Subgroups — #93442

Order 512.

258 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12898	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊C₁₆	—	2	1	no	—
7	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
8	C₂×C₄×C₁₆	—	2	1	no	—
9	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈²	—	2	1	no	—
13	#1374	C₄	1	2	yes	—
14	#1273	—	2	1	no	—
15	C₂²×C₁₆	—	2	2	no	—
16	C₂²×C₁₆	—	2	2	no	—
17	C₂×C₄⋊C₈	—	2	1	no	—
18	C₂×C₄⋊C₈	C₈	1	2	yes	—
19	C₂×C₄×C₈	—	2	1	no	—
20	C₂×C₄×C₈	—	2	1	no	—
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
27	C₄³	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₁₆	—	2	2	no	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄⋊C₁₆	—	4	2	no	—
31	C₄⋊C₁₆	—	4	1	no	—
32	C₄⋊C₁₆	—	4	1	no	—
33	C₄⋊C₁₆	—	4	1	no	—
34	C₄⋊C₁₆	—	4	1	no	—
35	C₄×C₁₆	—	4	1	no	—
36	C₄×C₁₆	—	4	1	no	—
37	#336	—	2	1	no	—
38	#336	C₈	1	2	yes	—
39	C₈²	—	4	1	no	—
40	C₈²	—	4	1	no	—
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄²	C₄²	1	1	yes	characteristic
48	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄²	M₁₆	1	1	yes	characteristic
53	C₂×C₄²	M₁₆	1	1	yes	characteristic
54	C₂×C₁₆	—	2	4	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	—	4	2	no	—
57	C₂×C₁₆	—	4	2	no	—
58	C₂×C₁₆	—	4	2	no	—
59	C₂×C₁₆	—	4	2	no	—
60	C₄⋊C₈	—	2	4	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	C₁₆	1	4	yes	complemented
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	4	1	no	—
70	C₄×C₈	—	4	1	no	—
71	C₄×C₈	—	4	1	no	—
72	C₄×C₈	—	4	1	no	—
73	C₄×C₈	—	4	1	no	—
74	C₄×C₈	D₈	1	1	yes	characteristic
75	C₄×C₈	D₈	1	1	yes	characteristic
76	C₄×C₈	SD₁₆	1	1	yes	characteristic
77	C₄×C₈	SD₁₆	1	1	yes	characteristic
78	C₄×C₈	SD₁₆	1	1	yes	characteristic
79	C₄×C₈	SD₁₆	1	1	yes	characteristic
80	C₄×C₈	Q₁₆	1	1	yes	characteristic
81	C₄×C₈	Q₁₆	1	1	yes	characteristic
82	C₂²×C₄	#95	1	1	yes	characteristic
83	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
84	C₂²×C₄	#97	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
87	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	#100	1	1	yes	characteristic
103	C₂×C₈	#101	1	1	yes	characteristic
104	C₂×C₈	#102	1	1	yes	characteristic
105	C₂×C₈	#102	1	1	yes	characteristic
106	C₂×C₈	#102	1	1	yes	characteristic
107	C₂×C₈	#103	1	1	yes	characteristic
108	C₂×C₈	#103	1	1	yes	characteristic
109	C₂×C₈	#103	1	1	yes	characteristic
110	C₂×C₈	#106	1	1	yes	characteristic
111	C₂×C₈	#107	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	4	1	no	—
117	C₄²	#102	1	1	yes	characteristic
118	C₄²	#103	1	1	yes	characteristic
119	C₄²	#104	1	1	yes	characteristic
120	C₄²	#104	1	1	yes	characteristic
121	C₄²	#106	1	1	yes	characteristic
122	C₄²	#107	1	1	yes	characteristic
123	C₄²	#108	1	1	yes	characteristic
124	C₄²	#108	1	1	yes	characteristic
125	C₄²	C₂×C₁₆	1	2	yes	—
126	C₄²	M₃₂	1	2	yes	—
127	C₁₆	—	4	4	no	—
128	C₁₆	—	8	2	no	—
129	C₂³	#336	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
140	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
141	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
142	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
143	C₂×C₄	#334	1	1	yes	characteristic
144	C₂×C₄	#334	1	1	yes	characteristic
145	C₂×C₄	#335	1	1	yes	characteristic
146	C₂×C₄	#335	1	1	yes	characteristic
147	C₂×C₄	#339	1	1	yes	characteristic
148	C₂×C₄	#340	1	1	yes	characteristic
149	C₂×C₄	#340	1	1	yes	characteristic
150	C₂×C₄	#341	1	1	yes	characteristic
151	C₂×C₄	#343	1	1	yes	characteristic
152	C₂×C₄	#344	1	1	yes	characteristic
153	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
154	C₂×C₄	#346	1	1	yes	characteristic
155	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
156	C₂×C₄	#350	1	2	yes	—
157	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
158	C₂×C₄	#364	1	2	yes	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	4	no	—
164	C₈	—	4	1	no	—
165	C₈	—	4	4	no	—
166	C₈	#331	1	1	yes	characteristic
167	C₈	#332	1	1	yes	characteristic
168	C₈	#332	1	1	yes	characteristic
169	C₈	#333	1	1	yes	characteristic
170	C₂²	#1277	1	1	yes	characteristic
171	C₂²	#1279	1	1	yes	characteristic
172	C₂²	#1283	1	1	yes	characteristic
173	C₂²	#1374	1	1	yes	characteristic
174	C₂²	#1376	1	1	yes	characteristic
175	C₂²	#1378	1	1	yes	characteristic
176	C₂²	#1379	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	4	1	no	—
184	C₄	#1296	1	1	yes	characteristic
185	C₄	#1299	1	1	yes	characteristic
186	C₄	#1301	1	1	yes	characteristic
187	C₄	#1303	1	1	yes	characteristic
188	C₄	D₄⋊C₁₆	1	2	yes	—
189	C₄	Q₈⋊C₁₆	1	2	yes	—
190	C₄	#1367	1	2	yes	—
191	C₄	#1370	1	2	yes	—
192	C₂	#7031	1	1	yes	characteristic
193	C₂	#7151	1	1	yes	characteristic
194	C₂	#7153	1	1	yes	characteristic
195	C₂	#7167	1	1	yes	characteristic
196	C₂	#7169	1	1	yes	characteristic
197	C₂	#7209	1	1	yes	characteristic
198	C₂	#7212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
62. 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
#12898	2
C₄×C₄⋊C₁₆	2
C₈×C₄⋊C₈	2