Subgroups — #141855

Order 512.

368 conjugacy classes of subgroups.

The subgroup lattice has 215 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#335	C₂	1	1	yes	maximal, characteristic
4	#7171	C₂	1	4	yes	maximal
5	C₂×#500	C₄	1	1	yes	characteristic
6	C₄×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₄⋊C₁₆	—	2	1	no	—
8	C₂×C₄×C₁₆	C₂²	1	2	yes	—
9	#1990	—	2	1	no	—
10	#1942	—	2	1	no	—
11	#1942	C₄	1	2	yes	—
12	#1845	C₄	1	1	yes	characteristic
13	C₄×#107	—	2	1	no	—
14	C₄×#107	C₂²	1	2	yes	—
15	C₄×C₄⋊C₈	—	2	1	no	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#335	C₂²	1	2	yes	—
18	#1374	—	2	1	no	—
19	#1277	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	1	no	—
21	C₂×C₄×Q₈	—	4	1	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	#500	—	2	2	no	—
24	#500	C₈	1	4	yes	complemented
25	C₂×#107	—	2	1	no	—
26	C₂×#107	—	2	1	no	—
27	C₂×#107	C₂×C₄	1	2	yes	—
28	C₂×#107	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄×C₈	—	4	1	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#391	—	4	1	no	—
38	#389	—	2	2	no	—
39	#389	—	4	1	no	—
40	#384	—	2	1	no	—
41	C₄×C₄⋊C₄	—	2	2	no	—
42	C₄×C₄⋊C₄	—	4	1	no	—
43	C₄³	D₄	1	1	yes	characteristic
44	C₄⋊C₁₆	—	2	4	no	—
45	C₄×C₁₆	—	2	4	no	—
46	#340	—	2	2	no	—
47	#340	—	2	2	no	—
48	#336	—	4	1	no	—
49	#335	—	2	2	no	—
50	#335	Q₈	1	4	yes	complemented
51	C₂²×Q₈	—	4	1	no	—
52	C₂²×C₈	—	4	2	no	—
53	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	C₄⋊Q₈	—	2	4	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	C₄×Q₈	—	8	2	no	—
60	C₄×Q₈	—	8	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	2	2	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄⋊C₄	—	4	2	no	—
66	C₂×C₄⋊C₄	—	4	1	no	—
67	C₂×C₄²	—	4	2	no	—
68	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	D₈	1	1	yes	characteristic
71	C₂×C₄²	D₈	1	1	yes	characteristic
72	C₂×C₄²	SD₁₆	1	1	yes	characteristic
73	C₂×C₄²	SD₁₆	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	—	4	2	no	—
77	C₂×C₁₆	—	4	2	no	—
78	#107	—	2	4	no	—
79	#107	—	2	2	no	—
80	#107	—	2	4	no	—
81	#107	—	4	2	no	—
82	#107	C₂×C₈	1	4	yes	—
83	C₄⋊C₈	—	2	4	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	4	1	no	—
89	C₄×C₈	—	8	2	no	—
90	C₄×C₈	C₂×C₈	1	2	yes	—
91	C₄×C₈	M₁₆	1	2	yes	—
92	C₄×C₈	C₂×Q₈	1	2	yes	—
93	C₄×C₈	#41	1	2	yes	—
94	#95	—	4	2	no	—
95	#95	—	4	2	no	—
96	C₂×Q₈	—	4	4	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂×Q₈	—	8	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	#102	1	1	yes	characteristic
102	C₂²×C₄	#102	1	1	yes	characteristic
103	C₂²×C₄	#102	1	1	yes	characteristic
104	C₂²×C₄	#102	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	4	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	C₂²⋊C₈	1	2	yes	—
119	C₂×C₈	C₄×Q₈	1	2	yes	—
120	C₂×C₈	#124	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₈	#131	1	2	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	2	no	—
131	C₄⋊C₄	—	8	4	no	—
132	C₄⋊C₄	—	8	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	1	no	—
138	C₄²	—	8	2	no	—
139	C₄²	—	8	2	no	—
140	C₄²	C₂²⋊C₈	1	2	yes	—
141	C₄²	#104	1	2	yes	—
142	C₄²	D₁₆	1	2	yes	—
143	C₄²	SD₃₂	1	2	yes	—
144	C₄²	C₂²⋊Q₈	1	2	yes	—
145	C₄²	#137	1	2	yes	—
146	C₁₆	—	4	4	no	—
147	C₂³	C₂×#102	1	1	yes	characteristic
148	Q₈	—	8	8	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	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	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	D₄⋊C₈	1	2	yes	—
165	C₂×C₄	D₄⋊C₈	1	2	yes	—
166	C₂×C₄	#357	1	2	yes	—
167	C₂×C₄	#357	1	2	yes	—
168	C₂×C₄	#359	1	2	yes	—
169	C₂×C₄	#391	1	1	yes	characteristic
170	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂×C₄	#408	1	1	yes	characteristic
172	C₂×C₄	#419	1	1	yes	characteristic
173	C₂×C₄	C₂×#104	1	1	yes	characteristic
174	C₂×C₄	#421	1	1	yes	characteristic
175	C₂×C₄	#474	1	2	yes	—
176	C₂×C₄	#475	1	2	yes	—
177	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
179	C₂×C₄	#510	1	2	yes	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	2	no	—
183	C₈	—	4	1	no	—
184	C₈	—	8	4	no	—
185	C₈	C₈×Q₈	1	2	yes	—
186	C₈	#446	1	2	yes	—
187	C₂²	C₂×D₄⋊C₈	1	1	yes	characteristic
188	C₂²	#1480	1	1	yes	characteristic
189	C₂²	#1988	1	1	yes	characteristic
190	C₂²	C₂×#357	1	1	yes	characteristic
191	C₂²	#2141	1	1	yes	characteristic
192	C₂²	C₂×#359	1	1	yes	characteristic
193	C₂²	#2143	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	1	no	—
198	C₄	—	8	4	no	—
199	C₄	—	16	2	no	—
200	C₄	#1331	1	4	yes	—
201	C₄	#1479	1	1	yes	characteristic
202	C₄	#1485	1	1	yes	characteristic
203	C₄	#1984	1	1	yes	characteristic
204	C₄	#1986	1	1	yes	characteristic
205	C₄	#2224	1	2	yes	—
206	C₄	#2225	1	2	yes	—
207	C₂	#9865	1	1	yes	characteristic
208	C₂	C₂×#1331	1	1	yes	characteristic
209	C₂	#11594	1	1	yes	characteristic
210	C₂	#11645	1	1	yes	characteristic
211	C₂	#11650	1	1	yes	characteristic
212	C₂	#13274	1	1	yes	characteristic
213	C₂	#13275	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ 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
24. #500	C₈	semidirect
50. #335	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#500	2
C₄×C₄⋊C₁₆	2
C₄×#335	2
#7171	2