Subgroups — #213365

Order 512.

588 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	#22307	C₂	1	2	yes	maximal
2	C₄×#498	C₂	1	1	yes	maximal, characteristic
3	#9873	C₂	1	2	yes	maximal
4	#9596	C₂	1	2	yes	maximal
5	#9593	C₂	1	2	yes	maximal
6	#8807	C₂	1	1	yes	maximal, characteristic
7	#8285	C₂	1	2	yes	maximal
8	C₈×#106	C₂	1	2	yes	maximal
9	#7841	C₂	1	1	yes	maximal, characteristic
10	C₂×#498	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
12	#1990	C₄	1	2	yes	—
13	#1984	—	2	4	no	—
14	#1941	C₄	1	2	yes	—
15	#1941	C₄	1	2	yes	—
16	#1939	C₂²	1	4	yes	—
17	#1916	—	2	4	no	—
18	#1844	C₄	1	1	yes	characteristic
19	#1831	C₂²	1	2	yes	—
20	C₄×#106	C₂²	1	2	yes	—
21	C₄×#106	C₂²	1	2	yes	—
22	C₈×C₄⋊C₄	C₂²	1	4	yes	—
23	#1744	C₂²	1	1	yes	characteristic
24	#1744	C₂²	1	1	yes	characteristic
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	C₂×#335	C₂²	1	2	yes	—
27	C₂×#334	C₂²	1	2	yes	—
28	C₂×#322	C₂²	1	1	yes	characteristic
29	C₂×#322	C₂²	1	1	yes	characteristic
30	C₂×C₈²	C₂²	1	1	yes	characteristic
31	C₂×C₈²	C₂²	1	1	yes	characteristic
32	#1279	C₂²	1	4	yes	—
33	#1279	C₂²	1	4	yes	—
34	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×Q₈	—	2	2	no	—
36	#498	C₈	1	8	yes	complemented
37	C₂×#106	C₂×C₄	1	2	yes	—
38	C₂×#106	C₂×C₄	1	2	yes	—
39	C₂×#106	C₂×C₄	1	2	yes	—
40	C₂×#106	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₈	—	2	4	no	—
42	C₂×C₄⋊C₈	C₂³	1	4	yes	—
43	C₂×C₄×C₈	—	2	4	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#391	—	2	2	no	—
55	#389	—	2	2	no	—
56	#389	C₂×C₄	1	4	yes	—
57	#384	C₂×C₄	1	2	yes	—
58	C₄×C₄⋊C₄	—	2	2	no	—
59	C₄×C₄⋊C₄	C₂³	1	4	yes	—
60	C₄³	C₂³	1	1	yes	characteristic
61	#340	C₂×C₄	1	4	yes	—
62	#340	C₂×C₄	1	4	yes	—
63	#336	—	2	4	no	—
64	#336	—	2	4	no	—
65	#336	—	2	2	no	—
66	#336	—	2	2	no	—
67	#336	—	2	2	no	—
68	#336	D₄	1	4	yes	—
69	#335	—	2	4	no	—
70	#334	—	2	4	no	—
71	#322	—	2	2	no	—
72	#322	Q₈	1	4	yes	complemented
73	C₈²	—	2	2	no	—
74	C₈²	Q₈	1	4	yes	complemented
75	C₂²×Q₈	—	2	2	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₄	1	2	yes	—
84	C₂²×C₈	C₂×D₄	1	2	yes	—
85	C₄⋊Q₈	—	2	4	no	—
86	C₄⋊Q₈	C₂×C₈	1	8	yes	—
87	C₄×Q₈	—	8	4	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₂×C₄⋊C₄	C₂²×C₄	1	4	yes	—
94	C₂×C₄²	—	2	4	no	—
95	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₄	1	2	yes	—
98	C₂×C₄²	#41	1	2	yes	—
99	C₂×C₄²	C₂⁴	1	1	yes	characteristic
100	#106	—	2	4	no	—
101	#106	—	2	4	no	—
102	#106	C₂×C₈	1	8	yes	—
103	#106	C₂×C₈	1	8	yes	—
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	2	4	no	—
106	C₄⋊C₈	—	8	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	8	4	no	—
115	C₄×C₈	C₂×C₈	1	4	yes	—
116	C₄×C₈	C₂×Q₈	1	2	yes	—
117	C₄×C₈	C₂×Q₈	1	2	yes	—
118	C₄×C₈	C₂×Q₈	1	2	yes	—
119	C₄×C₈	C₂×Q₈	1	2	yes	—
120	C₄×C₈	C₂×Q₈	1	2	yes	—
121	C₄×C₈	C₂×Q₈	1	2	yes	—
122	C₄×C₈	#41	1	2	yes	—
123	C₄×C₈	#41	1	2	yes	—
124	#95	—	2	4	no	—
125	#95	—	2	4	no	—
126	C₂×Q₈	—	2	8	no	—
127	C₂×Q₈	—	8	2	no	—
128	C₂×Q₈	—	8	2	no	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	C₄×D₄	1	2	yes	—
132	C₂²×C₄	C₄×D₄	1	2	yes	—
133	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₂×#41	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	4	4	no	—
149	C₂×C₈	—	4	4	no	—
150	C₂×C₈	C₄×Q₈	1	2	yes	—
151	C₂×C₈	C₄×Q₈	1	2	yes	—
152	C₂×C₈	C₄×Q₈	1	2	yes	—
153	C₂×C₈	C₄×Q₈	1	2	yes	—
154	C₂×C₈	C₄⋊Q₈	1	4	yes	—
155	C₂×C₈	C₂²×C₈	1	2	yes	—
156	C₂×C₈	C₂²×C₈	1	2	yes	—
157	C₂×C₈	#131	1	2	yes	—
158	C₂×C₈	#131	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	8	2	no	—
164	C₄⋊C₄	—	8	2	no	—
165	C₄⋊C₄	—	8	4	no	—
166	C₄⋊C₄	C₂²×C₈	1	8	yes	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	4	no	—
172	C₄²	—	4	4	no	—
173	C₄²	C₂²×C₈	1	2	yes	—
174	C₄²	#131	1	2	yes	—
175	C₄²	C₂²×Q₈	1	1	yes	characteristic
176	C₄²	C₂²×Q₈	1	1	yes	characteristic
177	C₄²	C₂×#41	1	1	yes	characteristic
178	C₄²	C₂×#41	1	1	yes	characteristic
179	C₂³	C₂×C₄×D₄	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₈×D₄	1	4	yes	—
194	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₄	#532	1	1	yes	characteristic
198	C₂×C₄	#533	1	1	yes	characteristic
199	C₂×C₄	#533	1	1	yes	characteristic
200	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
201	C₂×C₄	C₂×#131	1	1	yes	characteristic
202	C₂×C₄	C₂×#131	1	1	yes	characteristic
203	C₂×C₄	C₂×#131	1	1	yes	characteristic
204	C₂×C₄	#577	1	4	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	4	2	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	8	4	no	—
212	C₈	C₈×Q₈	1	4	yes	—
213	C₈	C₈×Q₈	1	4	yes	—
214	C₂²	C₄×C₄⋊Q₈	1	1	yes	characteristic
215	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
216	C₂²	#2931	1	1	yes	characteristic
217	C₂²	#2949	1	2	yes	—
218	C₂²	#2957	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	8	no	—
224	C₄	—	8	4	no	—
225	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
226	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
227	C₄	#2960	1	1	yes	characteristic
228	C₄	#2960	1	1	yes	characteristic
229	C₄	#3163	1	2	yes	—
230	C₄	#3173	1	2	yes	—
231	C₂	#17855	1	2	yes	—
232	C₂	#17971	1	2	yes	—
233	C₂	C₈×C₄⋊Q₈	1	1	yes	characteristic
234	C₂	#21724	1	1	yes	characteristic
235	C₂	#21733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
36. #498	C₈	semidirect
72. #322	Q₈	semidirect
74. C₈²	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#22307	2
C₄×#498	2
#9873	2
#9596	2
#9593	2
#8807	2
#8285	2
C₈×#106	2
#7841	2