Subgroups — #140394

Order 512.

278 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12879	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1370	C₂	1	1	yes	maximal, characteristic
4	C₂×#1367	C₂	1	1	yes	maximal, characteristic
5	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#7151	C₂	1	1	yes	maximal, characteristic
7	#7151	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	—	2	1	no	—
13	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1374	—	2	1	no	—
19	#1374	—	2	1	no	—
20	#1374	—	2	1	no	—
21	#1370	—	2	2	no	—
22	#1367	—	2	2	no	—
23	#1273	C₄	1	1	yes	characteristic
24	#1273	C₄	1	1	yes	characteristic
25	C₂²×C₁₆	—	2	1	no	—
26	C₂²×C₁₆	—	2	1	no	—
27	C₂²×C₁₆	—	2	1	no	—
28	C₂²×C₁₆	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	1	no	—
30	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
31	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₄³	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₁₆	—	2	1	no	—
44	C₄⋊C₁₆	—	2	1	no	—
45	C₄⋊C₁₆	—	2	1	no	—
46	C₄⋊C₁₆	—	2	1	no	—
47	C₄⋊C₁₆	—	2	1	no	—
48	C₄⋊C₁₆	—	2	1	no	—
49	C₄⋊C₁₆	D₄	1	2	yes	—
50	C₄⋊C₁₆	Q₈	1	2	yes	—
51	C₄×C₁₆	—	4	2	no	—
52	#336	—	2	1	no	—
53	#336	C₈	1	1	yes	characteristic
54	#336	C₈	1	1	yes	characteristic
55	C₈²	—	2	1	no	—
56	C₈²	—	2	1	no	—
57	C₈²	—	2	1	no	—
58	C₈²	—	2	1	no	—
59	C₂²×C₈	—	2	1	no	—
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄²	M₁₆	1	1	yes	characteristic
74	C₂×C₄²	M₁₆	1	1	yes	characteristic
75	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	4	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	—	4	1	no	—
83	C₂×C₁₆	—	4	1	no	—
84	C₂×C₁₆	—	4	1	no	—
85	C₂×C₁₆	—	4	1	no	—
86	C₂×C₁₆	—	4	1	no	—
87	C₂×C₁₆	—	4	1	no	—
88	C₄⋊C₈	—	2	4	no	—
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	C₁₆	1	4	yes	complemented
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	C₂×D₄	1	1	yes	characteristic
108	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₄×C₈	#41	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
113	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
117	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	#108	1	2	yes	—
142	C₂×C₈	C₂×C₁₆	1	2	yes	—
143	C₂×C₈	C₂×C₁₆	1	2	yes	—
144	C₂×C₈	#121	1	1	yes	characteristic
145	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₈	#125	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₈	#135	1	1	yes	characteristic
149	C₂×C₈	#135	1	1	yes	characteristic
150	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₈	#137	1	1	yes	characteristic
152	C₄²	—	2	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	C₂×C₁₆	1	2	yes	—
161	C₄²	M₃₂	1	2	yes	—
162	C₄²	C₄×D₄	1	1	yes	characteristic
163	C₄²	C₄×Q₈	1	1	yes	characteristic
164	C₄²	#131	1	1	yes	characteristic
165	C₄²	#131	1	1	yes	characteristic
166	C₁₆	—	4	2	no	—
167	C₁₆	—	4	2	no	—
168	C₁₆	—	8	1	no	—
169	C₁₆	—	8	1	no	—
170	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
183	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
184	C₂×C₄	#364	1	2	yes	—
185	C₂×C₄	#389	1	1	yes	characteristic
186	C₂×C₄	#424	1	1	yes	characteristic
187	C₂×C₄	#427	1	1	yes	characteristic
188	C₂×C₄	#428	1	1	yes	characteristic
189	C₂×C₄	C₂×#108	1	1	yes	characteristic
190	C₂×C₄	#430	1	1	yes	characteristic
191	C₂×C₄	C₈×D₄	1	1	yes	characteristic
192	C₂×C₄	#436	1	1	yes	characteristic
193	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
194	C₂×C₄	#446	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
196	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
197	C₂×C₄	#504	1	1	yes	characteristic
198	C₂×C₄	#504	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	4	1	no	—
207	C₈	—	4	1	no	—
208	C₈	—	4	1	no	—
209	C₈	#463	1	1	yes	characteristic
210	C₈	#464	1	1	yes	characteristic
211	C₈	#478	1	1	yes	characteristic
212	C₈	#479	1	1	yes	characteristic
213	C₂²	#1564	1	1	yes	characteristic
214	C₂²	#1565	1	1	yes	characteristic
215	C₂²	#1939	1	1	yes	characteristic
216	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
217	C₂²	#2151	1	1	yes	characteristic
218	C₂²	C₂×#364	1	1	yes	characteristic
219	C₂²	#2153	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	1	no	—
226	C₄	#1572	1	1	yes	characteristic
227	C₄	#1573	1	1	yes	characteristic
228	C₄	#1949	1	1	yes	characteristic
229	C₄	#1950	1	1	yes	characteristic
230	C₄	D₄×C₁₆	1	1	yes	characteristic
231	C₄	#2169	1	1	yes	characteristic
232	C₄	Q₈×C₁₆	1	1	yes	characteristic
233	C₄	#2183	1	1	yes	characteristic
234	C₂	#9603	1	1	yes	characteristic
235	C₂	#12012	1	1	yes	characteristic
236	C₂	#12014	1	1	yes	characteristic
237	C₂	#12015	1	1	yes	characteristic
238	C₂	#12016	1	1	yes	characteristic
239	C₂	#13159	1	1	yes	characteristic
240	C₂	#13163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×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
90. C₄⋊C₈	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#12879	2
C₄×C₄⋊C₁₆	2
C₂×#1370	2
C₂×#1367	2
C₈×C₄⋊C₈	2
#7151	2