Subgroups — #493181

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 281 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2166	C₂	1	2	yes	maximal
2	#20331	C₂	1	1	yes	maximal, characteristic
3	#12659	C₂	1	2	yes	maximal, complemented
4	#12656	C₂	1	2	yes	maximal, complemented
5	#12608	C₂	1	2	yes	maximal
6	C₄×#349	C₂	1	2	yes	maximal
7	#12595	C₂	1	2	yes	maximal, complemented
8	#12541	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	C₂²	1	1	yes	characteristic
14	C₂²×#97	C₄	1	1	yes	characteristic
15	C₄×#117	C₄	1	1	yes	characteristic
16	#2166	C₄	1	16	yes	complemented
17	C₂×#364	C₂²	1	4	yes	—
18	C₂×#349	C₂²	1	4	yes	—
19	C₂×#347	C₂²	1	4	yes	—
20	C₂×#347	C₂²	1	4	yes	—
21	C₄×M₃₂	C₂²	1	4	yes	—
22	#1726	C₄	1	1	yes	characteristic
23	#1726	C₄	1	1	yes	characteristic
24	#1726	C₂²	1	1	yes	characteristic
25	#1726	C₂²	1	1	yes	characteristic
26	C₄×#97	C₄	1	1	yes	characteristic
27	C₄×#97	C₄	1	1	yes	characteristic
28	C₄×#97	C₂²	1	1	yes	characteristic
29	C₄×#97	C₂²	1	1	yes	characteristic
30	#1377	C₂²	1	4	yes	—
31	#1376	C₂²	1	4	yes	—
32	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×M₁₆	C₂³	1	1	yes	characteristic
34	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×#117	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂×M₃₂	C₂³	1	8	yes	—
40	#405	—	2	4	no	—
41	#405	—	2	4	no	—
42	#405	—	2	4	no	—
43	#405	C₂×C₄	1	8	yes	—
44	C₄×M₁₆	—	2	4	no	—
45	C₂×#97	—	2	4	no	—
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×#97	C₂×C₄	1	1	yes	characteristic
55	C₂×#97	C₂×C₄	1	1	yes	characteristic
56	C₂×#97	C₂×C₄	1	1	yes	characteristic
57	C₂×#97	C₂³	1	1	yes	characteristic
58	C₂×#97	C₂³	1	1	yes	characteristic
59	C₂×#97	C₂³	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	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₄⋊C₄	—	2	2	no	—
71	C₄×C₂²⋊C₄	—	2	2	no	—
72	#376	C₂×C₄	1	1	yes	characteristic
73	#376	C₂×C₄	1	1	yes	characteristic
74	C₄³	C₂×C₄	1	1	yes	characteristic
75	C₄³	C₂×C₄	1	1	yes	characteristic
76	#364	C₂×C₄	1	16	yes	complemented
77	#349	C₂×C₄	1	16	yes	—
78	#347	—	2	8	no	—
79	#347	C₂×C₄	1	16	yes	complemented
80	#346	—	2	4	no	—
81	C₄×C₁₆	—	2	4	no	—
82	C₂³×C₄	M₁₆	1	2	yes	—
83	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	M₁₆	1	4	yes	—
89	C₂×M₁₆	C₂²×C₄	1	4	yes	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	C₄²	1	4	yes	—
96	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂⁴	1	1	yes	characteristic
104	#117	—	2	2	no	—
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	#117	C₄²	1	4	yes	—
108	C₂×C₄⋊C₄	—	2	1	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	—	2	2	no	—
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	M₁₆	1	2	yes	—
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
123	M₃₂	—	2	8	no	—
124	M₃₂	C₂²×C₄	1	16	yes	—
125	C₂×C₁₆	—	2	8	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	#97	—	2	2	no	—
133	#97	—	2	2	no	—
134	#97	—	2	2	no	—
135	#97	C₄²	1	4	yes	—
136	#97	M₁₆	1	4	yes	—
137	#97	M₁₆	1	4	yes	—
138	#97	C₂²×C₄	1	4	yes	—
139	#97	#41	1	4	yes	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	C₄²	1	4	yes	—
148	C₄×C₈	C₂²×C₄	1	4	yes	—
149	C₄×C₈	#41	1	4	yes	—
150	#95	—	2	2	no	—
151	#95	—	2	2	no	—
152	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	8	2	no	—
163	C₂²×C₄	C₂×C₄²	1	2	yes	—
164	C₂²×C₄	C₂×M₁₆	1	2	yes	—
165	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×M₁₆	1	2	yes	—
167	C₂²×C₄	C₂×M₁₆	1	2	yes	—
168	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
169	M₁₆	—	2	8	no	—
170	M₁₆	—	2	8	no	—
171	C₂×C₈	—	2	8	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	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	C₂×C₄²	1	2	yes	—
184	C₂×C₈	C₂×C₄²	1	2	yes	—
185	C₂×C₈	C₂×C₄²	1	2	yes	—
186	C₂×C₈	#117	1	2	yes	—
187	C₂×C₈	#117	1	2	yes	—
188	C₂×C₈	C₂×M₁₆	1	2	yes	—
189	C₂×C₈	C₂×M₁₆	1	2	yes	—
190	C₂×C₈	C₂×M₁₆	1	2	yes	—
191	C₂×C₈	C₂×M₁₆	1	2	yes	—
192	C₂×C₈	C₂×M₁₆	1	2	yes	—
193	C₂×C₈	C₂×M₁₆	1	2	yes	—
194	C₂×C₈	C₂³×C₄	1	2	yes	—
195	C₂×C₈	C₂×#41	1	2	yes	—
196	C₂×C₈	C₂×#41	1	2	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	8	2	no	—
209	C₄²	—	8	2	no	—
210	C₄²	C₂×C₄²	1	2	yes	—
211	C₄²	C₂×C₄²	1	2	yes	—
212	C₄²	#117	1	2	yes	—
213	C₄²	#117	1	2	yes	—
214	C₄²	C₂×M₁₆	1	2	yes	—
215	C₄²	C₂×M₁₆	1	2	yes	—
216	C₁₆	—	2	16	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₄×M₁₆	1	2	yes	—
221	C₂³	C₂²×M₁₆	1	1	yes	characteristic
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	C₄×M₁₆	1	2	yes	—
241	C₂×C₄	#432	1	4	yes	—
242	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
243	C₂×C₄	C₂×#117	1	1	yes	characteristic
244	C₂×C₄	C₂×#117	1	1	yes	characteristic
245	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
246	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
247	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	C₄×M₁₆	1	4	yes	—
254	C₈	#432	1	4	yes	—
255	C₈	C₄×#41	1	4	yes	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	C₂×C₄×M₁₆	1	1	yes	characteristic
262	C₂²	C₂×#432	1	1	yes	characteristic
263	C₂²	#2920	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	2	no	—
271	C₄	C₄×#117	1	1	yes	characteristic
272	C₄	C₂×C₄×M₁₆	1	1	yes	characteristic
273	C₄	C₂×#432	1	1	yes	characteristic
274	C₄	#2920	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	4	2	no	—
278	C₂	C₄×#432	1	1	yes	characteristic
279	C₂	#33427	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12659	C₂	semidirect
4. #12656	C₂	semidirect
7. #12595	C₂	semidirect
8. #12541	C₂	semidirect
16. #2166	C₄	semidirect
76. #364	C₂×C₄	semidirect
79. #347	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2166	2
#20331	2
#12659	2
#12656	2
#12608	2
C₄×#349	2
#12595	2
#12541	2