Subgroups — #72319252

Order 1024.

304 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#151979	C₂	1	1	yes	maximal, characteristic
2	C₂×#7457	C₂	1	2	yes	maximal
3	#135718	C₂	1	1	yes	maximal, characteristic
4	C₄×#1368	C₂	1	1	yes	maximal, characteristic
5	#94030	C₂	1	2	yes	maximal
6	#13627	C₄	1	1	yes	characteristic
7	#13627	C₄	1	1	yes	characteristic
8	C₂×#1398	C₂²	1	2	yes	—
9	#12900	—	2	1	no	—
10	C₄×#346	C₂²	1	1	yes	characteristic
11	C₂×#1369	—	2	1	no	—
12	C₂×#1369	C₂²	1	2	yes	—
13	C₂×#1368	C₂²	1	1	yes	characteristic
14	C₂×#1368	C₂²	1	1	yes	characteristic
15	C₄×#334	—	2	1	no	—
16	#7466	C₄	1	2	yes	—
17	#7457	—	2	4	no	—
18	#7156	—	2	1	no	—
19	C₂×M₆₄	C₂×C₄	1	2	yes	—
20	C₂×M₆₄	C₂×C₄	1	2	yes	—
21	C₂×#364	—	2	2	no	—
22	C₂×#364	—	4	1	no	—
23	#2150	—	4	1	no	—
24	C₂×#346	C₂×C₄	1	1	yes	characteristic
25	C₂×#346	D₄	1	1	yes	characteristic
26	C₂×#346	Q₈	1	1	yes	characteristic
27	C₂×#346	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₆	Q₈	1	1	yes	characteristic
30	C₄×C₄⋊C₈	—	4	1	no	—
31	C₄²×C₈	D₄	1	1	yes	characteristic
32	C₂×#334	—	2	1	no	—
33	C₂×#334	—	2	1	no	—
34	C₄⋊C₃₂	—	2	1	no	—
35	C₄⋊C₃₂	—	2	1	no	—
36	C₄⋊C₃₂	C₈	1	2	yes	complemented
37	C₄⋊C₃₂	C₈	1	2	yes	complemented
38	#1398	—	2	2	no	—
39	#1398	—	2	2	no	—
40	#1369	—	2	2	no	—
41	#1369	—	2	2	no	—
42	#1369	—	4	2	no	—
43	#1368	—	2	1	no	—
44	#1368	—	2	1	no	—
45	#1368	D₄	1	2	yes	—
46	#1368	Q₈	1	2	yes	—
47	#1277	—	4	1	no	—
48	C₂×M₃₂	—	4	2	no	—
49	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₁₆	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	4	1	no	—
54	C₂×C₄⋊C₈	—	4	1	no	—
55	C₂×C₄×C₈	—	8	1	no	—
56	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄³	SD₁₆	1	1	yes	characteristic
63	M₆₄	—	2	2	no	—
64	M₆₄	—	2	2	no	—
65	M₆₄	—	2	2	no	—
66	M₆₄	—	2	2	no	—
67	C₂×C₃₂	—	2	2	no	—
68	C₂×C₃₂	C₂×C₈	1	4	yes	—
69	#364	—	4	2	no	—
70	#364	—	4	2	no	—
71	#364	—	8	2	no	—
72	C₄⋊C₁₆	—	8	1	no	—
73	C₄⋊C₁₆	—	8	1	no	—
74	#346	—	2	1	no	—
75	#346	—	2	1	no	—
76	#346	—	2	1	no	—
77	#346	—	2	1	no	—
78	#346	—	4	1	no	—
79	#346	C₂×D₄	1	1	yes	characteristic
80	#346	C₂×Q₈	1	1	yes	characteristic
81	#346	#41	1	1	yes	characteristic
82	#346	#41	1	1	yes	characteristic
83	C₄×C₁₆	—	2	1	no	—
84	C₄×C₁₆	—	2	1	no	—
85	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
86	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
87	C₄×C₁₆	M₁₆	1	1	yes	characteristic
88	C₄×C₁₆	M₁₆	1	1	yes	characteristic
89	#336	—	8	1	no	—
90	#334	—	2	2	no	—
91	#334	—	2	2	no	—
92	#334	—	4	1	no	—
93	#334	—	4	1	no	—
94	C₂²×C₈	—	8	1	no	—
95	C₂²×C₈	—	8	1	no	—
96	C₂²×C₈	#106	1	1	yes	characteristic
97	C₂²×C₈	#106	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄²	—	2	1	no	—
101	C₂×C₄²	—	2	1	no	—
102	C₂×C₄²	#106	1	1	yes	characteristic
103	C₂×C₄²	#106	1	1	yes	characteristic
104	C₂×C₄²	C₂×SD₁₆	1	1	yes	characteristic
105	M₃₂	—	8	2	no	—
106	M₃₂	—	8	2	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	2	1	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	—	8	2	no	—
115	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
116	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₁₆	C₄×D₄	1	1	yes	characteristic
118	C₂×C₁₆	C₄×Q₈	1	1	yes	characteristic
119	C₂×C₁₆	#125	1	1	yes	characteristic
120	C₂×C₁₆	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₁₆	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₁₆	#131	1	1	yes	characteristic
124	C₂×C₁₆	#131	1	1	yes	characteristic
125	C₄⋊C₈	—	4	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	8	1	no	—
128	C₄⋊C₈	—	8	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	—	16	2	no	—
136	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₈	#108	1	1	yes	characteristic
139	C₄×C₈	#108	1	1	yes	characteristic
140	C₄×C₈	#121	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊Q₈	1	1	yes	characteristic
142	C₄×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₄×C₈	#137	1	1	yes	characteristic
144	C₃₂	—	2	4	no	—
145	C₃₂	—	4	2	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	#365	1	1	yes	characteristic
149	C₂²×C₄	#365	1	1	yes	characteristic
150	C₂²×C₄	C₂×#106	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	8	2	no	—
159	C₂×C₈	—	8	2	no	—
160	C₂×C₈	—	16	1	no	—
161	C₂×C₈	—	16	1	no	—
162	C₂×C₈	—	16	1	no	—
163	C₂×C₈	—	16	1	no	—
164	C₂×C₈	#334	1	1	yes	characteristic
165	C₂×C₈	#334	1	1	yes	characteristic
166	C₂×C₈	#389	1	1	yes	characteristic
167	C₂×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
168	C₂×C₈	#424	1	1	yes	characteristic
169	C₂×C₈	#428	1	1	yes	characteristic
170	C₂×C₈	C₂×#108	1	1	yes	characteristic
171	C₂×C₈	#430	1	1	yes	characteristic
172	C₂×C₈	#460	1	1	yes	characteristic
173	C₂×C₈	#475	1	1	yes	characteristic
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	4	2	no	—
177	C₄²	—	8	1	no	—
178	C₄²	#334	1	1	yes	characteristic
179	C₄²	#334	1	1	yes	characteristic
180	C₄²	#365	1	2	yes	—
181	C₄²	#365	1	1	yes	characteristic
182	C₄²	#365	1	1	yes	characteristic
183	C₄²	#461	1	1	yes	characteristic
184	C₄²	#476	1	1	yes	characteristic
185	C₁₆	—	2	1	no	—
186	C₁₆	—	2	1	no	—
187	C₁₆	—	2	1	no	—
188	C₁₆	—	2	1	no	—
189	C₁₆	—	4	1	no	—
190	C₁₆	—	16	2	no	—
191	C₁₆	C₈×D₄	1	1	yes	characteristic
192	C₁₆	#436	1	1	yes	characteristic
193	C₁₆	C₈×Q₈	1	1	yes	characteristic
194	C₁₆	#446	1	1	yes	characteristic
195	C₂³	C₂×#365	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	#1368	1	1	yes	characteristic
206	C₂×C₄	#1368	1	1	yes	characteristic
207	C₂×C₄	#1368	1	1	yes	characteristic
208	C₂×C₄	#1368	1	1	yes	characteristic
209	C₂×C₄	C₂×#334	1	1	yes	characteristic
210	C₂×C₄	#1570	1	1	yes	characteristic
211	C₂×C₄	#1948	1	1	yes	characteristic
212	C₂×C₄	C₂×#365	1	1	yes	characteristic
213	C₂×C₄	C₂×#365	1	1	yes	characteristic
214	C₂×C₄	#2155	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	4	1	no	—
220	C₈	—	16	2	no	—
221	C₈	—	16	2	no	—
222	C₈	#1565	1	1	yes	characteristic
223	C₈	#1574	1	1	yes	characteristic
224	C₈	#1939	1	1	yes	characteristic
225	C₈	#1950	1	1	yes	characteristic
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	C₂×#1368	1	1	yes	characteristic
229	C₂²	#12030	1	1	yes	characteristic
230	C₂²	#13185	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	#7457	1	2	yes	complemented
238	C₄	#9602	1	1	yes	characteristic
239	C₄	#12000	1	1	yes	characteristic
240	C₄	#12029	1	1	yes	characteristic
241	C₄	#13183	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#140388	1	1	yes	characteristic
245	C₂	C₂×#7457	1	1	yes	characteristic
246	C₂	#151024	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#1368
commutator	C₂×C₁₆	C₂²×C₈
Frattini, cosocle	C₂×#346	C₂³
prime radical	C₂³	C₂×#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
36. C₄⋊C₃₂	C₈	semidirect
37. C₄⋊C₃₂	C₈	semidirect
237. C₄	#7457	semidirect

Maximal subgroups

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

subgroup	index
#151979	2
C₂×#7457	2
#135718	2
C₄×#1368	2
#94030	2