Subgroups — #129087

Order 512.

300 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12653	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1367	C₂	1	2	yes	maximal
4	#7841	C₂	1	1	yes	maximal, characteristic
5	#7151	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	C₂×#346	—	2	1	no	—
9	C₂×C₄×C₁₆	—	2	1	no	—
10	#1744	C₄	1	1	yes	characteristic
11	#1744	C₄	1	1	yes	characteristic
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	C₂×#322	C₄	1	1	yes	characteristic
14	C₂×#322	C₄	1	1	yes	characteristic
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	#1374	—	2	1	no	—
18	#1374	—	2	1	no	—
19	#1367	C₄	1	8	yes	complemented
20	C₂²×C₁₆	—	2	2	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
30	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
32	C₄³	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₁₆	—	2	2	no	—
34	C₄⋊C₁₆	—	2	2	no	—
35	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
36	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
37	#346	—	4	2	no	—
38	C₄×C₁₆	—	4	2	no	—
39	#336	—	2	2	no	—
40	#336	C₈	1	4	yes	—
41	#322	—	2	2	no	—
42	#322	C₈	1	4	yes	—
43	C₈²	—	2	2	no	—
44	C₈²	C₂×C₄	1	4	yes	—
45	C₂²×C₈	—	2	1	no	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂×C₈	1	2	yes	—
50	C₂²×C₈	C₂×C₈	1	2	yes	—
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄²	C₂×C₈	1	2	yes	—
56	C₂×C₄²	M₁₆	1	2	yes	—
57	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₁₆	—	2	4	no	—
59	C₂×C₁₆	—	2	4	no	—
60	C₂×C₁₆	—	4	2	no	—
61	C₂×C₁₆	—	4	2	no	—
62	C₂×C₁₆	—	4	2	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	C₄²	1	2	yes	—
72	C₄×C₈	C₄²	1	2	yes	—
73	C₄×C₈	C₂×C₈	1	2	yes	—
74	C₄×C₈	C₂×C₈	1	2	yes	—
75	C₄×C₈	C₂×C₈	1	2	yes	—
76	C₄×C₈	M₁₆	1	2	yes	—
77	C₄×C₈	SD₁₆	1	2	yes	—
78	C₄×C₈	SD₁₆	1	2	yes	—
79	C₄×C₈	C₂²×C₄	1	2	yes	—
80	C₄×C₈	#41	1	2	yes	—
81	C₂²×C₄	C₄⋊C₈	1	2	yes	—
82	C₂²×C₄	C₄⋊C₈	1	2	yes	—
83	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	C₄×C₈	1	4	yes	—
98	C₂×C₈	C₄⋊C₈	1	2	yes	—
99	C₂×C₈	C₄⋊C₈	1	2	yes	—
100	C₂×C₈	#106	1	2	yes	—
101	C₂×C₈	#108	1	2	yes	—
102	C₂×C₈	C₄×D₄	1	2	yes	—
103	C₂×C₈	C₄×Q₈	1	2	yes	—
104	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₈	#135	1	2	yes	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	4	no	—
110	C₄²	—	4	2	no	—
111	C₄²	#106	1	2	yes	—
112	C₄²	#108	1	2	yes	—
113	C₄²	C₂×C₄²	1	1	yes	characteristic
114	C₄²	#117	1	1	yes	characteristic
115	C₄²	C₂²×C₈	1	1	yes	characteristic
116	C₄²	C₂×M₁₆	1	1	yes	characteristic
117	C₁₆	—	4	4	no	—
118	C₁₆	—	4	4	no	—
119	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	#334	1	4	yes	—
127	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
129	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₄	C₂×#106	1	1	yes	characteristic
132	C₂×C₄	#427	1	1	yes	characteristic
133	C₂×C₄	C₂×#108	1	1	yes	characteristic
134	C₂×C₄	C₂×#108	1	1	yes	characteristic
135	C₂×C₄	#431	1	1	yes	characteristic
136	C₂×C₄	#431	1	1	yes	characteristic
137	C₂×C₄	#504	1	4	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	2	4	no	—
140	C₈	—	2	2	no	—
141	C₈	—	4	2	no	—
142	C₈	—	4	2	no	—
143	C₈	—	4	2	no	—
144	C₈	#334	1	4	yes	—
145	C₈	C₄×SD₁₆	1	2	yes	—
146	C₈	C₄×SD₁₆	1	2	yes	—
147	C₂²	C₂×#334	1	1	yes	characteristic
148	C₂²	#1564	1	1	yes	characteristic
149	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
150	C₂²	#2151	1	2	yes	—
151	C₂²	#2153	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	4	no	—
155	C₄	—	4	2	no	—
156	C₄	C₂×#334	1	1	yes	characteristic
157	C₄	#1564	1	1	yes	characteristic
158	C₄	C₄×#106	1	1	yes	characteristic
159	C₄	C₄×#108	1	1	yes	characteristic
160	C₄	#2108	1	2	yes	—
161	C₄	#2164	1	2	yes	—
162	C₂	C₄×#334	1	1	yes	characteristic
163	C₂	#11997	1	2	yes	—
164	C₂	#12014	1	2	yes	—
165	C₂	#12654	1	1	yes	characteristic
166	C₂	#12656	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
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
19. #1367	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12653	2
C₄×C₄⋊C₁₆	2
C₂×#1367	2
#7841	2
#7151	2