Subgroups — #10694944

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×#4682	C₂	1	1	yes	maximal, characteristic
2	C₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#69582	C₂	1	2	yes	maximal
5	#69579	C₂	1	2	yes	maximal
6	#9823	—	3	1	no	maximal, Sylow
7	C₂×#869	C₂²	1	2	yes	—
8	C₂×C₃×#107	—	2	2	no	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×#384	C₂²	1	2	yes	—
11	C₂×#818	—	2	2	no	—
12	C₂×#810	—	2	4	no	—
13	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	#4682	—	2	2	no	—
16	#4682	C₄	1	4	yes	complemented
17	#4529	—	2	4	no	—
18	C₈×C₃⋊C₈	—	2	2	no	—
19	C₂×#500	—	3	1	no	—
20	#1990	—	3	2	no	—
21	#1956	—	3	2	no	—
22	#1845	D₃	1	1	yes	characteristic
23	C₂×C₈²	—	3	1	no	—
24	C₂²×Dic₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂²×C₃⋊C₈	D₄	1	2	yes	—
33	C₂²×C₃⋊C₈	D₄	1	2	yes	—
34	#869	—	2	2	no	—
35	#869	—	2	2	no	—
36	#869	—	2	2	no	—
37	#869	C₂×C₄	1	4	yes	—
38	C₃×#107	—	4	4	no	—
39	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
40	C₃×C₄×C₈	D₄	1	2	yes	—
41	#818	—	2	8	no	—
42	#810	—	4	8	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	—	3	2	no	—
48	#500	—	3	4	no	—
49	#500	—	6	2	no	—
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	C₂×D₃	1	2	yes	—
57	#340	—	6	4	no	—
58	C₈²	—	6	2	no	—
59	C₂²×C₃×C₄	—	4	2	no	—
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₃	—	4	2	no	—
64	C₂×Dic₆	—	2	4	no	—
65	C₂×Dic₆	—	4	4	no	—
66	C₂×Dic₆	—	4	4	no	—
67	C₂×Dic₆	—	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	—	4	4	no	—
72	C₃×C₄⋊C₄	—	4	4	no	—
73	C₃×C₄⋊C₄	—	4	4	no	—
74	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄²	#41	1	1	yes	characteristic
77	C₃×C₄²	#41	1	1	yes	characteristic
78	C₄⋊Dic₃	—	2	4	no	—
79	C₄⋊Dic₃	—	2	4	no	—
80	C₄⋊Dic₃	—	4	2	no	—
81	C₄⋊Dic₃	—	4	2	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	—	6	2	no	—
89	C₂²×C₈	—	3	2	no	—
90	C₂²×C₈	—	3	2	no	—
91	C₂²×C₈	#67	1	2	yes	—
92	C₄⋊Q₈	—	3	4	no	—
93	C₄⋊Q₈	—	6	2	no	—
94	C₄⋊Q₈	—	6	2	no	—
95	C₄⋊Q₈	—	6	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	2	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	—	4	4	no	—
103	#107	—	6	8	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	C₄×D₃	1	2	yes	—
110	C₄×C₈	D₁₂	1	2	yes	—
111	C₂³×C₃	#127	1	1	yes	characteristic
112	C₂×C₃×C₄	—	8	2	no	—
113	C₂×C₃×C₄	—	8	2	no	—
114	C₂×C₃×C₄	—	8	2	no	—
115	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	#121	1	1	yes	characteristic
119	C₂×C₃×C₄	#121	1	1	yes	characteristic
120	C₂×C₃×C₄	#124	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
122	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
123	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
124	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
125	C₂×C₃×C₄	#135	1	2	yes	—
126	C₂×C₃×C₄	#135	1	2	yes	—
127	C₂×Dic₃	—	4	4	no	—
128	C₂×Dic₃	—	8	2	no	—
129	C₂×Dic₃	—	8	2	no	—
130	Dic₆	—	4	8	no	—
131	Dic₆	—	8	4	no	—
132	C₃×C₈	#103	1	8	yes	complemented
133	C₃⋊C₈	—	2	4	no	—
134	C₃⋊C₈	—	2	4	no	—
135	C₂×Q₈	—	6	4	no	—
136	C₂×Q₈	—	12	2	no	—
137	C₂×Q₈	—	12	4	no	—
138	C₂×Q₈	—	12	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	(C₂×D₃)⋊C₄	1	4	yes	—
151	C₂×C₈	#214	1	4	yes	—
152	C₂×C₈	#216	1	4	yes	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	6	4	no	—
157	C₄⋊C₄	—	6	4	no	—
158	C₄⋊C₄	—	12	2	no	—
159	C₄⋊C₄	—	12	2	no	—
160	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
161	C₄²	C₂×D₁₂	1	1	yes	characteristic
162	C₄²	#239	1	1	yes	characteristic
163	C₄²	#239	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	—	16	2	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	—	8	4	no	—
179	C₂³	#940	1	1	yes	characteristic
180	Q₈	—	12	8	no	—
181	Q₈	—	24	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	12	4	no	—
186	C₂×C₄	—	24	2	no	—
187	C₂×C₄	—	24	2	no	—
188	C₂×C₄	#893	1	1	yes	characteristic
189	C₂×C₄	#893	1	1	yes	characteristic
190	C₂×C₄	#895	1	1	yes	characteristic
191	C₂×C₄	#895	1	1	yes	characteristic
192	C₂×C₄	D₃×D₈	1	2	yes	—
193	C₂×C₄	#912	1	2	yes	—
194	C₂×C₄	D₃×Q₁₆	1	2	yes	—
195	C₂×C₄	#919	1	2	yes	—
196	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₂×#214	1	2	yes	—
198	C₂×C₄	C₂×#216	1	2	yes	—
199	C₂×C₄	#947	1	1	yes	characteristic
200	C₈	—	6	4	no	—
201	C₈	—	6	4	no	—
202	C₈	#810	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	#4988	1	1	yes	characteristic
211	C₂²	#5076	1	1	yes	characteristic
212	C₂²	#5092	1	1	yes	characteristic
213	C₂²	#5151	1	1	yes	characteristic
214	C₂²	#5155	1	1	yes	characteristic
215	C₂²	#5183	1	1	yes	characteristic
216	C₂²	#5191	1	1	yes	characteristic
217	C₄	—	16	2	no	—
218	C₄	—	24	4	no	—
219	C₄	#4872	1	2	yes	—
220	C₄	#4873	1	2	yes	—
221	C₄	#4883	1	2	yes	—
222	C₄	#4887	1	2	yes	—
223	C₄	C₂×#810	1	2	yes	—
224	C₄	#5093	1	2	yes	—
225	C₃	#9823	1	1	yes	characteristic, complemented, Sylow
226	C₂	#69579	1	1	yes	characteristic
227	C₂	#70627	1	1	yes	characteristic
228	C₂	#70628	1	1	yes	characteristic
229	C₂	#71274	1	1	yes	characteristic
230	C₂	#71296	1	1	yes	characteristic
231	C₂	#71361	1	1	yes	characteristic
232	C₂	#71362	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#940
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1845	C₂
socle, prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1845	D₃
hypocenter	C₃	#9823
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
16. #4682	C₄	semidirect
132. C₃×C₈	#103	semidirect
202. C₈	#810	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×#4682	2
C₃×#1845	2
C₂×C₈×C₃⋊C₈	2
#69582	2
#69579	2
#9823	3