Subgroups — #10713351

Order 768.

450 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂×#4561	C₂	1	4	yes	maximal, complemented
3	#69743	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#13191	—	3	1	no	maximal, Sylow
6	C₂×C₃×#131	C₂²	1	2	yes	—
7	C₃×#532	C₄	1	1	yes	characteristic
8	C₃×#436	—	2	2	no	—
9	C₃×C₈×D₄	—	2	2	no	—
10	C₃×#423	C₄	1	1	yes	characteristic
11	C₃×#408	C₄	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#812	C₂²	1	2	yes	—
14	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
15	#4561	—	2	8	no	—
16	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
17	C₄×C₃⋊C₁₆	—	2	2	no	—
18	#2931	D₃	1	1	yes	characteristic
19	#2150	—	3	1	no	—
20	C₂×#350	—	3	4	no	—
21	C₂×C₄×C₁₆	—	3	1	no	—
22	C₂×C₃×#41	C₂×C₄	1	2	yes	—
23	C₃×#131	—	2	4	no	—
24	C₃×#131	—	2	4	no	—
25	C₂×C₃×M₁₆	—	2	2	no	—
26	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₈	—	2	2	no	—
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
31	C₃×#127	C₈	1	1	yes	characteristic
32	C₃×#124	C₈	1	2	yes	—
33	C₃×#121	—	2	2	no	—
34	C₃×C₄×D₄	—	2	2	no	—
35	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄⋊C₈	—	2	2	no	—
37	C₃×C₂²⋊C₈	—	2	4	no	—
38	C₃×C₄×C₈	D₄	1	2	yes	—
39	C₃×C₄×C₈	D₄	1	2	yes	—
40	#812	—	2	4	no	—
41	C₂×C₃⋊C₁₆	—	2	2	no	—
42	C₂×C₃⋊C₁₆	—	2	2	no	—
43	C₂×C₃⋊C₁₆	—	2	2	no	—
44	C₂×C₃⋊C₁₆	D₄	1	4	yes	complemented
45	C₂×#131	C₂×D₃	1	2	yes	—
46	#532	Dic₃	1	1	yes	characteristic
47	C₂×M₃₂	—	3	2	no	—
48	C₂²×C₁₆	—	3	2	no	—
49	#436	—	2	2	no	—
50	C₈×D₄	—	2	2	no	—
51	#423	Dic₃	1	1	yes	characteristic
52	#408	Dic₃	1	2	yes	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄⋊C₁₆	—	6	2	no	—
55	#350	—	6	8	no	—
56	C₄×C₁₆	—	6	2	no	—
57	C₃×#41	—	2	4	no	—
58	C₃×#41	—	2	4	no	—
59	C₂×C₃×Q₈	C₂×C₈	1	2	yes	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	2	no	—
62	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	M₁₆	1	2	yes	—
65	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃×M₁₆	—	2	4	no	—
67	C₃×M₁₆	—	4	4	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	4	2	no	—
71	C₂×C₃×C₈	—	4	2	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₈	#41	1	2	yes	—
77	C₃×C₄⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	2	4	no	—
79	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
80	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
81	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
82	C₃⋊C₁₆	—	2	4	no	—
83	C₃⋊C₁₆	—	4	2	no	—
84	C₂×#41	C₂×Dic₃	1	2	yes	—
85	#131	—	2	4	no	—
86	#131	—	2	4	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₄⋊Q₈	C₃⋊C₈	1	1	yes	characteristic
93	#127	C₃⋊C₈	1	1	yes	characteristic
94	#124	C₃⋊C₈	1	2	yes	—
95	#121	—	2	2	no	—
96	C₄×D₄	—	2	2	no	—
97	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
98	M₃₂	—	6	4	no	—
99	C₂×C₁₆	—	3	4	no	—
100	C₂×C₁₆	—	6	2	no	—
101	C₂×C₁₆	—	6	2	no	—
102	C₂×C₁₆	—	6	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	—	2	4	no	—
105	C₄×C₈	#67	1	2	yes	—
106	C₄×C₈	#67	1	2	yes	—
107	C₂³×C₃	—	2	2	no	—
108	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
109	C₃×Q₈	—	2	4	no	—
110	C₃×D₄	—	2	4	no	—
111	C₃×D₄	—	4	4	no	—
112	C₃×D₄	—	4	4	no	—
113	C₂×C₃×C₄	—	2	4	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
118	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
119	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₃×C₄	#131	1	2	yes	—
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	—	4	2	no	—
125	C₃×C₈	#121	1	2	yes	—
126	C₃×C₈	#124	1	1	yes	characteristic
127	C₃×C₈	#127	1	1	yes	characteristic
128	#41	—	2	4	no	—
129	#41	—	2	4	no	—
130	C₂×Q₈	C₂×C₃⋊C₈	1	2	yes	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	#208	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	—	4	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	#217	1	2	yes	—
144	C₂×C₈	D₃×D₄	1	2	yes	—
145	C₂×C₈	#237	1	2	yes	—
146	C₂×C₈	C₂×#67	1	1	yes	characteristic
147	C₂×C₈	C₂×#67	1	1	yes	characteristic
148	C₄⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
151	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
152	C₄²	#217	1	2	yes	—
153	C₁₆	—	6	4	no	—
154	C₁₆	—	12	2	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	4	4	no	—
157	C₂²×C₃	—	4	2	no	—
158	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₃	#436	1	2	yes	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	4	2	no	—
162	C₃×C₄	#350	1	4	yes	complemented
163	C₃×C₄	#390	1	1	yes	characteristic
164	C₃×C₄	#408	1	1	yes	characteristic
165	C₃×C₄	C₈×D₄	1	2	yes	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂×#208	1	1	yes	characteristic
168	Q₈	—	2	4	no	—
169	D₄	—	2	4	no	—
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#830	1	2	yes	—
177	C₂×C₄	#830	1	2	yes	—
178	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₃	1	2	yes	—
180	C₂×C₄	#948	1	2	yes	—
181	C₂×C₄	C₂×#217	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	4	2	no	—
184	C₈	#936	1	1	yes	characteristic
185	C₈	#938	1	2	yes	—
186	C₈	#940	1	1	yes	characteristic
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	—	8	2	no	—
189	C₂×C₃	#1954	1	1	yes	characteristic
190	C₂×C₃	C₂×#350	1	2	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	4	2	no	—
194	C₂²	#5012	1	2	yes	—
195	C₂²	C₂×#830	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	#4561	1	4	yes	complemented
199	C₄	D₄×C₃⋊C₈	1	2	yes	—
200	C₄	#5221	1	1	yes	characteristic
201	C₄	#5234	1	1	yes	characteristic
202	C₃	#13191	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	2	2	no	—
204	C₂	—	8	2	no	—
205	C₂	#71614	1	1	yes	characteristic
206	C₂	C₂×#4561	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#2931	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₃×#532	C₄
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#2931	D₃
hypocenter	C₃	#13191
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#4561	C₂	semidirect
3. #69743	C₂	semidirect
4. C₂×C₄×C₃⋊C₁₆	C₂	semidirect
44. C₂×C₃⋊C₁₆	D₄	semidirect
162. C₃×C₄	#350	semidirect
198. C₄	#4561	semidirect
202. C₃	#13191	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2931	2
C₂×#4561	2
#69743	2
C₂×C₄×C₃⋊C₁₆	2
#13191	3