Subgroups — #72319368

Order 1024.

343 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₄×#1398	C₂	1	1	yes	maximal, characteristic
3	C₂×#7333	C₂	1	2	yes	maximal
4	#129049	C₂	1	1	yes	maximal, characteristic
5	#93997	C₂	1	2	yes	maximal
6	#13627	C₄	1	2	yes	—
7	C₂×#1398	C₄	1	2	yes	—
8	C₂×#1398	C₂²	1	1	yes	characteristic
9	C₂×#1398	C₂²	1	2	yes	—
10	C₂×#1398	C₂²	1	1	yes	characteristic
11	C₄×M₆₄	C₄	1	2	yes	—
12	#12653	C₄	1	1	yes	characteristic
13	C₄×#346	C₂²	1	1	yes	characteristic
14	C₂×#1313	C₂²	1	2	yes	—
15	#8260	C₄	1	1	yes	characteristic
16	#7466	C₄	1	2	yes	—
17	#7333	—	2	4	no	—
18	#7137	C₄	1	2	yes	—
19	C₂×M₆₄	C₂×C₄	1	2	yes	—
20	C₂×M₆₄	C₂×C₄	1	2	yes	—
21	C₂×M₆₄	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊C₁₆	—	2	1	no	—
23	C₂×#346	C₂×C₄	1	1	yes	characteristic
24	C₂×#346	C₂×C₄	1	1	yes	characteristic
25	C₂×#346	C₂×C₄	1	2	yes	—
26	C₂×#346	C₂×C₄	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	C₂×C₄	1	1	yes	characteristic
30	C₄×C₄⋊C₈	—	2	1	no	—
31	C₄²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×#322	C₂×C₄	1	2	yes	—
33	C₄⋊C₃₂	C₈	1	4	yes	complemented
34	C₄⋊C₃₂	C₈	1	4	yes	complemented
35	#1398	—	2	4	no	—
36	#1398	—	2	2	no	—
37	#1398	—	2	1	no	—
38	#1398	—	2	1	no	—
39	#1398	—	2	2	no	—
40	#1398	D₄	1	2	yes	—
41	#1398	Q₈	1	2	yes	—
42	#1397	—	2	2	no	—
43	C₄×C₃₂	—	2	2	no	—
44	#1374	—	2	1	no	—
45	#1313	—	2	2	no	—
46	#1313	—	2	2	no	—
47	#1273	—	2	1	no	—
48	C₂²×C₁₆	—	2	2	no	—
49	C₂²×C₁₆	C₄²	1	1	yes	characteristic
50	C₂²×C₁₆	C₄²	1	1	yes	characteristic
51	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	4	1	no	—
54	C₂×C₄×C₈	—	2	2	no	—
55	C₂×C₄×C₈	C₄²	1	1	yes	characteristic
56	C₂×C₄×C₈	C₄²	1	1	yes	characteristic
57	C₂×C₄×C₈	M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈	M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₈	M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₄³	M₁₆	1	1	yes	characteristic
62	M₆₄	—	2	2	no	—
63	M₆₄	—	2	4	no	—
64	M₆₄	—	2	2	no	—
65	M₆₄	—	2	4	no	—
66	M₆₄	C₄⋊C₄	1	4	yes	—
67	M₆₄	M₁₆	1	4	yes	—
68	C₂×C₃₂	—	2	2	no	—
69	C₂×C₃₂	—	2	2	no	—
70	C₂×C₃₂	C₂×C₈	1	8	yes	—
71	C₄⋊C₁₆	—	4	1	no	—
72	C₄⋊C₁₆	—	4	1	no	—
73	#346	—	2	2	no	—
74	#346	—	2	2	no	—
75	#346	—	2	2	no	—
76	#346	—	2	1	no	—
77	#346	—	2	1	no	—
78	#346	—	2	2	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	C₂×C₈	1	2	yes	—
84	C₄×C₁₆	C₂×C₈	1	2	yes	—
85	C₄×C₁₆	M₁₆	1	2	yes	—
86	C₄×C₁₆	M₁₆	1	2	yes	—
87	#336	—	4	1	no	—
88	#322	—	2	2	no	—
89	#322	—	2	2	no	—
90	C₂²×C₈	—	4	2	no	—
91	C₂²×C₈	#97	1	1	yes	characteristic
92	C₂²×C₈	#97	1	1	yes	characteristic
93	C₂²×C₈	C₂×C₄²	1	1	yes	characteristic
94	C₂²×C₈	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	#97	1	1	yes	characteristic
98	C₂×C₄²	#97	1	1	yes	characteristic
99	C₂×C₄²	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	4	2	no	—
105	C₂×C₁₆	—	4	2	no	—
106	C₂×C₁₆	—	4	2	no	—
107	C₂×C₁₆	C₄×C₈	1	2	yes	—
108	C₂×C₁₆	#97	1	2	yes	—
109	C₂×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₁₆	#117	1	1	yes	characteristic
111	C₂×C₁₆	C₄×D₄	1	1	yes	characteristic
112	C₂×C₁₆	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₁₆	C₂²×C₈	1	2	yes	—
114	C₂×C₁₆	C₂×M₁₆	1	2	yes	—
115	C₂×C₁₆	C₂×M₁₆	1	2	yes	—
116	C₂×C₁₆	#131	1	2	yes	—
117	C₄⋊C₈	—	8	2	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	—	4	2	no	—
125	C₄×C₈	—	4	2	no	—
126	C₄×C₈	C₄×C₈	1	1	yes	characteristic
127	C₄×C₈	C₄×C₈	1	1	yes	characteristic
128	C₄×C₈	#97	1	1	yes	characteristic
129	C₄×C₈	#97	1	1	yes	characteristic
130	C₄×C₈	C₄×D₄	1	1	yes	characteristic
131	C₄×C₈	C₄×Q₈	1	1	yes	characteristic
132	C₄×C₈	#131	1	1	yes	characteristic
133	C₄×C₈	#131	1	1	yes	characteristic
134	C₃₂	—	2	4	no	—
135	C₃₂	—	2	4	no	—
136	C₃₂	—	2	8	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	#347	1	1	yes	characteristic
140	C₂²×C₄	#347	1	1	yes	characteristic
141	C₂²×C₄	C₂×#97	1	1	yes	characteristic
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	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	8	2	no	—
150	C₂×C₈	—	8	2	no	—
151	C₂×C₈	—	8	2	no	—
152	C₂×C₈	#322	1	1	yes	characteristic
153	C₂×C₈	#322	1	1	yes	characteristic
154	C₂×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
156	C₂×C₈	C₂×#97	1	1	yes	characteristic
157	C₂×C₈	C₄×M₁₆	1	1	yes	characteristic
158	C₂×C₈	#405	1	1	yes	characteristic
159	C₂×C₈	#405	1	1	yes	characteristic
160	C₂×C₈	#436	1	1	yes	characteristic
161	C₂×C₈	#446	1	1	yes	characteristic
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	8	1	no	—
167	C₄²	#322	1	1	yes	characteristic
168	C₄²	#322	1	1	yes	characteristic
169	C₄²	#347	1	1	yes	characteristic
170	C₄²	#347	1	1	yes	characteristic
171	C₄²	#436	1	1	yes	characteristic
172	C₄²	#446	1	1	yes	characteristic
173	C₁₆	—	2	2	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	8	2	no	—
176	C₁₆	#423	1	2	yes	—
177	C₁₆	#432	1	2	yes	—
178	C₁₆	C₈×D₄	1	2	yes	—
179	C₁₆	C₈×Q₈	1	2	yes	—
180	C₂³	C₂×#347	1	1	yes	characteristic
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	#1313	1	1	yes	characteristic
191	C₂×C₄	#1313	1	1	yes	characteristic
192	C₂×C₄	#1313	1	1	yes	characteristic
193	C₂×C₄	#1313	1	1	yes	characteristic
194	C₂×C₄	C₂×#322	1	1	yes	characteristic
195	C₂×C₄	#1451	1	1	yes	characteristic
196	C₂×C₄	#1770	1	1	yes	characteristic
197	C₂×C₄	C₂×#347	1	1	yes	characteristic
198	C₂×C₄	#2110	1	1	yes	characteristic
199	C₂×C₄	#2110	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	16	2	no	—
206	C₈	C₈×M₁₆	1	1	yes	characteristic
207	C₈	#1452	1	1	yes	characteristic
208	C₈	#1767	1	1	yes	characteristic
209	C₈	C₈×C₄⋊C₄	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×#1313	1	1	yes	characteristic
213	C₂²	#11414	1	1	yes	characteristic
214	C₂²	#12666	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	1	no	—
220	C₄	—	8	1	no	—
221	C₄	#7333	1	2	yes	complemented
222	C₄	#8261	1	1	yes	characteristic
223	C₄	#11406	1	1	yes	characteristic
224	C₄	#11414	1	1	yes	characteristic
225	C₄	#12666	1	1	yes	characteristic
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	#129062	1	1	yes	characteristic
229	C₂	C₂×#7333	1	1	yes	characteristic
230	C₂	#149504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#322
commutator	C₂×C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×#346	C₂³
socle	C₂²	C₂×#1313
prime radical	C₂³	C₂×#347
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
33. C₄⋊C₃₂	C₈	semidirect
34. C₄⋊C₃₂	C₈	semidirect
221. C₄	#7333	semidirect

Maximal subgroups

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

subgroup	index
#151979	2
C₄×#1398	2
C₂×#7333	2
#129049	2
#93997	2