Subgroups — #494111

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#405	C₂	1	1	yes	maximal, characteristic
2	#12895	C₂	1	2	yes	maximal, complemented
3	#12880	C₂	1	2	yes	maximal, complemented
4	#12845	C₂	1	2	yes	maximal
5	#12784	C₂	1	2	yes	maximal
6	#12653	C₂	1	2	yes	maximal, complemented
7	#12599	C₂	1	2	yes	maximal
8	#12591	C₂	1	2	yes	maximal, complemented
9	C₂×#405	C₄	1	1	yes	characteristic
10	C₂×#405	C₄	1	1	yes	characteristic
11	C₂×#405	C₂²	1	1	yes	characteristic
12	C₂×#405	C₂²	1	1	yes	characteristic
13	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×#117	C₄	1	1	yes	characteristic
16	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₁₆	C₂²	1	4	yes	—
18	C₂×#346	C₂²	1	4	yes	—
19	#1726	C₄	1	1	yes	characteristic
20	#1726	C₄	1	1	yes	characteristic
21	#1726	C₂²	1	1	yes	characteristic
22	#1726	C₂²	1	1	yes	characteristic
23	C₄×#97	C₄	1	1	yes	characteristic
24	C₄×#97	C₄	1	1	yes	characteristic
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1374	C₂²	1	4	yes	—
28	#1374	C₂²	1	4	yes	—
29	#1374	C₂²	1	4	yes	—
30	#1374	C₂²	1	4	yes	—
31	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	C₂×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×#117	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₁₆	C₂³	1	8	yes	—
39	#405	—	2	4	no	—
40	#405	—	2	4	no	—
41	#405	—	2	4	no	—
42	#405	—	2	4	no	—
43	C₄×M₁₆	—	2	4	no	—
44	C₂×#97	C₂×C₄	1	1	yes	characteristic
45	C₂×#97	C₂×C₄	1	1	yes	characteristic
46	C₂×#97	C₂×C₄	1	1	yes	characteristic
47	C₂×#97	C₂×C₄	1	1	yes	characteristic
48	C₂×#97	C₂×C₄	1	1	yes	characteristic
49	C₂×#97	C₂×C₄	1	1	yes	characteristic
50	C₂×#97	C₂×C₄	1	1	yes	characteristic
51	C₂×#97	C₂×C₄	1	1	yes	characteristic
52	C₂×#97	C₂×C₄	1	1	yes	characteristic
53	C₂×#97	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	—	2	4	no	—
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₄⋊C₄	C₈	1	4	yes	—
70	C₄×C₂²⋊C₄	C₈	1	4	yes	—
71	#376	C₂×C₄	1	1	yes	characteristic
72	#376	C₂×C₄	1	1	yes	characteristic
73	C₄³	C₂×C₄	1	1	yes	characteristic
74	C₄³	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₁₆	—	2	8	no	—
76	C₂²⋊C₁₆	—	2	8	no	—
77	#346	—	2	8	no	—
78	C₂³×C₄	C₂×C₈	1	2	yes	—
79	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×M₁₆	—	2	2	no	—
81	C₂×M₁₆	—	2	2	no	—
82	C₂×M₁₆	—	2	2	no	—
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂²×C₈	—	2	2	no	—
87	C₂²×C₈	—	2	2	no	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	M₁₆	1	4	yes	—
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	#41	1	4	yes	—
99	C₂²×C₈	C₂⁴	1	1	yes	characteristic
100	#117	—	2	2	no	—
101	#117	—	2	2	no	—
102	#117	—	2	2	no	—
103	#117	—	2	2	no	—
104	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
105	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
106	C₂×C₂²⋊C₄	C₂×C₈	1	2	yes	—
107	C₂×C₂²⋊C₄	C₂×C₈	1	2	yes	—
108	C₂×C₄²	—	2	2	no	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂×C₈	1	2	yes	—
111	C₂×C₄²	C₂×C₈	1	4	yes	—
112	C₂×C₄²	C₂×C₈	1	4	yes	—
113	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₁₆	—	2	8	no	—
120	C₂×C₁₆	—	2	8	no	—
121	C₂×C₁₆	—	2	8	no	—
122	#97	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	#97	—	2	2	no	—
127	#97	—	2	2	no	—
128	#97	—	2	2	no	—
129	#97	—	2	2	no	—
130	#97	—	2	2	no	—
131	#97	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	M₁₆	1	4	yes	—
143	C₄×C₈	M₁₆	1	4	yes	—
144	C₄×C₈	#41	1	4	yes	—
145	C₄×C₈	#41	1	4	yes	—
146	#95	C₂×C₈	1	4	yes	—
147	#95	C₂×C₈	1	4	yes	—
148	C₂⁴	C₂²×C₈	1	1	yes	characteristic
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	#117	1	2	yes	—
156	C₂²×C₄	C₂²×C₈	1	2	yes	—
157	C₂²×C₄	C₂²×C₈	1	2	yes	—
158	C₂²×C₄	C₂²×C₈	1	2	yes	—
159	C₂²×C₄	C₂²×C₈	1	2	yes	—
160	C₂²×C₄	C₂²×C₈	1	2	yes	—
161	C₂²×C₄	C₂²×C₈	1	2	yes	—
162	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂×M₁₆	1	2	yes	—
164	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
165	M₁₆	—	2	8	no	—
166	M₁₆	—	2	8	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	#117	1	2	yes	—
182	C₂×C₈	#117	1	2	yes	—
183	C₂×C₈	#117	1	2	yes	—
184	C₂×C₈	C₂×M₁₆	1	2	yes	—
185	C₂×C₈	C₂×M₁₆	1	2	yes	—
186	C₂×C₈	C₂×M₁₆	1	2	yes	—
187	C₂×C₈	C₂×M₁₆	1	2	yes	—
188	C₂×C₈	C₂×M₁₆	1	2	yes	—
189	C₂×C₈	C₂×M₁₆	1	2	yes	—
190	C₂×C₈	C₂×#41	1	2	yes	—
191	C₂×C₈	C₂×#41	1	2	yes	—
192	C₂×C₈	C₂×#41	1	2	yes	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	4	4	no	—
207	C₄²	—	4	4	no	—
208	C₄²	#117	1	2	yes	—
209	C₄²	#117	1	2	yes	—
210	C₄²	C₂×M₁₆	1	2	yes	—
211	C₄²	C₂×M₁₆	1	2	yes	—
212	C₁₆	—	4	8	no	—
213	C₂³	—	2	4	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	#431	1	2	yes	—
217	C₂³	C₂³×C₈	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	#431	1	4	yes	—
236	C₂×C₄	#431	1	4	yes	—
237	C₂×C₄	#431	1	2	yes	—
238	C₂×C₄	C₂×#117	1	1	yes	characteristic
239	C₂×C₄	C₂×#117	1	1	yes	characteristic
240	C₂×C₄	C₂×#117	1	1	yes	characteristic
241	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
242	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
243	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	#432	1	4	yes	—
251	C₈	#529	1	4	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₂×#431	1	1	yes	characteristic
257	C₂²	C₂×#431	1	1	yes	characteristic
258	C₂²	C₂×#431	1	1	yes	characteristic
259	C₂²	#3407	1	4	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	#2283	1	1	yes	characteristic
268	C₄	C₂×#432	1	1	yes	characteristic
269	C₄	#2920	1	1	yes	characteristic
270	C₄	#2920	1	1	yes	characteristic
271	C₂	—	4	2	no	—
272	C₂	#20777	1	1	yes	characteristic
273	C₂	#33419	1	2	yes	—
274	C₂	#33421	1	2	yes	—
275	C₂	#33421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #12895	C₂	semidirect
3. #12880	C₂	semidirect
6. #12653	C₂	semidirect
8. #12591	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#405	2
#12895	2
#12880	2
#12845	2
#12784	2
#12653	2
#12599	2
#12591	2