Subgroups — #142116

Order 512.

320 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33258	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#8806	C₂	1	1	yes	maximal, characteristic
4	#7274	C₂	1	2	yes	maximal
5	#7274	C₂	1	2	yes	maximal
6	#2859	—	2	1	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	—	2	1	no	—
9	#1942	—	2	1	no	—
10	#1942	—	2	1	no	—
11	#1942	C₄	1	2	yes	—
12	#1942	C₄	1	2	yes	—
13	#1845	C₂²	1	1	yes	characteristic
14	#1845	C₂²	1	1	yes	characteristic
15	#1842	—	2	1	no	—
16	#1834	—	2	1	no	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#335	—	2	1	no	—
19	C₂×#335	C₂²	1	2	yes	—
20	#1374	—	2	1	no	—
21	C₂×#125	—	4	1	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	C₂×#107	—	2	1	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	C₂×C₄	1	2	yes	—
26	C₂×#107	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×C₄⋊C₈	—	4	1	no	—
29	C₂×C₄⋊C₈	—	4	1	no	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	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	C₂³	1	1	yes	characteristic
36	#389	—	2	2	no	—
37	#389	—	2	2	no	—
38	#389	—	4	1	no	—
39	#389	—	4	1	no	—
40	#384	—	2	1	no	—
41	#384	—	2	1	no	—
42	#382	—	4	1	no	—
43	C₄³	D₄	1	1	yes	characteristic
44	C₄⋊C₁₆	—	2	2	no	—
45	C₄⋊C₁₆	Q₈	1	4	yes	—
46	C₄×C₁₆	—	4	2	no	—
47	#340	—	2	2	no	—
48	#340	—	2	2	no	—
49	#335	—	2	4	no	—
50	#335	—	4	2	no	—
51	C₂²×C₈	—	4	2	no	—
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	#125	—	8	2	no	—
57	#125	—	8	2	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	2	no	—
62	C₂×C₄⋊C₄	—	4	1	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄⋊C₄	—	4	1	no	—
66	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄²	D₈	1	1	yes	characteristic
69	C₂×C₄²	D₈	1	1	yes	characteristic
70	C₂×C₄²	SD₁₆	1	1	yes	characteristic
71	C₂×C₄²	SD₁₆	1	1	yes	characteristic
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₁₆	—	2	4	no	—
74	C₂×C₁₆	—	4	2	no	—
75	C₂×C₁₆	—	4	2	no	—
76	#107	—	4	2	no	—
77	#107	—	4	1	no	—
78	#107	—	4	1	no	—
79	#107	—	4	2	no	—
80	#107	—	4	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	8	2	no	—
84	C₄⋊C₈	—	8	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	4	1	no	—
89	C₄×C₈	—	4	1	no	—
90	C₄×C₈	SD₁₆	1	2	yes	—
91	C₄×C₈	Q₁₆	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	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	#102	1	1	yes	characteristic
99	C₂²×C₄	#102	1	1	yes	characteristic
100	C₂²×C₄	#102	1	1	yes	characteristic
101	C₂²×C₄	#102	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
104	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	8	2	no	—
112	C₂×C₈	—	8	2	no	—
113	C₂×C₈	—	8	2	no	—
114	C₂×C₈	#103	1	2	yes	—
115	C₂×C₈	#103	1	2	yes	—
116	C₂×C₈	C₄×Q₈	1	2	yes	—
117	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₈	#135	1	2	yes	—
119	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₈	#137	1	1	yes	characteristic
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	8	2	no	—
126	C₄⋊C₄	—	8	2	no	—
127	C₄⋊C₄	—	8	4	no	—
128	C₄⋊C₄	—	8	4	no	—
129	C₄⋊C₄	—	8	1	no	—
130	C₄⋊C₄	—	8	1	no	—
131	C₄⋊C₄	—	8	1	no	—
132	C₄⋊C₄	—	8	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	—	4	1	no	—
139	C₄²	D₁₆	1	2	yes	—
140	C₄²	SD₃₂	1	2	yes	—
141	C₄²	SD₃₂	1	2	yes	—
142	C₄²	Q₃₂	1	2	yes	—
143	C₄²	#124	1	1	yes	characteristic
144	C₄²	C₄⋊Q₈	1	1	yes	characteristic
145	C₄²	C₂×SD₁₆	1	1	yes	characteristic
146	C₄²	C₂×Q₁₆	1	1	yes	characteristic
147	C₁₆	—	4	4	no	—
148	C₂³	C₂×#102	1	1	yes	characteristic
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	—	2	1	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	—	8	2	no	—
165	C₂×C₄	#357	1	2	yes	—
166	C₂×C₄	#357	1	2	yes	—
167	C₂×C₄	#358	1	2	yes	—
168	C₂×C₄	#358	1	2	yes	—
169	C₂×C₄	#391	1	1	yes	characteristic
170	C₂×C₄	#413	1	1	yes	characteristic
171	C₂×C₄	#413	1	1	yes	characteristic
172	C₂×C₄	C₂×#103	1	1	yes	characteristic
173	C₂×C₄	#416	1	1	yes	characteristic
174	C₂×C₄	#417	1	1	yes	characteristic
175	C₂×C₄	#486	1	1	yes	characteristic
176	C₂×C₄	#487	1	1	yes	characteristic
177	C₂×C₄	#498	1	1	yes	characteristic
178	C₂×C₄	#500	1	1	yes	characteristic
179	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
180	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
181	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
182	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₈	—	16	2	no	—
187	C₈	#478	1	2	yes	—
188	C₈	#479	1	2	yes	—
189	C₂²	#1540	1	1	yes	characteristic
190	C₂²	#1547	1	1	yes	characteristic
191	C₂²	#1990	1	1	yes	characteristic
192	C₂²	C₂×#357	1	1	yes	characteristic
193	C₂²	C₂×#358	1	1	yes	characteristic
194	C₂²	#2148	1	1	yes	characteristic
195	C₂²	#2148	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	16	2	no	—
203	C₄	—	16	2	no	—
204	C₄	#1542	1	1	yes	characteristic
205	C₄	#1550	1	1	yes	characteristic
206	C₄	#1989	1	1	yes	characteristic
207	C₄	#1998	1	1	yes	characteristic
208	C₄	#2240	1	1	yes	characteristic
209	C₄	#2241	1	1	yes	characteristic
210	C₄	#2252	1	1	yes	characteristic
211	C₄	#2254	1	1	yes	characteristic
212	C₂	#9900	1	1	yes	characteristic
213	C₂	#11947	1	1	yes	characteristic
214	C₂	#11949	1	1	yes	characteristic
215	C₂	#11981	1	1	yes	characteristic
216	C₂	#11983	1	1	yes	characteristic
217	C₂	#13282	1	1	yes	characteristic
218	C₂	#13289	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#33258	2
C₄×C₄⋊C₁₆	2
#8806	2
#7274	2