Subgroups — #494079

Order 512.

664 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	#20797	C₂	1	1	yes	maximal, characteristic
4	#12887	C₂	1	4	yes	maximal, complemented
5	#12886	C₂	1	4	yes	maximal, complemented
6	#12563	C₂	1	4	yes	maximal
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	C₄×M₃₂	C₄	1	2	yes	complemented
16	C₄×M₃₂	C₄	1	2	yes	complemented
17	C₄×M₃₂	C₂²	1	2	yes	—
18	C₄×M₃₂	C₂²	1	2	yes	—
19	C₂×#346	C₄	1	2	yes	complemented
20	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
21	#1848	—	2	1	no	—
22	#1848	—	2	1	no	—
23	#1848	C₂²	1	2	yes	—
24	#1848	C₂²	1	2	yes	—
25	#1846	—	2	1	no	—
26	#1846	C₂²	1	2	yes	—
27	#1845	—	2	1	no	—
28	#1844	C₂²	1	2	yes	—
29	#1383	C₂²	1	16	yes	—
30	C₂²×M₁₆	—	2	2	no	—
31	C₂³×C₈	D₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	—	2	2	no	—
34	C₂²×C₄²	D₄	1	1	yes	characteristic
35	C₂×M₃₂	—	2	4	no	—
36	C₂×M₃₂	—	2	4	no	—
37	C₂×M₃₂	C₂×C₄	1	4	yes	—
38	C₂×M₃₂	C₂³	1	4	yes	—
39	C₂²×C₁₆	C₂×C₄	1	4	yes	—
40	C₂×#108	—	2	8	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	—	4	4	no	—
45	C₂×#108	C₂³	1	4	yes	—
46	#427	—	2	8	no	—
47	#427	—	4	4	no	—
48	C₂×#107	—	2	2	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#106	—	2	2	no	—
51	C₂×#106	C₂³	1	4	yes	—
52	#423	—	2	2	no	—
53	#423	—	2	2	no	—
54	#423	—	4	2	no	—
55	C₂×C₄⋊C₈	—	4	2	no	—
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	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	Q₈	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	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	#384	—	2	2	no	—
69	#382	—	2	2	no	—
70	#380	—	4	2	no	—
71	C₄×C₄⋊C₄	—	4	2	no	—
72	#346	C₂×C₄	1	4	yes	—
73	#346	C₂×C₄	1	4	yes	—
74	C₄×C₁₆	C₂×C₄	1	4	yes	—
75	C₄×C₁₆	C₂×C₄	1	4	yes	—
76	C₂³×C₄	SD₁₆	1	2	yes	—
77	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×M₁₆	—	2	4	no	—
79	C₂×M₁₆	—	4	2	no	—
80	C₂×M₁₆	—	4	4	no	—
81	C₂×M₁₆	—	4	4	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	D₈	1	4	yes	—
88	C₂²×C₈	Q₁₆	1	4	yes	—
89	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₈	#41	1	4	yes	—
97	C₂²×C₈	#41	1	4	yes	—
98	C₂²×C₈	C₂⁴	1	1	yes	characteristic
99	#117	—	4	4	no	—
100	#117	—	4	4	no	—
101	C₂×C₄⋊C₄	—	2	4	no	—
102	C₂×C₄⋊C₄	—	4	4	no	—
103	C₂×C₂²⋊C₄	—	4	2	no	—
104	C₂×C₄²	—	4	2	no	—
105	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄²	SD₁₆	1	2	yes	—
110	C₂×C₄²	SD₁₆	1	2	yes	—
111	C₂×C₄²	SD₁₆	1	2	yes	—
112	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
114	M₃₂	—	2	8	no	—
115	M₃₂	—	2	4	no	—
116	M₃₂	—	2	4	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	C₂²×C₄	1	8	yes	—
119	#108	—	2	8	no	—
120	#108	—	4	4	no	—
121	#108	—	4	4	no	—
122	#108	—	4	4	no	—
123	#107	—	4	4	no	—
124	#107	—	4	4	no	—
125	#106	—	2	8	no	—
126	#106	—	4	4	no	—
127	C₄⋊C₈	—	4	4	no	—
128	C₄⋊C₈	—	4	4	no	—
129	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₈	C₂²×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	C₂×Q₈	1	1	yes	characteristic
142	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
143	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
144	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
145	#95	—	4	4	no	—
146	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	#106	1	2	yes	—
150	C₂²×C₄	#106	1	2	yes	—
151	C₂²×C₄	#106	1	2	yes	—
152	C₂²×C₄	#106	1	2	yes	—
153	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	#124	1	2	yes	—
156	C₂²×C₄	#124	1	2	yes	—
157	C₂²×C₄	C₂×D₈	1	2	yes	—
158	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
159	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
160	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
162	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
163	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
164	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
165	M₁₆	—	4	4	no	—
166	M₁₆	—	4	4	no	—
167	M₁₆	—	8	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	8	4	no	—
170	C₂×C₈	#107	1	4	yes	—
171	C₂×C₈	#107	1	4	yes	—
172	C₂×C₈	#107	1	4	yes	—
173	C₂×C₈	#107	1	4	yes	—
174	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₈	#117	1	4	yes	—
181	C₂×C₈	#117	1	4	yes	—
182	C₂×C₈	#124	1	2	yes	—
183	C₂×C₈	#124	1	2	yes	—
184	C₂×C₈	#125	1	4	yes	—
185	C₂×C₈	#125	1	4	yes	—
186	C₂×C₈	C₂×D₈	1	2	yes	—
187	C₂×C₈	C₂×D₈	1	2	yes	—
188	C₂×C₈	C₂×Q₁₆	1	2	yes	—
189	C₂×C₈	C₂×Q₁₆	1	2	yes	—
190	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
192	C₂×C₈	C₂×#41	1	2	yes	—
193	C₂×C₈	C₂×#41	1	2	yes	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	8	8	no	—
197	C₂²⋊C₄	—	8	4	no	—
198	C₄²	—	8	4	no	—
199	C₄²	#106	1	2	yes	—
200	C₄²	#106	1	2	yes	—
201	C₄²	#106	1	2	yes	—
202	C₄²	#106	1	2	yes	—
203	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₄²	C₂×SD₁₆	1	2	yes	—
208	C₄²	C₂×SD₁₆	1	2	yes	—
209	C₁₆	—	2	8	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×#106	1	1	yes	characteristic
212	C₂³	C₂×#106	1	1	yes	characteristic
213	C₂³	#486	1	2	yes	—
214	C₂³	#487	1	2	yes	—
215	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	8	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	#382	1	2	yes	—
225	C₂×C₄	#382	1	2	yes	—
226	C₂×C₄	C₂×#106	1	2	yes	—
227	C₂×C₄	C₂×#106	1	2	yes	—
228	C₂×C₄	C₂×#106	1	2	yes	—
229	C₂×C₄	C₂×#106	1	1	yes	characteristic
230	C₂×C₄	C₂×#106	1	2	yes	—
231	C₂×C₄	C₂×#106	1	1	yes	characteristic
232	C₂×C₄	C₂×#107	1	2	yes	—
233	C₂×C₄	C₂×#107	1	2	yes	—
234	C₂×C₄	#486	1	2	yes	—
235	C₂×C₄	#486	1	2	yes	—
236	C₂×C₄	#486	1	2	yes	—
237	C₂×C₄	#487	1	2	yes	—
238	C₂×C₄	#487	1	2	yes	—
239	C₂×C₄	#487	1	2	yes	—
240	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₂×#124	1	1	yes	characteristic
242	C₂×C₄	C₂×#124	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
244	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
246	C₈	—	8	4	no	—
247	C₈	#382	1	2	yes	—
248	C₈	#382	1	2	yes	—
249	C₈	C₂×#107	1	2	yes	—
250	C₈	C₂×#107	1	2	yes	—
251	C₈	C₂×#107	1	2	yes	—
252	C₈	C₂×#107	1	2	yes	—
253	C₈	C₂×#117	1	2	yes	—
254	C₈	C₂×#125	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#1834	1	2	yes	—
261	C₂²	#1834	1	2	yes	—
262	C₂²	C₂²×#106	1	1	yes	characteristic
263	C₂²	C₂×#486	1	1	yes	characteristic
264	C₂²	C₂×#487	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	16	4	no	—
269	C₄	#1834	1	2	yes	—
270	C₄	#1834	1	2	yes	—
271	C₄	#2155	1	4	yes	—
272	C₄	C₂×#382	1	1	yes	characteristic
273	C₄	C₂²×#107	1	1	yes	characteristic
274	C₄	C₂×#486	1	1	yes	characteristic
275	C₄	C₂×#487	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	2	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	1	no	—
280	C₂	C₂×#1834	1	1	yes	characteristic
281	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
4. #12887	C₂	semidirect
5. #12886	C₂	semidirect
15. C₄×M₃₂	C₄	semidirect
16. C₄×M₃₂	C₄	semidirect
19. C₂×#346	C₄	semidirect
20. C₂×C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×#1848	2
#20797	2
#12887	2
#12886	2
#12563	2