Subgroups — #494152

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2155	C₂	1	4	yes	maximal
2	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#1848	C₂	1	1	yes	maximal, characteristic
4	#20797	C₂	1	1	yes	maximal, characteristic
5	#12904	C₂	1	4	yes	maximal, complemented
6	#12903	C₂	1	4	yes	maximal, complemented
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂²×#108	—	2	1	no	—
9	C₂²×#108	C₂²	1	2	yes	—
10	C₂×#427	—	2	1	no	—
11	C₂×#427	C₂²	1	2	yes	—
12	C₂×#423	—	2	1	no	—
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#2285	—	2	1	no	—
15	#2155	—	2	16	no	—
16	C₂×#365	C₂²	1	16	yes	—
17	C₄×M₃₂	C₄	1	2	yes	complemented
18	C₄×M₃₂	C₄	1	2	yes	complemented
19	C₄×M₃₂	C₂²	1	2	yes	—
20	C₄×M₃₂	C₂²	1	2	yes	—
21	C₂×#346	C₄	1	2	yes	complemented
22	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
23	#1848	—	2	1	no	—
24	#1848	—	2	1	no	—
25	#1848	C₂²	1	2	yes	—
26	#1848	C₂²	1	2	yes	—
27	#1846	—	2	1	no	—
28	#1846	C₂²	1	2	yes	—
29	#1845	—	2	1	no	—
30	#1844	C₂²	1	2	yes	—
31	C₂²×M₁₆	—	2	2	no	—
32	C₂³×C₈	D₄	1	1	yes	characteristic
33	C₂³×C₈	C₂³	1	1	yes	characteristic
34	C₂×#117	—	2	2	no	—
35	C₂²×C₄²	D₄	1	1	yes	characteristic
36	C₂×M₃₂	C₂×C₄	1	4	yes	—
37	C₂×M₃₂	D₄	1	8	yes	—
38	C₂×M₃₂	Q₈	1	8	yes	—
39	C₂×M₃₂	C₂³	1	4	yes	—
40	C₂²×C₁₆	C₂×C₄	1	4	yes	—
41	C₂×#108	—	2	8	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	4	4	no	—
46	C₂×#108	C₂³	1	4	yes	—
47	#427	—	2	8	no	—
48	#427	—	4	4	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#107	—	2	2	no	—
51	C₂×#106	—	2	2	no	—
52	C₂×#106	C₂³	1	4	yes	—
53	#423	—	2	2	no	—
54	#423	—	2	2	no	—
55	#423	—	4	2	no	—
56	C₂×C₄⋊C₈	—	4	2	no	—
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	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	D₄	1	1	yes	characteristic
63	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	#384	—	2	2	no	—
70	#382	—	2	2	no	—
71	#380	—	4	2	no	—
72	C₄×C₄⋊C₄	—	4	2	no	—
73	#365	—	2	32	no	—
74	#346	C₂×C₄	1	4	yes	—
75	#346	C₂×C₄	1	4	yes	—
76	C₄×C₁₆	C₂×C₄	1	4	yes	—
77	C₄×C₁₆	C₂×C₄	1	4	yes	—
78	C₂³×C₄	SD₁₆	1	2	yes	—
79	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×M₁₆	—	2	4	no	—
81	C₂×M₁₆	—	4	2	no	—
82	C₂×M₁₆	—	4	4	no	—
83	C₂×M₁₆	—	4	4	no	—
84	C₂²×C₈	—	4	2	no	—
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	SD₁₆	1	4	yes	—
90	C₂²×C₈	SD₁₆	1	4	yes	—
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	4	yes	—
95	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×Q₈	1	4	yes	—
98	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₈	C₂⁴	1	1	yes	characteristic
101	#117	—	4	4	no	—
102	#117	—	4	4	no	—
103	C₂×C₄⋊C₄	—	2	4	no	—
104	C₂×C₄⋊C₄	—	4	4	no	—
105	C₂×C₂²⋊C₄	—	4	2	no	—
106	C₂×C₄²	—	4	2	no	—
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄²	SD₁₆	1	2	yes	—
112	C₂×C₄²	SD₁₆	1	2	yes	—
113	C₂×C₄²	SD₁₆	1	2	yes	—
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
116	M₃₂	C₄⋊C₄	1	16	yes	complemented
117	M₃₂	C₂×D₄	1	8	yes	—
118	M₃₂	C₂×Q₈	1	8	yes	—
119	C₂×C₁₆	C₄⋊C₄	1	16	yes	complemented
120	C₂×C₁₆	C₂²×C₄	1	8	yes	—
121	#108	—	2	8	no	—
122	#108	—	4	4	no	—
123	#108	—	4	4	no	—
124	#108	—	4	4	no	—
125	#107	—	4	4	no	—
126	#107	—	4	4	no	—
127	#106	—	2	8	no	—
128	#106	—	4	4	no	—
129	C₄⋊C₈	—	4	4	no	—
130	C₄⋊C₈	—	4	4	no	—
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
144	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
145	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
146	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
147	#95	—	4	4	no	—
148	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	#106	1	2	yes	—
152	C₂²×C₄	#106	1	2	yes	—
153	C₂²×C₄	#106	1	2	yes	—
154	C₂²×C₄	#106	1	2	yes	—
155	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	#127	1	2	yes	—
158	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
159	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
160	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
161	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
162	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
163	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
165	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
167	M₁₆	—	4	4	no	—
168	M₁₆	—	4	4	no	—
169	M₁₆	—	8	4	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	8	4	no	—
172	C₂×C₈	#106	1	4	yes	—
173	C₂×C₈	#106	1	4	yes	—
174	C₂×C₈	#106	1	4	yes	—
175	C₂×C₈	#106	1	4	yes	—
176	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
177	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
178	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₈	#127	1	2	yes	—
185	C₂×C₈	C₄⋊Q₈	1	4	yes	—
186	C₂×C₈	C₄⋊Q₈	1	4	yes	—
187	C₂×C₈	C₄⋊Q₈	1	2	yes	—
188	C₂×C₈	C₂×SD₁₆	1	2	yes	—
189	C₂×C₈	C₂×SD₁₆	1	2	yes	—
190	C₂×C₈	C₂×SD₁₆	1	2	yes	—
191	C₂×C₈	C₂×SD₁₆	1	2	yes	—
192	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₄	1	2	yes	—
194	C₂×C₈	C₂²×Q₈	1	2	yes	—
195	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	8	8	no	—
199	C₂²⋊C₄	—	8	4	no	—
200	C₄²	—	8	4	no	—
201	C₄²	#106	1	2	yes	—
202	C₄²	#106	1	2	yes	—
203	C₄²	#106	1	2	yes	—
204	C₄²	#106	1	2	yes	—
205	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₄²	C₂×SD₁₆	1	2	yes	—
210	C₄²	C₂×SD₁₆	1	2	yes	—
211	C₁₆	C₂×C₄⋊C₄	1	16	yes	—
212	C₂³	—	4	2	no	—
213	C₂³	C₂×#106	1	1	yes	characteristic
214	C₂³	C₂×#106	1	1	yes	characteristic
215	C₂³	#492	1	2	yes	—
216	C₂³	#498	1	2	yes	—
217	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	8	8	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	#384	1	2	yes	—
227	C₂×C₄	#384	1	2	yes	—
228	C₂×C₄	C₂×#106	1	2	yes	—
229	C₂×C₄	C₂×#106	1	2	yes	—
230	C₂×C₄	C₂×#106	1	2	yes	—
231	C₂×C₄	C₂×#106	1	1	yes	characteristic
232	C₂×C₄	C₂×#106	1	2	yes	—
233	C₂×C₄	C₂×#106	1	2	yes	—
234	C₂×C₄	C₂×#106	1	2	yes	—
235	C₂×C₄	C₂×#106	1	1	yes	characteristic
236	C₂×C₄	#492	1	2	yes	—
237	C₂×C₄	#492	1	2	yes	—
238	C₂×C₄	#492	1	2	yes	—
239	C₂×C₄	#498	1	2	yes	—
240	C₂×C₄	#498	1	2	yes	—
241	C₂×C₄	#498	1	2	yes	—
242	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×#127	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
246	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
247	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
248	C₈	—	8	4	no	—
249	C₈	#384	1	2	yes	—
250	C₈	#384	1	2	yes	—
251	C₈	C₂×#106	1	2	yes	—
252	C₈	C₂×#106	1	2	yes	—
253	C₈	C₂×#106	1	2	yes	—
254	C₈	C₂×#106	1	2	yes	—
255	C₈	C₂²×C₄⋊C₄	1	2	yes	—
256	C₈	C₂×C₄⋊Q₈	1	2	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	#1844	1	2	yes	—
263	C₂²	#1844	1	2	yes	—
264	C₂²	C₂²×#106	1	1	yes	characteristic
265	C₂²	C₂×#492	1	1	yes	characteristic
266	C₂²	C₂×#498	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	16	4	no	—
271	C₄	#1844	1	2	yes	—
272	C₄	#1844	1	2	yes	—
273	C₄	#2155	1	4	yes	complemented
274	C₄	C₂×#384	1	1	yes	characteristic
275	C₄	C₂²×#106	1	1	yes	characteristic
276	C₄	C₂×#492	1	1	yes	characteristic
277	C₄	C₂×#498	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	2	no	—
281	C₂	—	2	1	no	—
282	C₂	C₂×#1844	1	1	yes	characteristic
283	C₂	C₂×#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×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
5. #12904	C₂	semidirect
6. #12903	C₂	semidirect
17. C₄×M₃₂	C₄	semidirect
18. C₄×M₃₂	C₄	semidirect
21. C₂×#346	C₄	semidirect
22. C₂×C₄×C₁₆	C₄	semidirect
116. M₃₂	C₄⋊C₄	semidirect
119. C₂×C₁₆	C₄⋊C₄	semidirect
273. C₄	#2155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2155	2
C₂×C₄×M₃₂	2
C₂×#1848	2
#20797	2
#12904	2
#12903	2