Subgroups — H(3,𝔽₃)×#485 #13145214

Order 1728.

528 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×#117	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)×#103	C₂	1	1	yes	maximal, characteristic, complemented
6	H(3,𝔽₃)×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,𝔽₃)×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₃²×#485	C₃	1	4	yes	maximal, complemented
9	C₂×Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
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)	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×H(3,𝔽₃)	—	2	1	no	—
18	C₂²⋊C₄×H(3,𝔽₃)	—	2	1	no	—
19	C₄²×H(3,𝔽₃)	—	2	1	no	—
20	C₃²×C₂²⋊Q₈	C₂×C₃	1	4	yes	—
21	C₃²×#117	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×#103	C₂×C₃	1	4	yes	complemented
25	C₃²×#103	C₂×C₃	1	4	yes	complemented
26	C₃²×C₂²⋊C₈	C₂×C₃	1	4	yes	—
27	C₂³×H(3,𝔽₃)	D₄	1	1	yes	characteristic
28	Q₈×H(3,𝔽₃)	—	2	1	no	—
29	Q₈×H(3,𝔽₃)	—	2	1	no	—
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)	—	2	1	no	—
34	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
35	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
36	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
37	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
38	C₈×H(3,𝔽₃)	—	2	1	no	—
39	C₈×H(3,𝔽₃)	—	2	1	no	—
40	C₃×#485	—	3	4	no	—
41	C₃×#485	C₃²	1	1	yes	characteristic
42	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
43	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
44	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
45	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
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×C₂²⋊C₄	—	2	4	no	—
52	C₃²×C₄²	—	2	4	no	—
53	C₂²×H(3,𝔽₃)	—	2	1	no	—
54	C₂²×H(3,𝔽₃)	—	2	1	no	—
55	C₂²×H(3,𝔽₃)	—	2	1	no	—
56	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
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)	—	4	1	no	—
62	C₄×H(3,𝔽₃)	#41	1	1	yes	characteristic
63	C₄×H(3,𝔽₃)	#41	1	1	yes	characteristic
64	C₃×C₂²⋊Q₈	—	3	4	no	—
65	C₃×C₂²⋊Q₈	C₂×C₃²	1	1	yes	characteristic
66	C₃×#117	—	3	4	no	—
67	C₃×#117	C₂×C₃²	1	1	yes	characteristic
68	C₃×#107	—	3	4	no	—
69	C₃×#107	C₂×C₃²	1	1	yes	characteristic
70	C₃×#106	—	3	4	no	—
71	C₃×#106	C₂×C₃²	1	1	yes	characteristic
72	C₃×#103	—	3	4	no	—
73	C₃×#103	—	3	4	no	—
74	C₃×#103	C₂×C₃²	1	1	yes	characteristic
75	C₃×#103	C₂×C₃²	1	1	yes	characteristic
76	C₃×C₂²⋊C₈	—	3	4	no	—
77	C₃×C₂²⋊C₈	C₂×C₃²	1	1	yes	characteristic
78	C₂³×C₃²	C₃×D₄	1	4	yes	—
79	C₃²×Q₈	—	2	4	no	—
80	C₃²×Q₈	—	2	4	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
88	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
89	C₃²×C₈	—	2	4	no	—
90	C₃²×C₈	—	2	4	no	—
91	#485	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×H(3,𝔽₃)	—	4	1	no	—
93	C₂×H(3,𝔽₃)	#123	1	1	yes	characteristic
94	C₂×H(3,𝔽₃)	#135	1	1	yes	characteristic
95	C₂×H(3,𝔽₃)	#137	1	1	yes	characteristic
96	C₂×C₃×Q₈	—	3	4	no	—
97	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	3	4	no	—
99	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
100	C₂×C₃×C₈	—	3	4	no	—
101	C₂×C₃×C₈	—	3	4	no	—
102	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
104	C₃×C₄⋊C₄	—	2	1	no	—
105	C₃×C₄⋊C₄	—	3	4	no	—
106	C₃×C₄⋊C₄	—	3	4	no	—
107	C₃×C₄⋊C₄	—	3	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	—	2	1	no	—
114	C₃×C₂²⋊C₄	—	6	4	no	—
115	C₃×C₂²⋊C₄	—	6	4	no	—
116	C₃×C₄²	—	2	1	no	—
117	C₃×C₄²	—	6	4	no	—
118	C₂²×C₃²	—	2	4	no	—
119	C₂²×C₃²	—	2	4	no	—
120	C₂²×C₃²	—	2	4	no	—
121	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
122	C₃²×C₄	—	2	4	no	—
123	C₃²×C₄	—	4	4	no	—
124	C₃²×C₄	—	4	4	no	—
125	C₃²×C₄	—	4	4	no	—
126	C₃²×C₄	—	4	4	no	—
127	C₃²×C₄	C₃×#41	1	4	yes	—
128	C₃²×C₄	C₃×#41	1	4	yes	—
129	C₂²⋊Q₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic
130	#117	C₂×H(3,𝔽₃)	1	1	yes	characteristic
131	#107	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
132	#106	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
133	#103	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
134	#103	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
135	C₂²⋊C₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic
136	H(3,𝔽₃)	#485	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³×C₃	—	3	4	no	—
138	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
139	C₃×Q₈	—	2	1	no	—
140	C₃×Q₈	—	2	1	no	—
141	C₃×Q₈	—	6	4	no	—
142	C₃×Q₈	—	6	4	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	3	4	no	—
150	C₂×C₃×C₄	—	3	4	no	—
151	C₂×C₃×C₄	—	6	4	no	—
152	C₂×C₃×C₄	—	6	4	no	—
153	C₂×C₃×C₄	—	6	4	no	—
154	C₂×C₃×C₄	—	6	4	no	—
155	C₂×C₃×C₄	—	6	4	no	—
156	C₂×C₃×C₄	—	6	4	no	—
157	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	6	4	no	—
162	C₃×C₈	—	6	4	no	—
163	C₂×C₃²	—	4	4	no	—
164	C₂×C₃²	C₃×#123	1	4	yes	—
165	C₂×C₃²	C₃×#135	1	4	yes	—
166	C₂×C₃²	C₃×#137	1	4	yes	—
167	C₂×Q₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
168	C₂²×C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
169	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
170	C₂×C₈	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
173	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
174	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
175	C₂²⋊C₄	—	2	1	no	—
176	C₂²⋊C₄	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	3	4	no	—
182	C₂²×C₃	—	6	4	no	—
183	C₂²×C₃	—	6	4	no	—
184	C₂²×C₃	—	6	4	no	—
185	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	3	4	no	—
188	C₃×C₄	—	3	4	no	—
189	C₃×C₄	—	4	1	no	—
190	C₃×C₄	—	4	1	no	—
191	C₃×C₄	—	4	1	no	—
192	C₃×C₄	—	4	1	no	—
193	C₃×C₄	—	6	4	no	—
194	C₃×C₄	—	12	4	no	—
195	C₃×C₄	—	12	4	no	—
196	C₃×C₄	—	12	4	no	—
197	C₃×C₄	—	12	4	no	—
198	C₃×C₄	C₃²×#41	1	1	yes	characteristic
199	C₃×C₄	C₃²×#41	1	1	yes	characteristic
200	C₃²	C₃×#485	1	4	yes	complemented
201	C₂³	D₄×H(3,𝔽₃)	1	1	yes	characteristic
202	Q₈	—	2	1	no	—
203	Q₈	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	D₄×H(3,𝔽₃)	1	1	yes	characteristic
211	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₂×C₃	—	3	4	no	—
215	C₂×C₃	—	3	4	no	—
216	C₂×C₃	—	3	4	no	—
217	C₂×C₃	—	4	1	no	—
218	C₂×C₃	—	12	4	no	—
219	C₂×C₃	C₃²×#123	1	1	yes	characteristic
220	C₂×C₃	C₃²×#135	1	1	yes	characteristic
221	C₂×C₃	C₃²×#137	1	1	yes	characteristic
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	1	no	—
231	C₄	#41×H(3,𝔽₃)	1	1	yes	characteristic
232	C₄	#41×H(3,𝔽₃)	1	1	yes	characteristic
233	C₃	—	3	4	no	—
234	C₃	C₃²×#485	1	1	yes	characteristic
235	C₂	—	4	1	no	—
236	C₂	H(3,𝔽₃)×#123	1	1	yes	characteristic
237	C₂	H(3,𝔽₃)×#135	1	1	yes	characteristic
238	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	C₂³×H(3,𝔽₃)	D₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₄×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
3. H(3,𝔽₃)×#107	C₂	semidirect
4. H(3,𝔽₃)×#106	C₂	semidirect
5. H(3,𝔽₃)×#103	C₂	semidirect
6. H(3,𝔽₃)×#103	C₂	semidirect
8. C₃²×#485	C₃	semidirect
22. C₃²×#107	C₂×C₃	semidirect
23. C₃²×#106	C₂×C₃	semidirect
24. C₃²×#103	C₂×C₃	semidirect
25. C₃²×#103	C₂×C₃	semidirect
91. #485	H(3,𝔽₃)	direct (N × H)
131. #107	C₂×H(3,𝔽₃)	semidirect
132. #106	C₂×H(3,𝔽₃)	semidirect
133. #103	C₂×H(3,𝔽₃)	semidirect
134. #103	C₂×H(3,𝔽₃)	semidirect
136. H(3,𝔽₃)	#485	direct (N × H)
200. C₃²	C₃×#485	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)×C₂²⋊Q₈	2
H(3,𝔽₃)×#117	2
H(3,𝔽₃)×#107	2
H(3,𝔽₃)×#106	2
H(3,𝔽₃)×#103	2
H(3,𝔽₃)×C₂²⋊C₈	2
C₃²×#485	3

Subgroups — H(3,𝔽3)×#485 #13145214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#485 #13145214