Subgroups — #140382

Order 512.

236 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12900	C₂	1	1	yes	maximal, characteristic
2	C₄×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#1373	C₂	1	2	yes	maximal
5	#7160	C₂	1	2	yes	maximal
6	C₂×#364	—	2	1	no	—
7	C₂×#364	C₂²	1	2	yes	—
8	C₂×#364	C₂²	1	1	yes	characteristic
9	C₂×#364	C₂²	1	1	yes	characteristic
10	#2150	—	2	1	no	—
11	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
12	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
13	C₄×M₃₂	—	2	1	no	—
14	C₂×C₄×C₁₆	C₂²	1	2	yes	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	#1376	—	2	1	no	—
17	#1374	C₄	1	2	yes	—
18	#1373	—	2	4	no	—
19	C₂×M₃₂	—	2	1	no	—
20	C₂×M₃₂	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂²×C₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₄³	Q₈	1	1	yes	characteristic
31	#364	—	2	2	no	—
32	#364	—	2	1	no	—
33	#364	—	2	1	no	—
34	#364	—	2	2	no	—
35	#364	—	4	2	no	—
36	#364	D₄	1	2	yes	—
37	#364	Q₈	1	2	yes	—
38	C₄⋊C₁₆	—	2	1	no	—
39	C₄⋊C₁₆	—	2	1	no	—
40	C₄⋊C₁₆	—	4	1	no	—
41	C₄⋊C₁₆	—	4	1	no	—
42	C₄⋊C₁₆	C₈	1	2	yes	—
43	C₄⋊C₁₆	C₈	1	2	yes	—
44	#346	—	4	1	no	—
45	C₄×C₁₆	—	2	2	no	—
46	C₄×C₁₆	—	2	2	no	—
47	C₄×C₁₆	—	4	1	no	—
48	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄²	—	2	1	no	—
53	C₂×C₄²	—	2	1	no	—
54	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
57	M₃₂	—	2	2	no	—
58	M₃₂	—	2	2	no	—
59	M₃₂	—	4	1	no	—
60	M₃₂	—	4	1	no	—
61	M₃₂	—	4	2	no	—
62	M₃₂	—	4	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	4	1	no	—
69	C₂×C₁₆	—	4	1	no	—
70	C₂×C₁₆	—	4	2	no	—
71	C₂×C₁₆	C₂×C₈	1	4	yes	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	4	1	no	—
77	C₄×C₈	C₂×C₈	1	1	yes	characteristic
78	C₄×C₈	C₂×C₈	1	1	yes	characteristic
79	C₄×C₈	M₁₆	1	1	yes	characteristic
80	C₄×C₈	M₁₆	1	1	yes	characteristic
81	C₄×C₈	D₈	1	1	yes	characteristic
82	C₄×C₈	D₈	1	1	yes	characteristic
83	C₄×C₈	Q₁₆	1	1	yes	characteristic
84	C₄×C₈	Q₁₆	1	1	yes	characteristic
85	C₄×C₈	C₂×D₄	1	1	yes	characteristic
86	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	#108	1	1	yes	characteristic
92	C₂²×C₄	#108	1	1	yes	characteristic
93	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₈	#107	1	1	yes	characteristic
101	C₂×C₈	#107	1	1	yes	characteristic
102	C₂×C₈	C₄×D₄	1	1	yes	characteristic
103	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₈	#121	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₈	#131	1	1	yes	characteristic
109	C₂×C₈	#131	1	1	yes	characteristic
110	C₂×C₈	C₂×D₈	1	1	yes	characteristic
111	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₈	#137	1	1	yes	characteristic
114	C₄²	—	2	4	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	4	2	no	—
119	C₄²	C₄⋊C₈	1	1	yes	characteristic
120	C₄²	C₄⋊C₈	1	1	yes	characteristic
121	C₄²	#107	1	1	yes	characteristic
122	C₄²	#107	1	1	yes	characteristic
123	C₄²	#125	1	1	yes	characteristic
124	C₄²	C₄⋊Q₈	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₁₆	—	4	2	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	4	2	no	—
129	C₁₆	—	8	2	no	—
130	C₂³	C₂×#108	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	#335	1	1	yes	characteristic
141	C₂×C₄	#335	1	1	yes	characteristic
142	C₂×C₄	#335	1	1	yes	characteristic
143	C₂×C₄	#335	1	1	yes	characteristic
144	C₂×C₄	#389	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₂×C₄	#424	1	1	yes	characteristic
147	C₂×C₄	C₂×#107	1	1	yes	characteristic
148	C₂×C₄	#428	1	1	yes	characteristic
149	C₂×C₄	#430	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	C₈×D₄	1	1	yes	characteristic
155	C₈	#436	1	1	yes	characteristic
156	C₈	C₈×Q₈	1	1	yes	characteristic
157	C₈	#446	1	1	yes	characteristic
158	C₈	#459	1	1	yes	characteristic
159	C₈	#462	1	1	yes	characteristic
160	C₈	#474	1	1	yes	characteristic
161	C₈	#477	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×#335	1	1	yes	characteristic
165	C₂²	#1574	1	1	yes	characteristic
166	C₂²	#1950	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	#1373	1	2	yes	complemented
174	C₄	#1565	1	1	yes	characteristic
175	C₄	#1571	1	1	yes	characteristic
176	C₄	#1939	1	1	yes	characteristic
177	C₄	#1947	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	#9601	1	1	yes	characteristic
181	C₂	C₂×#1373	1	1	yes	characteristic
182	C₂	#12011	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#335
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×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
173. C₄	#1373	semidirect

Maximal subgroups

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

subgroup	index
#12900	2
C₄×#364	2
C₄×C₄⋊C₁₆	2
C₂×#1373	2
#7160	2