Subgroups — #134621

Order 512.

350 conjugacy classes of subgroups.

The subgroup lattice has 237 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	#12879	C₂	1	1	yes	maximal, characteristic
3	C₂×#1365	C₂	1	2	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	#7274	C₂	1	2	yes	maximal
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₄×C₄⋊Q₈	—	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	C₂²	1	1	yes	characteristic
13	#1842	—	2	1	no	—
14	C₄×#107	—	2	1	no	—
15	C₄×#107	C₄	1	2	yes	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#335	—	2	1	no	—
18	C₂×#335	C₂²	1	2	yes	—
19	#1374	—	2	1	no	—
20	#1374	—	2	1	no	—
21	#1365	—	2	4	no	—
22	C₂×C₄⋊Q₈	—	2	1	no	—
23	C₂×C₄×Q₈	—	4	1	no	—
24	C₂²×C₁₆	—	2	2	no	—
25	#500	—	2	1	no	—
26	#500	—	2	1	no	—
27	#500	D₄	1	2	yes	—
28	#500	Q₈	1	2	yes	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	C₂×C₄	1	2	yes	—
32	C₂×#107	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	—	4	1	no	—
35	C₂×C₄⋊C₈	—	4	1	no	—
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#391	—	4	1	no	—
43	#389	—	2	2	no	—
44	#389	—	4	1	no	—
45	#384	—	2	1	no	—
46	C₄×C₄⋊C₄	—	2	2	no	—
47	C₄×C₄⋊C₄	—	4	1	no	—
48	C₄³	D₄	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	2	no	—
50	C₄⋊C₁₆	—	2	2	no	—
51	#340	—	2	2	no	—
52	#340	—	2	2	no	—
53	#335	—	2	2	no	—
54	#335	—	2	2	no	—
55	#335	—	4	2	no	—
56	C₂²×Q₈	—	4	1	no	—
57	C₂²×C₈	—	4	2	no	—
58	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	1	no	—
66	C₄×Q₈	—	8	4	no	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	4	1	no	—
70	C₂×C₄⋊C₄	—	4	2	no	—
71	C₂×C₄⋊C₄	—	4	1	no	—
72	C₂×C₄⋊C₄	—	4	1	no	—
73	C₂×C₄²	—	4	2	no	—
74	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	D₈	1	1	yes	characteristic
77	C₂×C₄²	D₈	1	1	yes	characteristic
78	C₂×C₄²	SD₁₆	1	1	yes	characteristic
79	C₂×C₄²	SD₁₆	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	—	4	2	no	—
82	C₂×C₁₆	—	4	2	no	—
83	C₂×C₁₆	—	4	2	no	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	2	2	no	—
87	#107	—	4	2	no	—
88	#107	—	4	2	no	—
89	#107	C₄⋊C₄	1	4	yes	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	8	2	no	—
93	C₄⋊C₈	—	8	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	#95	—	4	2	no	—
105	#95	—	4	2	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	—	8	2	no	—
109	C₂×Q₈	—	8	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	#102	1	1	yes	characteristic
113	C₂²×C₄	#102	1	1	yes	characteristic
114	C₂²×C₄	#102	1	1	yes	characteristic
115	C₂²×C₄	#102	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
118	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	8	2	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	—	8	2	no	—
129	C₂×C₈	#101	1	2	yes	—
130	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈	#117	1	1	yes	characteristic
132	C₂×C₈	#121	1	1	yes	characteristic
133	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₈	#135	1	2	yes	—
135	C₂×C₈	#137	1	2	yes	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	8	2	no	—
143	C₄⋊C₄	—	8	4	no	—
144	C₄⋊C₄	—	8	1	no	—
145	C₄⋊C₄	—	8	1	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	—	8	4	no	—
155	C₄²	SD₃₂	1	2	yes	—
156	C₄²	Q₃₂	1	2	yes	—
157	C₄²	#121	1	1	yes	characteristic
158	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
159	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄²	#137	1	1	yes	characteristic
161	C₁₆	—	8	2	no	—
162	C₂³	C₂×#102	1	1	yes	characteristic
163	Q₈	—	8	4	no	—
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	#358	1	2	yes	—
183	C₂×C₄	#358	1	2	yes	—
184	C₂×C₄	#361	1	2	yes	—
185	C₂×C₄	#380	1	1	yes	characteristic
186	C₂×C₄	C₂×#101	1	1	yes	characteristic
187	C₂×C₄	#413	1	1	yes	characteristic
188	C₂×C₄	#416	1	1	yes	characteristic
189	C₂×C₄	#417	1	1	yes	characteristic
190	C₂×C₄	#419	1	1	yes	characteristic
191	C₂×C₄	#459	1	1	yes	characteristic
192	C₂×C₄	#461	1	1	yes	characteristic
193	C₂×C₄	#474	1	1	yes	characteristic
194	C₂×C₄	#476	1	1	yes	characteristic
195	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
196	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
197	C₂×C₄	#509	1	1	yes	characteristic
198	C₂×C₄	#510	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	2	2	no	—
202	C₈	—	4	1	no	—
203	C₈	—	16	2	no	—
204	C₈	#464	1	2	yes	—
205	C₈	#479	1	2	yes	—
206	C₂²	#1540	1	1	yes	characteristic
207	C₂²	#1546	1	1	yes	characteristic
208	C₂²	#1802	1	1	yes	characteristic
209	C₂²	C₂×#358	1	1	yes	characteristic
210	C₂²	#2139	1	1	yes	characteristic
211	C₂²	C₂×#361	1	1	yes	characteristic
212	C₂²	#2148	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	4	1	no	—
219	C₄	—	8	4	no	—
220	C₄	—	16	2	no	—
221	C₄	#1552	1	1	yes	characteristic
222	C₄	#1559	1	1	yes	characteristic
223	C₄	#1801	1	1	yes	characteristic
224	C₄	#1805	1	1	yes	characteristic
225	C₄	#2207	1	1	yes	characteristic
226	C₄	#2209	1	1	yes	characteristic
227	C₄	#2225	1	1	yes	characteristic
228	C₄	#2227	1	1	yes	characteristic
229	C₂	#8589	1	1	yes	characteristic
230	C₂	#11949	1	1	yes	characteristic
231	C₂	#11952	1	1	yes	characteristic
232	C₂	#11972	1	1	yes	characteristic
233	C₂	#11976	1	1	yes	characteristic
234	C₂	#12812	1	1	yes	characteristic
235	C₂	#12818	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
C₄×#500	2
#12879	2
C₂×#1365	2
#8806	2
#7274	2