₃×C₂²⋊C₄	—	2	1	no	—
113	C₃×C₂²⋊C₄	—	6	4	no	—
114	C₃×C₄²	—	3	4	no	—
115	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
116	C₂²×C₃²	—	2	8	no	—
117	C₂²×C₃²	—	4	4	no	—
118	C₂²×C₃²	—	4	4	no	—
119	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
120	C₃²×C₄	—	2	4	no	—
121	C₃²×C₄	—	2	4	no	—
122	C₃²×C₄	—	4	4	no	—
123	C₃²×C₄	—	4	4	no	—
124	C₃²×C₄	—	4	4	no	—
125	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
126	C₃²×C₄	C₃×#41	1	4	yes	—
127	#125	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
128	C₄×D₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
129	#107	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
130	#106	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
131	C₄⋊C₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
132	#102	C₂×H(3,𝔽₃)	1	1	yes	characteristic
133	#102	C₂×H(3,𝔽₃)	1	1	yes	characteristic
134	H(3,𝔽₃)	#478	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³×C₃	—	2	1	no	—
136	C₂³×C₃	—	6	4	no	—
137	C₃×D₄	—	2	1	no	—
138	C₃×D₄	—	3	8	no	—
139	C₃×D₄	—	6	4	no	—
140	C₃×D₄	C₃²×Q₈	1	2	yes	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	3	4	no	—
145	C₂×C₃×C₄	—	3	4	no	—
146	C₂×C₃×C₄	—	3	4	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	6	4	no	—
149	C₂×C₃×C₄	—	6	4	no	—
150	C₂×C₃×C₄	—	6	4	no	—
151	C₂×C₃×C₄	—	12	8	no	—
152	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
155	C₃×C₈	—	2	1	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	6	4	no	—
158	C₃×C₈	—	6	4	no	—
159	C₂×C₃²	—	4	8	no	—
160	C₂×C₃²	C₃×C₂²⋊Q₈	1	4	yes	—
161	C₂×C₃²	C₃×#135	1	4	yes	—
162	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	4	yes	—
163	C₂×D₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
164	C₂²×C₄	—	2	1	no	—
165	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
166	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
170	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
171	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
172	C₂²⋊C₄	—	2	1	no	—
173	C₄²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	3	4	no	—
176	C₂²×C₃	—	4	1	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	6	8	no	—
179	C₂²×C₃	—	12	4	no	—
180	C₂²×C₃	—	12	4	no	—
181	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	3	4	no	—
185	C₃×C₄	—	3	4	no	—
186	C₃×C₄	—	4	1	no	—
187	C₃×C₄	—	4	1	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	6	4	no	—
190	C₃×C₄	—	6	4	no	—
191	C₃×C₄	—	12	4	no	—
192	C₃×C₄	—	12	4	no	—
193	C₃×C₄	—	12	4	no	—
194	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
195	C₃×C₄	C₃²×#41	1	1	yes	characteristic
196	C₃²	C₃×#478	1	4	yes	complemented
197	C₂³	—	2	1	no	—
198	D₄	—	2	1	no	—
199	D₄	Q₈×H(3,𝔽₃)	1	2	yes	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
205	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
206	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₂×C₃	—	3	4	no	—
210	C₂×C₃	—	3	4	no	—
211	C₂×C₃	—	3	4	no	—
212	C₂×C₃	—	4	2	no	—
213	C₂×C₃	—	12	8	no	—
214	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₃	C₃²×#135	1	1	yes	characteristic
216	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂²	—	2	2	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	1	no	—
226	C₄	C₂×Q₈×H(3,𝔽₃)	1	1	yes	characteristic
227	C₄	#41×H(3,𝔽₃)	1	1	yes	characteristic
228	C₃	—	3	4	no	—
229	C₃	C₃²×#478	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	H(3,𝔽₃)×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂	H(3,𝔽₃)×#135	1	1	yes	characteristic
233	C₂	H(3,𝔽₃)×AGL(1,ℤ/8)	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. H(3,𝔽₃)×#125	C₂	semidirect
3. H(3,𝔽₃)×#107	C₂	semidirect
4. H(3,𝔽₃)×#106	C₂	semidirect
5. H(3,𝔽₃)×C₄⋊C₈	C₂	semidirect
8. C₃²×#478	C₃	semidirect
20. C₃²×#125	C₂×C₃	semidirect
22. C₃²×#107	C₂×C₃	semidirect
23. C₃²×#106	C₂×C₃	semidirect
24. C₃²×C₄⋊C₈	C₂×C₃	semidirect
89. #478	H(3,𝔽₃)	direct (N × H)
127. #125	C₂×H(3,𝔽₃)	semidirect
129. #107	C₂×H(3,𝔽₃)	semidirect
130. #106	C₂×H(3,𝔽₃)	semidirect
131. C₄⋊C₈	C₂×H(3,𝔽₃)	semidirect
134. H(3,𝔽₃)	#478	direct (N × H)
196. C₃²	C₃×#478	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)×#125	2
C₄×D₄×H(3,𝔽₃)	2
H(3,𝔽₃)×#107	2
H(3,𝔽₃)×#106	2
H(3,𝔽₃)×C₄⋊C₈	2
H(3,𝔽₃)×#102	2
C₃²×#478	3

Subgroups — H(3,𝔽3)×#478 #13145193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#478 #13145193