Subgroups — #134380

Order 512.

348 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20516	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	C₄×#364	C₂	1	1	yes	maximal, characteristic
4	#11611	C₂	1	2	yes	maximal
5	#7168	C₂	1	2	yes	maximal
6	C₂×#446	C₂²	1	1	yes	characteristic
7	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
8	#2295	C₄	1	1	yes	characteristic
9	C₂×#364	—	2	1	no	—
10	C₂×#364	—	2	1	no	—
11	C₂×#364	—	2	1	no	—
12	#2150	C₂²	1	2	yes	—
13	C₄×M₃₂	C₂²	1	2	yes	—
14	#1771	—	2	1	no	—
15	#1770	C₂²	1	1	yes	characteristic
16	C₈×C₄⋊C₄	—	2	1	no	—
17	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
18	#1768	—	2	1	no	—
19	#1767	C₄	1	1	yes	characteristic
20	#1726	—	2	1	no	—
21	C₄²×C₈	C₂²	1	1	yes	characteristic
22	#1376	—	2	1	no	—
23	#1338	—	2	4	no	—
24	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	#446	—	2	1	no	—
28	#446	—	2	1	no	—
29	#446	D₄	1	2	yes	—
30	#446	Q₈	1	2	yes	—
31	C₈×Q₈	—	2	2	no	—
32	C₈×Q₈	C₈	1	4	yes	—
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#97	—	2	1	no	—
38	C₂×#97	—	2	1	no	—
39	C₂×#97	—	2	1	no	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	#391	—	2	1	no	—
49	#389	—	2	1	no	—
50	#387	—	2	2	no	—
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	#376	—	2	1	no	—
54	C₄³	C₂×C₄	1	1	yes	characteristic
55	#364	—	2	4	no	—
56	#364	—	4	2	no	—
57	#364	—	4	2	no	—
58	C₄⋊C₁₆	—	2	2	no	—
59	C₄⋊C₁₆	—	2	2	no	—
60	#346	—	2	2	no	—
61	C₄×C₁₆	—	2	2	no	—
62	#336	—	2	1	no	—
63	#336	—	2	1	no	—
64	#336	—	2	1	no	—
65	C₂²×Q₈	M₁₆	1	1	yes	characteristic
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×Q₈	—	2	1	no	—
73	C₄×Q₈	—	2	1	no	—
74	C₄×Q₈	C₄⋊C₄	1	2	yes	—
75	C₄×Q₈	C₂×C₈	1	2	yes	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	—	2	1	no	—
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
87	M₃₂	—	2	4	no	—
88	M₃₂	—	4	2	no	—
89	M₃₂	—	4	2	no	—
90	M₃₂	—	4	2	no	—
91	C₂×C₁₆	—	4	2	no	—
92	C₂×C₁₆	—	4	2	no	—
93	C₄⋊C₈	—	2	4	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
99	C₄⋊C₈	C₂×C₈	1	2	yes	—
100	#97	—	2	2	no	—
101	#97	—	2	2	no	—
102	#97	—	4	2	no	—
103	#97	—	4	1	no	—
104	#97	—	4	1	no	—
105	C₄×C₈	—	2	4	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	—	4	2	no	—
111	C₄×C₈	C₂×C₈	1	2	yes	—
112	C₄×C₈	M₁₆	1	2	yes	—
113	C₄×C₈	SD₁₆	1	2	yes	—
114	C₄×C₈	Q₁₆	1	2	yes	—
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
117	C₄×C₈	#41	1	1	yes	characteristic
118	C₄×C₈	#41	1	1	yes	characteristic
119	#95	—	2	2	no	—
120	#95	—	2	2	no	—
121	#95	—	2	2	no	—
122	#95	M₁₆	1	2	yes	—
123	C₂×Q₈	—	2	1	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	C₄⋊C₈	1	2	yes	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
133	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	C₂²⋊C₈	1	2	yes	—
145	C₂×C₈	C₂²⋊C₈	1	2	yes	—
146	C₂×C₈	#103	1	2	yes	—
147	C₂×C₈	#104	1	2	yes	—
148	C₂×C₈	#121	1	1	yes	characteristic
149	C₂×C₈	#121	1	1	yes	characteristic
150	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
154	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₈	#137	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	4	2	no	—
168	C₄²	—	4	2	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	#103	1	2	yes	—
172	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₄²	#117	1	1	yes	characteristic
174	C₄²	C₂²×C₈	1	1	yes	characteristic
175	C₄²	C₂×M₁₆	1	1	yes	characteristic
176	C₁₆	—	4	4	no	—
177	C₁₆	—	8	2	no	—
178	C₂³	#407	1	1	yes	characteristic
179	Q₈	—	4	2	no	—
180	Q₈	—	4	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#380	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#103	1	1	yes	characteristic
199	C₂×C₄	#417	1	1	yes	characteristic
200	C₂×C₄	C₂×#104	1	1	yes	characteristic
201	C₂×C₄	#421	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
203	C₂×C₄	#423	1	1	yes	characteristic
204	C₂×C₄	#431	1	1	yes	characteristic
205	C₂×C₄	#432	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	2	no	—
211	C₈	Q₈⋊C₈	1	4	yes	—
212	C₈	#460	1	1	yes	characteristic
213	C₈	#462	1	1	yes	characteristic
214	C₈	#475	1	1	yes	characteristic
215	C₈	#477	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	#1487	1	1	yes	characteristic
219	C₂²	#1489	1	1	yes	characteristic
220	C₂²	#1797	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	8	2	no	—
229	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
230	C₄	#1477	1	1	yes	characteristic
231	C₄	#1803	1	1	yes	characteristic
232	C₄	#1806	1	1	yes	characteristic
233	C₄	#2166	1	2	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	#8564	1	1	yes	characteristic
237	C₂	#12790	1	1	yes	characteristic
238	C₂	#12792	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂³	#407
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×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
#20516	2
#12900	2
C₄×#364	2
#11611	2
#7168	2