Subgroups — #203720

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20826	C₂	1	1	yes	maximal, characteristic
2	#17809	C₂	1	2	yes	maximal
3	C₂×#1566	C₂	1	1	yes	maximal, characteristic
4	C₂×#1452	C₂	1	1	yes	maximal, characteristic
5	#9609	C₂	1	8	yes	maximal, complemented
6	#7934	C₂	1	2	yes	maximal
7	#2920	C₂²	1	2	yes	—
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#431	C₂²	1	1	yes	characteristic
10	C₂×#423	C₂²	1	2	yes	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#1848	C₄	1	2	yes	—
13	#1837	C₄	1	2	yes	—
14	#1810	C₄	1	4	yes	—
15	C₄×#108	C₂²	1	4	yes	—
16	#1574	—	2	4	no	—
17	#1571	—	2	2	no	—
18	#1570	—	2	2	no	—
19	#1566	—	2	2	no	—
20	#1566	—	2	2	no	—
21	#1565	C₂²	1	8	yes	—
22	C₂×#334	C₂²	1	4	yes	—
23	#1452	—	2	2	no	—
24	#1452	C₂²	1	4	yes	—
25	C₂×#322	—	2	2	no	—
26	#1283	C₂²	1	8	yes	—
27	C₂²×M₁₆	—	2	2	no	—
28	C₂³×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₄²	C₂³	1	1	yes	characteristic
30	#432	—	2	2	no	—
31	#432	—	2	2	no	—
32	#431	—	2	2	no	—
33	#431	—	2	2	no	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	D₄	1	2	yes	—
39	#431	Q₈	1	2	yes	—
40	#431	C₂³	1	2	yes	—
41	#431	C₂³	1	2	yes	—
42	C₂×#108	C₂×C₄	1	8	yes	—
43	#423	—	2	2	no	—
44	#423	—	2	2	no	—
45	#423	—	2	2	no	—
46	#423	—	2	2	no	—
47	#423	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₈	—	2	1	no	—
49	C₂×C₄⋊C₈	—	2	2	no	—
50	C₂×C₄⋊C₈	D₄	1	2	yes	—
51	C₂×C₄⋊C₈	C₂³	1	4	yes	—
52	#407	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₈	—	2	1	no	—
54	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
55	C₄×M₁₆	C₂³	1	4	yes	—
56	C₂×C₄×C₈	—	2	1	no	—
57	C₂×C₄×C₈	—	2	1	no	—
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	D₄	1	2	yes	—
61	C₂×C₄×C₈	D₄	1	2	yes	—
62	C₂×C₄×C₈	Q₈	1	2	yes	—
63	C₂×C₄×C₈	Q₈	1	2	yes	—
64	C₂×C₄×C₈	C₂³	1	2	yes	—
65	#341	C₂×C₄	1	8	yes	—
66	#335	—	2	8	no	—
67	#334	—	2	8	no	—
68	#334	—	2	8	no	—
69	#322	—	2	8	no	—
70	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
72	C₂³×C₄	#41	1	1	yes	characteristic
73	C₂×M₁₆	—	4	2	no	—
74	C₂×M₁₆	—	4	2	no	—
75	C₂×M₁₆	—	8	4	no	—
76	C₂×M₁₆	C₂²×C₄	1	8	yes	—
77	C₂²×C₈	—	2	2	no	—
78	C₂²×C₈	—	2	2	no	—
79	C₂²×C₈	—	2	4	no	—
80	C₂²×C₈	—	4	4	no	—
81	C₂²×C₈	C₄⋊C₄	1	4	yes	—
82	C₂²×C₈	C₄⋊C₄	1	4	yes	—
83	C₂²×C₈	C₂²×C₄	1	2	yes	—
84	C₂²×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₄²	#41	1	1	yes	characteristic
94	C₂×C₄²	#41	1	1	yes	characteristic
95	C₂×C₄²	#41	1	1	yes	characteristic
96	C₂×C₄²	C₂⁴	1	1	yes	characteristic
97	#108	—	4	8	no	—
98	C₄⋊C₈	—	2	4	no	—
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	4	no	—
103	C₄⋊C₈	—	2	4	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	4	2	no	—
106	C₄⋊C₈	—	4	2	no	—
107	C₄⋊C₈	C₂×D₄	1	4	yes	—
108	C₂²⋊C₈	—	2	4	no	—
109	C₂²⋊C₈	—	2	4	no	—
110	C₂²⋊C₈	—	2	2	no	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	C₂×Q₈	1	4	yes	—
114	#97	—	2	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	4	no	—
120	C₄×C₈	C₄⋊C₄	1	2	yes	—
121	C₄×C₈	C₄⋊C₄	1	2	yes	—
122	C₄×C₈	C₄⋊C₄	1	2	yes	—
123	C₄×C₈	C₄⋊C₄	1	2	yes	—
124	C₄×C₈	C₂×D₄	1	4	yes	—
125	C₄×C₈	C₂×Q₈	1	4	yes	—
126	C₂⁴	C₄×Q₈	1	1	yes	characteristic
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
134	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
135	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
136	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
137	C₂²×C₄	#121	1	1	yes	characteristic
138	C₂²×C₄	#121	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₄	#125	1	1	yes	characteristic
142	C₂²×C₄	#125	1	1	yes	characteristic
143	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
147	C₂²×C₄	C₂×#41	1	1	yes	characteristic
148	M₁₆	—	2	16	no	—
149	M₁₆	—	8	4	no	—
150	M₁₆	—	8	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	4	4	no	—
158	C₂×C₈	—	4	4	no	—
159	C₂×C₈	—	4	4	no	—
160	C₂×C₈	—	4	4	no	—
161	C₂×C₈	—	8	4	no	—
162	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
163	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
164	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₄²	C₄×D₄	1	1	yes	characteristic
167	C₄²	C₄×D₄	1	1	yes	characteristic
168	C₄²	#121	1	1	yes	characteristic
169	C₄²	#121	1	1	yes	characteristic
170	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
171	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
172	C₄²	#125	1	1	yes	characteristic
173	C₄²	#125	1	1	yes	characteristic
174	C₄²	C₄⋊Q₈	1	1	yes	characteristic
175	C₄²	C₄⋊Q₈	1	1	yes	characteristic
176	C₄²	C₂²×D₄	1	1	yes	characteristic
177	C₄²	C₂²×D₄	1	1	yes	characteristic
178	C₄²	C₂²×Q₈	1	1	yes	characteristic
179	C₄²	C₂×#41	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	4	1	no	—
183	C₂³	#389	1	1	yes	characteristic
184	C₂³	#389	1	1	yes	characteristic
185	C₂³	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	#389	1	1	yes	characteristic
203	C₂×C₄	#389	1	1	yes	characteristic
204	C₂×C₄	#389	1	1	yes	characteristic
205	C₂×C₄	#389	1	1	yes	characteristic
206	C₂×C₄	#389	1	1	yes	characteristic
207	C₂×C₄	#389	1	1	yes	characteristic
208	C₂×C₄	#471	1	4	yes	—
209	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂×#121	1	1	yes	characteristic
212	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
213	C₂×C₄	C₂×#125	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
215	C₂×C₄	#568	1	4	yes	—
216	C₈	—	2	4	no	—
217	C₈	—	2	4	no	—
218	C₈	—	4	4	no	—
219	C₈	—	4	8	no	—
220	C₈	—	4	4	no	—
221	C₈	—	8	4	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	#1951	1	2	yes	—
231	C₂²	C₂×#389	1	1	yes	characteristic
232	C₂²	#2961	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	#1951	1	2	yes	—
241	C₄	#1953	1	2	yes	—
242	C₄	#2905	1	2	yes	—
243	C₄	#2932	1	2	yes	—
244	C₄	C₂×#471	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	—	2	2	no	—
247	C₂	—	4	1	no	—
248	C₂	#21144	1	1	yes	characteristic
249	C₂	C₂×#1951	1	1	yes	characteristic
250	C₂	C₂×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #9609	C₂	semidirect

Maximal subgroups

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

subgroup	index
#20826	2
#17809	2
C₂×#1566	2
C₂×#1452	2
#9609	2
#7934	2