Subgroups — #10948830

Order 768.

486 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#2162	C₂	1	1	yes	maximal, characteristic
3	#69744	C₂	1	1	yes	maximal, characteristic
4	#69281	C₂	1	4	yes	maximal, complemented
5	#69093	C₂	1	4	yes	maximal
6	#69048	C₂	1	4	yes	maximal, complemented
7	#33497	—	3	1	no	maximal, Sylow
8	C₂×C₄×Dic₆	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₃⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₃	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
13	C₈×Dic₆	C₄	1	2	yes	—
14	C₈×Dic₆	C₄	1	2	yes	—
15	C₈×Dic₆	C₂²	1	2	yes	—
16	C₈×Dic₆	C₂²	1	2	yes	—
17	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
19	C₃×C₄×C₁₆	C₂²	1	2	yes	—
20	#4549	C₂²	1	2	yes	—
21	#4502	C₂²	1	8	yes	—
22	Dic₃⋊C₁₆	C₂²	1	8	yes	—
23	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
24	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
25	C₂×C₈×Q₈	—	3	1	no	—
26	#2183	—	3	4	no	—
27	#2183	—	3	4	no	—
28	#2163	—	3	4	no	—
29	#2162	—	3	1	no	—
30	#2162	D₃	1	1	yes	characteristic
31	C₂²×Dic₆	C₈	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₃	C₈	1	1	yes	characteristic
36	C₂×Dic₃⋊C₄	C₈	1	2	yes	—
37	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
38	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×Dic₆	C₈	1	2	yes	—
40	C₄×Dic₆	C₈	1	2	yes	—
41	C₄×Dic₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₆	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₁₆	C₂³	1	4	yes	—
44	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
48	Dic₃⋊C₈	C₂×C₄	1	4	yes	—
49	Dic₃⋊C₈	C₂×C₄	1	4	yes	—
50	C₈×Dic₃	C₂×C₄	1	4	yes	—
51	C₈×Dic₃	C₂³	1	4	yes	—
52	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
53	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
54	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
55	C₂×C₄×Q₈	—	3	1	no	—
56	C₈×Q₈	—	3	2	no	—
57	C₈×Q₈	—	3	2	no	—
58	C₈×Q₈	—	3	2	no	—
59	C₈×Q₈	—	3	2	no	—
60	C₂×C₄⋊C₈	—	3	2	no	—
61	C₂×C₄⋊C₈	—	3	1	no	—
62	C₂×C₄×C₈	—	3	2	no	—
63	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
64	C₄⋊C₁₆	—	3	8	no	—
65	C₄⋊C₁₆	—	3	2	no	—
66	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
67	C₂²⋊C₁₆	—	3	2	no	—
68	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
69	#346	—	3	8	no	—
70	C₄×C₁₆	—	3	2	no	—
71	C₄×C₁₆	C₂×D₃	1	2	yes	—
72	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₃	C₂×C₈	1	2	yes	—
76	C₂²×Dic₃	C₂×C₈	1	2	yes	—
77	C₂×Dic₆	—	2	4	no	—
78	C₂×Dic₆	C₂×C₈	1	2	yes	—
79	C₂×Dic₆	C₂×C₈	1	2	yes	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	4	1	no	—
82	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
86	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
87	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
88	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
92	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
93	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
94	C₄×Dic₃	C₂×C₈	1	4	yes	—
95	C₄×Dic₃	C₂²×C₄	1	4	yes	—
96	C₂×C₃⋊C₈	—	4	2	no	—
97	C₂×C₃⋊C₈	C₂²×C₄	1	4	yes	—
98	C₃×C₁₆	—	2	4	no	—
99	C₃⋊C₁₆	—	2	4	no	—
100	C₂²×Q₈	—	3	1	no	—
101	C₂²×C₈	—	3	2	no	—
102	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
104	C₄×Q₈	—	3	2	no	—
105	C₄×Q₈	—	3	2	no	—
106	C₄×Q₈	—	3	2	no	—
107	C₄×Q₈	—	3	2	no	—
108	C₂×C₄⋊C₄	—	3	2	no	—
109	C₂×C₄⋊C₄	—	3	1	no	—
110	C₂×C₄²	—	3	2	no	—
111	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
112	C₂×C₁₆	—	3	4	no	—
113	C₂×C₁₆	C₂²×D₃	1	4	yes	—
114	C₄⋊C₈	—	3	4	no	—
115	C₄⋊C₈	—	3	4	no	—
116	C₄⋊C₈	—	3	2	no	—
117	C₄⋊C₈	—	3	2	no	—
118	C₄×C₈	—	3	4	no	—
119	C₄×C₈	—	3	4	no	—
120	C₄×C₈	C₄×D₃	1	1	yes	characteristic
121	C₄×C₈	C₄×D₃	1	1	yes	characteristic
122	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
124	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	4	1	no	—
128	C₂×C₃×C₄	M₃₂	1	4	yes	—
129	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
135	C₂×Dic₃	—	2	4	no	—
136	C₂×Dic₃	—	4	2	no	—
137	C₂×Dic₃	C₂²×C₈	1	4	yes	—
138	C₂×Dic₃	C₂²×C₈	1	4	yes	—
139	Dic₆	—	2	4	no	—
140	Dic₆	—	2	4	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	2	1	no	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
145	C₃⋊C₈	—	2	4	no	—
146	C₂×Q₈	—	3	2	no	—
147	C₂×Q₈	—	3	2	no	—
148	C₂×Q₈	—	6	4	no	—
149	C₂²×C₄	—	3	2	no	—
150	C₂²×C₄	—	3	2	no	—
151	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
152	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	3	4	no	—
156	C₂×C₈	—	4	1	no	—
157	C₂×C₈	—	12	2	no	—
158	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
162	C₄⋊C₄	—	3	4	no	—
163	C₄⋊C₄	—	3	4	no	—
164	C₄⋊C₄	—	3	2	no	—
165	C₄⋊C₄	—	3	2	no	—
166	C₄²	—	3	4	no	—
167	C₄²	—	3	4	no	—
168	C₄²	D₃×C₈	1	1	yes	characteristic
169	C₄²	D₃×C₈	1	1	yes	characteristic
170	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
172	C₁₆	—	2	4	no	—
173	C₁₆	—	6	4	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	C₂×M₃₂	1	2	yes	—
177	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	C₂×M₃₂	1	2	yes	—
182	C₃×C₄	C₂×M₃₂	1	2	yes	—
183	C₃×C₄	#519	1	1	yes	characteristic
184	C₃×C₄	#568	1	1	yes	characteristic
185	Dic₃	—	2	4	no	—
186	Dic₃	—	2	4	no	—
187	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
188	Q₈	—	6	4	no	—
189	Q₈	—	6	4	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	3	4	no	—
193	C₂×C₄	—	3	4	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	6	4	no	—
196	C₂×C₄	—	12	2	no	—
197	C₂×C₄	#798	1	4	yes	—
198	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
199	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
201	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
202	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	6	4	no	—
208	C₈	#1010	1	2	yes	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	#2982	1	1	yes	characteristic
211	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
212	C₂×C₃	#3407	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#798	1	2	yes	—
216	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	6	4	no	—
221	C₄	—	6	4	no	—
222	C₄	C₂×#798	1	2	yes	—
223	C₄	C₂×#798	1	2	yes	—
224	C₄	#5516	1	1	yes	characteristic
225	C₄	#5745	1	1	yes	characteristic
226	C₃	#33497	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	2	2	no	—
228	C₂	#79263	1	1	yes	characteristic
229	C₂	C₂²×#798	1	1	yes	characteristic
230	C₂	#82007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2162	C₂
prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2162	D₃
hypocenter	C₃	#33497
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
4. #69281	C₂	semidirect
6. #69048	C₂	semidirect
226. C₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₆	2
C₃×#2162	2
#69744	2
#69281	2
#69093	2
#69048	2
#33497	3