Subgroups — #128681

Order 512.

294 conjugacy classes of subgroups.

The subgroup lattice has 221 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33474	C₂	1	1	yes	maximal, characteristic
2	#12662	C₂	1	1	yes	maximal, characteristic
3	#11610	C₂	1	1	yes	maximal, characteristic
4	#11399	C₂	1	1	yes	maximal, characteristic
5	#8252	C₂	1	1	yes	maximal, characteristic
6	#7169	C₂	1	1	yes	maximal, characteristic
7	#7153	C₂	1	1	yes	maximal, characteristic
8	#2960	C₂²	1	1	yes	characteristic
9	#2183	—	2	1	no	—
10	#2183	C₄	1	2	yes	complemented
11	Q₈×C₁₆	—	2	1	no	—
12	Q₈×C₁₆	C₄	1	2	yes	complemented
13	#2151	—	2	1	no	—
14	#2150	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	#2108	—	2	1	no	—
17	C₄×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×#346	—	2	1	no	—
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#1542	C₄	1	1	yes	characteristic
22	#1503	C₄	1	1	yes	characteristic
23	#1450	C₂²	1	1	yes	characteristic
24	#1374	—	2	1	no	—
25	#1338	C₄	1	4	yes	complemented
26	#1311	—	2	1	no	—
27	C₈×C₁₆	—	2	1	no	—
28	#1303	C₄	1	1	yes	characteristic
29	#1301	C₄	1	1	yes	characteristic
30	#533	C₂×C₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	1	no	—
32	C₂×M₃₂	—	2	1	no	—
33	C₂²×C₁₆	—	2	1	no	—
34	C₂²×C₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₁₆	D₄	1	1	yes	characteristic
36	#446	—	2	1	no	—
37	#446	C₂×C₄	1	2	yes	—
38	C₈×Q₈	—	2	1	no	—
39	C₈×Q₈	C₂×C₄	1	2	yes	—
40	#431	C₂×C₄	1	1	yes	characteristic
41	#431	C₂×C₄	1	1	yes	characteristic
42	#424	—	2	1	no	—
43	#423	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	#405	—	2	1	no	—
46	C₂×C₄×C₈	—	2	1	no	—
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	2	no	—
50	C₄⋊C₁₆	—	2	2	no	—
51	C₄⋊C₁₆	—	2	1	no	—
52	C₄⋊C₁₆	—	2	1	no	—
53	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
54	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
55	#346	—	2	2	no	—
56	#346	—	4	2	no	—
57	#346	C₂×C₄	1	2	yes	—
58	C₄×C₁₆	—	2	2	no	—
59	C₄×C₁₆	—	2	1	no	—
60	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₆	D₄	1	1	yes	characteristic
63	C₄×C₁₆	D₄	1	1	yes	characteristic
64	#336	—	2	1	no	—
65	#333	C₈	1	2	yes	—
66	#332	C₈	1	2	yes	—
67	#328	C₈	1	4	yes	—
68	#322	—	2	1	no	—
69	C₈²	—	2	1	no	—
70	C₂×M₁₆	—	2	1	no	—
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	#41	1	1	yes	characteristic
74	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
75	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
76	#125	—	2	1	no	—
77	C₂²⋊Q₈	—	2	1	no	—
78	C₂²⋊Q₈	C₂×C₈	1	2	yes	—
79	C₄×Q₈	—	2	1	no	—
80	C₄×Q₈	C₄²	1	2	yes	—
81	#117	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
85	M₃₂	—	2	2	no	—
86	M₃₂	—	2	2	no	—
87	M₃₂	—	4	1	no	—
88	M₃₂	—	4	1	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	4	1	no	—
95	C₂×C₁₆	—	4	1	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	SD₁₆	1	2	yes	—
101	C₂×C₁₆	Q₁₆	1	2	yes	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	C₄²	1	2	yes	—
108	C₄⋊C₈	C₂×C₈	1	2	yes	—
109	C₄⋊C₈	C₂×C₈	1	2	yes	—
110	C₂²⋊C₈	—	2	1	no	—
111	C₂²⋊C₈	C₂×C₈	1	2	yes	—
112	#97	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₄×C₈	C₂×D₄	1	1	yes	characteristic
121	C₄×C₈	#41	1	1	yes	characteristic
122	#95	C₂×C₈	1	2	yes	—
123	C₂×Q₈	—	2	1	no	—
124	C₂×Q₈	C₄×C₈	1	2	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
127	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
128	C₂²×C₄	#131	1	1	yes	characteristic
129	M₁₆	—	4	1	no	—
130	M₁₆	—	4	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₈	#117	1	1	yes	characteristic
141	C₂×C₈	C₄×D₄	1	1	yes	characteristic
142	C₂×C₈	C₄×D₄	1	1	yes	characteristic
143	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₈	#135	1	1	yes	characteristic
146	C₂×C₈	#135	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	C₄×C₈	1	2	yes	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	4	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	C₂×C₄²	1	1	yes	characteristic
159	C₄²	C₄×D₄	1	1	yes	characteristic
160	C₄²	C₂²×C₈	1	1	yes	characteristic
161	C₄²	#131	1	1	yes	characteristic
162	C₁₆	—	2	4	no	—
163	C₁₆	—	2	1	no	—
164	C₁₆	—	2	1	no	—
165	C₁₆	—	4	2	no	—
166	C₁₆	—	4	2	no	—
167	C₁₆	#103	1	2	yes	—
168	C₁₆	#103	1	2	yes	—
169	C₂³	C₈×D₄	1	1	yes	characteristic
170	Q₈	—	4	2	no	—
171	Q₈	—	4	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
181	C₂×C₄	#405	1	1	yes	characteristic
182	C₂×C₄	C₈×D₄	1	1	yes	characteristic
183	C₂×C₄	C₈×D₄	1	1	yes	characteristic
184	C₂×C₄	C₈×D₄	1	1	yes	characteristic
185	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
186	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
187	C₂×C₄	#443	1	1	yes	characteristic
188	C₂×C₄	#443	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	4	2	no	—
192	C₈	—	4	2	no	—
193	C₈	—	4	2	no	—
194	C₈	—	4	2	no	—
195	C₈	C₂×#103	1	1	yes	characteristic
196	C₈	#416	1	1	yes	characteristic
197	C₈	C₄×SD₁₆	1	1	yes	characteristic
198	C₈	C₄×Q₁₆	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₈×SD₁₆	1	1	yes	characteristic
202	C₂²	C₈×Q₁₆	1	1	yes	characteristic
203	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	8	2	no	—
210	C₄	C₈×SD₁₆	1	1	yes	characteristic
211	C₄	C₈×Q₁₆	1	1	yes	characteristic
212	C₄	C₄×#103	1	1	yes	characteristic
213	C₄	#1764	1	1	yes	characteristic
214	C₄	#2171	1	2	yes	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	C₈×#103	1	1	yes	characteristic
218	C₂	#12613	1	1	yes	characteristic
219	C₂	#12615	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
10. #2183	C₄	semidirect
12. Q₈×C₁₆	C₄	semidirect
25. #1338	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33474	2
#12662	2
#11610	2
#11399	2
#8252	2
#7169	2
#7153	2