Subgroups — #72319363

Order 1024.

306 conjugacy classes of subgroups.

The subgroup lattice has 219 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	#135715	C₂	1	1	yes	maximal, characteristic
4	C₄×#1369	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	C₄×#364	—	2	1	no	—
10	C₄×#346	C₂²	1	1	yes	characteristic
11	C₂×#1369	C₂²	1	1	yes	characteristic
12	C₂×#1369	C₂²	1	1	yes	characteristic
13	C₂×#1368	—	2	1	no	—
14	C₂×#1368	C₂²	1	2	yes	—
15	#8805	—	2	1	no	—
16	#7466	C₄	1	2	yes	—
17	#7457	—	2	4	no	—
18	#7161	—	2	1	no	—
19	C₂×M₆₄	C₂×C₄	1	2	yes	—
20	C₂×M₆₄	C₂×C₄	1	2	yes	—
21	C₂×#364	—	2	1	no	—
22	C₂×#364	—	2	1	no	—
23	C₄×M₃₂	—	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	#1842	—	4	1	no	—
31	C₄²×C₈	D₄	1	1	yes	characteristic
32	C₂×#334	—	2	2	no	—
33	C₂×#334	—	4	1	no	—
34	C₄⋊C₃₂	—	2	2	no	—
35	C₄⋊C₃₂	C₈	1	4	yes	complemented
36	#1398	—	2	2	no	—
37	#1398	—	2	2	no	—
38	#1376	—	4	1	no	—
39	#1369	—	2	1	no	—
40	#1369	—	2	1	no	—
41	#1369	D₄	1	2	yes	—
42	#1369	Q₈	1	2	yes	—
43	#1368	—	2	2	no	—
44	#1368	—	2	2	no	—
45	#1368	—	4	2	no	—
46	C₂×M₃₂	—	4	1	no	—
47	C₂×M₃₂	—	4	1	no	—
48	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₁₆	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄⋊C₈	—	4	2	no	—
53	C₂×C₄⋊C₈	—	8	2	no	—
54	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
60	C₄³	SD₁₆	1	1	yes	characteristic
61	M₆₄	—	2	2	no	—
62	M₆₄	—	2	2	no	—
63	M₆₄	—	2	2	no	—
64	M₆₄	—	2	2	no	—
65	C₂×C₃₂	—	2	2	no	—
66	C₂×C₃₂	C₂×C₈	1	4	yes	—
67	#364	—	2	2	no	—
68	#364	—	2	2	no	—
69	#364	—	4	1	no	—
70	#364	—	4	1	no	—
71	#346	—	2	1	no	—
72	#346	—	2	1	no	—
73	#346	—	2	1	no	—
74	#346	—	2	1	no	—
75	#346	—	4	1	no	—
76	#346	—	8	1	no	—
77	#346	C₂×D₄	1	1	yes	characteristic
78	#346	C₂×Q₈	1	1	yes	characteristic
79	#346	#41	1	1	yes	characteristic
80	#346	#41	1	1	yes	characteristic
81	C₄×C₁₆	—	2	2	no	—
82	C₄×C₁₆	—	8	1	no	—
83	C₄×C₁₆	C₂×C₈	1	2	yes	—
84	C₄×C₁₆	M₁₆	1	2	yes	—
85	#334	—	4	2	no	—
86	#334	—	4	2	no	—
87	#334	—	8	2	no	—
88	C₂²×C₈	—	8	2	no	—
89	C₂²×C₈	#106	1	1	yes	characteristic
90	C₂²×C₈	#106	1	1	yes	characteristic
91	C₂²×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄²	—	2	2	no	—
94	C₂×C₄²	#106	1	1	yes	characteristic
95	C₂×C₄²	#106	1	1	yes	characteristic
96	C₂×C₄²	C₂×SD₁₆	1	1	yes	characteristic
97	M₃₂	—	4	2	no	—
98	M₃₂	—	4	2	no	—
99	M₃₂	—	8	1	no	—
100	M₃₂	—	8	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	8	1	no	—
109	C₂×C₁₆	—	8	1	no	—
110	C₂×C₁₆	C₄⋊C₈	1	2	yes	—
111	C₂×C₁₆	C₄×D₄	1	1	yes	characteristic
112	C₂×C₁₆	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₁₆	#125	1	1	yes	characteristic
114	C₂×C₁₆	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₁₆	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₁₆	#131	1	1	yes	characteristic
118	C₂×C₁₆	#131	1	1	yes	characteristic
119	C₄⋊C₈	—	8	2	no	—
120	C₄⋊C₈	—	8	2	no	—
121	C₄⋊C₈	—	16	4	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	4	1	no	—
128	C₄×C₈	C₄⋊C₈	1	2	yes	—
129	C₄×C₈	#108	1	2	yes	—
130	C₄×C₈	#121	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊Q₈	1	1	yes	characteristic
132	C₄×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₄×C₈	#137	1	1	yes	characteristic
134	C₃₂	—	2	4	no	—
135	C₃₂	—	4	2	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	#365	1	2	yes	—
139	C₂²×C₄	C₂×#106	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	16	2	no	—
147	C₂×C₈	—	16	4	no	—
148	C₂×C₈	#334	1	2	yes	—
149	C₂×C₈	#365	1	2	yes	—
150	C₂×C₈	#389	1	1	yes	characteristic
151	C₂×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₈	#424	1	1	yes	characteristic
153	C₂×C₈	#428	1	1	yes	characteristic
154	C₂×C₈	C₂×#108	1	1	yes	characteristic
155	C₂×C₈	#430	1	1	yes	characteristic
156	C₂×C₈	#461	1	1	yes	characteristic
157	C₂×C₈	#476	1	1	yes	characteristic
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	4	4	no	—
162	C₄²	#334	1	2	yes	—
163	C₄²	#365	1	1	yes	characteristic
164	C₄²	#365	1	1	yes	characteristic
165	C₄²	#460	1	1	yes	characteristic
166	C₄²	#475	1	1	yes	characteristic
167	C₁₆	—	2	2	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	2	1	no	—
170	C₁₆	—	4	1	no	—
171	C₁₆	—	8	2	no	—
172	C₁₆	—	8	2	no	—
173	C₁₆	C₈×D₄	1	1	yes	characteristic
174	C₁₆	#436	1	1	yes	characteristic
175	C₁₆	C₈×Q₈	1	1	yes	characteristic
176	C₁₆	#446	1	1	yes	characteristic
177	C₂³	C₂×#365	1	1	yes	characteristic
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	#1368	1	2	yes	—
185	C₂×C₄	#1368	1	2	yes	—
186	C₂×C₄	C₂×#334	1	1	yes	characteristic
187	C₂×C₄	#1570	1	1	yes	characteristic
188	C₂×C₄	#1948	1	1	yes	characteristic
189	C₂×C₄	C₂×#365	1	1	yes	characteristic
190	C₂×C₄	C₂×#365	1	1	yes	characteristic
191	C₂×C₄	#2155	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	32	2	no	—
197	C₈	#1565	1	1	yes	characteristic
198	C₈	#1574	1	1	yes	characteristic
199	C₈	#1939	1	1	yes	characteristic
200	C₈	#1950	1	1	yes	characteristic
201	C₂²	—	2	2	no	—
202	C₂²	C₂×#1368	1	1	yes	characteristic
203	C₂²	#12029	1	1	yes	characteristic
204	C₂²	#13183	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	4	1	no	—
207	C₄	—	4	1	no	—
208	C₄	—	8	1	no	—
209	C₄	#7457	1	4	yes	complemented
210	C₄	#9602	1	1	yes	characteristic
211	C₄	#12000	1	1	yes	characteristic
212	C₄	#12030	1	1	yes	characteristic
213	C₄	#13185	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#140388	1	1	yes	characteristic
217	C₂	C₂×#7457	1	2	yes	—

	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
35. C₄⋊C₃₂	C₈	semidirect
209. 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
#135715	2
C₄×#1369	2
#94030	2