Subgroups — #148727

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1946	C₂	1	1	yes	maximal, characteristic
2	#20799	C₂	1	1	yes	maximal, characteristic
3	#12569	C₂	1	2	yes	maximal
4	#12567	C₂	1	2	yes	maximal
5	C₂×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×#108	C₂²	1	1	yes	characteristic
8	C₂×#427	—	2	1	no	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂×#424	—	2	1	no	—
12	C₂²×#97	C₂²	1	1	yes	characteristic
13	#2285	—	2	1	no	—
14	#1946	—	2	2	no	—
15	#1946	C₄	1	4	yes	complemented
16	#1847	—	2	4	no	—
17	C₂×#339	—	2	1	no	—
18	#1387	—	2	4	no	—
19	#1386	—	2	4	no	—
20	#1375	—	2	2	no	—
21	C₂²×M₁₆	—	2	1	no	—
22	C₂³×C₈	D₄	1	1	yes	characteristic
23	C₂³×C₈	C₂³	1	1	yes	characteristic
24	C₂×#117	—	2	1	no	—
25	C₂×#117	—	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	—	2	2	no	—
32	C₂×M₃₂	Q₈	1	4	yes	—
33	C₂²×C₁₆	—	2	2	no	—
34	C₂×#108	—	2	2	no	—
35	C₂×#108	—	2	2	no	—
36	C₂×#108	—	2	1	no	—
37	C₂×#108	C₂×C₄	1	2	yes	—
38	#427	—	2	1	no	—
39	#427	—	2	1	no	—
40	#427	—	2	2	no	—
41	#427	—	2	2	no	—
42	#427	—	2	2	no	—
43	#427	—	2	2	no	—
44	#427	—	4	2	no	—
45	#427	—	4	2	no	—
46	#427	C₂×C₄	1	2	yes	—
47	#427	Q₈	1	2	yes	—
48	C₂×#107	—	2	1	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#107	D₄	1	4	yes	—
51	C₂×#106	—	2	2	no	—
52	C₂×#106	—	2	2	no	—
53	C₂×#106	D₄	1	2	yes	—
54	#424	—	2	4	no	—
55	#424	—	4	2	no	—
56	C₂×C₄⋊C₈	—	4	2	no	—
57	C₂×#97	—	2	2	no	—
58	C₂×#97	—	2	2	no	—
59	C₂×#97	C₂×C₄	1	2	yes	—
60	C₂×#97	D₄	1	2	yes	—
61	#384	—	2	2	no	—
62	#382	—	2	2	no	—
63	#380	—	4	2	no	—
64	C₄×C₄⋊C₄	—	4	2	no	—
65	#339	—	2	4	no	—
66	C₂³×C₄	—	2	1	no	—
67	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×M₁₆	—	2	2	no	—
69	C₂×M₁₆	—	4	1	no	—
70	C₂×M₁₆	—	4	1	no	—
71	C₂×M₁₆	—	4	1	no	—
72	C₂×M₁₆	—	4	1	no	—
73	C₂×M₁₆	—	4	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	4	1	no	—
79	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₈	D₈	1	2	yes	—
81	C₂²×C₈	SD₁₆	1	2	yes	—
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₄	1	2	yes	—
85	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₈	C₂×Q₈	1	2	yes	—
88	C₂²×C₈	#41	1	1	yes	characteristic
89	C₂²×C₈	#41	1	1	yes	characteristic
90	C₂²×C₈	#41	1	2	yes	—
91	C₂²×C₈	#41	1	2	yes	—
92	#117	—	2	2	no	—
93	#117	—	2	2	no	—
94	#117	—	4	1	no	—
95	#117	—	4	1	no	—
96	#117	—	4	2	no	—
97	#117	—	4	2	no	—
98	#117	—	4	2	no	—
99	#117	—	4	2	no	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	4	no	—
102	C₂×C₄⋊C₄	—	4	4	no	—
103	C₂×C₂²⋊C₄	—	4	1	no	—
104	C₂×C₂²⋊C₄	—	4	2	no	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	—	2	1	no	—
107	C₂×C₄²	—	2	1	no	—
108	C₂×C₄²	—	4	1	no	—
109	C₂×C₄²	—	4	2	no	—
110	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
111	C₂×C₄²	SD₁₆	1	2	yes	—
112	C₂×C₄²	Q₁₆	1	2	yes	—
113	M₃₂	—	2	4	no	—
114	M₃₂	—	2	4	no	—
115	M₃₂	—	4	2	no	—
116	M₃₂	—	4	2	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	4	2	no	—
119	C₂×C₁₆	—	4	2	no	—
120	#108	—	2	4	no	—
121	#108	—	4	2	no	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	4	no	—
125	#107	—	2	4	no	—
126	#107	—	4	4	no	—
127	#106	—	2	2	no	—
128	#106	—	4	4	no	—
129	#106	—	4	2	no	—
130	#106	—	4	2	no	—
131	#106	C₄⋊C₄	1	4	yes	complemented
132	C₄⋊C₈	—	4	8	no	—
133	#97	—	2	4	no	—
134	#97	—	2	4	no	—
135	#95	—	4	4	no	—
136	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	4	4	no	—
142	C₂²×C₄	—	8	2	no	—
143	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	#120	1	1	yes	characteristic
145	C₂²×C₄	#121	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂²×C₄	#123	1	1	yes	characteristic
148	C₂²×C₄	#124	1	1	yes	characteristic
149	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
151	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
152	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
153	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
154	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂²×C₄	#137	1	1	yes	characteristic
156	C₂²×C₄	#137	1	1	yes	characteristic
157	M₁₆	—	4	2	no	—
158	M₁₆	—	4	2	no	—
159	M₁₆	—	8	1	no	—
160	M₁₆	—	8	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	8	1	no	—
169	C₂×C₈	—	8	1	no	—
170	C₂×C₈	#102	1	4	yes	—
171	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₈	#117	1	1	yes	characteristic
173	C₂×C₈	C₄×D₄	1	2	yes	—
174	C₂×C₈	C₄×Q₈	1	2	yes	—
175	C₂×C₈	#121	1	2	yes	—
176	C₂×C₈	#121	1	1	yes	characteristic
177	C₂×C₈	#121	1	2	yes	—
178	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
179	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
180	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
181	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
182	C₂×C₈	#123	1	2	yes	—
183	C₂×C₈	#125	1	2	yes	—
184	C₂×C₈	#125	1	2	yes	—
185	C₂×C₈	#126	1	2	yes	—
186	C₂×C₈	C₄⋊Q₈	1	2	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	8	8	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	2	no	—
194	C₂²⋊C₄	—	8	2	no	—
195	C₂²⋊C₄	—	8	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	4	2	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	8	2	no	—
202	C₄²	—	8	2	no	—
203	C₄²	#103	1	4	yes	—
204	C₁₆	—	4	4	no	—
205	C₂³	—	4	2	no	—
206	C₂³	#417	1	1	yes	characteristic
207	C₂³	#450	1	1	yes	characteristic
208	C₂³	#459	1	1	yes	characteristic
209	C₂³	#476	1	1	yes	characteristic
210	C₂³	#484	1	1	yes	characteristic
211	C₂³	#490	1	1	yes	characteristic
212	C₂³	#501	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	8	8	no	—
220	C₂×C₄	—	8	4	no	—
221	C₂×C₄	#380	1	1	yes	characteristic
222	C₂×C₄	#386	1	1	yes	characteristic
223	C₂×C₄	#391	1	1	yes	characteristic
224	C₂×C₄	#394	1	1	yes	characteristic
225	C₂×C₄	#395	1	1	yes	characteristic
226	C₂×C₄	#398	1	1	yes	characteristic
227	C₂×C₄	#400	1	1	yes	characteristic
228	C₂×C₄	C₂×#102	1	1	yes	characteristic
229	C₂×C₄	C₂×#103	1	1	yes	characteristic
230	C₂×C₄	#417	1	1	yes	characteristic
231	C₂×C₄	#447	1	1	yes	characteristic
232	C₂×C₄	#448	1	1	yes	characteristic
233	C₂×C₄	#451	1	1	yes	characteristic
234	C₂×C₄	#460	1	1	yes	characteristic
235	C₂×C₄	#461	1	1	yes	characteristic
236	C₂×C₄	#462	1	1	yes	characteristic
237	C₂×C₄	#474	1	1	yes	characteristic
238	C₂×C₄	#475	1	1	yes	characteristic
239	C₂×C₄	#477	1	1	yes	characteristic
240	C₂×C₄	#480	1	1	yes	characteristic
241	C₂×C₄	#481	1	1	yes	characteristic
242	C₂×C₄	#483	1	1	yes	characteristic
243	C₂×C₄	#486	1	1	yes	characteristic
244	C₂×C₄	#487	1	1	yes	characteristic
245	C₂×C₄	#491	1	1	yes	characteristic
246	C₂×C₄	#498	1	1	yes	characteristic
247	C₂×C₄	#500	1	1	yes	characteristic
248	C₂×C₄	#501	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	—	8	2	no	—
251	C₈	#387	1	2	yes	—
252	C₈	#389	1	2	yes	—
253	C₈	#396	1	2	yes	—
254	C₈	#398	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	#1802	1	1	yes	characteristic
263	C₂²	#1872	1	1	yes	characteristic
264	C₂²	#1992	1	1	yes	characteristic
265	C₂²	#2041	1	1	yes	characteristic
266	C₂²	#2058	1	1	yes	characteristic
267	C₂²	#2083	1	1	yes	characteristic
268	C₂²	#2100	1	1	yes	characteristic
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	—	16	4	no	—
272	C₄	#1441	1	1	yes	characteristic
273	C₄	#1803	1	1	yes	characteristic
274	C₄	#1874	1	1	yes	characteristic
275	C₄	#1990	1	1	yes	characteristic
276	C₄	#2043	1	1	yes	characteristic
277	C₄	#2057	1	1	yes	characteristic
278	C₄	#2084	1	1	yes	characteristic
279	C₄	#2099	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#11207	1	1	yes	characteristic
287	C₂	#13416	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#486
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2041
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ 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. #1946	C₄	semidirect
131. #106	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1946	2
#20799	2
#12569	2
#12567	2
C₂×#1375	2