Subgroups — #10713402

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#4562	C₂	1	4	yes	maximal, complemented
3	#69765	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#13202	—	3	1	no	maximal, Sylow
6	C₂²×C₃×D₈	—	2	1	no	—
7	C₂²×C₃×D₈	C₂²	1	2	yes	—
8	C₂×C₃×#127	—	2	1	no	—
9	C₃×#493	—	2	2	no	—
10	C₃×#493	C₄	1	4	yes	complemented
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#818	—	2	1	no	—
13	C₂×#818	C₂²	1	2	yes	—
14	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
15	#4935	—	2	1	no	—
16	#4562	—	2	8	no	—
17	C₄×C₃⋊C₁₆	—	2	2	no	—
18	C₂×#493	D₃	1	1	yes	characteristic
19	C₂×#357	—	3	4	no	—
20	C₂×C₄×C₁₆	—	3	1	no	—
21	#1845	—	3	1	no	—
22	C₂²×C₃×D₄	—	2	2	no	—
23	C₂²×C₃×D₄	—	4	2	no	—
24	C₂×C₃×D₈	—	2	4	no	—
25	C₂×C₃×D₈	—	2	4	no	—
26	C₂×C₃×D₈	—	2	2	no	—
27	C₂×C₃×D₈	—	2	4	no	—
28	C₂×C₃×D₈	—	4	2	no	—
29	C₂×C₃×D₈	—	4	2	no	—
30	C₂×C₃×D₈	C₂×C₄	1	4	yes	—
31	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
33	C₃×#127	—	2	4	no	—
34	C₃×#127	—	4	2	no	—
35	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₃	—	2	2	no	—
37	C₂×C₄⋊Dic₃	—	4	2	no	—
38	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₃×C₄×C₈	D₄	1	2	yes	—
40	#818	—	2	4	no	—
41	#818	—	4	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	C₂²×D₈	—	2	1	no	—
46	C₂²×D₈	C₂×D₃	1	2	yes	—
47	C₂×#127	—	2	1	no	—
48	C₂²×C₁₆	—	3	2	no	—
49	#493	—	2	2	no	—
50	#493	Dic₃	1	4	yes	complemented
51	C₂×#107	—	3	2	no	—
52	C₂×#107	—	6	1	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	#384	—	6	1	no	—
55	#357	—	6	8	no	—
56	C₄×C₁₆	—	6	2	no	—
57	C₂⁴×C₃	—	4	2	no	—
58	C₂×C₃×D₄	—	2	4	no	—
59	C₂×C₃×D₄	—	4	8	no	—
60	C₂×C₃×D₄	—	4	2	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂×C₃×D₄	—	4	2	no	—
63	C₂×C₃×D₄	—	4	2	no	—
64	C₂×C₃×D₄	—	4	2	no	—
65	C₂×C₃×D₄	—	8	4	no	—
66	C₂×C₃×D₄	—	8	4	no	—
67	C₂²×C₃×C₄	D₈	1	2	yes	—
68	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₃	—	4	2	no	—
70	C₃×D₈	—	2	8	no	—
71	C₃×D₈	—	4	8	no	—
72	C₃×D₈	—	4	4	no	—
73	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃×C₈	D₈	1	2	yes	—
75	C₂×C₃×C₈	SD₁₆	1	2	yes	—
76	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₈	#41	1	2	yes	—
80	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
81	C₃×C₄²	SD₁₆	1	2	yes	—
82	C₄⋊Dic₃	—	4	2	no	—
83	C₄⋊Dic₃	—	4	2	no	—
84	C₄⋊Dic₃	—	8	4	no	—
85	C₃⋊C₁₆	—	2	4	no	—
86	C₂²×D₄	—	2	2	no	—
87	C₂²×D₄	—	4	2	no	—
88	C₂×D₈	—	2	4	no	—
89	C₂×D₈	—	2	4	no	—
90	C₂×D₈	—	2	4	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	4	2	no	—
93	C₂×D₈	—	4	2	no	—
94	C₂×D₈	C₂×Dic₃	1	4	yes	—
95	C₂²×C₈	#67	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
97	#127	—	2	4	no	—
98	#127	—	4	2	no	—
99	C₂×C₄⋊C₄	—	6	2	no	—
100	C₂×C₄⋊C₄	—	12	2	no	—
101	C₂×C₄²	#67	1	1	yes	characteristic
102	C₂×C₁₆	—	3	4	no	—
103	C₂×C₁₆	—	6	2	no	—
104	C₂×C₁₆	—	6	2	no	—
105	#107	—	6	4	no	—
106	#107	—	12	2	no	—
107	C₄×C₈	C₂×Dic₃	1	2	yes	—
108	C₄×C₈	#67	1	2	yes	—
109	C₂³×C₃	—	4	4	no	—
110	C₂³×C₃	—	8	2	no	—
111	C₂³×C₃	—	8	2	no	—
112	C₂³×C₃	—	8	4	no	—
113	C₂³×C₃	—	8	4	no	—
114	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
115	C₃×D₄	—	4	4	no	—
116	C₃×D₄	—	4	4	no	—
117	C₃×D₄	—	8	16	no	—
118	C₃×D₄	—	8	2	no	—
119	C₃×D₄	—	8	2	no	—
120	C₂×C₃×C₄	#102	1	4	yes	—
121	C₂×C₃×C₄	D₁₆	1	4	yes	—
122	C₂×C₃×C₄	SD₃₂	1	4	yes	—
123	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	#124	1	1	yes	characteristic
125	C₂×C₃×C₄	#127	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×Dic₃	—	8	2	no	—
130	C₂×Dic₃	—	8	4	no	—
131	C₃×C₈	#102	1	4	yes	—
132	C₃×C₈	C₄×D₄	1	2	yes	—
133	C₃×C₈	#121	1	2	yes	—
134	C₂⁴	—	4	2	no	—
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	—	4	8	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	8	4	no	—
143	C₂×D₄	—	8	4	no	—
144	C₂²×C₄	—	12	2	no	—
145	C₂²×C₄	#213	1	2	yes	—
146	C₂²×C₄	C₂×#67	1	1	yes	characteristic
147	D₈	—	2	8	no	—
148	D₈	—	4	8	no	—
149	D₈	—	4	4	no	—
150	C₂×C₈	#213	1	2	yes	—
151	C₂×C₈	#214	1	2	yes	—
152	C₂×C₈	#217	1	2	yes	—
153	C₂×C₈	D₃×D₄	1	2	yes	—
154	C₂×C₈	#237	1	2	yes	—
155	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
156	C₂×C₈	C₂×#67	1	1	yes	characteristic
157	C₄⋊C₄	—	12	2	no	—
158	C₄⋊C₄	—	12	2	no	—
159	C₄⋊C₄	—	24	4	no	—
160	C₄²	#214	1	2	yes	—
161	C₄²	#217	1	2	yes	—
162	C₁₆	—	6	4	no	—
163	C₂²×C₃	—	8	8	no	—
164	C₂²×C₃	—	8	4	no	—
165	C₂²×C₃	—	16	4	no	—
166	C₂²×C₃	—	16	2	no	—
167	C₂²×C₃	—	16	2	no	—
168	C₂²×C₃	C₂×#102	1	1	yes	characteristic
169	C₂²×C₃	#486	1	1	yes	characteristic
170	C₂²×C₃	#493	1	1	yes	characteristic
171	C₂²×C₃	C₂×D₁₆	1	2	yes	—
172	C₂²×C₃	C₂×SD₃₂	1	2	yes	—
173	C₃×C₄	#357	1	8	yes	complemented
174	C₃×C₄	#390	1	1	yes	characteristic
175	C₃×C₄	C₂×#102	1	1	yes	characteristic
176	C₃×C₄	#486	1	1	yes	characteristic
177	C₃×C₄	#492	1	1	yes	characteristic
178	Dic₃	—	16	2	no	—
179	C₂³	—	4	4	no	—
180	C₂³	—	8	4	no	—
181	C₂³	—	8	4	no	—
182	C₂³	—	8	2	no	—
183	C₂³	—	8	2	no	—
184	C₂³	C₂×#213	1	1	yes	characteristic
185	D₄	—	4	4	no	—
186	D₄	—	4	4	no	—
187	D₄	—	8	2	no	—
188	D₄	—	8	16	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	24	4	no	—
191	C₂×C₄	—	24	2	no	—
192	C₂×C₄	#826	1	4	yes	—
193	C₂×C₄	#827	1	4	yes	—
194	C₂×C₄	#832	1	4	yes	—
195	C₂×C₄	C₂×#213	1	1	yes	characteristic
196	C₂×C₄	C₂×#214	1	1	yes	characteristic
197	C₂×C₄	C₂×#214	1	1	yes	characteristic
198	C₂×C₄	#936	1	1	yes	characteristic
199	C₂×C₄	#940	1	1	yes	characteristic
200	C₂×C₄	C₂×#217	1	1	yes	characteristic
201	C₈	#832	1	4	yes	—
202	C₈	D₄×Dic₃	1	2	yes	—
203	C₈	#938	1	2	yes	—
204	C₂×C₃	—	16	4	no	—
205	C₂×C₃	#1957	1	1	yes	characteristic
206	C₂×C₃	C₂×#357	1	2	yes	—
207	C₂×C₃	#2240	1	2	yes	—
208	C₂×C₃	#2246	1	1	yes	characteristic
209	C₂×C₃	#2248	1	1	yes	characteristic
210	C₂²	—	8	4	no	—
211	C₂²	—	8	8	no	—
212	C₂²	—	16	4	no	—
213	C₂²	—	16	2	no	—
214	C₂²	—	16	2	no	—
215	C₂²	#5070	1	1	yes	characteristic
216	C₂²	#5073	1	1	yes	characteristic
217	C₂²	C₂×#826	1	2	yes	—
218	C₂²	C₂×#827	1	2	yes	—
219	C₂²	C₂×#832	1	1	yes	characteristic
220	C₄	—	48	2	no	—
221	C₄	#4562	1	8	yes	complemented
222	C₄	#5070	1	1	yes	characteristic
223	C₄	#5076	1	1	yes	characteristic
224	C₄	C₂×#832	1	1	yes	characteristic
225	C₄	#5221	1	1	yes	characteristic
226	C₃	#13202	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	16	4	no	—
228	C₂	#71288	1	2	yes	—
229	C₂	#71290	1	1	yes	characteristic
230	C₂	#71291	1	1	yes	characteristic
231	C₂	#71764	1	1	yes	characteristic
232	C₂	C₂×#4562	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#213
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×#493	C₂
socle	C₂³×C₃	C₂×D₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#493	D₃
hypocenter	C₃	#13202
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
2. C₂×#4562	C₂	semidirect
3. #69765	C₂	semidirect
4. C₂×C₄×C₃⋊C₁₆	C₂	semidirect
10. C₃×#493	C₄	semidirect
44. C₂×C₃⋊C₁₆	D₄	semidirect
50. #493	Dic₃	semidirect
173. C₃×C₄	#357	semidirect
221. C₄	#4562	semidirect
226. C₃	#13202	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#493	2
C₂×#4562	2
#69765	2
C₂×C₄×C₃⋊C₁₆	2
#13202	3