Subgroups — #10694481

Order 768.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 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	C₂×C₃×#334	C₂	1	1	yes	maximal, characteristic
3	#70460	C₂	1	2	yes	maximal
4	C₂×#4486	C₂	1	2	yes	maximal
5	#69757	C₂	1	1	yes	maximal, characteristic
6	#9742	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	—	2	1	no	—
8	C₂²×Dic₁₂	—	2	1	no	—
9	C₂×#869	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#815	—	2	1	no	—
13	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	#4699	—	2	2	no	—
16	#4699	C₄	1	4	yes	complemented
17	C₃×#334	—	2	2	no	—
18	#4551	—	2	4	no	—
19	#4550	—	2	1	no	—
20	#4486	—	2	4	no	—
21	C₂×#494	—	3	1	no	—
22	#1988	—	3	2	no	—
23	#1770	—	3	1	no	—
24	C₂×#334	D₃	1	1	yes	characteristic
25	C₂×#332	—	3	2	no	—
26	C₂²×Dic₆	—	2	2	no	—
27	C₂²×C₃×C₈	—	2	2	no	—
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	—	2	2	no	—
33	C₂×C₄⋊Dic₃	D₄	1	2	yes	—
34	C₂×C₄×Dic₃	—	2	1	no	—
35	C₂²×C₃⋊C₈	—	2	1	no	—
36	C₂×Dic₁₂	—	2	4	no	—
37	C₂×Dic₁₂	—	2	4	no	—
38	C₂×Dic₁₂	—	4	2	no	—
39	C₂×Dic₁₂	—	4	2	no	—
40	C₂×Dic₁₂	—	4	2	no	—
41	C₂×Dic₁₂	—	4	2	no	—
42	#869	—	2	2	no	—
43	#869	—	2	2	no	—
44	#869	—	2	2	no	—
45	#869	C₂×C₄	1	4	yes	—
46	C₃×C₄⋊C₈	—	2	2	no	—
47	C₃×C₄⋊C₈	—	2	2	no	—
48	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
49	C₃×C₄×C₈	D₄	1	2	yes	—
50	C₃⋊#95	—	2	1	no	—
51	#815	—	4	2	no	—
52	C₄⋊(C₃⋊C₈)	—	2	1	no	—
53	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
54	C₂²×Q₁₆	—	6	1	no	—
55	C₂²×Q₁₆	—	6	1	no	—
56	C₂×C₄⋊Q₈	—	3	2	no	—
57	#494	—	3	4	no	—
58	#494	—	6	2	no	—
59	C₂×C₄⋊C₈	—	3	1	no	—
60	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
61	C₂×#97	—	6	1	no	—
62	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	C₄×C₄⋊C₄	—	3	1	no	—
64	#340	—	6	4	no	—
65	#336	—	6	1	no	—
66	#334	—	2	2	no	—
67	#332	—	6	4	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	C₂²×Dic₃	—	4	2	no	—
73	C₂×Dic₆	—	2	4	no	—
74	C₂×Dic₆	—	4	2	no	—
75	C₂×Dic₆	—	4	2	no	—
76	C₂×Dic₆	—	4	2	no	—
77	C₂×Dic₆	—	4	2	no	—
78	C₂×Dic₆	—	4	2	no	—
79	C₂×C₃×C₈	—	2	1	no	—
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	—	4	2	no	—
82	C₂×C₃×C₈	—	4	4	no	—
83	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₈	D₈	1	2	yes	—
86	C₂×C₃×C₈	SD₁₆	1	2	yes	—
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	—	2	4	no	—
94	C₄⋊Dic₃	—	4	2	no	—
95	C₄⋊Dic₃	—	4	2	no	—
96	C₄×Dic₃	—	4	1	no	—
97	C₄×Dic₃	—	4	1	no	—
98	C₂×C₃⋊C₈	—	2	2	no	—
99	C₂×C₃⋊C₈	—	4	1	no	—
100	C₂×C₃⋊C₈	—	4	1	no	—
101	Dic₁₂	—	4	8	no	—
102	Dic₁₂	—	4	8	no	—
103	C₂²×Q₈	—	6	2	no	—
104	C₂×Q₁₆	—	6	4	no	—
105	C₂×Q₁₆	—	6	4	no	—
106	C₂×Q₁₆	—	12	2	no	—
107	C₂×Q₁₆	—	12	2	no	—
108	C₂×Q₁₆	—	12	2	no	—
109	C₂×Q₁₆	—	12	2	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	6	1	no	—
112	C₂²×C₈	D₁₂	1	1	yes	characteristic
113	C₂²×C₈	D₁₂	1	1	yes	characteristic
114	C₄⋊Q₈	—	3	4	no	—
115	C₄⋊Q₈	—	6	2	no	—
116	C₄⋊Q₈	—	6	2	no	—
117	C₄⋊Q₈	—	6	2	no	—
118	C₂×C₄⋊C₄	—	3	2	no	—
119	C₂×C₄⋊C₄	—	6	2	no	—
120	C₂×C₄⋊C₄	—	6	2	no	—
121	C₂×C₄²	—	6	1	no	—
122	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	3	2	no	—
126	C₄⋊C₈	—	6	1	no	—
127	#97	—	12	2	no	—
128	C₄×C₈	C₄×D₃	1	2	yes	—
129	C₄×C₈	#67	1	2	yes	—
130	#95	—	6	1	no	—
131	C₂³×C₃	#121	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	#121	1	1	yes	characteristic
137	C₂×C₃×C₄	#124	1	1	yes	characteristic
138	C₂×C₃×C₄	#127	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₃×C₄	#137	1	4	yes	—
144	C₂×Dic₃	—	4	4	no	—
145	C₂×Dic₃	—	4	2	no	—
146	C₂×Dic₃	—	4	2	no	—
147	C₂×Dic₃	—	4	2	no	—
148	C₂×Dic₃	—	8	2	no	—
149	C₂×Dic₃	—	8	2	no	—
150	Dic₆	—	4	4	no	—
151	Dic₆	—	4	4	no	—
152	Dic₆	—	8	2	no	—
153	Dic₆	—	8	2	no	—
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	8	2	no	—
156	C₃×C₈	#102	1	4	yes	—
157	C₃⋊C₈	—	4	2	no	—
158	C₂×Q₈	—	6	4	no	—
159	C₂×Q₈	—	12	2	no	—
160	C₂×Q₈	—	12	2	no	—
161	C₂×Q₈	—	12	2	no	—
162	C₂×Q₈	—	12	2	no	—
163	C₂×Q₈	—	12	2	no	—
164	C₂²×C₄	—	6	2	no	—
165	C₂²×C₄	—	12	2	no	—
166	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
167	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
168	C₂²×C₄	#239	1	1	yes	characteristic
169	Q₁₆	—	12	8	no	—
170	Q₁₆	—	12	8	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	6	2	no	—
176	C₂×C₈	—	12	1	no	—
177	C₂×C₈	—	12	1	no	—
178	C₂×C₈	#204	1	2	yes	—
179	C₂×C₈	D₂₄	1	2	yes	—
180	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
181	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
182	C₄⋊C₄	—	6	4	no	—
183	C₄⋊C₄	—	6	4	no	—
184	C₄⋊C₄	—	6	4	no	—
185	C₄⋊C₄	—	12	2	no	—
186	C₄⋊C₄	—	12	2	no	—
187	C₄²	—	12	1	no	—
188	C₄²	—	12	1	no	—
189	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
190	C₄²	D₃×D₄	1	1	yes	characteristic
191	C₄²	#239	1	1	yes	characteristic
192	C₄²	C₂×#67	1	1	yes	characteristic
193	C₂²×C₃	#390	1	1	yes	characteristic
194	C₂²×C₃	#459	1	1	yes	characteristic
195	C₂²×C₃	#460	1	1	yes	characteristic
196	C₂²×C₃	#470	1	2	yes	—
197	C₂²×C₃	#473	1	2	yes	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	C₂×#102	1	1	yes	characteristic
201	C₃×C₄	#418	1	1	yes	characteristic
202	C₃×C₄	#441	1	2	yes	—
203	C₃×C₄	#459	1	1	yes	characteristic
204	C₃×C₄	#460	1	1	yes	characteristic
205	C₃×C₄	#497	1	2	yes	—
206	Dic₃	—	8	2	no	—
207	Dic₃	—	8	4	no	—
208	C₂³	#895	1	1	yes	characteristic
209	Q₈	—	12	4	no	—
210	Q₈	—	12	4	no	—
211	Q₈	—	24	2	no	—
212	Q₈	—	24	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	12	2	no	—
215	C₂×C₄	—	12	2	no	—
216	C₂×C₄	—	12	4	no	—
217	C₂×C₄	—	12	2	no	—
218	C₂×C₄	—	24	2	no	—
219	C₂×C₄	—	24	2	no	—
220	C₂×C₄	#893	1	1	yes	characteristic
221	C₂×C₄	#893	1	1	yes	characteristic
222	C₂×C₄	#895	1	1	yes	characteristic
223	C₂×C₄	C₂×#204	1	1	yes	characteristic
224	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
225	C₂×C₄	#908	1	1	yes	characteristic
226	C₂×C₄	#908	1	1	yes	characteristic
227	C₂×C₄	#915	1	4	yes	—
228	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	#940	1	1	yes	characteristic
230	C₂×C₄	#947	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	8	2	no	—
233	C₈	—	12	2	no	—
234	C₈	#821	1	4	yes	—
235	C₂×C₃	#1689	1	2	yes	—
236	C₂×C₃	#1690	1	2	yes	—
237	C₂×C₃	#1971	1	1	yes	characteristic
238	C₂×C₃	#1979	1	1	yes	characteristic
239	C₂×C₃	#1982	1	1	yes	characteristic
240	C₂²	#4843	1	1	yes	characteristic
241	C₂²	#4848	1	1	yes	characteristic
242	C₂²	#4867	1	2	yes	—
243	C₂²	#4896	1	2	yes	—
244	C₂²	#4988	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	24	4	no	—
248	C₄	—	24	2	no	—
249	C₄	#4843	1	1	yes	characteristic
250	C₄	#4848	1	1	yes	characteristic
251	C₄	#4853	1	2	yes	—
252	C₄	C₂×#821	1	1	yes	characteristic
253	C₄	#5131	1	1	yes	characteristic
254	C₄	#5167	1	2	yes	—
255	C₃	#9742	1	1	yes	characteristic, complemented, Sylow
256	C₂	#68541	1	2	yes	—
257	C₂	#68551	1	2	yes	—
258	C₂	#70457	1	1	yes	characteristic
259	C₂	#70575	1	1	yes	characteristic
260	C₂	#70710	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#895
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#334	C₂
socle, prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#334	D₃
hypocenter	C₃	#9742
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
16. #4699	C₄	semidirect
255. C₃	#9742	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
C₂×C₃×#334	2
#70460	2
C₂×#4486	2
#69757	2
#9742	3