Subgroups — #129185

Order 512.

274 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#1371	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	#8282	C₂	1	1	yes	maximal, characteristic
5	C₈×#107	C₂	1	1	yes	maximal, characteristic
6	#7171	C₂	1	1	yes	maximal, characteristic
7	#7171	C₂	1	1	yes	maximal, characteristic
8	C₂×#366	C₄	1	2	yes	—
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₄×#107	C₂²	1	1	yes	characteristic
12	C₄×#107	C₂²	1	1	yes	characteristic
13	#1770	—	2	1	no	—
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₂×#335	C₂²	1	1	yes	characteristic
16	C₂×#335	C₂²	1	1	yes	characteristic
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1380	C₄	1	2	yes	—
19	#1371	C₄	1	4	yes	complemented
20	#1312	C₄	1	4	yes	complemented
21	#1279	—	2	1	no	—
22	#1279	—	2	1	no	—
23	C₂²×C₁₆	—	2	1	no	—
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×#107	C₂×C₄	1	2	yes	—
27	C₂×#107	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	1	no	—
30	C₂×#97	—	4	1	no	—
31	C₂×C₄×C₈	—	4	1	no	—
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₄⋊C₄	—	2	1	no	—
36	C₄×C₄⋊C₄	—	2	1	no	—
37	#366	—	2	4	no	—
38	C₄×C₁₆	—	2	1	no	—
39	C₄×C₁₆	C₂×C₄	1	2	yes	—
40	C₄×C₁₆	C₂×C₄	1	2	yes	—
41	C₄×C₁₆	C₂×C₄	1	2	yes	—
42	#340	—	2	1	no	—
43	#340	—	2	1	no	—
44	#336	—	4	1	no	—
45	#336	—	4	1	no	—
46	#335	—	2	1	no	—
47	#335	—	2	1	no	—
48	#335	C₂×C₄	1	2	yes	—
49	#335	C₂×C₄	1	2	yes	—
50	C₈²	C₂×C₄	1	2	yes	—
51	C₈²	Q₈	1	2	yes	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	—	4	1	no	—
54	C₂²×C₈	—	4	1	no	—
55	C₂²×C₈	D₈	1	1	yes	characteristic
56	C₂²×C₈	D₈	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	#41	1	1	yes	characteristic
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄²	—	4	1	no	—
62	C₂×C₄²	—	4	1	no	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	4	2	no	—
68	C₂×C₁₆	C₄²	1	2	yes	—
69	C₂×C₁₆	C₄²	1	2	yes	—
70	#107	—	2	2	no	—
71	#107	—	2	2	no	—
72	#107	—	2	2	no	—
73	#107	M₁₆	1	4	yes	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	4	1	no	—
77	C₄⋊C₈	—	4	1	no	—
78	#97	—	8	2	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	8	2	no	—
81	C₄×C₈	C₄⋊C₄	1	2	yes	—
82	C₄×C₈	C₄⋊C₄	1	2	yes	—
83	C₄×C₈	Q₁₆	1	2	yes	—
84	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	#95	—	4	1	no	—
89	#95	—	4	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
94	C₂²×C₄	#135	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	—	8	1	no	—
104	C₂×C₈	—	8	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	#107	1	2	yes	—
107	C₂×C₈	#107	1	2	yes	—
108	C₂×C₈	D₁₆	1	2	yes	—
109	C₂×C₈	Q₃₂	1	2	yes	—
110	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₈	#117	1	1	yes	characteristic
112	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₈	C₂×M₁₆	1	2	yes	—
115	C₂×C₈	#131	1	2	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄²	—	8	1	no	—
121	C₄²	—	8	1	no	—
122	C₄²	—	8	1	no	—
123	C₄²	—	8	1	no	—
124	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₄²	C₄×D₄	1	1	yes	characteristic
126	C₄²	C₂×Q₁₆	1	1	yes	characteristic
127	C₄²	#135	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	—	2	2	no	—
130	C₁₆	—	4	2	no	—
131	C₂³	C₄×D₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×#107	1	1	yes	characteristic
141	C₂×C₄	#427	1	1	yes	characteristic
142	C₂×C₄	#435	1	2	yes	—
143	C₂×C₄	C₄×D₈	1	1	yes	characteristic
144	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
145	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₄	#444	1	2	yes	—
147	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
149	C₂×C₄	#508	1	1	yes	characteristic
150	C₂×C₄	#508	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	4	2	no	—
155	C₈	—	8	2	no	—
156	C₈	—	8	2	no	—
157	C₈	#366	1	4	yes	complemented
158	C₈	C₄×M₁₆	1	1	yes	characteristic
159	C₈	#405	1	1	yes	characteristic
160	C₈	#446	1	2	yes	—
161	C₂²	#1581	1	2	yes	—
162	C₂²	C₄×#107	1	1	yes	characteristic
163	C₂²	C₄×D₁₆	1	1	yes	characteristic
164	C₂²	C₄×Q₃₂	1	1	yes	characteristic
165	C₂²	#2179	1	2	yes	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	8	4	no	—
169	C₄	—	16	2	no	—
170	C₄	#1584	1	2	yes	—
171	C₄	#1771	1	1	yes	characteristic
172	C₄	#1773	1	1	yes	characteristic
173	C₄	C₂×#366	1	1	yes	characteristic
174	C₄	#2158	1	1	yes	characteristic
175	C₄	C₄×D₁₆	1	1	yes	characteristic
176	C₄	C₄×Q₃₂	1	1	yes	characteristic
177	C₂	#8291	1	1	yes	characteristic
178	C₂	#12069	1	2	yes	—
179	C₂	#12072	1	2	yes	—
180	C₂	#12671	1	1	yes	characteristic
181	C₂	C₄×#366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ 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
19. #1371	C₄	semidirect
20. #1312	C₄	semidirect
157. C₈	#366	semidirect

Maximal subgroups

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

subgroup	index
C₄×#366	2
C₂×#1371	2
C₂×#1312	2
#8282	2
C₈×#107	2
#7171	2