Subgroups — #126606

Order 512.

476 conjugacy classes of subgroups.

The subgroup lattice has 239 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	2	yes	maximal
2	C₂²×#322	C₂	1	1	yes	maximal, characteristic
3	#7146	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	#2163	C₄	1	4	yes	—
8	#2163	C₂²	1	4	yes	—
9	#2162	—	2	2	no	—
10	#2162	—	2	2	no	—
11	#2150	—	2	2	no	—
12	#2112	—	2	2	no	—
13	C₄×M₃₂	C₄	1	4	yes	—
14	C₂×#322	—	2	2	no	—
15	C₂×#322	—	2	2	no	—
16	C₂×#322	C₄	1	2	yes	—
17	C₂×#322	C₂²	1	2	yes	—
18	C₂³×C₈	—	2	1	no	—
19	C₂³×C₈	C₈	1	2	yes	—
20	C₂³×C₈	D₄	1	1	yes	characteristic
21	C₂³×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×M₃₂	—	2	4	no	—
24	C₂×C₄×C₈	—	2	1	no	—
25	C₂×C₄×C₈	—	2	1	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	—	2	1	no	—
28	C₂×C₄×C₈	—	2	1	no	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₈	1	2	yes	—
32	C₂×C₄×C₈	C₈	1	2	yes	—
33	C₂×C₄×C₈	C₈	1	2	yes	—
34	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	2	yes	—
41	C₂×C₄×C₈	D₄	1	2	yes	—
42	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₄⋊C₁₆	—	2	4	no	—
45	C₄⋊C₁₆	—	2	4	no	—
46	C₂²⋊C₁₆	—	2	4	no	—
47	C₂²⋊C₁₆	—	2	4	no	—
48	#346	C₂×C₄	1	8	yes	—
49	C₄×C₁₆	—	2	4	no	—
50	#322	—	2	4	no	—
51	#322	—	2	4	no	—
52	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
54	C₂³×C₄	M₁₆	1	1	yes	characteristic
55	C₂²×C₈	—	2	8	no	—
56	C₂²×C₈	—	2	2	no	—
57	C₂²×C₈	—	2	2	no	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	4	2	no	—
62	C₂²×C₈	—	4	2	no	—
63	C₂²×C₈	—	4	4	no	—
64	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₈	C₄⋊C₄	1	2	yes	—
67	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	C₂×C₈	1	4	yes	—
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄²	C₄²	1	1	yes	characteristic
74	C₂×C₄²	C₄²	1	1	yes	characteristic
75	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄²	M₁₆	1	1	yes	characteristic
82	C₂×C₄²	M₁₆	1	1	yes	characteristic
83	C₂×C₄²	M₁₆	1	1	yes	characteristic
84	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
85	M₃₂	—	4	8	no	—
86	C₂×C₁₆	—	2	8	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	C₄²	1	2	yes	—
95	C₄×C₈	C₂²⋊C₄	1	2	yes	—
96	C₄×C₈	C₂²⋊C₄	1	2	yes	—
97	C₄×C₈	C₂²⋊C₄	1	2	yes	—
98	C₄×C₈	C₄⋊C₄	1	2	yes	—
99	C₄×C₈	C₄⋊C₄	1	2	yes	—
100	C₄×C₈	C₂×C₈	1	4	yes	—
101	C₄×C₈	C₂²×C₄	1	2	yes	—
102	C₄×C₈	C₂×D₄	1	2	yes	—
103	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	#95	1	1	yes	characteristic
109	C₂²×C₄	#95	1	1	yes	characteristic
110	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
111	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
112	C₂²×C₄	#97	1	1	yes	characteristic
113	C₂²×C₄	#97	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	—	2	8	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	#95	1	2	yes	—
143	C₂×C₈	#95	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
145	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
146	C₄²	#95	1	1	yes	characteristic
147	C₄²	#95	1	1	yes	characteristic
148	C₄²	C₄×C₈	1	1	yes	characteristic
149	C₄²	C₄×C₈	1	1	yes	characteristic
150	C₄²	#97	1	1	yes	characteristic
151	C₄²	#97	1	1	yes	characteristic
152	C₄²	C₂²⋊C₈	1	1	yes	characteristic
153	C₄²	C₂²⋊C₈	1	1	yes	characteristic
154	C₄²	C₂²⋊C₈	1	1	yes	characteristic
155	C₄²	C₂²⋊C₈	1	1	yes	characteristic
156	C₄²	C₄⋊C₈	1	1	yes	characteristic
157	C₄²	C₄⋊C₈	1	1	yes	characteristic
158	C₄²	C₂×C₄²	1	1	yes	characteristic
159	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₄²	C₂²×C₈	1	1	yes	characteristic
161	C₄²	C₂×M₁₆	1	1	yes	characteristic
162	C₁₆	—	4	8	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	#336	1	1	yes	characteristic
167	C₂³	#336	1	1	yes	characteristic
168	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	#336	1	1	yes	characteristic
187	C₂×C₄	#336	1	1	yes	characteristic
188	C₂×C₄	#336	1	1	yes	characteristic
189	C₂×C₄	#336	1	1	yes	characteristic
190	C₂×C₄	#336	1	1	yes	characteristic
191	C₂×C₄	#336	1	1	yes	characteristic
192	C₂×C₄	H(3,ℤ/4)	1	2	yes	—
193	C₂×C₄	#338	1	2	yes	—
194	C₂×C₄	#349	1	4	yes	—
195	C₂×C₄	#364	1	4	yes	—
196	C₂×C₄	C₂×#95	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#97	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
202	C₈	—	2	8	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	4	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#1275	1	2	yes	—
216	C₂²	#1377	1	2	yes	—
217	C₂²	C₂×#336	1	1	yes	characteristic
218	C₂²	C₂×#364	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	#1275	1	2	yes	—
226	C₄	#1376	1	2	yes	—
227	C₄	#1376	1	2	yes	—
228	C₄	#1377	1	2	yes	—
229	C₄	C₂×H(3,ℤ/4)	1	1	yes	characteristic
230	C₄	C₂×#338	1	1	yes	characteristic
231	C₄	C₂×#349	1	2	yes	—
232	C₂	—	2	2	no	—
233	C₂	—	2	2	no	—
234	C₂	—	2	2	no	—
235	C₂	C₂×#1275	1	1	yes	characteristic
236	C₂	C₂×#1376	1	1	yes	characteristic
237	C₂	C₂×#1377	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #7146	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
C₂²×#322	2
#7146	2