Subgroups — #483740

Order 512.

386 conjugacy classes of subgroups.

The subgroup lattice has 195 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33499	C₂	1	2	yes	maximal
2	#17958	C₂	1	1	yes	maximal, characteristic
3	#12060	C₂	1	4	yes	maximal, complemented
4	#12053	C₂	1	4	yes	maximal, complemented
5	#12013	C₂	1	1	yes	maximal, characteristic
6	#11640	C₂	1	2	yes	maximal
7	#11411	C₂	1	1	yes	maximal, characteristic
8	#3170	C₄	1	1	yes	characteristic
9	#2960	C₂²	1	2	yes	—
10	#2183	—	2	4	no	—
11	#2183	C₂²	1	4	yes	—
12	Q₈×C₁₆	C₂²	1	4	yes	—
13	#2164	—	2	4	no	—
14	#2163	C₂²	1	2	yes	—
15	#2162	C₂²	1	1	yes	characteristic
16	#2162	C₂²	1	1	yes	characteristic
17	#1591	C₄	1	2	yes	—
18	#1591	C₄	1	2	yes	—
19	C₈×Q₁₆	C₂²	1	2	yes	—
20	C₈×Q₁₆	C₂²	1	2	yes	—
21	#1567	C₄	1	1	yes	characteristic
22	#1493	C₄	1	2	yes	—
23	C₂×C₈²	C₂²	1	1	yes	characteristic
24	#1370	C₂²	1	2	yes	—
25	#1367	C₂²	1	2	yes	—
26	Q₈⋊C₁₆	C₂²	1	8	yes	—
27	#1312	C₂²	1	2	yes	—
28	#1311	C₂²	1	2	yes	—
29	#533	C₂×C₄	1	2	yes	—
30	#500	C₈	1	1	yes	characteristic
31	#494	C₈	1	1	yes	characteristic
32	#487	C₈	1	2	yes	—
33	#467	C₈	1	2	yes	—
34	#467	C₈	1	2	yes	—
35	#446	—	2	2	no	—
36	#446	C₂×C₄	1	4	yes	—
37	C₈×Q₈	—	2	2	no	—
38	C₈×Q₈	C₂³	1	4	yes	—
39	C₄×Q₁₆	C₂×C₄	1	2	yes	—
40	C₄×Q₁₆	C₂×C₄	1	2	yes	—
41	#424	—	2	2	no	—
42	#423	C₂×C₄	1	2	yes	—
43	#405	—	2	2	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₄⋊C₁₆	—	2	4	no	—
48	C₄⋊C₁₆	—	2	4	no	—
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	D₄	1	2	yes	—
51	C₄⋊C₁₆	C₂³	1	4	yes	—
52	C₂²⋊C₁₆	—	2	2	no	—
53	C₂²⋊C₁₆	—	2	1	no	—
54	C₂²⋊C₁₆	D₄	1	2	yes	—
55	#346	—	2	4	no	—
56	#346	—	2	2	no	—
57	C₄×C₁₆	—	2	4	no	—
58	C₄×C₁₆	C₂³	1	2	yes	—
59	C₄×C₁₆	C₂³	1	2	yes	—
60	#335	C₂×C₄	1	2	yes	—
61	#335	C₂×C₄	1	2	yes	—
62	Q₈⋊C₈	C₂×C₄	1	4	yes	—
63	Q₈⋊C₈	C₂×C₄	1	4	yes	—
64	C₈²	C₂×C₄	1	1	yes	characteristic
65	C₈²	C₂×C₄	1	1	yes	characteristic
66	C₈²	C₂³	1	1	yes	characteristic
67	C₈²	C₂³	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×C₈	1	2	yes	—
69	C₂×Q₁₆	C₂×C₈	1	2	yes	—
70	C₂×M₁₆	—	2	2	no	—
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	#41	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×C₈	1	2	yes	—
77	C₄⋊Q₈	C₂×C₈	1	2	yes	—
78	#125	—	2	2	no	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₂²⋊Q₈	C₂×C₈	1	4	yes	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	C₂²×C₄	1	4	yes	—
83	#117	—	2	2	no	—
84	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₁₆	—	2	4	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	C₂×D₄	1	4	yes	—
88	#107	C₂×C₈	1	2	yes	—
89	#107	C₂×C₈	1	2	yes	—
90	C₄⋊C₈	—	2	4	no	—
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	C₂²×C₄	1	4	yes	—
94	#103	C₂×C₈	1	4	yes	—
95	#103	C₂×C₈	1	4	yes	—
96	#97	—	2	4	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	C₂×C₈	1	1	yes	characteristic
100	C₄×C₈	C₂×C₈	1	1	yes	characteristic
101	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₄×C₈	C₂×D₄	1	1	yes	characteristic
107	C₄×C₈	#41	1	1	yes	characteristic
108	C₄×C₈	C₂⁴	1	1	yes	characteristic
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	C₂²×C₈	1	4	yes	—
111	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
112	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
113	C₂²×C₄	#131	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	M₁₆	—	8	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₈	C₄×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₈	C₂×#41	1	1	yes	characteristic
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	C₂²×C₈	1	4	yes	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₄²	—	2	4	no	—
141	C₄²	C₄×D₄	1	1	yes	characteristic
142	C₄²	C₂²×C₈	1	1	yes	characteristic
143	C₄²	#131	1	1	yes	characteristic
144	C₄²	C₂³×C₄	1	1	yes	characteristic
145	C₁₆	—	2	4	no	—
146	C₁₆	—	4	2	no	—
147	C₁₆	—	4	4	no	—
148	C₂³	C₈×D₄	1	1	yes	characteristic
149	Q₈	—	2	4	no	—
150	Q₈	—	2	4	no	—
151	Q₈	—	8	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	C₈×D₄	1	1	yes	characteristic
160	C₂×C₄	C₈×D₄	1	1	yes	characteristic
161	C₂×C₄	C₈×D₄	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
164	C₂×C₄	C₂×#131	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	4	no	—
175	C₈	#578	1	2	yes	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	4	no	—
186	C₄	#2950	1	1	yes	characteristic
187	C₄	#2957	1	1	yes	characteristic
188	C₄	#3407	1	1	yes	characteristic
189	C₄	#3407	1	1	yes	characteristic
190	C₂	—	4	1	no	—
191	C₂	#17856	1	1	yes	characteristic
192	C₂	#33435	1	1	yes	characteristic
193	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ 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. #12060	C₂	semidirect
4. #12053	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33499	2
#17958	2
#12060	2
#12053	2
#12013	2
#11640	2
#11411	2