Subgroups — #128621

Order 512.

370 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#131	C₂	1	1	yes	maximal, characteristic
2	#11613	C₂	1	2	yes	maximal
3	#11411	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#7163	C₂	1	2	yes	maximal, complemented
6	C₈×#41	C₄	1	1	yes	characteristic
7	C₈×#41	C₂²	1	1	yes	characteristic
8	C₄×#131	C₄	1	1	yes	characteristic
9	#2162	C₂²	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	#1450	—	2	1	no	—
12	#1450	C₄	1	1	yes	characteristic
13	C₈×M₁₆	—	2	1	no	—
14	C₈×M₁₆	—	2	1	no	—
15	C₈×M₁₆	C₄	1	1	yes	characteristic
16	C₈×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₈²	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	Q₈⋊C₁₆	C₄	1	4	yes	—
20	D₄⋊C₁₆	C₄	1	4	yes	—
21	#1312	—	2	1	no	—
22	#1311	—	2	1	no	—
23	#1311	—	2	1	no	—
24	#1311	—	2	1	no	—
25	C₂×#131	C₈	1	2	yes	—
26	C₄×#41	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	—	2	2	no	—
28	C₈×Q₈	C₈	1	2	yes	—
29	C₈×Q₈	C₂×C₄	1	2	yes	—
30	C₈×D₄	—	2	2	no	—
31	C₈×D₄	—	2	2	no	—
32	C₈×D₄	C₈	1	2	yes	—
33	C₈×D₄	C₂×C₄	1	2	yes	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	C₂×C₄	1	1	yes	characteristic
37	#431	C₂×C₄	1	1	yes	characteristic
38	#405	—	2	2	no	—
39	#405	C₈	1	2	yes	—
40	C₄×M₁₆	—	2	1	no	—
41	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	—	2	1	no	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
49	C₂²⋊C₁₆	—	2	2	no	—
50	C₄×C₁₆	—	2	2	no	—
51	C₄×C₁₆	—	2	2	no	—
52	C₄×C₁₆	C₂×C₄	1	4	yes	—
53	#322	—	2	1	no	—
54	#322	—	2	1	no	—
55	#322	—	2	1	no	—
56	C₈²	—	2	1	no	—
57	C₈²	—	2	1	no	—
58	C₈²	—	2	1	no	—
59	C₈²	D₄	1	1	yes	characteristic
60	C₈²	D₄	1	1	yes	characteristic
61	C₈²	D₄	1	1	yes	characteristic
62	C₈²	D₄	1	1	yes	characteristic
63	C₂×#41	C₂×C₈	1	1	yes	characteristic
64	#131	—	2	4	no	—
65	#131	—	2	4	no	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	C₂×C₈	1	2	yes	—
68	C₂²×C₈	—	2	2	no	—
69	C₂²×C₈	—	2	2	no	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	C₂×C₈	1	2	yes	—
72	C₂²×C₈	M₁₆	1	2	yes	—
73	C₂²×C₈	#41	1	2	yes	—
74	C₄×Q₈	C₄²	1	1	yes	characteristic
75	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₄×D₄	—	2	1	no	—
77	C₄×D₄	—	2	1	no	—
78	C₄×D₄	C₄²	1	1	yes	characteristic
79	C₄×D₄	C₂×C₈	1	1	yes	characteristic
80	#117	—	2	1	no	—
81	#117	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₁₆	—	2	8	no	—
85	C₂×C₁₆	—	2	4	no	—
86	C₂×C₁₆	—	2	4	no	—
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	2	2	no	—
89	C₄⋊C₈	C₄²	1	2	yes	—
90	C₄⋊C₈	C₂×C₈	1	2	yes	—
91	C₂²⋊C₈	—	2	1	no	—
92	C₂²⋊C₈	—	2	1	no	—
93	C₂²⋊C₈	—	2	2	no	—
94	C₂²⋊C₈	—	2	2	no	—
95	#97	—	2	2	no	—
96	#97	—	2	1	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₈	C₂×C₈	1	2	yes	—
108	C₄×C₈	M₁₆	1	2	yes	—
109	C₄×C₈	C₂²×C₄	1	2	yes	—
110	C₄×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄×C₈	C₂×D₄	1	1	yes	characteristic
112	#41	—	2	2	no	—
113	#41	M₃₂	1	4	yes	—
114	C₂×Q₈	C₄×C₈	1	1	yes	characteristic
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	C₄×C₈	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	#117	1	1	yes	characteristic
121	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
122	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
123	M₁₆	—	2	8	no	—
124	M₁₆	—	2	2	no	—
125	M₁₆	M₃₂	1	4	yes	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	C₂²⋊C₈	1	2	yes	—
137	C₂×C₈	C₂²⋊C₈	1	2	yes	—
138	C₂×C₈	C₂²⋊C₈	1	2	yes	—
139	C₂×C₈	C₂²⋊C₈	1	2	yes	—
140	C₂×C₈	C₄×D₄	1	2	yes	—
141	C₂×C₈	C₄×D₄	1	2	yes	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
145	C₄⋊C₄	C₄×C₈	1	1	yes	characteristic
146	C₂²⋊C₄	—	2	1	no	—
147	C₂²⋊C₄	—	2	1	no	—
148	C₂²⋊C₄	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	C₂×C₄²	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₄²	C₂²×C₈	1	1	yes	characteristic
155	C₄²	C₂×M₁₆	1	1	yes	characteristic
156	C₁₆	—	2	8	no	—
157	C₁₆	—	4	4	no	—
158	C₂³	—	2	1	no	—
159	C₂³	#431	1	1	yes	characteristic
160	Q₈	—	2	1	no	—
161	Q₈	#346	1	2	yes	—
162	D₄	—	2	4	no	—
163	D₄	—	2	1	no	—
164	D₄	#346	1	2	yes	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
175	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
177	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
178	C₂×C₄	#431	1	1	yes	characteristic
179	C₂×C₄	C₂×M₃₂	1	2	yes	—
180	C₂×C₄	C₂×M₃₂	1	2	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	2	4	no	—
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₈	—	4	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#443	1	4	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	C₄×C₂²⋊C₈	1	1	yes	characteristic
196	C₂²	#2163	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	1	no	—
202	C₄	#1764	1	2	yes	—
203	C₄	C₂×#346	1	1	yes	characteristic
204	C₄	C₄×M₃₂	1	1	yes	characteristic
205	C₄	#2112	1	2	yes	—
206	C₄	#2117	1	2	yes	—
207	C₂	—	2	2	no	—
208	C₂	—	4	2	no	—
209	C₂	#8184	1	1	yes	characteristic
210	C₂	#12599	1	1	yes	characteristic
211	C₂	#12604	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×C₂²⋊C₈
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈² ◁ 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
3. #11411	C₂	semidirect
4. C₂×#1311	C₂	semidirect
5. #7163	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×#131	2
#11613	2
#11411	2
C₂×#1311	2
#7163	2