Subgroups — #493261

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 253 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₃₂	C₂	1	2	yes	maximal
2	#20521	C₂	1	1	yes	maximal, characteristic
3	#12666	C₂	1	2	yes	maximal
4	#12666	C₂	1	4	yes	maximal
5	#12658	C₂	1	1	yes	maximal, characteristic
6	#12657	C₂	1	2	yes	maximal
7	#12541	C₂	1	2	yes	maximal
8	C₄×#347	C₂	1	1	yes	maximal, characteristic
9	C₂×#446	C₄	1	1	yes	characteristic
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	#2295	C₄	1	1	yes	characteristic
12	#2183	C₄	1	4	yes	complemented
13	#2183	C₄	1	8	yes	complemented
14	Q₈×C₁₆	C₄	1	4	yes	complemented
15	#2150	C₂²	1	4	yes	—
16	#2150	C₂²	1	2	yes	—
17	C₂×#347	C₂²	1	1	yes	characteristic
18	C₂×#347	C₂²	1	2	yes	—
19	C₂×#347	C₂²	1	2	yes	—
20	C₂×#347	C₂²	1	4	yes	—
21	C₂×#347	C₂²	1	1	yes	characteristic
22	C₄×M₃₂	C₂²	1	2	yes	—
23	C₄×M₃₂	C₂²	1	4	yes	—
24	#1771	C₄	1	2	yes	—
25	#1770	C₂²	1	2	yes	—
26	#1770	C₂²	1	1	yes	characteristic
27	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
28	#1768	C₄	1	2	yes	—
29	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
30	#1726	C₂²	1	2	yes	—
31	C₄×#97	C₂²	1	1	yes	characteristic
32	#1375	C₂²	1	4	yes	—
33	#1375	C₂²	1	2	yes	—
34	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×M₃₂	C₂³	1	2	yes	—
36	C₂×M₃₂	C₂³	1	4	yes	—
37	C₂×M₃₂	C₂³	1	2	yes	—
38	#446	—	2	2	no	—
39	#446	—	2	2	no	—
40	C₈×Q₈	C₂×C₄	1	2	yes	—
41	C₈×Q₈	C₂×C₄	1	2	yes	—
42	C₈×Q₈	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₂×#97	C₂×C₄	1	2	yes	—
48	C₂×#97	C₂×C₄	1	1	yes	characteristic
49	C₂×#97	C₂×C₄	1	1	yes	characteristic
50	C₂×#97	C₂³	1	1	yes	characteristic
51	C₂×#97	C₂³	1	2	yes	—
52	C₂×#97	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	2	yes	—
58	#391	—	2	1	no	—
59	#389	—	2	1	no	—
60	#387	—	2	2	no	—
61	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	#376	C₂×C₄	1	2	yes	—
64	C₄³	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
66	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
67	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
68	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
69	#347	—	2	4	no	—
70	#347	—	2	8	no	—
71	#347	—	2	2	no	—
72	#347	—	2	4	no	—
73	#347	Q₈	1	4	yes	complemented
74	#346	C₂×C₄	1	4	yes	—
75	#346	C₂×C₄	1	8	yes	—
76	C₄×C₁₆	C₂×C₄	1	4	yes	—
77	C₄×C₁₆	C₂×C₄	1	8	yes	—
78	#336	C₂×C₄	1	2	yes	—
79	#336	C₂×C₄	1	2	yes	—
80	#336	C₂×C₄	1	2	yes	—
81	C₂²×Q₈	M₁₆	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	2	yes	—
84	C₂²×C₈	C₂²×C₄	1	2	yes	—
85	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂⁴	1	1	yes	characteristic
88	C₄×Q₈	C₄²	1	2	yes	—
89	C₄×Q₈	C₄²	1	2	yes	—
90	C₄×Q₈	M₁₆	1	2	yes	—
91	C₄×Q₈	M₁₆	1	2	yes	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	M₁₆	1	2	yes	—
95	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₄	1	2	yes	—
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄²	C₂²×C₄	1	2	yes	—
103	M₃₂	—	2	2	no	—
104	M₃₂	—	2	4	no	—
105	M₃₂	—	2	2	no	—
106	M₃₂	—	2	4	no	—
107	M₃₂	C₂×Q₈	1	4	yes	—
108	M₃₂	#41	1	4	yes	—
109	C₂×C₁₆	C₂²×C₄	1	4	yes	—
110	C₂×C₁₆	C₂²×C₄	1	8	yes	—
111	C₂×C₁₆	C₂²×C₄	1	4	yes	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	C₄²	1	1	yes	characteristic
116	C₄⋊C₈	C₄²	1	2	yes	—
117	C₄⋊C₈	C₄²	1	1	yes	characteristic
118	C₄⋊C₈	C₄²	1	4	yes	—
119	C₄⋊C₈	C₄²	1	4	yes	—
120	#97	—	2	2	no	—
121	#97	—	2	4	no	—
122	#97	—	2	4	no	—
123	#97	—	4	1	no	—
124	#97	—	4	1	no	—
125	#97	C₂×Q₈	1	2	yes	—
126	#97	#41	1	2	yes	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	4	2	no	—
130	C₄×C₈	—	4	2	no	—
131	C₄×C₈	C₂²×C₄	1	2	yes	—
132	C₄×C₈	C₂²×C₄	1	4	yes	—
133	C₄×C₈	C₂²×C₄	1	2	yes	—
134	C₄×C₈	C₂²×C₄	1	4	yes	—
135	#95	—	2	2	no	—
136	#95	—	2	2	no	—
137	#95	—	2	1	no	—
138	#95	—	2	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	#97	1	2	yes	—
141	C₂×Q₈	#97	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
146	C₂²×C₄	C₂×M₁₆	1	2	yes	—
147	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×M₁₆	1	2	yes	—
149	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	C₂×C₄²	1	2	yes	—
163	C₂×C₈	C₂×C₄²	1	2	yes	—
164	C₂×C₈	C₂×C₄²	1	2	yes	—
165	C₂×C₈	C₄×Q₈	1	2	yes	—
166	C₂×C₈	C₂³×C₄	1	2	yes	—
167	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₈	C₂×#41	1	2	yes	—
169	C₂×C₈	C₂×#41	1	1	yes	characteristic
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	8	2	no	—
172	C₄⋊C₄	#97	1	4	yes	—
173	C₄⋊C₄	#97	1	4	yes	—
174	C₄⋊C₄	#97	1	2	yes	—
175	C₄⋊C₄	#97	1	2	yes	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	4	2	no	—
182	C₄²	—	8	1	no	—
183	C₄²	C₂×C₄²	1	4	yes	—
184	C₄²	C₂×C₄²	1	1	yes	characteristic
185	C₄²	C₂×C₄²	1	1	yes	characteristic
186	C₄²	C₄×Q₈	1	2	yes	—
187	C₄²	C₂×M₁₆	1	1	yes	characteristic
188	C₄²	C₂×M₁₆	1	1	yes	characteristic
189	C₄²	C₂×M₁₆	1	4	yes	—
190	C₁₆	—	2	4	no	—
191	C₁₆	—	2	8	no	—
192	C₁₆	C₄×Q₈	1	8	yes	—
193	C₂³	C₂²×M₁₆	1	1	yes	characteristic
194	Q₈	—	2	4	no	—
195	Q₈	—	4	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	C₂×#97	1	1	yes	characteristic
210	C₂×C₄	C₂×#97	1	1	yes	characteristic
211	C₂×C₄	C₂×#97	1	4	yes	—
212	C₂×C₄	C₂×#97	1	1	yes	characteristic
213	C₂×C₄	C₂×#97	1	4	yes	—
214	C₂×C₄	C₂×#97	1	1	yes	characteristic
215	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
217	C₂×C₄	C₄×#41	1	1	yes	characteristic
218	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
219	C₂×C₄	#568	1	1	yes	characteristic
220	C₂×C₄	#568	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	4	1	no	—
226	C₈	—	4	2	no	—
227	C₈	C₂×C₄×Q₈	1	2	yes	—
228	C₈	C₄×#41	1	2	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂²×#97	1	1	yes	characteristic
232	C₂²	Q₈×M₁₆	1	1	yes	characteristic
233	C₂²	#2971	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	4	no	—
239	C₄	—	4	2	no	—
240	C₄	—	8	1	no	—
241	C₄	—	8	2	no	—
242	C₄	#2110	1	2	yes	—
243	C₄	C₄²×Q₈	1	1	yes	characteristic
244	C₄	#2875	1	1	yes	characteristic
245	C₄	Q₈×M₁₆	1	1	yes	characteristic
246	C₄	#2971	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	Q₈×#97	1	1	yes	characteristic
250	C₂	C₂×#2110	1	1	yes	characteristic
251	C₂	#33329	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂³	C₂²×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #2183	C₄	semidirect
13. #2183	C₄	semidirect
14. Q₈×C₁₆	C₄	semidirect
73. #347	Q₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₃₂	2
#20521	2
#12666	2
#12658	2
#12657	2
#12541	2
C₄×#347	2