Subgroups — #494209

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2108	C₂	1	1	yes	maximal, characteristic
2	#20828	C₂	1	2	yes	maximal
3	#12909	C₂	1	2	yes	maximal, complemented
4	#12892	C₂	1	2	yes	maximal, complemented
5	#12838	C₂	1	2	yes	maximal
6	#12837	C₂	1	2	yes	maximal
7	#12676	C₂	1	2	yes	maximal, complemented
8	C₄×#368	C₂	1	2	yes	maximal, complemented
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂³×C₁₆	C₂²	1	1	yes	characteristic
11	#2921	—	2	2	no	—
12	C₂×#405	C₂²	1	1	yes	characteristic
13	C₂×#368	C₂²	1	4	yes	—
14	C₂×#368	C₂²	1	4	yes	—
15	#2108	C₄	1	8	yes	—
16	C₂×#346	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	#1848	C₂²	1	4	yes	—
21	#1838	—	2	2	no	—
22	#1811	—	2	2	no	—
23	#1810	C₂²	1	4	yes	—
24	#1778	—	2	2	no	—
25	C₄×#108	C₂²	1	4	yes	—
26	#1381	C₂²	1	4	yes	—
27	#1381	C₂²	1	4	yes	—
28	C₂²×M₁₆	D₄	1	1	yes	characteristic
29	C₂³×C₈	C₂³	1	1	yes	characteristic
30	C₂×#117	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	4	no	—
32	C₂×M₃₂	C₂×C₄	1	4	yes	—
33	C₂²×C₁₆	—	2	4	no	—
34	C₂²×C₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	C₂²×C₁₆	C₂³	1	1	yes	characteristic
38	C₂²×C₁₆	C₂³	1	1	yes	characteristic
39	#433	—	4	8	no	—
40	C₂×#108	—	2	4	no	—
41	C₂×#108	C₂³	1	8	yes	—
42	#423	—	2	4	no	—
43	#407	—	2	4	no	—
44	#405	C₂×C₄	1	4	yes	—
45	#405	Q₈	1	4	yes	—
46	C₄×M₁₆	—	2	4	no	—
47	C₂×#97	D₄	1	1	yes	characteristic
48	C₂×#97	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	#368	—	2	8	no	—
52	#368	—	2	8	no	—
53	#346	C₂×C₄	1	4	yes	—
54	#346	C₂×C₄	1	4	yes	—
55	C₄×C₁₆	C₂×C₄	1	4	yes	—
56	C₄×C₁₆	C₂×C₄	1	4	yes	—
57	#341	—	2	4	no	—
58	#341	—	2	4	no	—
59	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×M₁₆	—	2	4	no	—
61	C₂×M₁₆	—	2	8	no	—
62	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
63	C₂²×C₈	—	2	2	no	—
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	C₂²×C₄	1	2	yes	—
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×Q₈	1	2	yes	—
70	C₂²×C₈	C₂⁴	1	1	yes	characteristic
71	#117	C₄⋊C₄	1	4	yes	—
72	#117	D₈	1	4	yes	—
73	C₂×C₄⋊C₄	Q₁₆	1	2	yes	—
74	C₂×C₂²⋊C₄	Q₁₆	1	2	yes	—
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
77	M₃₂	—	2	8	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	8	no	—
81	C₂×C₁₆	C₂²×C₄	1	2	yes	—
82	C₂×C₁₆	C₂²×C₄	1	2	yes	—
83	C₂×C₁₆	C₂²×C₄	1	2	yes	—
84	C₂×C₁₆	C₂²×C₄	1	2	yes	—
85	C₂×C₁₆	#41	1	4	yes	—
86	C₂×C₁₆	#41	1	4	yes	—
87	#108	—	2	8	no	—
88	#108	—	2	8	no	—
89	#108	—	4	8	no	—
90	C₄⋊C₈	—	4	8	no	—
91	C₂²⋊C₈	—	4	8	no	—
92	#97	—	4	4	no	—
93	#97	C₄⋊C₄	1	4	yes	—
94	#97	C₄⋊C₄	1	4	yes	—
95	C₄×C₈	—	4	4	no	—
96	C₄×C₈	C₂²×C₄	1	2	yes	—
97	C₄×C₈	C₂²×C₄	1	2	yes	—
98	C₄×C₈	C₂×Q₈	1	2	yes	—
99	C₄×C₈	C₂×Q₈	1	2	yes	—
100	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
104	C₂²×C₄	C₂×D₈	1	2	yes	—
105	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
106	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
107	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
108	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
109	M₁₆	—	2	8	no	—
110	M₁₆	—	4	8	no	—
111	M₁₆	—	4	8	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	4	8	no	—
118	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₈	#135	1	4	yes	—
122	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₈	C₂×#41	1	2	yes	—
125	C₂×C₈	C₂×#41	1	2	yes	—
126	C₄⋊C₄	#107	1	8	yes	—
127	C₂²⋊C₄	#107	1	8	yes	—
128	C₄²	C₂×C₄⋊C₄	1	2	yes	—
129	C₄²	C₂×C₄⋊C₄	1	2	yes	—
130	C₄²	C₂×D₈	1	2	yes	—
131	C₄²	C₂×D₈	1	2	yes	—
132	C₁₆	—	2	2	no	—
133	C₁₆	—	2	2	no	—
134	C₁₆	#117	1	4	yes	—
135	C₁₆	#117	1	4	yes	—
136	C₂³	—	2	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂×#107	1	2	yes	—
139	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	C₂×#107	1	4	yes	—
147	C₂×C₄	C₂×#107	1	4	yes	—
148	C₂×C₄	C₂×#107	1	2	yes	—
149	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄	#532	1	2	yes	—
151	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
152	C₂×C₄	C₂×#135	1	2	yes	—
153	C₈	—	2	2	no	—
154	C₈	—	2	2	no	—
155	C₈	—	8	8	no	—
156	C₈	#427	1	4	yes	—
157	C₈	C₂×#117	1	1	yes	characteristic
158	C₈	C₂×#117	1	1	yes	characteristic
159	C₈	#533	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂²×#107	1	1	yes	characteristic
165	C₂²	#3170	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	#2285	1	1	yes	characteristic
170	C₄	C₂×#427	1	1	yes	characteristic
171	C₄	#3169	1	2	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	4	2	no	—
174	C₂	#20817	1	1	yes	characteristic
175	C₂	#33414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12909	C₂	semidirect
4. #12892	C₂	semidirect
7. #12676	C₂	semidirect
8. C₄×#368	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2108	2
#20828	2
#12909	2
#12892	2
#12838	2
#12837	2
#12676	2
C₄×#368	2