Subgroups — #498531

Order 512.

446 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33726	C₂	1	1	yes	maximal, characteristic
2	#23170	C₂	1	2	yes	maximal
3	#13274	C₂	1	2	yes	maximal
4	#13169	C₂	1	2	yes	maximal
5	#13113	C₂	1	4	yes	maximal
6	C₄×#368	C₂	1	1	yes	maximal, characteristic
7	#12670	C₂	1	2	yes	maximal
8	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	#3160	C₂²	1	2	yes	—
10	#2962	—	2	2	no	—
11	#2254	C₄	1	2	yes	complemented
12	#2253	C₄	1	2	yes	complemented
13	#2253	C₄	1	2	yes	complemented
14	#2252	C₄	1	1	yes	characteristic, complemented
15	#2252	C₄	1	1	yes	characteristic, complemented
16	C₂×#368	C₂²	1	1	yes	characteristic
17	C₂×#368	C₂²	1	1	yes	characteristic
18	#2158	C₂²	1	2	yes	—
19	#2158	C₂²	1	2	yes	—
20	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	2	yes	—
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1986	—	2	2	no	—
24	#1945	C₂²	1	4	yes	—
25	#1922	—	2	4	no	—
26	C₄×#108	C₂²	1	2	yes	—
27	#1772	C₂²	1	4	yes	—
28	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	#1382	C₂²	1	4	yes	—
30	#1382	C₂²	1	4	yes	—
31	#1381	C₂²	1	2	yes	—
32	#1374	C₂²	1	2	yes	—
33	#533	—	2	2	no	—
34	C₂²×C₁₆	C₂³	1	2	yes	—
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	#500	C₂×C₄	1	2	yes	—
38	#500	C₂×C₄	1	2	yes	—
39	#499	—	2	2	no	—
40	#499	C₂×C₄	1	4	yes	—
41	#498	—	2	2	no	—
42	#446	—	4	4	no	—
43	#446	—	4	2	no	—
44	C₈×Q₈	—	4	2	no	—
45	C₂×#108	C₂³	1	2	yes	—
46	C₂×#108	C₂³	1	2	yes	—
47	#427	—	2	2	no	—
48	#427	C₂³	1	4	yes	—
49	#424	—	2	4	no	—
50	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
51	#405	—	2	4	no	—
52	C₄×M₁₆	—	2	2	no	—
53	C₂×C₄×C₈	C₂³	1	2	yes	—
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
56	#368	—	2	2	no	—
57	#368	Q₈	1	4	yes	complemented
58	#367	C₂×C₄	1	4	yes	—
59	#367	C₂×C₄	1	4	yes	—
60	#366	C₂×C₄	1	2	yes	—
61	#366	C₂×C₄	1	2	yes	—
62	#366	C₂×C₄	1	2	yes	—
63	#366	C₂×C₄	1	2	yes	—
64	C₄⋊C₁₆	—	2	1	no	—
65	C₄⋊C₁₆	—	2	1	no	—
66	C₄×C₁₆	—	2	4	no	—
67	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
68	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
69	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
70	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
71	#341	—	2	2	no	—
72	#339	—	2	4	no	—
73	#339	—	2	4	no	—
74	#336	D₄	1	2	yes	—
75	C₂×M₁₆	—	2	2	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₄	1	2	yes	—
79	C₂²×C₈	C₂⁴	1	1	yes	characteristic
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	2	2	no	—
82	#125	—	2	4	no	—
83	C₂²⋊Q₈	—	4	2	no	—
84	C₂²⋊Q₈	—	4	2	no	—
85	C₄×Q₈	—	4	2	no	—
86	C₄×Q₈	—	4	2	no	—
87	#117	—	2	4	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄²	C₂×D₄	1	2	yes	—
90	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₁₆	C₂×Q₈	1	2	yes	—
101	C₂×C₁₆	#41	1	2	yes	—
102	#108	—	2	2	no	—
103	#108	—	2	2	no	—
104	#108	—	2	2	no	—
105	#108	C₂×Q₈	1	4	yes	—
106	#107	—	2	2	no	—
107	#107	—	2	2	no	—
108	#107	C₂²×C₄	1	4	yes	—
109	#107	C₂²×C₄	1	4	yes	—
110	#106	—	2	4	no	—
111	#106	—	2	4	no	—
112	C₄⋊C₈	—	4	4	no	—
113	C₄⋊C₈	—	4	4	no	—
114	C₄⋊C₈	D₈	1	1	yes	characteristic
115	C₄⋊C₈	D₈	1	1	yes	characteristic
116	C₄⋊C₈	D₈	1	2	yes	—
117	#97	—	4	4	no	—
118	#97	—	4	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	4	4	no	—
122	C₄×C₈	—	4	2	no	—
123	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₄×C₈	#41	1	1	yes	characteristic
126	C₄×C₈	#41	1	1	yes	characteristic
127	#95	—	2	1	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	#135	1	1	yes	characteristic
132	C₂²×C₄	#135	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
134	M₁₆	—	2	4	no	—
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	C₄×D₄	1	1	yes	characteristic
145	C₂×C₈	C₄×D₄	1	1	yes	characteristic
146	C₂×C₈	C₂×D₈	1	2	yes	—
147	C₂×C₈	C₂×D₈	1	2	yes	—
148	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₈	C₂×#41	1	1	yes	characteristic
151	C₂×C₈	C₂×#41	1	1	yes	characteristic
152	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	4	no	—
165	C₄²	C₄×D₄	1	1	yes	characteristic
166	C₄²	C₄×D₄	1	1	yes	characteristic
167	C₄²	C₂×D₈	1	1	yes	characteristic
168	C₄²	C₂×D₈	1	1	yes	characteristic
169	C₁₆	—	2	1	no	—
170	C₁₆	—	2	1	no	—
171	C₁₆	—	4	2	no	—
172	C₁₆	C₄×Q₈	1	2	yes	—
173	C₁₆	C₄×Q₈	1	2	yes	—
174	C₂³	C₂×#135	1	1	yes	characteristic
175	Q₈	—	8	4	no	—
176	Q₈	—	16	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₄×D₈	1	1	yes	characteristic
185	C₂×C₄	C₄×D₈	1	1	yes	characteristic
186	C₂×C₄	C₄×D₈	1	1	yes	characteristic
187	C₂×C₄	C₄×D₈	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂×C₄	#547	1	1	yes	characteristic
190	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
191	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
192	C₂×C₄	#578	1	1	yes	characteristic
193	C₂×C₄	#578	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	4	4	no	—
199	C₈	—	8	2	no	—
200	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
201	C₈	#519	1	1	yes	characteristic
202	C₈	#556	1	1	yes	characteristic
203	C₈	#557	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
207	C₂²	#3342	1	1	yes	characteristic
208	C₂²	#3383	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	4	4	no	—
213	C₄	—	8	4	no	—
214	C₄	—	8	4	no	—
215	C₄	#2178	1	2	yes	—
216	C₄	#2365	1	1	yes	characteristic
217	C₄	#2950	1	1	yes	characteristic
218	C₄	#3334	1	1	yes	characteristic
219	C₄	Q₈×D₈	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#23157	1	1	yes	characteristic
223	C₂	C₂×#2178	1	1	yes	characteristic
224	C₂	#33465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ 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
11. #2254	C₄	semidirect
12. #2253	C₄	semidirect
13. #2253	C₄	semidirect
14. #2252	C₄	semidirect
15. #2252	C₄	semidirect
57. #368	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33726	2
#23170	2
#13274	2
#13169	2
#13113	2
C₄×#368	2
#12670	2
C₁₆×C₄⋊C₄	2