Subgroups — #72226974

Order 1024.

981 conjugacy classes of subgroups.

The subgroup lattice has 227 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#479942	C₂	1	2	yes	maximal
2	C₈×#504	C₂	1	1	yes	maximal, characteristic
3	#122211	C₂	1	2	yes	maximal, complemented
4	#122209	C₂	1	2	yes	maximal, complemented
5	#122207	C₂	1	2	yes	maximal, complemented
6	#122207	C₂	1	4	yes	maximal, complemented
7	#122206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#7052	C₂	1	1	yes	maximal, characteristic, complemented
9	#33479	C₄	1	4	yes	—
10	C₄×#504	C₄	1	2	yes	—
11	C₈×#131	C₂²	1	1	yes	characteristic
12	#11410	C₂²	1	2	yes	—
13	#11410	C₂²	1	4	yes	—
14	#11409	C₄	1	8	yes	—
15	#11408	C₄	1	4	yes	—
16	#11399	C₄	1	4	yes	—
17	#11399	C₄	1	2	yes	—
18	C₈×M₃₂	C₂²	1	1	yes	characteristic
19	C₈×M₃₂	C₂²	1	4	yes	—
20	C₈×M₃₂	C₂²	1	2	yes	—
21	C₂×C₈×C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₈×C₁₆	C₂²	1	2	yes	—
23	C₂×C₈×C₁₆	C₂²	1	2	yes	—
24	#7052	C₂²	1	2	yes	—
25	#7052	C₂²	1	1	yes	characteristic
26	#7052	C₂²	1	1	yes	characteristic
27	#7052	C₂²	1	2	yes	—
28	#7052	C₂²	1	4	yes	—
29	#7052	C₂²	1	4	yes	—
30	C₁₆²	C₂²	1	2	yes	—
31	C₂×#504	C₈	1	4	yes	—
32	C₈×#41	C₂×C₄	1	2	yes	—
33	C₄×#131	C₂×C₄	1	1	yes	characteristic
34	#2183	C₈	1	8	yes	—
35	#2169	C₈	1	8	yes	—
36	#2168	C₈	1	16	yes	—
37	#2162	C₂×C₄	1	4	yes	—
38	#2162	C₂×C₄	1	8	yes	—
39	#2108	C₈	1	8	yes	—
40	#2108	C₈	1	4	yes	—
41	C₄×M₃₂	C₂×C₄	1	2	yes	—
42	C₄×M₃₂	C₂×C₄	1	4	yes	—
43	C₄×M₃₂	C₂×C₄	1	8	yes	—
44	C₂×C₄×C₁₆	C₂×C₄	1	4	yes	—
45	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₄×C₁₆	C₂×C₄	1	4	yes	—
47	#1450	C₂×C₄	1	2	yes	—
48	#1450	C₂×C₄	1	1	yes	characteristic
49	C₈×M₁₆	C₂×C₄	1	2	yes	—
50	C₈×M₁₆	C₂×C₄	1	4	yes	—
51	C₂×C₈²	C₂³	1	1	yes	characteristic
52	C₂×C₈²	C₂³	1	2	yes	—
53	#1312	C₂×C₄	1	8	yes	—
54	#1312	C₂×C₄	1	4	yes	—
55	#1311	C₂×C₄	1	2	yes	—
56	#1311	C₂×C₄	1	4	yes	—
57	#1311	C₂×C₄	1	8	yes	—
58	C₈×C₁₆	C₂×C₄	1	2	yes	—
59	C₈×C₁₆	C₂×C₄	1	4	yes	—
60	C₈×C₁₆	C₂×C₄	1	4	yes	—
61	C₈×C₁₆	C₂³	1	1	yes	characteristic
62	C₈×C₁₆	C₂³	1	1	yes	characteristic
63	C₈×C₁₆	C₂³	1	2	yes	—
64	C₈×C₁₆	C₂³	1	2	yes	—
65	C₈×C₁₆	C₂³	1	2	yes	—
66	C₈×C₁₆	C₂³	1	4	yes	—
67	C₂×#131	C₂×C₈	1	2	yes	—
68	C₄×#41	C₄²	1	1	yes	characteristic
69	#504	—	2	16	no	—
70	C₂×M₃₂	C₂×C₈	1	4	yes	—
71	C₂×M₃₂	C₂×C₈	1	8	yes	—
72	C₂×M₃₂	C₂×C₈	1	16	yes	—
73	C₂²×C₁₆	C₂×C₈	1	8	yes	—
74	C₂²×C₁₆	C₂×C₈	1	8	yes	—
75	C₂²×C₁₆	C₂×C₈	1	4	yes	—
76	C₈×Q₈	C₂×C₈	1	4	yes	—
77	C₈×D₄	C₂×C₈	1	8	yes	—
78	C₈×D₄	C₂×C₈	1	4	yes	—
79	#431	C₄²	1	2	yes	—
80	#431	C₄²	1	4	yes	—
81	#405	C₂×C₈	1	4	yes	—
82	#405	C₂×C₈	1	2	yes	—
83	C₄×M₁₆	C₄²	1	1	yes	characteristic
84	C₄×M₁₆	C₄²	1	2	yes	—
85	C₂×C₄×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₄	1	4	yes	—
89	C₄⋊C₁₆	C₂×C₈	1	16	yes	—
90	C₄⋊C₁₆	C₂×C₈	1	8	yes	—
91	C₂²⋊C₁₆	C₂×C₈	1	8	yes	—
92	C₂²⋊C₁₆	C₂×C₈	1	16	yes	—
93	#346	C₂×C₈	1	4	yes	—
94	#346	C₂×C₈	1	8	yes	—
95	#346	C₂×C₈	1	16	yes	—
96	C₄×C₁₆	C₂×C₈	1	8	yes	—
97	C₄×C₁₆	C₂×C₈	1	4	yes	—
98	C₄×C₁₆	C₂×C₈	1	8	yes	—
99	C₄×C₁₆	C₂²×C₄	1	2	yes	—
100	C₄×C₁₆	C₂²×C₄	1	4	yes	—
101	C₄×C₁₆	C₂²×C₄	1	4	yes	—
102	C₄×C₁₆	C₂²×C₄	1	2	yes	—
103	C₄×C₁₆	C₂²×C₄	1	4	yes	—
104	C₄×C₁₆	C₂²×C₄	1	8	yes	—
105	#322	C₄²	1	4	yes	—
106	#322	C₄²	1	2	yes	—
107	C₈²	C₂²×C₄	1	1	yes	characteristic
108	C₈²	C₂²×C₄	1	2	yes	—
109	C₈²	C₂²×C₄	1	2	yes	—
110	C₈²	C₂²×C₄	1	4	yes	—
111	C₈²	C₂⁴	1	1	yes	characteristic
112	C₂×#41	C₄×C₈	1	1	yes	characteristic
113	#131	—	2	8	no	—
114	C₂×M₁₆	C₄×C₈	1	2	yes	—
115	C₂×M₁₆	C₄×C₈	1	4	yes	—
116	C₂²×C₈	C₂²×C₈	1	4	yes	—
117	C₂²×C₈	C₂²×C₈	1	4	yes	—
118	C₂²×C₈	C₂²×C₈	1	8	yes	—
119	C₂²×C₈	C₂²×C₈	1	2	yes	—
120	C₄×Q₈	C₄×C₈	1	2	yes	—
121	C₄×D₄	C₄×C₈	1	4	yes	—
122	C₄×D₄	C₄×C₈	1	2	yes	—
123	#117	C₄×C₈	1	2	yes	—
124	#117	C₄×C₈	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄²	1	2	yes	—
126	C₂×C₄²	C₂×C₄²	1	1	yes	characteristic
127	M₃₂	—	2	8	no	—
128	M₃₂	—	2	16	no	—
129	M₃₂	—	4	16	no	—
130	C₂×C₁₆	—	2	16	no	—
131	C₂×C₁₆	—	4	8	no	—
132	C₂×C₁₆	M₃₂	1	16	yes	complemented
133	C₂×C₁₆	C₂²×C₈	1	8	yes	—
134	C₂×C₁₆	C₂²×C₈	1	8	yes	—
135	C₂×C₁₆	C₂²×C₈	1	8	yes	—
136	C₂×C₁₆	C₂²×C₈	1	16	yes	—
137	C₂×C₁₆	C₂²×C₈	1	4	yes	—
138	C₂×C₁₆	C₂²×C₈	1	4	yes	—
139	C₄⋊C₈	C₄×C₈	1	8	yes	—
140	C₄⋊C₈	C₄×C₈	1	4	yes	—
141	C₂²⋊C₈	C₄×C₈	1	4	yes	—
142	C₂²⋊C₈	C₄×C₈	1	8	yes	—
143	#97	C₄×C₈	1	2	yes	—
144	#97	C₄×C₈	1	4	yes	—
145	C₄×C₈	C₂×C₄²	1	2	yes	—
146	C₄×C₈	C₂×C₄²	1	1	yes	characteristic
147	C₄×C₈	C₂×C₄²	1	2	yes	—
148	C₄×C₈	C₂×C₄²	1	4	yes	—
149	C₄×C₈	C₂²×C₈	1	4	yes	—
150	C₄×C₈	C₂²×C₈	1	2	yes	—
151	C₄×C₈	C₂²×C₈	1	4	yes	—
152	C₄×C₈	C₂²×C₈	1	8	yes	—
153	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
154	C₄×C₈	C₂³×C₄	1	2	yes	—
155	#41	—	2	4	no	—
156	C₂×Q₈	C₈²	1	1	yes	characteristic
157	C₂×D₄	C₈²	1	2	yes	—
158	C₂×D₄	C₈²	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×C₈	1	4	yes	—
160	C₂²×C₄	C₂×C₄×C₈	1	2	yes	—
161	C₂²×C₄	C₂×C₄×C₈	1	2	yes	—
162	C₂²×C₄	C₂×C₄×C₈	1	1	yes	characteristic
163	M₁₆	—	2	4	no	—
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	16	no	—
168	C₂×C₈	C₂×C₄×C₈	1	4	yes	—
169	C₂×C₈	C₂×C₄×C₈	1	4	yes	—
170	C₂×C₈	C₂×C₄×C₈	1	8	yes	—
171	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
172	C₂×C₈	C₂×M₃₂	1	8	yes	—
173	C₂×C₈	C₂³×C₈	1	2	yes	—
174	C₂×C₈	C₂³×C₈	1	4	yes	—
175	C₄⋊C₄	C₈²	1	4	yes	—
176	C₄⋊C₄	C₈²	1	2	yes	—
177	C₂²⋊C₄	C₈²	1	4	yes	—
178	C₂²⋊C₄	C₈²	1	2	yes	—
179	C₄²	C₂×C₄×C₈	1	2	yes	—
180	C₄²	C₂×C₄×C₈	1	1	yes	characteristic
181	C₄²	C₂×C₄×C₈	1	4	yes	—
182	C₄²	C₂×C₄×C₈	1	2	yes	—
183	C₄²	C₂²×C₄²	1	1	yes	characteristic
184	C₁₆	—	2	8	no	—
185	C₁₆	—	2	8	no	—
186	C₁₆	—	2	8	no	—
187	C₁₆	—	2	16	no	—
188	C₁₆	C₂×M₃₂	1	8	yes	—
189	C₁₆	C₂×M₃₂	1	8	yes	—
190	C₂³	C₂×C₈²	1	2	yes	—
191	C₂³	C₂×C₈²	1	1	yes	characteristic
192	Q₈	—	2	2	no	—
193	D₄	—	2	4	no	—
194	D₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	C₂×C₈²	1	4	yes	—
199	C₂×C₄	C₂×C₈²	1	2	yes	—
200	C₂×C₄	C₂×C₈²	1	2	yes	—
201	C₂×C₄	C₂×C₈²	1	1	yes	characteristic
202	C₂×C₄	C₄×M₃₂	1	4	yes	—
203	C₂×C₄	C₂²×C₄×C₈	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₄×C₈	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	2	4	no	—
207	C₈	—	2	8	no	—
208	C₈	C₄×M₃₂	1	4	yes	—
209	C₈	C₂²×M₃₂	1	4	yes	—
210	C₈	#3407	1	8	yes	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	4	no	—
214	C₂²	C₈×M₃₂	1	2	yes	—
215	C₂²	C₂²×C₈²	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	C₈×M₃₂	1	2	yes	—
220	C₄	C₂×C₄×M₃₂	1	2	yes	—
221	C₄	#33326	1	4	yes	—
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	C₂×C₈×M₃₂	1	1	yes	characteristic
225	C₂	#479899	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₈²	C₂⁴
commutator, socle	C₂²	C₂²×C₈²
prime radical	C₂×#41	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #122211	C₂	semidirect
4. #122209	C₂	semidirect
5. #122207	C₂	semidirect
6. #122207	C₂	semidirect
7. #122206	C₂	semidirect
8. C₂×#7052	C₂	semidirect
132. C₂×C₁₆	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#479942	2
C₈×#504	2
#122211	2
#122209	2
#122207	2
#122206	2
C₂×#7052	2