Subgroups — #72234377

Order 2048.

400 conjugacy classes of subgroups.

The subgroup lattice has 221 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234363	C₂	1	2	yes	maximal
2	#72234237	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₂₈	C₄	1	4	yes	—
4	C₄×#1397	C₄	1	1	yes	characteristic
5	C₄×#1397	C₂²	1	1	yes	characteristic
6	C₄×#1311	C₄	1	1	yes	characteristic
7	#94756	—	2	2	no	—
8	#94756	—	2	2	no	—
9	#93990	—	2	2	no	—
10	C₂×M₁₂₈	—	2	4	no	—
11	C₂×#1397	—	2	1	no	—
12	C₂×#1397	—	2	1	no	—
13	C₂×#1397	D₄	1	2	yes	—
14	C₂×#1397	D₄	1	1	yes	characteristic
15	C₂×#1397	Q₈	1	1	yes	characteristic
16	C₂×C₄×C₃₂	C₈	1	2	yes	—
17	C₂×C₄×C₃₂	C₂×C₄	1	2	yes	—
18	C₄²×C₁₆	C₈	1	1	yes	characteristic
19	C₄²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×#1311	—	2	1	no	—
21	C₂×#1311	—	2	1	no	—
22	C₄×C₈²	C₈	1	1	yes	characteristic
23	#7510	—	2	4	no	—
24	C₄×C₆₄	C₈	1	8	yes	complemented
25	C₂²×C₃₂	—	2	2	no	—
26	C₂²×C₃₂	C₂²⋊C₄	1	2	yes	—
27	C₂²×C₃₂	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₆	—	2	1	no	—
29	C₂×C₄×C₁₆	—	2	1	no	—
30	C₂×C₄×C₁₆	C₁₆	1	2	yes	—
31	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₆	M₁₆	1	1	yes	characteristic
36	C₂×C₄×C₁₆	M₁₆	1	1	yes	characteristic
37	C₄²×C₈	—	2	1	no	—
38	C₄²×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈²	—	2	1	no	—
40	C₂×C₈²	C₁₆	1	2	yes	—
41	M₁₂₈	—	4	4	no	—
42	M₁₂₈	—	4	4	no	—
43	C₂×C₆₄	—	2	8	no	—
44	#1397	—	4	2	no	—
45	#1397	—	4	1	no	—
46	#1397	—	4	1	no	—
47	#1397	—	4	2	no	—
48	#1397	—	4	2	no	—
49	C₄×C₃₂	—	2	2	no	—
50	C₄×C₃₂	C₁₆	1	4	yes	—
51	C₄×C₃₂	C₂×C₈	1	4	yes	—
52	C₄×C₃₂	M₁₆	1	4	yes	—
53	#1311	—	4	2	no	—
54	#1311	—	4	2	no	—
55	C₂²×C₁₆	—	2	2	no	—
56	C₂²×C₁₆	#95	1	1	yes	characteristic
57	C₂²×C₁₆	#97	1	1	yes	characteristic
58	C₂²×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
59	C₂²×C₁₆	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₈	—	2	2	no	—
61	C₂×C₄×C₈	—	4	2	no	—
62	C₂×C₄×C₈	C₄×C₈	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
67	C₂×C₄×C₈	M₃₂	1	1	yes	characteristic
68	C₄³	M₃₂	1	1	yes	characteristic
69	C₂×C₃₂	—	2	2	no	—
70	C₂×C₃₂	—	2	2	no	—
71	C₂×C₃₂	—	2	2	no	—
72	C₂×C₃₂	—	2	2	no	—
73	C₂×C₃₂	—	2	4	no	—
74	C₂×C₃₂	—	4	2	no	—
75	C₂×C₃₂	—	4	2	no	—
76	C₂×C₃₂	C₂²⋊C₈	1	4	yes	—
77	C₂×C₃₂	C₄⋊C₈	1	4	yes	—
78	C₄×C₁₆	—	2	2	no	—
79	C₄×C₁₆	—	2	1	no	—
80	C₄×C₁₆	—	2	1	no	—
81	C₄×C₁₆	—	2	2	no	—
82	C₄×C₁₆	—	2	2	no	—
83	C₄×C₁₆	—	2	2	no	—
84	C₄×C₁₆	—	4	1	no	—
85	C₄×C₁₆	—	4	1	no	—
86	C₄×C₁₆	—	4	2	no	—
87	C₄×C₁₆	C₃₂	1	4	yes	—
88	C₄×C₁₆	C₄×C₈	1	2	yes	—
89	C₄×C₁₆	#97	1	2	yes	—
90	C₄×C₁₆	C₂×C₁₆	1	2	yes	—
91	C₄×C₁₆	M₃₂	1	2	yes	—
92	C₈²	—	2	2	no	—
93	C₈²	—	4	1	no	—
94	C₈²	—	4	1	no	—
95	C₈²	C₃₂	1	4	yes	—
96	C₆₄	—	4	8	no	—
97	C₂²×C₈	—	4	2	no	—
98	C₂²×C₈	#336	1	1	yes	characteristic
99	C₂²×C₈	C₄×C₁₆	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₂²×C₈	C₄⋊C₁₆	1	1	yes	characteristic
102	C₂×C₄²	—	2	2	no	—
103	C₂×C₄²	#346	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊C₁₆	1	1	yes	characteristic
105	C₂×C₄²	C₄⋊C₁₆	1	1	yes	characteristic
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	2	1	no	—
110	C₂×C₁₆	—	2	4	no	—
111	C₂×C₁₆	—	4	2	no	—
112	C₂×C₁₆	—	4	2	no	—
113	C₂×C₁₆	#322	1	2	yes	—
114	C₂×C₁₆	#336	1	2	yes	—
115	C₂×C₁₆	H(3,ℤ/4)	1	1	yes	characteristic
116	C₂×C₁₆	H(3,ℤ/4)	1	2	yes	—
117	C₂×C₁₆	#338	1	1	yes	characteristic
118	C₂×C₁₆	#347	1	2	yes	—
119	C₂×C₁₆	#347	1	2	yes	—
120	C₂×C₁₆	C₂²⋊C₁₆	1	2	yes	—
121	C₂×C₁₆	C₄⋊C₁₆	1	2	yes	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	4	no	—
128	C₄×C₈	—	4	2	no	—
129	C₄×C₈	—	8	4	no	—
130	C₄×C₈	C₈²	1	1	yes	characteristic
131	C₄×C₈	#322	1	1	yes	characteristic
132	C₄×C₈	C₄×C₁₆	1	1	yes	characteristic
133	C₄×C₈	#346	1	1	yes	characteristic
134	C₄×C₈	C₂×C₃₂	1	2	yes	—
135	C₄×C₈	C₂×C₃₂	1	2	yes	—
136	C₄×C₈	M₆₄	1	2	yes	—
137	C₄×C₈	M₆₄	1	2	yes	—
138	C₃₂	—	2	4	no	—
139	C₃₂	—	2	4	no	—
140	C₃₂	—	4	4	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	#1374	1	1	yes	characteristic
144	C₂²×C₄	#1398	1	2	yes	—
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	—	4	4	no	—
150	C₂×C₈	—	8	4	no	—
151	C₂×C₈	#1273	1	1	yes	characteristic
152	C₂×C₈	#1274	1	1	yes	characteristic
153	C₂×C₈	#1275	1	1	yes	characteristic
154	C₂×C₈	C₈×C₁₆	1	1	yes	characteristic
155	C₂×C₈	#1312	1	1	yes	characteristic
156	C₂×C₈	#1374	1	1	yes	characteristic
157	C₂×C₈	#1375	1	1	yes	characteristic
158	C₂×C₈	#1375	1	1	yes	characteristic
159	C₂×C₈	C₄×C₃₂	1	1	yes	characteristic
160	C₂×C₈	C₄×C₃₂	1	1	yes	characteristic
161	C₂×C₈	#1397	1	1	yes	characteristic
162	C₂×C₈	#1397	1	1	yes	characteristic
163	C₂×C₈	C₂²⋊C₃₂	1	2	yes	—
164	C₂×C₈	C₄⋊C₃₂	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	4	4	no	—
168	C₄²	#1311	1	1	yes	characteristic
169	C₄²	#1312	1	1	yes	characteristic
170	C₄²	#1398	1	2	yes	—
171	C₄²	C₂²⋊C₃₂	1	2	yes	—
172	C₄²	C₄⋊C₃₂	1	2	yes	—
173	C₁₆	—	2	2	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	4	4	no	—
176	C₁₆	#1274	1	2	yes	—
177	C₁₆	#1275	1	2	yes	—
178	C₁₆	#1313	1	4	yes	—
179	C₂³	#7466	1	1	yes	characteristic
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	#7136	1	1	yes	characteristic
186	C₂×C₄	#7142	1	1	yes	characteristic
187	C₂×C₄	#7145	1	1	yes	characteristic
188	C₂×C₄	#7330	1	2	yes	—
189	C₂×C₄	#7332	1	2	yes	—
190	C₂×C₄	#7465	1	1	yes	characteristic
191	C₂×C₄	#7465	1	1	yes	characteristic
192	C₂×C₄	#7466	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	8	4	no	—
196	C₈	#7014	1	1	yes	characteristic
197	C₈	#7137	1	1	yes	characteristic
198	C₈	#7142	1	1	yes	characteristic
199	C₈	#7145	1	1	yes	characteristic
200	C₈	C₈×C₃₂	1	1	yes	characteristic
201	C₈	#7329	1	1	yes	characteristic
202	C₈	#7331	1	1	yes	characteristic
203	C₈	#7331	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	#93990	1	1	yes	characteristic
206	C₂²	#93998	1	1	yes	characteristic
207	C₂²	#94005	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	#93249	1	1	yes	characteristic
212	C₄	#93989	1	1	yes	characteristic
213	C₄	#93998	1	1	yes	characteristic
214	C₄	#94005	1	1	yes	characteristic
215	C₄	#94506	1	4	yes	—
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	#72234282	1	1	yes	characteristic
219	C₂	#72234375	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₈²
commutator	C₈	C₈×C₃₂
Frattini, cosocle	C₄×#1397	C₂²
socle	C₂²	#93990
prime radical	C₂³	#7466
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×#1397 ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
24. C₄×C₆₄	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72234363	2
#72234237	2