Subgroups — #506088

Order 512.

422 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27994	C₂	1	2	yes	maximal
2	#13537	C₂	1	2	yes	maximal
3	#13515	C₂	1	2	yes	maximal
4	#13168	C₂	1	1	yes	maximal, characteristic
5	#13167	C₂	1	1	yes	maximal, characteristic
6	#13167	C₂	1	1	yes	maximal, characteristic
7	#13166	C₂	1	1	yes	maximal, characteristic
8	#13112	C₂	1	2	yes	maximal
9	#12905	C₂	1	1	yes	maximal, characteristic
10	#12674	C₂	1	1	yes	maximal, characteristic
11	#12663	C₂	1	1	yes	maximal, characteristic
12	#2506	—	2	2	no	—
13	C₂×#367	C₂²	1	1	yes	characteristic
14	C₂×#367	C₂²	1	1	yes	characteristic
15	C₂×#366	C₂²	1	2	yes	—
16	C₂×#366	C₂²	1	1	yes	characteristic
17	C₂×#366	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×#346	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	#2099	—	2	2	no	—
23	#2086	—	2	2	no	—
24	#1942	C₂²	1	2	yes	—
25	#1942	C₂²	1	2	yes	—
26	#1942	C₂²	1	2	yes	—
27	#1942	C₂²	1	2	yes	—
28	#1918	—	2	2	no	—
29	#1845	C₂²	1	2	yes	—
30	C₄×#107	C₂²	1	2	yes	—
31	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	#1380	C₂²	1	2	yes	—
33	#1380	C₂²	1	2	yes	—
34	#1380	C₂²	1	2	yes	—
35	#1380	C₂²	1	2	yes	—
36	#1380	C₂²	1	2	yes	—
37	#1374	C₂²	1	1	yes	characteristic
38	#1374	C₂²	1	1	yes	characteristic
39	C₂×#125	—	4	2	no	—
40	C₂²×C₁₆	C₂³	1	1	yes	characteristic
41	C₂²×C₁₆	C₂³	1	1	yes	characteristic
42	C₂²×C₁₆	C₂³	1	1	yes	characteristic
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂×#107	—	2	2	no	—
45	C₂×#107	C₂³	1	2	yes	—
46	C₂×#107	C₂³	1	2	yes	—
47	C₂×#107	C₂³	1	2	yes	—
48	C₂×#107	C₂³	1	2	yes	—
49	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	#400	—	4	2	no	—
54	#398	—	4	2	no	—
55	#389	—	2	2	no	—
56	#389	—	2	2	no	—
57	#389	—	2	2	no	—
58	#389	—	2	2	no	—
59	#387	—	4	2	no	—
60	#384	—	2	2	no	—
61	C₄×C₄⋊C₄	—	2	2	no	—
62	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
63	#367	—	2	2	no	—
64	#367	Q₈	1	4	yes	—
65	#366	—	2	4	no	—
66	#366	—	2	2	no	—
67	#366	—	2	2	no	—
68	C₄⋊C₁₆	—	2	2	no	—
69	#346	—	2	2	no	—
70	#346	—	2	2	no	—
71	C₄×C₁₆	—	2	2	no	—
72	#340	—	2	2	no	—
73	#340	—	2	2	no	—
74	#340	—	2	2	no	—
75	#340	—	2	2	no	—
76	#340	—	2	2	no	—
77	#336	D₄	1	1	yes	characteristic
78	#336	D₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₈	C₂⁴	1	1	yes	characteristic
83	#125	—	8	8	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	4	2	no	—
89	C₂×C₄⋊C₄	—	4	2	no	—
90	C₂×C₄⋊C₄	—	4	2	no	—
91	C₂×C₄⋊C₄	—	4	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄⋊C₄	—	4	2	no	—
94	C₂×C₄⋊C₄	D₈	1	2	yes	—
95	C₂×C₄²	—	4	2	no	—
96	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	C₂×Q₈	1	2	yes	—
108	C₂×C₁₆	#41	1	2	yes	—
109	C₂×C₁₆	#41	1	2	yes	—
110	C₂×C₁₆	#41	1	2	yes	—
111	#107	—	2	2	no	—
112	#107	—	2	2	no	—
113	#107	—	2	2	no	—
114	#107	—	2	2	no	—
115	#107	—	2	2	no	—
116	#107	—	2	2	no	—
117	#107	—	2	2	no	—
118	#107	—	4	4	no	—
119	#107	C₂×Q₈	1	4	yes	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	1	no	—
127	#95	—	4	2	no	—
128	#95	—	4	2	no	—
129	#95	—	4	2	no	—
130	#95	—	4	2	no	—
131	#95	—	4	2	no	—
132	#95	D₈	1	2	yes	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₈	1	2	yes	—
139	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₈	1	2	yes	—
141	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	#135	1	2	yes	—
151	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₈	C₂×#41	1	1	yes	characteristic
153	C₂×C₈	C₂×#41	1	1	yes	characteristic
154	C₂×C₈	C₂×#41	1	1	yes	characteristic
155	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	8	4	no	—
167	C₄⋊C₄	—	8	4	no	—
168	C₄⋊C₄	—	8	4	no	—
169	C₄⋊C₄	—	8	4	no	—
170	C₄⋊C₄	—	8	4	no	—
171	C₄⋊C₄	—	8	4	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	8	4	no	—
179	C₁₆	—	2	2	no	—
180	C₁₆	—	4	1	no	—
181	C₁₆	—	4	1	no	—
182	C₁₆	#125	1	4	yes	complemented
183	C₂³	C₂²×D₈	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	#509	1	4	yes	—
206	C₂×C₄	#532	1	1	yes	characteristic
207	C₂×C₄	#547	1	1	yes	characteristic
208	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
209	C₂×C₄	C₂×#135	1	1	yes	characteristic
210	C₂×C₄	#578	1	1	yes	characteristic
211	C₂×C₄	#578	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	4	1	no	—
214	C₈	—	4	1	no	—
215	C₈	C₂×#125	1	1	yes	characteristic
216	C₈	#541	1	1	yes	characteristic
217	C₈	#552	1	1	yes	characteristic
218	C₈	#557	1	1	yes	characteristic
219	C₂²	#3169	1	1	yes	characteristic
220	C₂²	#3334	1	1	yes	characteristic
221	C₂²	Q₈×D₈	1	1	yes	characteristic
222	C₂²	C₂×#509	1	2	yes	—
223	C₂²	#3417	1	2	yes	—
224	C₄	—	2	4	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	8	4	no	—
229	C₄	—	16	2	no	—
230	C₄	—	16	2	no	—
231	C₄	—	16	2	no	—
232	C₄	#2506	1	1	yes	characteristic
233	C₄	#3177	1	1	yes	characteristic
234	C₄	#3336	1	1	yes	characteristic
235	C₄	#3383	1	1	yes	characteristic
236	C₂	#27994	1	1	yes	characteristic
237	C₂	#33741	1	1	yes	characteristic
238	C₂	#33748	1	1	yes	characteristic
239	C₂	#33910	1	2	yes	—
240	C₂	#33953	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
182. C₁₆	#125	semidirect

Maximal subgroups

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

subgroup	index
#27994	2
#13537	2
#13515	2
#13168	2
#13167	2
#13166	2
#13112	2
#12905	2
#12674	2
#12663	2