Subgroups — #10694947

Order 768.

526 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×#500	C₂	1	1	yes	maximal, characteristic
2	#71787	C₂	1	2	yes	maximal
3	#71785	C₂	1	2	yes	maximal
4	#69765	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#9823	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#500	—	2	2	no	—
9	C₃×#500	C₄	1	4	yes	complemented
10	C₂×C₃×#107	—	2	2	no	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#835	—	2	4	no	—
13	C₂×#818	—	2	2	no	—
14	#4935	C₂²	1	2	yes	—
15	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	#4529	—	2	4	no	—
18	C₈×C₃⋊C₈	—	2	2	no	—
19	C₂×#500	D₃	1	1	yes	characteristic
20	#1990	—	3	2	no	—
21	#1956	—	3	2	no	—
22	#1845	—	3	1	no	—
23	C₂×C₈²	—	3	1	no	—
24	C₂²×C₃×Q₈	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	2	yes	—
26	C₃×C₄⋊Q₈	—	2	2	no	—
27	C₃×C₄⋊Q₈	—	2	2	no	—
28	C₃×C₄⋊Q₈	—	2	2	no	—
29	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₃	—	2	2	no	—
34	C₂×C₄⋊Dic₃	—	2	2	no	—
35	C₂×C₄⋊Dic₃	—	2	2	no	—
36	C₂²×C₃⋊C₈	D₄	1	2	yes	—
37	C₂²×C₃⋊C₈	D₄	1	2	yes	—
38	C₃×#107	—	2	8	no	—
39	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
40	C₃×C₄×C₈	D₄	1	2	yes	—
41	#835	—	4	8	no	—
42	#818	—	4	4	no	—
43	C₄×C₃⋊C₈	—	2	1	no	—
44	C₄×C₃⋊C₈	—	2	1	no	—
45	C₄×C₃⋊C₈	—	2	1	no	—
46	C₄×C₃⋊C₈	—	2	1	no	—
47	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
48	#500	—	2	2	no	—
49	#500	Dic₃	1	4	yes	complemented
50	C₂×#107	—	2	2	no	—
51	C₂×#107	—	6	2	no	—
52	C₂×#103	—	6	4	no	—
53	C₂×C₄×C₈	—	3	1	no	—
54	C₂×C₄×C₈	—	3	1	no	—
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#384	—	3	2	no	—
57	#340	—	6	4	no	—
58	C₈²	—	6	2	no	—
59	C₂×C₃×Q₈	—	2	4	no	—
60	C₂×C₃×Q₈	—	4	2	no	—
61	C₂×C₃×Q₈	—	4	4	no	—
62	C₂×C₃×Q₈	—	4	4	no	—
63	C₂²×C₃×C₄	—	4	2	no	—
64	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₃	—	4	2	no	—
68	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
69	C₂×C₃×C₈	SD₁₆	1	4	yes	—
70	C₂×C₃×C₈	Q₁₆	1	4	yes	—
71	C₃×C₄⋊C₄	—	2	4	no	—
72	C₃×C₄⋊C₄	—	2	4	no	—
73	C₃×C₄⋊C₄	—	4	2	no	—
74	C₃×C₄⋊C₄	—	4	2	no	—
75	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄²	#41	1	1	yes	characteristic
78	C₃×C₄²	#41	1	1	yes	characteristic
79	C₄⋊Dic₃	—	4	4	no	—
80	C₄⋊Dic₃	—	4	4	no	—
81	C₄⋊Dic₃	—	4	4	no	—
82	C₂×C₃⋊C₈	—	2	2	no	—
83	C₂×C₃⋊C₈	—	2	2	no	—
84	C₂×C₃⋊C₈	—	2	2	no	—
85	C₂×C₃⋊C₈	—	2	2	no	—
86	C₂×C₃⋊C₈	D₈	1	4	yes	—
87	C₂×C₃⋊C₈	Q₁₆	1	4	yes	—
88	C₂²×Q₈	—	2	2	no	—
89	C₂²×C₈	—	3	2	no	—
90	C₂²×C₈	—	3	2	no	—
91	C₂²×C₈	#67	1	2	yes	—
92	C₄⋊Q₈	—	2	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	—	2	2	no	—
95	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	6	2	no	—
99	C₂×C₄⋊C₄	—	6	2	no	—
100	C₂×C₄⋊C₄	—	6	2	no	—
101	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
102	#107	—	2	8	no	—
103	#107	—	12	4	no	—
104	#103	—	12	8	no	—
105	C₄×C₈	—	6	1	no	—
106	C₄×C₈	—	6	1	no	—
107	C₄×C₈	—	6	1	no	—
108	C₄×C₈	—	6	1	no	—
109	C₄×C₈	C₂×Dic₃	1	2	yes	—
110	C₄×C₈	#67	1	2	yes	—
111	C₂³×C₃	#127	1	1	yes	characteristic
112	C₃×Q₈	—	4	8	no	—
113	C₃×Q₈	—	8	4	no	—
114	C₂×C₃×C₄	—	4	4	no	—
115	C₂×C₃×C₄	—	8	2	no	—
116	C₂×C₃×C₄	—	8	2	no	—
117	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	#121	1	1	yes	characteristic
121	C₂×C₃×C₄	#121	1	1	yes	characteristic
122	C₂×C₃×C₄	#124	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
124	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
125	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
126	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
127	C₂×C₃×C₄	#135	1	2	yes	—
128	C₂×C₃×C₄	#135	1	2	yes	—
129	C₂×Dic₃	—	8	2	no	—
130	C₂×Dic₃	—	8	2	no	—
131	C₂×Dic₃	—	8	2	no	—
132	C₃×C₈	#103	1	8	yes	complemented
133	C₃⋊C₈	—	2	4	no	—
134	C₃⋊C₈	—	2	4	no	—
135	C₂×Q₈	—	2	4	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	4	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	12	2	no	—
141	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
142	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×#67	1	1	yes	characteristic
144	C₂×C₈	—	3	4	no	—
145	C₂×C₈	—	3	4	no	—
146	C₂×C₈	—	6	2	no	—
147	C₂×C₈	—	6	2	no	—
148	C₂×C₈	—	6	2	no	—
149	C₂×C₈	—	6	2	no	—
150	C₂×C₈	#215	1	4	yes	—
151	C₂×C₈	#216	1	4	yes	—
152	C₂×C₈	#217	1	4	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	12	4	no	—
158	C₄⋊C₄	—	12	4	no	—
159	C₄⋊C₄	—	12	4	no	—
160	C₄²	#237	1	1	yes	characteristic
161	C₄²	#237	1	1	yes	characteristic
162	C₄²	C₂²×Dic₃	1	1	yes	characteristic
163	C₄²	C₂×#67	1	1	yes	characteristic
164	C₂²×C₃	#390	1	1	yes	characteristic
165	C₂²×C₃	#492	1	1	yes	characteristic
166	C₂²×C₃	#493	1	1	yes	characteristic
167	C₂²×C₃	#494	1	1	yes	characteristic
168	C₂²×C₃	#494	1	1	yes	characteristic
169	C₂²×C₃	#495	1	1	yes	characteristic
170	C₂²×C₃	#495	1	1	yes	characteristic
171	C₃×C₄	—	8	4	no	—
172	C₃×C₄	C₂×#103	1	2	yes	—
173	C₃×C₄	C₄×D₈	1	2	yes	—
174	C₃×C₄	C₄×Q₁₆	1	2	yes	—
175	C₃×C₄	#466	1	2	yes	—
176	C₃×C₄	#467	1	2	yes	—
177	C₃×C₄	#487	1	2	yes	—
178	Dic₃	—	16	2	no	—
179	C₂³	#940	1	1	yes	characteristic
180	Q₈	—	4	8	no	—
181	Q₈	—	8	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	24	2	no	—
186	C₂×C₄	—	24	2	no	—
187	C₂×C₄	—	24	2	no	—
188	C₂×C₄	D₃×D₈	1	2	yes	—
189	C₂×C₄	#912	1	2	yes	—
190	C₂×C₄	D₃×Q₁₆	1	2	yes	—
191	C₂×C₄	#919	1	2	yes	—
192	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
194	C₂×C₄	#936	1	1	yes	characteristic
195	C₂×C₄	#938	1	1	yes	characteristic
196	C₂×C₄	#938	1	1	yes	characteristic
197	C₂×C₄	C₂×#215	1	2	yes	—
198	C₂×C₄	C₂×#216	1	2	yes	—
199	C₂×C₄	C₂×#217	1	1	yes	characteristic
200	C₈	—	6	4	no	—
201	C₈	—	6	4	no	—
202	C₈	#835	1	8	yes	complemented
203	C₂×C₃	#1706	1	1	yes	characteristic
204	C₂×C₃	#1707	1	1	yes	characteristic
205	C₂×C₃	#1710	1	1	yes	characteristic
206	C₂×C₃	#1713	1	1	yes	characteristic
207	C₂×C₃	#1956	1	1	yes	characteristic
208	C₂×C₃	#1969	1	1	yes	characteristic
209	C₂×C₃	#1970	1	1	yes	characteristic
210	C₂²	#5085	1	1	yes	characteristic
211	C₂²	#5092	1	1	yes	characteristic
212	C₂²	#5151	1	1	yes	characteristic
213	C₂²	#5155	1	1	yes	characteristic
214	C₂²	#5183	1	1	yes	characteristic
215	C₂²	#5191	1	1	yes	characteristic
216	C₂²	#5221	1	1	yes	characteristic
217	C₄	—	8	4	no	—
218	C₄	—	48	2	no	—
219	C₄	#5079	1	2	yes	—
220	C₄	Dic₃×D₈	1	2	yes	—
221	C₄	#5157	1	2	yes	—
222	C₄	Dic₃×Q₁₆	1	2	yes	—
223	C₄	#5188	1	2	yes	—
224	C₄	C₂×#835	1	2	yes	—
225	C₃	#9823	1	1	yes	characteristic, complemented, Sylow
226	C₂	#71359	1	1	yes	characteristic
227	C₂	#71360	1	1	yes	characteristic
228	C₂	#71361	1	1	yes	characteristic
229	C₂	#71362	1	1	yes	characteristic
230	C₂	#71785	1	1	yes	characteristic
231	C₂	#72038	1	1	yes	characteristic
232	C₂	#72066	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#940
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#500	C₂
socle, prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#500	D₃
hypocenter	C₃	#9823
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
9. C₃×#500	C₄	semidirect
49. #500	Dic₃	semidirect
132. C₃×C₈	#103	semidirect
202. C₈	#835	semidirect
225. C₃	#9823	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#500	2
#71787	2
#71785	2
#69765	2
C₂×C₈×C₃⋊C₈	2
#9823	3