Subgroups — #10694416

Order 768.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4682	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#4486	C₂	1	2	yes	maximal
4	C₂×#4458	C₂	1	1	yes	maximal, characteristic
5	#69486	C₂	1	2	yes	maximal
6	#9731	—	3	1	no	maximal, Sylow
7	C₂×#869	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	—	2	1	no	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×#818	—	2	1	no	—
13	C₂×#818	—	2	1	no	—
14	C₂×#810	—	2	4	no	—
15	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	#4682	—	2	2	no	—
17	#4682	C₄	1	4	yes	complemented
18	C₃×#336	—	2	1	no	—
19	#4486	—	2	4	no	—
20	#4458	—	2	2	no	—
21	C₂×#500	—	3	1	no	—
22	#1981	—	3	2	no	—
23	C₈×C₄⋊C₄	D₃	1	1	yes	characteristic
24	C₂×#335	—	3	1	no	—
25	C₂×#332	—	3	2	no	—
26	C₂²×Dic₆	—	2	2	no	—
27	C₂²×C₃×C₈	—	2	1	no	—
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄⋊C₄	D₄	1	2	yes	—
31	C₂×C₃×C₄²	—	2	1	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	C₂²×C₃⋊C₈	—	2	2	no	—
36	#869	—	2	2	no	—
37	#869	—	2	2	no	—
38	#869	—	2	2	no	—
39	#869	C₂×C₄	1	4	yes	—
40	C₃×C₄⋊C₈	—	2	1	no	—
41	C₃×C₄⋊C₈	D₄	1	2	yes	—
42	C₃×C₄×C₈	—	4	2	no	—
43	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
44	C₃×C₄×C₈	D₄	1	2	yes	—
45	C₃×#95	—	2	1	no	—
46	#818	—	2	4	no	—
47	#818	—	2	4	no	—
48	#810	—	4	8	no	—
49	C₄⋊(C₃⋊C₈)	—	2	2	no	—
50	C₄⋊(C₃⋊C₈)	—	2	2	no	—
51	C₂×C₄⋊Q₈	—	3	2	no	—
52	#500	—	3	4	no	—
53	#500	—	6	2	no	—
54	C₂×#107	—	6	1	no	—
55	C₂×#107	—	6	1	no	—
56	C₂×C₄⋊C₈	—	3	2	no	—
57	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×#103	—	6	4	no	—
59	C₂×C₄×C₈	—	2	1	no	—
60	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
61	C₄×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
62	#336	—	2	1	no	—
63	#335	—	6	2	no	—
64	#332	—	6	4	no	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	#41	1	1	yes	characteristic
69	C₂²×Dic₃	—	4	2	no	—
70	C₂×Dic₆	—	2	4	no	—
71	C₂×Dic₆	—	4	4	no	—
72	C₂×Dic₆	—	4	4	no	—
73	C₂×Dic₆	—	4	2	no	—
74	C₂×C₃×C₈	—	2	1	no	—
75	C₂×C₃×C₈	—	2	1	no	—
76	C₂×C₃×C₈	—	2	2	no	—
77	C₂×C₃×C₈	—	4	1	no	—
78	C₂×C₃×C₈	—	4	1	no	—
79	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃×C₈	SD₁₆	1	2	yes	—
82	C₂×C₃×C₈	Q₁₆	1	2	yes	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	D₈	1	4	yes	—
85	C₃×C₄²	—	4	1	no	—
86	C₃×C₄²	—	4	1	no	—
87	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄²	#41	1	1	yes	characteristic
91	C₄⋊Dic₃	—	2	4	no	—
92	C₄⋊Dic₃	—	2	4	no	—
93	C₄⋊Dic₃	—	4	2	no	—
94	C₄⋊Dic₃	—	4	2	no	—
95	C₂×C₃⋊C₈	—	4	2	no	—
96	C₂×C₃⋊C₈	—	4	4	no	—
97	C₂²×Q₈	—	6	2	no	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	6	2	no	—
100	C₂²×C₈	#67	1	1	yes	characteristic
101	C₂²×C₈	#67	1	1	yes	characteristic
102	C₄⋊Q₈	—	3	4	no	—
103	C₄⋊Q₈	—	6	2	no	—
104	C₄⋊Q₈	—	6	2	no	—
105	C₄⋊Q₈	—	6	2	no	—
106	C₂×C₄⋊C₄	—	6	2	no	—
107	C₂×C₄⋊C₄	—	6	2	no	—
108	C₂×C₄⋊C₄	D₁₂	1	2	yes	—
109	C₂×C₄²	—	2	1	no	—
110	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
111	#107	—	6	4	no	—
112	#107	—	6	4	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	6	2	no	—
115	C₄⋊C₈	—	6	2	no	—
116	C₄⋊C₈	D₁₂	1	2	yes	—
117	#103	—	12	8	no	—
118	C₄×C₈	—	4	2	no	—
119	C₄×C₈	C₄×D₃	1	2	yes	—
120	C₄×C₈	D₁₂	1	2	yes	—
121	#95	—	2	1	no	—
122	C₂³×C₃	#121	1	1	yes	characteristic
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	4	2	no	—
126	C₂×C₃×C₄	—	4	2	no	—
127	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	#121	1	1	yes	characteristic
131	C₂×C₃×C₄	#124	1	1	yes	characteristic
132	C₂×C₃×C₄	#127	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
134	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
136	C₂×C₃×C₄	#135	1	2	yes	—
137	C₂×C₃×C₄	#137	1	1	yes	characteristic
138	C₂×C₃×C₄	#137	1	1	yes	characteristic
139	C₂×Dic₃	—	4	4	no	—
140	C₂×Dic₃	—	8	4	no	—
141	Dic₆	—	4	8	no	—
142	Dic₆	—	8	2	no	—
143	Dic₆	—	8	2	no	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃×C₈	#103	1	4	yes	complemented
147	C₃⋊C₈	—	8	2	no	—
148	C₂×Q₈	—	6	4	no	—
149	C₂×Q₈	—	12	2	no	—
150	C₂×Q₈	—	12	4	no	—
151	C₂×Q₈	—	12	4	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	12	2	no	—
154	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
155	C₂²×C₄	#235	1	1	yes	characteristic
156	C₂²×C₄	C₂×#67	1	1	yes	characteristic
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	12	2	no	—
163	C₂×C₈	—	12	4	no	—
164	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
165	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
166	C₂×C₈	#214	1	2	yes	—
167	C₂×C₈	#216	1	2	yes	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	6	4	no	—
170	C₄⋊C₄	—	6	4	no	—
171	C₄⋊C₄	—	12	2	no	—
172	C₄⋊C₄	—	12	2	no	—
173	C₄⋊C₄	D₂₄	1	4	yes	—
174	C₄²	—	4	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
177	C₄²	C₂×D₁₂	1	1	yes	characteristic
178	C₄²	C₂×D₁₂	1	1	yes	characteristic
179	C₄²	#235	1	1	yes	characteristic
180	C₂²×C₃	#390	1	1	yes	characteristic
181	C₂²×C₃	#461	1	1	yes	characteristic
182	C₂²×C₃	#462	1	1	yes	characteristic
183	C₂²×C₃	#466	1	2	yes	—
184	C₂²×C₃	#473	1	2	yes	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	4	4	no	—
188	C₃×C₄	C₂×#103	1	1	yes	characteristic
189	C₃×C₄	#419	1	1	yes	characteristic
190	C₃×C₄	C₄×D₈	1	2	yes	—
191	C₃×C₄	#461	1	1	yes	characteristic
192	C₃×C₄	#462	1	1	yes	characteristic
193	C₃×C₄	#493	1	1	yes	characteristic
194	C₃×C₄	#495	1	1	yes	characteristic
195	Dic₃	—	8	4	no	—
196	C₂³	#930	1	1	yes	characteristic
197	Q₈	—	12	8	no	—
198	Q₈	—	24	2	no	—
199	Q₈	—	24	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	12	4	no	—
205	C₂×C₄	—	24	4	no	—
206	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
207	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
208	C₂×C₄	#874	1	1	yes	characteristic
209	C₂×C₄	#875	1	1	yes	characteristic
210	C₂×C₄	C₂×D₂₄	1	2	yes	—
211	C₂×C₄	#904	1	2	yes	—
212	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₂×C₄	#930	1	1	yes	characteristic
214	C₂×C₄	C₂×#214	1	1	yes	characteristic
215	C₂×C₄	C₂×#216	1	1	yes	characteristic
216	C₂×C₄	#951	1	1	yes	characteristic
217	C₂×C₄	#951	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	4	2	no	—
220	C₈	—	24	2	no	—
221	C₈	#810	1	4	yes	complemented
222	C₂×C₃	#1673	1	2	yes	—
223	C₂×C₃	#1674	1	2	yes	—
224	C₂×C₃	#1970	1	1	yes	characteristic
225	C₂×C₃	#1979	1	1	yes	characteristic
226	C₂×C₃	#1981	1	1	yes	characteristic
227	C₂²	#4939	1	1	yes	characteristic
228	C₂²	#5019	1	1	yes	characteristic
229	C₂²	#5031	1	1	yes	characteristic
230	C₂²	#5115	1	2	yes	—
231	C₂²	#5143	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	4	no	—
235	C₄	—	24	4	no	—
236	C₄	C₄×D₂₄	1	2	yes	—
237	C₄	#4695	1	1	yes	characteristic
238	C₄	#4696	1	1	yes	characteristic
239	C₄	C₂×#810	1	1	yes	characteristic
240	C₄	#5004	1	1	yes	characteristic
241	C₄	#5019	1	1	yes	characteristic
242	C₄	#5031	1	1	yes	characteristic
243	C₃	#9731	1	1	yes	characteristic, complemented, Sylow
244	C₂	#69486	1	1	yes	characteristic
245	C₂	#69776	1	1	yes	characteristic
246	C₂	#69904	1	1	yes	characteristic
247	C₂	#70852	1	2	yes	—
248	C₂	#70944	1	2	yes	—

	subgroup	G/H
center	C₂³	#930
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₈×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₈×C₄⋊C₄	D₃
hypocenter	C₃	#9731
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. #4682	C₄	semidirect
146. C₃×C₈	#103	semidirect
221. C₈	#810	semidirect
243. C₃	#9731	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4682	2
C₃×C₈×C₄⋊C₄	2
C₂×#4486	2
C₂×#4458	2
#69486	2
#9731	3