Subgroups — #10694412

Order 768.

454 conjugacy classes of subgroups.

The subgroup lattice has 244 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	#71779	C₂	1	2	yes	maximal
3	C₂×#4546	C₂	1	2	yes	maximal
4	C₈×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#4458	C₂	1	1	yes	maximal, characteristic
6	#9730	—	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	1	no	—
11	C₂×C₃×#107	—	2	1	no	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₃	—	2	1	no	—
14	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	#4550	—	2	1	no	—
18	#4546	—	2	4	no	—
19	#4529	—	2	4	no	—
20	#4458	—	2	2	no	—
21	C₂×#500	D₃	1	1	yes	characteristic
22	#1987	—	3	2	no	—
23	C₈×C₄⋊C₄	—	3	1	no	—
24	C₂×#335	—	3	1	no	—
25	C₂×#333	—	3	2	no	—
26	C₂²×C₃×Q₈	—	2	2	no	—
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₃×C₄⋊Q₈	—	2	2	no	—
30	C₃×C₄⋊Q₈	—	2	2	no	—
31	C₃×C₄⋊Q₈	—	2	2	no	—
32	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
33	C₂×C₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₃	—	2	1	no	—
38	C₂²×C₃⋊C₈	—	2	1	no	—
39	C₂²×C₃⋊C₈	—	2	2	no	—
40	C₃×#107	—	2	4	no	—
41	C₃×#107	—	2	4	no	—
42	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₃×C₄×C₈	D₄	1	2	yes	—
44	C₃⋊#95	—	2	1	no	—
45	C₈×Dic₃	—	4	2	no	—
46	C₄⋊(C₃⋊C₈)	—	2	1	no	—
47	C₄⋊(C₃⋊C₈)	—	2	2	no	—
48	C₄⋊(C₃⋊C₈)	—	2	2	no	—
49	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
50	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
51	#500	—	2	2	no	—
52	#500	Dic₃	1	4	yes	complemented
53	C₂×#107	—	2	1	no	—
54	C₂×#107	—	2	1	no	—
55	C₂×C₄⋊C₈	—	3	1	no	—
56	C₂×C₄⋊C₈	—	3	2	no	—
57	C₂×C₄×C₈	—	6	1	no	—
58	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₄⋊C₄	—	3	1	no	—
60	#340	—	6	4	no	—
61	#336	—	6	1	no	—
62	#335	—	6	2	no	—
63	#333	—	6	4	no	—
64	C₂×C₃×Q₈	—	2	4	no	—
65	C₂×C₃×Q₈	—	4	2	no	—
66	C₂×C₃×Q₈	—	8	4	no	—
67	C₂²×C₃×C₄	—	4	2	no	—
68	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	#41	1	1	yes	characteristic
71	C₂²×Dic₃	—	2	2	no	—
72	C₂×C₃×C₈	—	2	1	no	—
73	C₂×C₃×C₈	—	2	1	no	—
74	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃×C₈	SD₁₆	1	2	yes	—
77	C₂×C₃×C₈	Q₁₆	1	2	yes	—
78	C₃×C₄⋊C₄	—	2	4	no	—
79	C₃×C₄⋊C₄	—	2	4	no	—
80	C₃×C₄⋊C₄	—	4	2	no	—
81	C₃×C₄⋊C₄	—	4	2	no	—
82	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
85	C₃×C₄²	#41	1	1	yes	characteristic
86	C₄⋊Dic₃	—	2	2	no	—
87	C₄⋊Dic₃	Q₁₆	1	4	yes	—
88	C₄×Dic₃	—	4	1	no	—
89	C₄×Dic₃	—	4	1	no	—
90	C₂×C₃⋊C₈	—	2	2	no	—
91	C₂×C₃⋊C₈	—	4	1	no	—
92	C₂×C₃⋊C₈	—	4	1	no	—
93	C₂×C₃⋊C₈	—	4	2	no	—
94	C₂×C₃⋊C₈	—	4	4	no	—
95	C₂²×Q₈	—	2	2	no	—
96	C₂²×C₈	—	6	1	no	—
97	C₂²×C₈	—	6	2	no	—
98	C₂²×C₈	#67	1	1	yes	characteristic
99	C₂²×C₈	#67	1	1	yes	characteristic
100	C₄⋊Q₈	—	2	2	no	—
101	C₄⋊Q₈	—	2	2	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	3	2	no	—
107	C₂×C₄²	—	6	1	no	—
108	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
109	#107	—	2	4	no	—
110	#107	—	2	4	no	—
111	C₄⋊C₈	—	3	2	no	—
112	C₄⋊C₈	—	6	2	no	—
113	C₄⋊C₈	—	6	1	no	—
114	C₄⋊C₈	—	6	2	no	—
115	C₄×C₈	—	12	2	no	—
116	C₄×C₈	C₂×Dic₃	1	2	yes	—
117	C₄×C₈	#67	1	2	yes	—
118	#95	—	6	1	no	—
119	C₂³×C₃	#121	1	1	yes	characteristic
120	C₃×Q₈	—	8	4	no	—
121	C₃×Q₈	—	8	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	4	4	no	—
124	C₂×C₃×C₄	—	8	4	no	—
125	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	#121	1	1	yes	characteristic
129	C₂×C₃×C₄	#124	1	1	yes	characteristic
130	C₂×C₃×C₄	#127	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
133	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₃×C₄	#135	1	2	yes	—
135	C₂×C₃×C₄	#137	1	1	yes	characteristic
136	C₂×C₃×C₄	#137	1	1	yes	characteristic
137	C₂×Dic₃	—	2	4	no	—
138	C₂×Dic₃	—	4	2	no	—
139	C₂×Dic₃	—	4	2	no	—
140	C₃×C₈	—	2	2	no	—
141	C₃×C₈	#103	1	4	yes	—
142	C₃⋊C₈	—	4	2	no	—
143	C₃⋊C₈	—	8	2	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	8	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	6	2	no	—
149	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
150	C₂²×C₄	#237	1	1	yes	characteristic
151	C₂²×C₄	C₂×#67	1	1	yes	characteristic
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	6	2	no	—
155	C₂×C₈	—	12	1	no	—
156	C₂×C₈	—	12	1	no	—
157	C₂×C₈	—	12	4	no	—
158	C₂×C₈	—	12	2	no	—
159	C₂×C₈	#215	1	2	yes	—
160	C₂×C₈	#216	1	2	yes	—
161	C₂×C₈	#217	1	2	yes	—
162	C₂×C₈	#217	1	2	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	3	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	6	2	no	—
169	C₄²	—	12	1	no	—
170	C₄²	—	12	1	no	—
171	C₄²	D₃×D₄	1	1	yes	characteristic
172	C₄²	#237	1	1	yes	characteristic
173	C₄²	C₂²×Dic₃	1	1	yes	characteristic
174	C₄²	C₂×#67	1	1	yes	characteristic
175	C₂²×C₃	#390	1	1	yes	characteristic
176	C₂²×C₃	#461	1	1	yes	characteristic
177	C₂²×C₃	#462	1	1	yes	characteristic
178	C₂²×C₃	#467	1	2	yes	—
179	C₂²×C₃	#472	1	2	yes	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	8	4	no	—
183	C₃×C₄	C₂×#103	1	1	yes	characteristic
184	C₃×C₄	#419	1	1	yes	characteristic
185	C₃×C₄	C₄×Q₁₆	1	2	yes	—
186	C₃×C₄	#461	1	1	yes	characteristic
187	C₃×C₄	#462	1	1	yes	characteristic
188	C₃×C₄	#494	1	1	yes	characteristic
189	C₃×C₄	#495	1	1	yes	characteristic
190	Dic₃	—	4	4	no	—
191	C₂³	#938	1	1	yes	characteristic
192	Q₈	—	8	4	no	—
193	Q₈	—	8	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	6	4	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	—	12	2	no	—
199	C₂×C₄	—	12	2	no	—
200	C₂×C₄	D₃×Q₁₆	1	2	yes	—
201	C₂×C₄	#919	1	2	yes	—
202	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
203	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
204	C₂×C₄	#936	1	1	yes	characteristic
205	C₂×C₄	#938	1	1	yes	characteristic
206	C₂×C₄	#940	1	1	yes	characteristic
207	C₂×C₄	C₂×#215	1	1	yes	characteristic
208	C₂×C₄	#942	1	1	yes	characteristic
209	C₂×C₄	#942	1	1	yes	characteristic
210	C₂×C₄	C₂×#216	1	1	yes	characteristic
211	C₂×C₄	C₂×#217	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	12	2	no	—
214	C₈	—	24	2	no	—
215	C₈	#835	1	4	yes	—
216	C₂×C₃	#1675	1	2	yes	—
217	C₂×C₃	#1676	1	2	yes	—
218	C₂×C₃	#1969	1	1	yes	characteristic
219	C₂×C₃	#1978	1	1	yes	characteristic
220	C₂×C₃	#1981	1	1	yes	characteristic
221	C₂²	#5083	1	1	yes	characteristic
222	C₂²	#5090	1	1	yes	characteristic
223	C₂²	#5160	1	2	yes	—
224	C₂²	#5188	1	2	yes	—
225	C₂²	#5221	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	8	4	no	—
229	C₄	—	12	4	no	—
230	C₄	#5083	1	1	yes	characteristic
231	C₄	#5090	1	1	yes	characteristic
232	C₄	Dic₃×Q₁₆	1	2	yes	—
233	C₄	#5183	1	1	yes	characteristic
234	C₄	#5191	1	1	yes	characteristic
235	C₄	C₂×#835	1	1	yes	characteristic
236	C₄	#5225	1	1	yes	characteristic
237	C₃	#9730	1	1	yes	characteristic, complemented, Sylow
238	C₂	#71372	1	2	yes	—
239	C₂	#71378	1	2	yes	—
240	C₂	#71778	1	1	yes	characteristic
241	C₂	#72041	1	1	yes	characteristic
242	C₂	#72066	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#938
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#500	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#500	D₃
hypocenter	C₃	#9730
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
52. #500	Dic₃	semidirect
237. C₃	#9730	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
#71779	2
C₂×#4546	2
C₈×C₄⋊Dic₃	2
C₂×#4458	2
#9730	3