Subgroups — #10713698

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#4563	C₂	1	2	yes	maximal, complemented
3	#72033	C₂	1	2	yes	maximal, complemented
4	#69779	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#4463	C₂	1	1	yes	maximal, characteristic, complemented
6	#13261	—	3	1	no	maximal, Sylow
7	C₂²×C₃×D₈	—	2	1	no	—
8	C₂²×C₃×D₈	C₂²	1	2	yes	—
9	C₂×C₃×#127	—	2	1	no	—
10	C₃×#493	—	2	2	no	—
11	C₃×#493	C₄	1	4	yes	complemented
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#819	C₂²	1	1	yes	characteristic
14	#5106	C₂²	1	1	yes	characteristic
15	#5101	—	2	1	no	—
16	C₂×#812	C₂²	1	2	yes	—
17	#4563	—	2	4	no	—
18	#4562	—	2	4	no	—
19	#4523	—	2	1	no	—
20	#4463	—	2	2	no	—
21	C₂×#493	D₃	1	1	yes	characteristic
22	C₂×#364	—	3	1	no	—
23	C₂×#361	—	3	2	no	—
24	#2140	—	3	2	no	—
25	#1772	—	3	1	no	—
26	C₂²×C₃×D₄	—	2	2	no	—
27	C₂²×C₃×D₄	—	4	2	no	—
28	C₂×C₃×D₈	—	2	4	no	—
29	C₂×C₃×D₈	—	2	2	no	—
30	C₂×C₃×D₈	—	2	2	no	—
31	C₂×C₃×D₈	—	2	2	no	—
32	C₂×C₃×D₈	—	2	2	no	—
33	C₂×C₃×D₈	—	2	2	no	—
34	C₂×C₃×D₈	—	4	1	no	—
35	C₂×C₃×D₈	—	4	1	no	—
36	C₂×C₃×D₈	—	4	2	no	—
37	C₂×C₃×D₈	C₂×C₄	1	4	yes	—
38	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
40	C₃×#127	—	2	4	no	—
41	C₃×#127	—	4	2	no	—
42	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
43	#926	—	2	1	no	—
44	C₂×#208	—	2	1	no	—
45	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
46	C₃×C₄×C₈	D₄	1	2	yes	—
47	#819	—	2	1	no	—
48	#819	D₄	1	2	yes	complemented
49	#818	D₄	1	2	yes	complemented
50	#817	—	2	1	no	—
51	#815	—	4	1	no	—
52	C₈×Dic₃	—	4	1	no	—
53	#812	—	2	2	no	—
54	#812	—	2	2	no	—
55	C₂×C₃⋊C₁₆	—	2	2	no	—
56	C₂²×D₈	—	2	1	no	—
57	C₂²×D₈	C₂×D₃	1	2	yes	—
58	C₂×#127	—	2	1	no	—
59	C₂×M₃₂	—	3	2	no	—
60	#493	—	2	2	no	—
61	#493	Dic₃	1	4	yes	complemented
62	C₂×#108	—	3	1	no	—
63	#427	—	3	1	no	—
64	#405	—	6	1	no	—
65	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
66	#364	—	6	2	no	—
67	#361	—	6	4	no	—
68	#357	—	6	4	no	—
69	#339	—	6	1	no	—
70	C₂⁴×C₃	—	4	2	no	—
71	C₂×C₃×D₄	—	2	4	no	—
72	C₂×C₃×D₄	—	4	8	no	—
73	C₂×C₃×D₄	—	4	2	no	—
74	C₂×C₃×D₄	—	4	2	no	—
75	C₂×C₃×D₄	—	4	2	no	—
76	C₂×C₃×D₄	—	4	2	no	—
77	C₂×C₃×D₄	—	4	2	no	—
78	C₂×C₃×D₄	—	8	8	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₃×D₈	—	2	4	no	—
82	C₃×D₈	—	2	4	no	—
83	C₃×D₈	—	4	4	no	—
84	C₃×D₈	—	4	4	no	—
85	C₃×D₈	—	4	2	no	—
86	C₃×D₈	—	4	2	no	—
87	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃×C₈	D₈	1	1	yes	characteristic
89	C₂×C₃×C₈	D₈	1	1	yes	characteristic
90	C₂×C₃×C₈	SD₁₆	1	2	yes	—
91	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃×C₈	#41	1	1	yes	characteristic
96	C₂×C₃×C₈	#41	1	1	yes	characteristic
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
99	#217	—	4	1	no	—
100	C₄⋊Dic₃	—	2	2	no	—
101	C₄×Dic₃	—	4	1	no	—
102	#208	—	2	2	no	—
103	#208	—	4	1	no	—
104	C₂×C₃⋊C₈	—	4	1	no	—
105	C₃⋊C₁₆	—	4	2	no	—
106	C₂²×D₄	—	2	2	no	—
107	C₂²×D₄	—	4	2	no	—
108	C₂×D₈	—	2	4	no	—
109	C₂×D₈	—	2	2	no	—
110	C₂×D₈	—	2	2	no	—
111	C₂×D₈	—	2	2	no	—
112	C₂×D₈	—	2	2	no	—
113	C₂×D₈	—	2	2	no	—
114	C₂×D₈	—	4	1	no	—
115	C₂×D₈	—	4	1	no	—
116	C₂×D₈	—	4	2	no	—
117	C₂×D₈	C₂×Dic₃	1	4	yes	—
118	C₂×M₁₆	—	6	1	no	—
119	C₂²×C₈	#67	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
121	#127	—	2	4	no	—
122	#127	—	4	2	no	—
123	#117	—	6	1	no	—
124	C₂×C₄²	#67	1	1	yes	characteristic
125	M₃₂	—	6	2	no	—
126	M₃₂	—	6	2	no	—
127	C₂×C₁₆	—	6	2	no	—
128	#108	—	3	2	no	—
129	#108	—	6	1	no	—
130	#107	—	3	2	no	—
131	#106	—	6	1	no	—
132	#97	—	12	1	no	—
133	C₄×C₈	—	12	1	no	—
134	C₄×C₈	C₂×Dic₃	1	2	yes	—
135	C₄×C₈	#67	1	2	yes	—
136	C₂³×C₃	—	4	4	no	—
137	C₂³×C₃	—	8	8	no	—
138	C₂³×C₃	—	8	2	no	—
139	C₂³×C₃	—	8	2	no	—
140	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₃×D₄	—	4	4	no	—
142	C₃×D₄	—	4	4	no	—
143	C₃×D₄	—	8	16	no	—
144	C₃×D₄	—	8	2	no	—
145	C₃×D₄	—	8	2	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₄	#121	1	1	yes	characteristic
150	C₂×C₃×C₄	#121	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×Dic₃	—	4	2	no	—
155	C₃×C₈	#102	1	2	yes	—
156	C₃×C₈	#102	1	2	yes	—
157	C₃×C₈	C₄×D₄	1	1	yes	characteristic
158	C₃×C₈	C₄×D₄	1	1	yes	characteristic
159	C₃×C₈	#124	1	1	yes	characteristic
160	C₃×C₈	#127	1	1	yes	characteristic
161	C₃⋊C₈	—	4	2	no	—
162	C₂⁴	—	4	2	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	8	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	8	8	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂×#67	1	1	yes	characteristic
173	D₈	—	2	4	no	—
174	D₈	—	2	4	no	—
175	D₈	—	4	4	no	—
176	D₈	—	4	4	no	—
177	D₈	—	4	2	no	—
178	D₈	—	4	2	no	—
179	M₁₆	—	6	2	no	—
180	M₁₆	—	12	1	no	—
181	C₂×C₈	—	12	1	no	—
182	C₂×C₈	#213	1	1	yes	characteristic
183	C₂×C₈	#213	1	1	yes	characteristic
184	C₂×C₈	#214	1	2	yes	—
185	C₂×C₈	#217	1	2	yes	—
186	C₂×C₈	D₃×D₄	1	1	yes	characteristic
187	C₂×C₈	D₃×D₄	1	1	yes	characteristic
188	C₂×C₈	#237	1	1	yes	characteristic
189	C₂×C₈	#237	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
191	C₂×C₈	C₂×#67	1	1	yes	characteristic
192	C₄⋊C₄	—	6	2	no	—
193	C₂²⋊C₄	—	12	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	12	1	no	—
196	C₄²	#217	1	2	yes	—
197	C₁₆	—	12	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	8	8	no	—
200	C₂²×C₃	—	8	4	no	—
201	C₂²×C₃	—	16	4	no	—
202	C₂²×C₃	—	16	4	no	—
203	C₂²×C₃	#418	1	1	yes	characteristic
204	C₂²×C₃	#459	1	1	yes	characteristic
205	C₂²×C₃	#460	1	1	yes	characteristic
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	#390	1	1	yes	characteristic
208	C₃×C₄	C₂×#102	1	1	yes	characteristic
209	C₃×C₄	#459	1	1	yes	characteristic
210	C₃×C₄	#460	1	1	yes	characteristic
211	Dic₃	—	8	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	8	8	no	—
214	C₂³	—	8	2	no	—
215	C₂³	—	8	2	no	—
216	C₂³	#932	1	1	yes	characteristic
217	D₄	—	4	4	no	—
218	D₄	—	4	4	no	—
219	D₄	—	8	16	no	—
220	D₄	—	8	2	no	—
221	D₄	—	8	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	12	2	no	—
225	C₂×C₄	C₂×#213	1	1	yes	characteristic
226	C₂×C₄	#932	1	1	yes	characteristic
227	C₂×C₄	C₂×#214	1	1	yes	characteristic
228	C₂×C₄	#938	1	1	yes	characteristic
229	C₂×C₄	#938	1	1	yes	characteristic
230	C₂×C₄	C₂×#217	1	1	yes	characteristic
231	C₈	—	12	2	no	—
232	C₈	#832	1	2	yes	—
233	C₈	#832	1	2	yes	—
234	C₈	D₄×Dic₃	1	1	yes	characteristic
235	C₈	D₄×Dic₃	1	1	yes	characteristic
236	C₈	#936	1	1	yes	characteristic
237	C₈	#940	1	1	yes	characteristic
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	16	4	no	—
241	C₂×C₃	#1982	1	1	yes	characteristic
242	C₂×C₃	#2213	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	8	8	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	16	4	no	—
247	C₂²	—	16	4	no	—
248	C₂²	#5072	1	1	yes	characteristic
249	C₂²	#5075	1	1	yes	characteristic
250	C₂²	#5215	1	1	yes	characteristic
251	C₄	—	2	4	no	—
252	C₄	—	24	2	no	—
253	C₄	#5072	1	1	yes	characteristic
254	C₄	#5075	1	1	yes	characteristic
255	C₄	C₂×#832	1	1	yes	characteristic
256	C₄	#5221	1	1	yes	characteristic
257	C₃	#13261	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	4	no	—
261	C₂	#71295	1	2	yes	—
262	C₂	#71761	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#5072
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×#493	C₂
socle	C₂²×C₃	#459
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#493	D₃
hypocenter	C₃	#13261
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
2. C₂×#4563	C₂	semidirect
3. #72033	C₂	semidirect
4. #69779	C₂	semidirect
5. C₂×#4463	C₂	semidirect
11. C₃×#493	C₄	semidirect
48. #819	D₄	semidirect
49. #818	D₄	semidirect
61. #493	Dic₃	semidirect
257. C₃	#13261	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#493	2
C₂×#4563	2
#72033	2
#69779	2
C₂×#4463	2
#13261	3