Subgroups — #146992

Order 512.

650 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	C₂×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂×#1383	C₂	1	4	yes	maximal
5	C₂²×M₃₂	C₂²	1	2	yes	—
6	C₂²×#108	—	2	1	no	—
7	C₂²×#108	C₂²	1	2	yes	—
8	C₂²×#107	—	2	1	no	—
9	C₂²×#106	C₂²	1	2	yes	—
10	C₂×#423	—	2	1	no	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#382	—	2	1	no	—
13	C₄×M₃₂	—	2	2	no	—
14	C₄×M₃₂	—	2	2	no	—
15	C₂×#346	C₄	1	2	yes	complemented
16	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
17	#1848	—	2	2	no	—
18	#1848	C₄	1	4	yes	complemented
19	#1846	—	2	2	no	—
20	#1846	C₄	1	4	yes	—
21	#1383	—	2	8	no	—
22	C₂²×M₁₆	—	2	2	no	—
23	C₂³×C₈	D₄	1	1	yes	characteristic
24	C₂³×C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₄⋊C₄	—	2	2	no	—
26	C₂²×C₄²	D₄	1	1	yes	characteristic
27	C₂×M₃₂	—	2	2	no	—
28	C₂×M₃₂	—	2	2	no	—
29	C₂×M₃₂	—	2	2	no	—
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	D₄	1	4	yes	—
32	C₂×M₃₂	Q₈	1	4	yes	—
33	C₂²×C₁₆	C₂×C₄	1	4	yes	—
34	C₂×#108	—	2	4	no	—
35	C₂×#108	—	2	4	no	—
36	C₂×#108	—	2	4	no	—
37	C₂×#108	—	2	2	no	—
38	C₂×#108	—	4	2	no	—
39	C₂×#108	—	4	2	no	—
40	C₂×#108	C₂×C₄	1	4	yes	—
41	C₂×#107	—	2	4	no	—
42	C₂×#107	—	4	2	no	—
43	C₂×#107	—	4	2	no	—
44	C₂×#106	—	2	4	no	—
45	C₂×#106	—	2	4	no	—
46	C₂×#106	—	2	2	no	—
47	C₂×#106	C₂×C₄	1	4	yes	—
48	#423	—	2	4	no	—
49	#423	—	4	2	no	—
50	C₂×C₄⋊C₈	—	4	2	no	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄×C₈	D₄	1	2	yes	—
56	C₂×C₄×C₈	D₄	1	2	yes	—
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
59	#382	—	2	4	no	—
60	#382	—	4	2	no	—
61	C₂×#95	—	4	2	no	—
62	#346	—	2	4	no	—
63	C₄×C₁₆	—	2	4	no	—
64	C₂³×C₄	—	4	2	no	—
65	C₂³×C₄	SD₁₆	1	2	yes	—
66	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×M₁₆	—	2	4	no	—
68	C₂×M₁₆	—	4	2	no	—
69	C₂×M₁₆	—	4	2	no	—
70	C₂×M₁₆	—	4	2	no	—
71	C₂×M₁₆	—	4	2	no	—
72	C₂×M₁₆	—	4	2	no	—
73	C₂²×C₈	—	4	2	no	—
74	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	D₈	1	2	yes	—
79	C₂²×C₈	SD₁₆	1	2	yes	—
80	C₂²×C₈	SD₁₆	1	2	yes	—
81	C₂²×C₈	Q₁₆	1	2	yes	—
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₄	1	2	yes	—
88	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₈	C₂×Q₈	1	2	yes	—
90	C₂²×C₈	#41	1	2	yes	—
91	C₂²×C₈	#41	1	2	yes	—
92	C₂×C₄⋊C₄	—	2	4	no	—
93	C₂×C₄⋊C₄	—	4	2	no	—
94	C₂×C₄⋊C₄	—	4	2	no	—
95	C₂×C₄⋊C₄	—	4	2	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
99	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
100	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	D₈	1	2	yes	—
103	C₂×C₄²	SD₁₆	1	2	yes	—
104	C₂×C₄²	Q₁₆	1	2	yes	—
105	M₃₂	—	2	4	no	—
106	M₃₂	—	2	4	no	—
107	M₃₂	—	2	4	no	—
108	M₃₂	—	2	4	no	—
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	2	4	no	—
111	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
112	#108	—	2	8	no	—
113	#108	—	4	8	no	—
114	#108	—	4	4	no	—
115	#107	—	4	8	no	—
116	#106	—	2	8	no	—
117	#106	—	4	4	no	—
118	C₄⋊C₈	—	4	8	no	—
119	C₄×C₈	C₄²	1	2	yes	—
120	C₄×C₈	C₄²	1	2	yes	—
121	C₄×C₈	C₂²⋊C₄	1	2	yes	—
122	C₄×C₈	C₂²⋊C₄	1	2	yes	—
123	C₄×C₈	C₄⋊C₄	1	2	yes	—
124	C₄×C₈	C₄⋊C₄	1	2	yes	—
125	C₄×C₈	C₄⋊C₄	1	2	yes	—
126	C₄×C₈	C₄⋊C₄	1	2	yes	—
127	#95	—	4	8	no	—
128	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
129	C₂²×C₄	—	4	4	no	—
130	C₂²×C₄	—	4	4	no	—
131	C₂²×C₄	—	8	8	no	—
132	C₂²×C₄	—	8	2	no	—
133	C₂²×C₄	—	8	2	no	—
134	C₂²×C₄	#102	1	4	yes	—
135	C₂²×C₄	#103	1	4	yes	—
136	C₂²×C₄	#106	1	2	yes	—
137	C₂²×C₄	#106	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₄	#124	1	1	yes	characteristic
142	C₂²×C₄	#124	1	1	yes	characteristic
143	C₂²×C₄	#127	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
151	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
152	M₁₆	—	4	4	no	—
153	M₁₆	—	4	4	no	—
154	M₁₆	—	8	2	no	—
155	M₁₆	—	8	2	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	8	2	no	—
158	C₂×C₈	—	8	2	no	—
159	C₂×C₈	#95	1	2	yes	—
160	C₂×C₈	#95	1	2	yes	—
161	C₂×C₈	#102	1	4	yes	—
162	C₂×C₈	#102	1	4	yes	—
163	C₂×C₈	#103	1	4	yes	—
164	C₂×C₈	#103	1	4	yes	—
165	C₂×C₈	#106	1	2	yes	—
166	C₂×C₈	#106	1	2	yes	—
167	C₂×C₈	#107	1	2	yes	—
168	C₂×C₈	#107	1	2	yes	—
169	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
172	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₈	#117	1	2	yes	—
175	C₂×C₈	C₄×D₄	1	2	yes	—
176	C₂×C₈	C₄×D₄	1	2	yes	—
177	C₂×C₈	C₄×Q₈	1	2	yes	—
178	C₂×C₈	C₄×Q₈	1	2	yes	—
179	C₂×C₈	#121	1	2	yes	—
180	C₂×C₈	#121	1	2	yes	—
181	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
182	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₈	#125	1	2	yes	—
184	C₂×C₈	C₄⋊Q₈	1	2	yes	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	8	2	no	—
188	C₄⋊C₄	—	8	2	no	—
189	C₄²	#95	1	2	yes	—
190	C₄²	#95	1	2	yes	—
191	C₄²	#102	1	4	yes	—
192	C₄²	#103	1	4	yes	—
193	C₄²	#107	1	2	yes	—
194	C₄²	#107	1	2	yes	—
195	C₁₆	—	2	8	no	—
196	C₂³	—	2	4	no	—
197	C₂³	C₂×#102	1	1	yes	characteristic
198	C₂³	C₂×#103	1	1	yes	characteristic
199	C₂³	C₂×#106	1	1	yes	characteristic
200	C₂³	#486	1	1	yes	characteristic
201	C₂³	#487	1	1	yes	characteristic
202	C₂³	#492	1	1	yes	characteristic
203	C₂³	#498	1	1	yes	characteristic
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	—	8	4	no	—
210	C₂×C₄	—	16	4	no	—
211	C₂×C₄	—	16	4	no	—
212	C₂×C₄	#340	1	4	yes	—
213	C₂×C₄	#340	1	4	yes	—
214	C₂×C₄	C₂×#95	1	1	yes	characteristic
215	C₂×C₄	#382	1	1	yes	characteristic
216	C₂×C₄	#384	1	1	yes	characteristic
217	C₂×C₄	#390	1	1	yes	characteristic
218	C₂×C₄	#390	1	1	yes	characteristic
219	C₂×C₄	#391	1	1	yes	characteristic
220	C₂×C₄	#391	1	1	yes	characteristic
221	C₂×C₄	C₂×#102	1	1	yes	characteristic
222	C₂×C₄	C₂×#102	1	1	yes	characteristic
223	C₂×C₄	C₂×#102	1	1	yes	characteristic
224	C₂×C₄	C₂×#103	1	1	yes	characteristic
225	C₂×C₄	C₂×#103	1	1	yes	characteristic
226	C₂×C₄	C₂×#103	1	1	yes	characteristic
227	C₂×C₄	C₂×#106	1	1	yes	characteristic
228	C₂×C₄	C₂×#107	1	1	yes	characteristic
229	C₂×C₄	C₂×#107	1	1	yes	characteristic
230	C₂×C₄	#486	1	1	yes	characteristic
231	C₂×C₄	#486	1	1	yes	characteristic
232	C₂×C₄	#486	1	1	yes	characteristic
233	C₂×C₄	#487	1	1	yes	characteristic
234	C₂×C₄	#487	1	1	yes	characteristic
235	C₂×C₄	#487	1	1	yes	characteristic
236	C₂×C₄	#492	1	1	yes	characteristic
237	C₂×C₄	#493	1	1	yes	characteristic
238	C₂×C₄	#494	1	1	yes	characteristic
239	C₂×C₄	#498	1	1	yes	characteristic
240	C₂×C₄	#500	1	1	yes	characteristic
241	C₂×C₄	#500	1	1	yes	characteristic
242	C₈	—	8	4	no	—
243	C₈	#340	1	4	yes	—
244	C₈	#340	1	4	yes	—
245	C₈	C₄×C₄⋊C₄	1	2	yes	—
246	C₈	#380	1	2	yes	—
247	C₈	#389	1	2	yes	—
248	C₈	#389	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	C₂×#340	1	1	yes	characteristic
256	C₂²	#1834	1	1	yes	characteristic
257	C₂²	#1844	1	1	yes	characteristic
258	C₂²	#1956	1	1	yes	characteristic
259	C₂²	#1957	1	1	yes	characteristic
260	C₂²	#1990	1	1	yes	characteristic
261	C₂²	#1990	1	1	yes	characteristic
262	C₂²	#2155	1	4	yes	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	16	4	no	—
266	C₄	C₄⋊#95	1	1	yes	characteristic
267	C₄	C₂×#340	1	1	yes	characteristic
268	C₄	#1834	1	1	yes	characteristic
269	C₄	#1845	1	1	yes	characteristic
270	C₄	#1956	1	1	yes	characteristic
271	C₄	#1957	1	1	yes	characteristic
272	C₄	#1990	1	1	yes	characteristic
273	C₄	#1990	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	#10841	1	1	yes	characteristic
279	C₂	#12563	1	2	yes	—
280	C₂	#12887	1	2	yes	—
281	C₂	#12904	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. C₂×#346	C₄	semidirect
16. C₂×C₄×C₁₆	C₄	semidirect
18. #1848	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
C₂×#1846	2
C₂×#1383	2