Subgroups — #10713703

Order 768.

462 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#4565	C₂	1	2	yes	maximal
3	#72061	C₂	1	2	yes	maximal
4	#69779	C₂	1	1	yes	maximal, characteristic
5	C₂×#4463	C₂	1	1	yes	maximal, characteristic
6	#13262	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₁₆	—	2	1	no	—
8	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
10	C₃×#494	—	2	2	no	—
11	C₃×#494	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	#4565	—	2	4	no	—
18	#4564	—	2	4	no	—
19	#4523	—	2	1	no	—
20	#4463	—	2	2	no	—
21	C₂×#494	D₃	1	1	yes	characteristic
22	C₂×#364	—	3	1	no	—
23	C₂×#362	—	3	2	no	—
24	#2141	—	3	2	no	—
25	#1772	—	3	1	no	—
26	C₂²×C₃×Q₈	—	2	2	no	—
27	C₂×C₃×Q₁₆	—	2	4	no	—
28	C₂×C₃×Q₁₆	—	2	2	no	—
29	C₂×C₃×Q₁₆	—	2	2	no	—
30	C₂×C₃×Q₁₆	—	2	2	no	—
31	C₂×C₃×Q₁₆	—	2	2	no	—
32	C₂×C₃×Q₁₆	—	2	2	no	—
33	C₂×C₃×Q₁₆	—	4	1	no	—
34	C₂×C₃×Q₁₆	—	4	1	no	—
35	C₂×C₃×Q₁₆	—	4	2	no	—
36	C₂×C₃×Q₁₆	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
39	C₃×C₄⋊Q₈	—	2	4	no	—
40	C₃×C₄⋊Q₈	—	4	2	no	—
41	C₂×C₃×C₄⋊C₄	—	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	—
49	#818	D₄	1	2	yes	—
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×Q₁₆	—	2	1	no	—
57	C₂²×Q₁₆	C₂×D₃	1	2	yes	—
58	C₂×C₄⋊Q₈	—	2	1	no	—
59	C₂×M₃₂	—	3	2	no	—
60	#494	—	2	2	no	—
61	#494	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	#362	—	6	4	no	—
68	#358	—	6	4	no	—
69	#339	—	6	1	no	—
70	C₂×C₃×Q₈	—	2	4	no	—
71	C₂×C₃×Q₈	—	4	2	no	—
72	C₂×C₃×Q₈	—	4	2	no	—
73	C₂×C₃×Q₈	—	4	2	no	—
74	C₂×C₃×Q₈	—	4	2	no	—
75	C₂×C₃×Q₈	—	4	2	no	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	—	4	2	no	—
78	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₃×Q₁₆	—	2	4	no	—
80	C₃×Q₁₆	—	2	4	no	—
81	C₃×Q₁₆	—	4	4	no	—
82	C₃×Q₁₆	—	4	4	no	—
83	C₃×Q₁₆	—	4	2	no	—
84	C₃×Q₁₆	—	4	2	no	—
85	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₈	D₈	1	1	yes	characteristic
87	C₂×C₃×C₈	D₈	1	1	yes	characteristic
88	C₂×C₃×C₈	SD₁₆	1	2	yes	—
89	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₈	#41	1	1	yes	characteristic
94	C₂×C₃×C₈	#41	1	1	yes	characteristic
95	C₃×C₄⋊C₄	—	4	8	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
98	#217	—	4	1	no	—
99	C₄⋊Dic₃	—	2	2	no	—
100	C₄×Dic₃	—	4	1	no	—
101	#208	—	2	2	no	—
102	#208	—	4	1	no	—
103	C₂×C₃⋊C₈	—	4	1	no	—
104	C₃⋊C₁₆	—	4	2	no	—
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	2	no	—
108	C₂×Q₁₆	—	2	2	no	—
109	C₂×Q₁₆	—	2	2	no	—
110	C₂×Q₁₆	—	2	2	no	—
111	C₂×Q₁₆	—	2	2	no	—
112	C₂×Q₁₆	—	4	1	no	—
113	C₂×Q₁₆	—	4	1	no	—
114	C₂×Q₁₆	—	4	2	no	—
115	C₂×Q₁₆	C₂×Dic₃	1	4	yes	—
116	C₂×M₁₆	—	6	1	no	—
117	C₂²×C₈	#67	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
119	C₄⋊Q₈	—	2	4	no	—
120	C₄⋊Q₈	—	4	2	no	—
121	#117	—	6	1	no	—
122	C₂×C₄⋊C₄	—	4	2	no	—
123	C₂×C₄²	#67	1	1	yes	characteristic
124	M₃₂	—	6	2	no	—
125	M₃₂	—	6	2	no	—
126	C₂×C₁₆	—	6	2	no	—
127	#108	—	3	2	no	—
128	#108	—	6	1	no	—
129	#107	—	3	2	no	—
130	#106	—	6	1	no	—
131	#97	—	12	1	no	—
132	C₄×C₈	—	12	1	no	—
133	C₄×C₈	C₂×Dic₃	1	2	yes	—
134	C₄×C₈	#67	1	2	yes	—
135	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₃×Q₈	—	4	4	no	—
137	C₃×Q₈	—	4	4	no	—
138	C₃×Q₈	—	8	2	no	—
139	C₃×Q₈	—	8	2	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	4	4	no	—
143	C₂×C₃×C₄	—	8	2	no	—
144	C₂×C₃×C₄	—	8	2	no	—
145	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₃×C₄	#121	1	1	yes	characteristic
147	C₂×C₃×C₄	#121	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×Dic₃	—	4	2	no	—
152	C₃×C₈	#102	1	2	yes	—
153	C₃×C₈	#102	1	2	yes	—
154	C₃×C₈	C₄×D₄	1	1	yes	characteristic
155	C₃×C₈	C₄×D₄	1	1	yes	characteristic
156	C₃×C₈	#124	1	1	yes	characteristic
157	C₃×C₈	#127	1	1	yes	characteristic
158	C₃⋊C₈	—	4	2	no	—
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂×#67	1	1	yes	characteristic
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	2	4	no	—
170	Q₁₆	—	4	4	no	—
171	Q₁₆	—	4	4	no	—
172	Q₁₆	—	4	2	no	—
173	Q₁₆	—	4	2	no	—
174	M₁₆	—	6	2	no	—
175	M₁₆	—	12	1	no	—
176	C₂×C₈	—	12	1	no	—
177	C₂×C₈	#213	1	1	yes	characteristic
178	C₂×C₈	#213	1	1	yes	characteristic
179	C₂×C₈	#214	1	2	yes	—
180	C₂×C₈	#217	1	2	yes	—
181	C₂×C₈	D₃×D₄	1	1	yes	characteristic
182	C₂×C₈	D₃×D₄	1	1	yes	characteristic
183	C₂×C₈	#237	1	1	yes	characteristic
184	C₂×C₈	#237	1	1	yes	characteristic
185	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
186	C₂×C₈	C₂×#67	1	1	yes	characteristic
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	6	2	no	—
189	C₂²⋊C₄	—	12	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	12	1	no	—
192	C₄²	#217	1	2	yes	—
193	C₁₆	—	12	2	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	#418	1	1	yes	characteristic
196	C₂²×C₃	#459	1	1	yes	characteristic
197	C₂²×C₃	#460	1	1	yes	characteristic
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	8	4	no	—
200	C₃×C₄	#390	1	1	yes	characteristic
201	C₃×C₄	C₂×#102	1	1	yes	characteristic
202	C₃×C₄	#459	1	1	yes	characteristic
203	C₃×C₄	#460	1	1	yes	characteristic
204	Dic₃	—	8	2	no	—
205	C₂³	#932	1	1	yes	characteristic
206	Q₈	—	4	4	no	—
207	Q₈	—	4	4	no	—
208	Q₈	—	8	2	no	—
209	Q₈	—	8	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	12	2	no	—
216	C₂×C₄	C₂×#213	1	1	yes	characteristic
217	C₂×C₄	#932	1	1	yes	characteristic
218	C₂×C₄	C₂×#214	1	1	yes	characteristic
219	C₂×C₄	#938	1	1	yes	characteristic
220	C₂×C₄	#938	1	1	yes	characteristic
221	C₂×C₄	C₂×#217	1	1	yes	characteristic
222	C₈	—	12	2	no	—
223	C₈	#832	1	2	yes	—
224	C₈	#832	1	2	yes	—
225	C₈	D₄×Dic₃	1	1	yes	characteristic
226	C₈	D₄×Dic₃	1	1	yes	characteristic
227	C₈	#936	1	1	yes	characteristic
228	C₈	#940	1	1	yes	characteristic
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	#1982	1	1	yes	characteristic
232	C₂×C₃	#2214	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	#5072	1	1	yes	characteristic
235	C₂²	#5075	1	1	yes	characteristic
236	C₂²	#5215	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	8	4	no	—
239	C₄	—	24	2	no	—
240	C₄	#5072	1	1	yes	characteristic
241	C₄	#5075	1	1	yes	characteristic
242	C₄	C₂×#832	1	1	yes	characteristic
243	C₄	#5221	1	1	yes	characteristic
244	C₃	#13262	1	1	yes	characteristic, complemented, Sylow
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#71317	1	2	yes	—
248	C₂	#71761	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#5072
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#494	C₂
socle	C₂²×C₃	#459
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#494	D₃
hypocenter	C₃	#13262
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
11. C₃×#494	C₄	semidirect
61. #494	Dic₃	semidirect
244. C₃	#13262	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#494	2
C₂×#4565	2
#72061	2
#69779	2
C₂×#4463	2
#13262	3