Subgroups — #484127

Order 512.

397 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33734	C₂	1	1	yes	maximal, characteristic
2	#18086	C₂	1	2	yes	maximal
3	#12085	C₂	1	2	yes	maximal, complemented
4	#12072	C₂	1	4	yes	maximal, complemented
5	#12044	C₂	1	2	yes	maximal, complemented
6	#12008	C₂	1	1	yes	maximal, characteristic
7	#11650	C₂	1	2	yes	maximal
8	#11408	C₂	1	1	yes	maximal, characteristic
9	#3170	C₂²	1	2	yes	—
10	#2986	—	2	2	no	—
11	#2252	C₄	1	1	yes	characteristic
12	#2247	C₄	1	1	yes	characteristic
13	#2241	C₄	1	2	yes	—
14	#2216	C₄	1	2	yes	—
15	#2216	C₄	1	2	yes	—
16	C₄×Q₃₂	C₂²	1	2	yes	—
17	C₄×Q₃₂	C₂²	1	2	yes	—
18	#2162	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	#1591	C₂²	1	4	yes	—
21	#1587	—	2	4	no	—
22	C₈×Q₁₆	C₂²	1	4	yes	—
23	#1564	C₂²	1	2	yes	—
24	#1485	—	2	2	no	—
25	C₈×M₁₆	C₂²	1	1	yes	characteristic
26	#1372	C₂²	1	1	yes	characteristic
27	#1372	C₂²	1	1	yes	characteristic
28	#1371	C₂²	1	2	yes	—
29	#1332	C₂²	1	4	yes	—
30	#1332	C₂²	1	4	yes	—
31	#1312	C₂²	1	2	yes	—
32	#1311	C₂²	1	2	yes	—
33	#533	—	2	2	no	—
34	C₂×Q₃₂	C₂×C₄	1	2	yes	—
35	C₂×Q₃₂	C₂×C₄	1	2	yes	—
36	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
38	#500	C₂×C₄	1	2	yes	—
39	#494	C₂×C₄	1	2	yes	—
40	#487	—	2	2	no	—
41	#467	—	2	2	no	—
42	#467	C₂×C₄	1	4	yes	—
43	#446	—	2	4	no	—
44	#446	—	4	2	no	—
45	C₈×Q₈	—	2	4	no	—
46	C₈×Q₈	—	4	2	no	—
47	C₄×Q₁₆	—	2	2	no	—
48	C₄×Q₁₆	C₂³	1	4	yes	—
49	#433	—	4	4	no	—
50	#431	—	2	2	no	—
51	#431	D₄	1	1	yes	characteristic
52	C₄×M₁₆	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#366	C₂×C₄	1	2	yes	—
55	#366	C₂×C₄	1	2	yes	—
56	C₄⋊C₁₆	—	2	1	no	—
57	#358	C₂×C₄	1	4	yes	—
58	#358	C₂×C₄	1	4	yes	—
59	C₂²⋊C₁₆	—	2	1	no	—
60	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₆	C₂³	1	2	yes	—
63	C₄×C₁₆	C₂³	1	1	yes	characteristic
64	C₄×C₁₆	C₂³	1	1	yes	characteristic
65	#335	C₂³	1	2	yes	—
66	#335	C₂³	1	2	yes	—
67	#334	—	2	2	no	—
68	Q₈⋊C₈	—	2	4	no	—
69	Q₈⋊C₈	—	2	4	no	—
70	#322	D₄	1	2	yes	—
71	C₈²	C₂³	1	2	yes	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₂×Q₁₆	C₂²×C₄	1	4	yes	—
74	C₂×M₁₆	—	2	1	no	—
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	#41	1	1	yes	characteristic
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	#125	—	4	2	no	—
80	C₂²⋊Q₈	—	2	4	no	—
81	C₂²⋊Q₈	—	4	2	no	—
82	C₄×Q₈	—	2	4	no	—
83	C₄×Q₈	—	4	2	no	—
84	#117	—	4	2	no	—
85	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
86	Q₃₂	—	2	4	no	—
87	Q₃₂	—	4	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	4	1	no	—
91	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	C₂²×C₄	1	4	yes	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	4	4	no	—
100	C₄⋊C₈	—	4	4	no	—
101	C₄⋊C₈	D₈	1	2	yes	—
102	#103	—	2	4	no	—
103	#103	—	2	4	no	—
104	C₂²⋊C₈	—	4	2	no	—
105	C₂²⋊C₈	D₈	1	2	yes	—
106	#97	—	2	1	no	—
107	#97	—	4	4	no	—
108	C₄×C₈	—	4	4	no	—
109	C₄×C₈	—	4	2	no	—
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂×D₄	1	2	yes	—
112	C₄×C₈	C₂×D₄	1	1	yes	characteristic
113	C₄×C₈	C₂×D₄	1	2	yes	—
114	C₄×C₈	#41	1	1	yes	characteristic
115	C₄×C₈	C₂⁴	1	1	yes	characteristic
116	C₂×Q₈	—	2	4	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
119	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
120	C₂²×C₄	#135	1	1	yes	characteristic
121	Q₁₆	—	2	4	no	—
122	Q₁₆	—	2	4	no	—
123	Q₁₆	—	8	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	C₄×D₄	1	1	yes	characteristic
132	C₂×C₈	C₄×D₄	1	1	yes	characteristic
133	C₂×C₈	C₂×D₈	1	4	yes	—
134	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₈	C₂×#41	1	1	yes	characteristic
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₂²⋊C₄	—	4	4	no	—
143	C₄²	—	4	4	no	—
144	C₄²	C₄×D₄	1	1	yes	characteristic
145	C₄²	C₂×D₈	1	1	yes	characteristic
146	C₄²	#135	1	1	yes	characteristic
147	C₄²	C₂²×D₄	1	1	yes	characteristic
148	C₁₆	—	2	1	no	—
149	C₁₆	—	2	1	no	—
150	C₁₆	—	2	1	no	—
151	C₁₆	—	4	2	no	—
152	C₁₆	#131	1	2	yes	—
153	C₂³	C₄×D₈	1	1	yes	characteristic
154	Q₈	—	4	4	no	—
155	Q₈	—	4	4	no	—
156	Q₈	—	8	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₄×D₈	1	1	yes	characteristic
164	C₂×C₄	C₄×D₈	1	1	yes	characteristic
165	C₂×C₄	C₄×D₈	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
168	C₂×C₄	C₂×#135	1	1	yes	characteristic
169	C₈	—	2	4	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	2	no	—
174	C₈	—	8	2	no	—
175	C₈	—	8	2	no	—
176	C₈	C₂×#131	1	1	yes	characteristic
177	C₈	#568	1	1	yes	characteristic
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	8	4	no	—
186	C₄	—	8	4	no	—
187	C₄	#2179	1	2	yes	—
188	C₄	#2935	1	1	yes	characteristic
189	C₄	#2957	1	1	yes	characteristic
190	C₄	#3417	1	2	yes	—
191	C₂	—	4	1	no	—
192	C₂	#18066	1	1	yes	characteristic
193	C₂	#33458	1	1	yes	characteristic
194	C₂	C₂×#2179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12085	C₂	semidirect
4. #12072	C₂	semidirect
5. #12044	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33734	2
#18086	2
#12085	2
#12072	2
#12044	2
#12008	2
#11650	2
#11408	2