Subgroups — #10714543

Order 768.

462 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4699	C₂	1	1	yes	maximal, characteristic
2	C₃×#1942	C₂	1	1	yes	maximal, characteristic
3	C₂×#4492	C₂	1	2	yes	maximal
4	C₂×#4492	C₂	1	2	yes	maximal
5	#67657	C₂	1	1	yes	maximal, characteristic
6	#13409	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	—	2	1	no	—
8	C₂²×Dic₁₂	C₂²	1	2	yes	—
9	C₂×#869	—	2	1	no	—
10	C₂×C₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#389	—	2	1	no	—
14	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
15	#4699	—	2	4	no	—
16	C₃×#340	—	2	1	no	—
17	#4492	—	2	4	no	—
18	#4492	—	2	4	no	—
19	C₂×#494	—	3	1	no	—
20	C₂×#358	—	3	2	no	—
21	C₂×#358	—	3	2	no	—
22	#1942	D₃	1	1	yes	characteristic
23	#1374	—	3	1	no	—
24	C₂²×Dic₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₃×C₄⋊C₄	—	4	1	no	—
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₃	—	4	1	no	—
32	C₂×C₄⋊Dic₃	—	4	1	no	—
33	C₂×Dic₁₂	—	2	2	no	—
34	C₂×Dic₁₂	—	2	2	no	—
35	C₂×Dic₁₂	—	2	2	no	—
36	C₂×Dic₁₂	—	2	2	no	—
37	C₂×Dic₁₂	—	2	4	no	—
38	C₂×Dic₁₂	—	2	2	no	—
39	C₂×Dic₁₂	—	4	2	no	—
40	C₂×Dic₁₂	—	4	2	no	—
41	C₂×Dic₁₂	D₄	1	4	yes	—
42	#869	—	2	4	no	—
43	#869	—	4	2	no	—
44	C₃×#107	—	2	1	no	—
45	C₃×#107	—	2	2	no	—
46	C₃×#107	D₄	1	2	yes	—
47	C₃×C₄×C₈	—	2	2	no	—
48	C₃×#95	—	4	1	no	—
49	C₂×C₃⋊C₁₆	—	2	2	no	—
50	C₂×C₃⋊C₁₆	—	2	2	no	—
51	C₂×C₃⋊C₁₆	—	2	2	no	—
52	C₂²×Q₁₆	—	3	2	no	—
53	C₂²×Q₁₆	—	6	1	no	—
54	C₂×C₄⋊Q₈	—	6	1	no	—
55	C₂²×C₁₆	—	3	2	no	—
56	#494	—	6	4	no	—
57	C₂×#107	C₂×D₃	1	1	yes	characteristic
58	C₂×#107	C₂×D₃	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	#389	—	2	1	no	—
61	#358	—	6	4	no	—
62	#358	—	6	4	no	—
63	#340	—	2	1	no	—
64	C₂²×C₃×C₄	—	4	1	no	—
65	C₂²×C₃×C₄	—	4	1	no	—
66	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
67	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₃	—	4	2	no	—
70	C₂×Dic₆	—	2	4	no	—
71	C₂×Dic₆	—	4	2	no	—
72	C₂×Dic₆	—	4	2	no	—
73	C₂×Dic₆	—	4	2	no	—
74	C₂×Dic₆	—	4	2	no	—
75	C₂×Dic₆	—	4	2	no	—
76	C₂×C₃×C₈	—	2	1	no	—
77	C₂×C₃×C₈	—	2	1	no	—
78	C₂×C₃×C₈	SD₁₆	1	2	yes	—
79	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₈	#41	1	1	yes	characteristic
82	C₂×C₃×C₈	#41	1	2	yes	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	—	4	1	no	—
85	C₃×C₄⋊C₄	—	4	1	no	—
86	C₃×C₄⋊C₄	—	4	1	no	—
87	C₃×C₄⋊C₄	—	8	2	no	—
88	C₃×C₄²	—	2	1	no	—
89	C₃×C₄²	—	2	1	no	—
90	C₄⋊Dic₃	—	4	4	no	—
91	C₄⋊Dic₃	—	4	4	no	—
92	Dic₁₂	—	2	4	no	—
93	Dic₁₂	—	2	4	no	—
94	Dic₁₂	—	4	8	no	—
95	Dic₁₂	—	4	2	no	—
96	Dic₁₂	—	4	2	no	—
97	C₃⋊C₁₆	—	4	2	no	—
98	C₂²×Q₈	—	6	2	no	—
99	C₂×Q₁₆	—	3	4	no	—
100	C₂×Q₁₆	—	6	4	no	—
101	C₂×Q₁₆	—	6	2	no	—
102	C₂×Q₁₆	—	6	2	no	—
103	C₂×Q₁₆	—	6	2	no	—
104	C₂×Q₁₆	—	6	2	no	—
105	C₂×Q₁₆	—	6	2	no	—
106	C₂×Q₁₆	—	12	2	no	—
107	C₂×Q₁₆	—	12	2	no	—
108	C₂²×C₈	#67	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
110	C₄⋊Q₈	—	6	4	no	—
111	C₄⋊Q₈	—	12	2	no	—
112	C₂×C₄⋊C₄	—	2	1	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄⋊C₄	—	4	1	no	—
115	C₂×C₄⋊C₄	—	12	1	no	—
116	C₂×C₄⋊C₄	—	12	1	no	—
117	C₂×C₄²	#67	1	1	yes	characteristic
118	C₂×C₁₆	—	6	2	no	—
119	C₂×C₁₆	—	6	2	no	—
120	C₂×C₁₆	—	6	2	no	—
121	#107	—	2	1	no	—
122	#107	—	2	2	no	—
123	#107	D₁₂	1	2	yes	—
124	C₄×C₈	—	2	2	no	—
125	#95	—	4	1	no	—
126	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	—	8	1	no	—
133	C₂×C₃×C₄	—	8	1	no	—
134	C₂×C₃×C₄	—	8	1	no	—
135	C₂×C₃×C₄	—	8	1	no	—
136	C₂×C₃×C₄	—	8	1	no	—
137	C₂×C₃×C₄	SD₃₂	1	2	yes	—
138	C₂×C₃×C₄	Q₃₂	1	2	yes	—
139	C₂×C₃×C₄	#120	1	1	yes	characteristic
140	C₂×C₃×C₄	#121	1	1	yes	characteristic
141	C₂×C₃×C₄	#124	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×Dic₃	—	4	4	no	—
146	C₂×Dic₃	—	8	2	no	—
147	C₂×Dic₃	—	8	2	no	—
148	Dic₆	—	4	4	no	—
149	Dic₆	—	4	4	no	—
150	Dic₆	—	8	2	no	—
151	Dic₆	—	8	2	no	—
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	#121	1	2	yes	—
154	C₃×C₈	#123	1	2	yes	—
155	C₂×Q₈	—	6	4	no	—
156	C₂×Q₈	—	12	2	no	—
157	C₂×Q₈	—	12	2	no	—
158	C₂×Q₈	—	12	2	no	—
159	C₂×Q₈	—	12	2	no	—
160	C₂×Q₈	—	12	2	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	12	2	no	—
164	C₂²×C₄	#213	1	1	yes	characteristic
165	C₂²×C₄	#213	1	1	yes	characteristic
166	C₂²×C₄	C₂×#67	1	1	yes	characteristic
167	Q₁₆	—	6	4	no	—
168	Q₁₆	—	6	4	no	—
169	Q₁₆	—	12	8	no	—
170	Q₁₆	—	12	2	no	—
171	Q₁₆	—	12	2	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	#215	1	2	yes	—
175	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
176	C₂×C₈	#235	1	1	yes	characteristic
177	C₂×C₈	D₃×D₄	1	2	yes	—
178	C₂×C₈	#239	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	8	2	no	—
184	C₄⋊C₄	—	12	4	no	—
185	C₄⋊C₄	—	12	4	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₁₆	—	12	2	no	—
189	C₂²×C₃	#447	1	1	yes	characteristic
190	C₂²×C₃	#459	1	1	yes	characteristic
191	C₂²×C₃	#486	1	1	yes	characteristic
192	C₂²×C₃	C₂×SD₃₂	1	1	yes	characteristic
193	C₂²×C₃	C₂×Q₃₂	1	1	yes	characteristic
194	C₂²×C₃	#510	1	1	yes	characteristic
195	C₂²×C₃	#510	1	1	yes	characteristic
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	8	2	no	—
199	C₃×C₄	—	16	1	no	—
200	C₃×C₄	#394	1	1	yes	characteristic
201	C₃×C₄	#450	1	1	yes	characteristic
202	C₃×C₄	#460	1	1	yes	characteristic
203	C₃×C₄	#490	1	1	yes	characteristic
204	Dic₃	—	8	4	no	—
205	C₂³	C₂×#213	1	1	yes	characteristic
206	Q₈	—	12	4	no	—
207	Q₈	—	12	4	no	—
208	Q₈	—	24	2	no	—
209	Q₈	—	24	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	12	4	no	—
221	C₂×C₄	—	24	2	no	—
222	C₂×C₄	—	24	2	no	—
223	C₂×C₄	#827	1	2	yes	—
224	C₂×C₄	#829	1	2	yes	—
225	C₂×C₄	#930	1	1	yes	characteristic
226	C₂×C₄	#937	1	1	yes	characteristic
227	C₂×C₄	C₂×#215	1	1	yes	characteristic
228	C₂×C₄	#947	1	1	yes	characteristic
229	C₂×C₄	#949	1	1	yes	characteristic
230	C₂×C₄	#949	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	#894	1	2	yes	—
233	C₈	#895	1	2	yes	—
234	C₂×C₃	#2041	1	1	yes	characteristic
235	C₂×C₃	#2185	1	1	yes	characteristic
236	C₂×C₃	#2188	1	1	yes	characteristic
237	C₂×C₃	#2208	1	1	yes	characteristic
238	C₂×C₃	#2209	1	1	yes	characteristic
239	C₂×C₃	#2241	1	1	yes	characteristic
240	C₂×C₃	#2245	1	1	yes	characteristic
241	C₂²	#5015	1	1	yes	characteristic
242	C₂²	#5036	1	1	yes	characteristic
243	C₂²	#5088	1	1	yes	characteristic
244	C₂²	C₂×#827	1	1	yes	characteristic
245	C₂²	C₂×#829	1	1	yes	characteristic
246	C₂²	#5195	1	1	yes	characteristic
247	C₂²	#5195	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	8	2	no	—
251	C₄	—	16	1	no	—
252	C₄	—	24	4	no	—
253	C₄	#4991	1	1	yes	characteristic
254	C₄	#5027	1	1	yes	characteristic
255	C₄	#5046	1	1	yes	characteristic
256	C₄	#5067	1	1	yes	characteristic
257	C₃	#13409	1	1	yes	characteristic, complemented, Sylow
258	C₂	#69627	1	1	yes	characteristic
259	C₂	#70892	1	1	yes	characteristic
260	C₂	#70916	1	1	yes	characteristic
261	C₂	#71077	1	1	yes	characteristic
262	C₂	#71087	1	1	yes	characteristic
263	C₂	#71353	1	1	yes	characteristic
264	C₂	#71382	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#213
commutator, cosocle	C₂²×C₃×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1942	C₂
socle, prime radical	C₂³×C₃	C₂×D₈
hypercenter	#1942	D₃
hypocenter	C₃	#13409
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
257. C₃	#13409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4699	2
C₃×#1942	2
C₂×#4492	2
#67657	2
#13409	3