Subgroups — #72321339

Order 2048.

1038 conjugacy classes of subgroups.

The subgroup lattice has 217 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321329	C₂	1	2	yes	maximal
2	C₂×#94506	C₂	1	6	yes	maximal, complemented
3	#72234379	C₂	1	6	yes	maximal, complemented
4	#72231854	C₂	1	1	yes	maximal, characteristic
5	#513308	C₄	1	4	yes	—
6	#512545	C₄	1	1	yes	characteristic
7	#512545	C₂²	1	1	yes	characteristic
8	#479899	C₄	1	1	yes	characteristic
9	#153425	C₄	1	12	yes	—
10	C₂×#7511	C₂²	1	12	yes	—
11	#149496	C₄	1	3	yes	—
12	#149496	C₂²	1	3	yes	—
13	#149495	C₄	1	3	yes	—
14	C₂×#7330	C₂²	1	3	yes	—
15	#94506	C₂²	1	4	yes	—
16	#94506	C₂²	1	12	yes	—
17	C₂×#2258	C₈	1	2	yes	—
18	C₂×#2258	C₂×C₄	1	2	yes	—
19	C₄×#504	C₈	1	1	yes	characteristic
20	C₄×#504	C₂×C₄	1	1	yes	characteristic
21	C₈×#131	C₈	1	1	yes	characteristic
22	#13737	C₈	1	32	yes	complemented
23	C₂×M₁₂₈	C₂×C₄	1	24	yes	—
24	#13549	C₈	1	6	yes	—
25	#13549	C₂×C₄	1	6	yes	—
26	C₄×M₆₄	C₂×C₄	1	3	yes	—
27	C₄×M₆₄	C₂×C₄	1	3	yes	—
28	C₂×#1397	C₂×C₄	1	3	yes	—
29	C₂×#1397	C₂³	1	3	yes	—
30	#11399	C₈	1	3	yes	—
31	#11399	C₂×C₄	1	3	yes	—
32	C₈×M₃₂	C₈	1	3	yes	—
33	C₂×#1311	C₂×C₄	1	3	yes	—
34	#7511	C₂×C₄	1	24	yes	—
35	#7511	C₂³	1	8	yes	—
36	#7330	C₂×C₄	1	3	yes	—
37	#7330	C₂×C₄	1	3	yes	—
38	#7330	C₂×C₄	1	6	yes	—
39	#7330	C₂³	1	1	yes	characteristic
40	#7330	C₂³	1	3	yes	—
41	C₂×#504	—	2	1	no	—
42	C₂×#504	C₄²	1	1	yes	characteristic
43	C₂×#504	C₂×C₈	1	1	yes	characteristic
44	C₈×#41	—	2	1	no	—
45	C₄×#131	C₂×C₈	1	1	yes	characteristic
46	#2258	—	2	8	no	—
47	#2258	C₂×C₈	1	16	yes	—
48	C₂×M₆₄	C₄²	1	6	yes	—
49	C₂×M₆₄	C₂×C₈	1	6	yes	—
50	C₂²×C₃₂	C₂×C₈	1	6	yes	—
51	C₂²×C₃₂	C₂²×C₄	1	6	yes	—
52	#2108	—	2	3	no	—
53	#2108	C₄²	1	3	yes	—
54	#2108	C₂×C₈	1	3	yes	—
55	C₄×M₃₂	C₂×C₈	1	3	yes	—
56	C₄×M₃₂	C₂×C₈	1	3	yes	—
57	C₂×C₄×C₁₆	C₂×C₈	1	3	yes	—
58	C₂×C₄×C₁₆	C₂²×C₄	1	3	yes	—
59	#1450	C₂×C₈	1	3	yes	—
60	C₈×M₁₆	—	2	3	no	—
61	C₂×C₈²	C₂×C₈	1	3	yes	—
62	M₁₂₈	C₂×C₈	1	48	yes	complemented
63	M₁₂₈	C₂²×C₄	1	16	yes	—
64	C₂×C₆₄	C₂×C₈	1	48	yes	complemented
65	#1397	C₂×C₈	1	6	yes	—
66	#1397	C₂²×C₄	1	6	yes	—
67	#1397	C₂²×C₄	1	3	yes	—
68	#1397	C₂²×C₄	1	1	yes	characteristic
69	#1397	C₂²×C₄	1	3	yes	—
70	#1397	C₂⁴	1	1	yes	characteristic
71	C₄×C₃₂	C₄²	1	6	yes	—
72	C₄×C₃₂	C₂×C₈	1	6	yes	—
73	#1311	C₂×C₈	1	3	yes	—
74	#1311	C₂×C₈	1	3	yes	—
75	#1311	C₂²×C₄	1	1	yes	characteristic
76	#1311	C₂²×C₄	1	3	yes	—
77	C₈×C₁₆	C₂×C₈	1	6	yes	—
78	C₂×#131	C₄×C₈	1	1	yes	characteristic
79	C₂×#131	M₃₂	1	1	yes	characteristic
80	C₄×#41	M₃₂	1	1	yes	characteristic
81	#504	—	4	4	no	—
82	#504	C₄×C₈	1	8	yes	—
83	#504	M₃₂	1	8	yes	—
84	C₂×M₃₂	—	2	3	no	—
85	C₂×M₃₂	C₄×C₈	1	3	yes	—
86	C₂×M₃₂	C₄×C₈	1	3	yes	—
87	C₂²×C₁₆	—	2	3	no	—
88	C₂²×C₁₆	C₂×C₄²	1	3	yes	—
89	C₂²×C₁₆	C₂²×C₈	1	3	yes	—
90	C₈×Q₈	—	4	1	no	—
91	C₈×D₄	—	4	3	no	—
92	#431	—	2	3	no	—
93	#405	C₄×C₈	1	3	yes	—
94	#405	M₃₂	1	3	yes	—
95	C₄×M₁₆	M₃₂	1	3	yes	—
96	C₂×C₄×C₈	—	2	3	no	—
97	C₂×C₄×C₈	C₂²×C₈	1	3	yes	—
98	M₆₄	—	2	12	no	—
99	M₆₄	C₄×C₈	1	24	yes	—
100	C₂×C₃₂	—	2	12	no	—
101	C₂×C₃₂	C₂×C₄²	1	6	yes	—
102	C₂×C₃₂	C₂²×C₈	1	6	yes	—
103	C₂×C₃₂	C₂²×C₈	1	2	yes	—
104	C₂×C₃₂	C₂²×C₈	1	24	yes	—
105	C₂×C₃₂	C₂³×C₄	1	2	yes	—
106	#346	—	2	3	no	—
107	#346	C₄×C₈	1	6	yes	—
108	C₄×C₁₆	—	2	3	no	—
109	C₄×C₁₆	C₂×C₄²	1	3	yes	—
110	C₄×C₁₆	C₂²×C₈	1	3	yes	—
111	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
112	C₄×C₁₆	C₂²×C₈	1	3	yes	—
113	C₄×C₁₆	C₂²×C₈	1	3	yes	—
114	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
115	#322	—	2	3	no	—
116	C₈²	—	2	3	no	—
117	C₈²	C₂²×C₈	1	3	yes	—
118	C₈²	C₂²×C₈	1	1	yes	characteristic
119	C₆₄	C₂²×C₈	1	32	yes	—
120	C₂×#41	#346	1	1	yes	characteristic
121	#131	—	2	4	no	—
122	#131	C₈²	1	4	yes	—
123	#131	#346	1	4	yes	—
124	C₂×M₁₆	#346	1	3	yes	—
125	C₂×M₁₆	#346	1	3	yes	—
126	C₂²×C₈	—	4	3	no	—
127	C₂²×C₈	C₂×C₄×C₈	1	3	yes	—
128	C₂²×C₈	C₂×M₃₂	1	3	yes	—
129	C₄×Q₈	—	2	1	no	—
130	C₄×D₄	—	2	3	no	—
131	#117	#346	1	3	yes	—
132	C₂×C₄²	C₂×M₃₂	1	3	yes	—
133	M₃₂	—	4	6	no	—
134	M₃₂	C₈²	1	12	yes	—
135	M₃₂	#346	1	12	yes	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	2	3	no	—
138	C₂×C₁₆	—	4	6	no	—
139	C₂×C₁₆	C₂×C₄×C₈	1	12	yes	—
140	C₂×C₁₆	C₂×C₄×C₈	1	3	yes	—
141	C₂×C₁₆	C₂×C₄×C₈	1	3	yes	—
142	C₂×C₁₆	C₂×M₃₂	1	12	yes	—
143	C₂×C₁₆	C₂²×C₄²	1	1	yes	characteristic
144	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
145	C₄⋊C₈	—	4	3	no	—
146	C₂²⋊C₈	—	4	3	no	—
147	#97	#346	1	6	yes	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	2	3	no	—
150	C₄×C₈	—	4	3	no	—
151	C₄×C₈	C₂×C₄×C₈	1	3	yes	—
152	C₄×C₈	C₂×M₃₂	1	3	yes	—
153	C₄×C₈	C₂×M₃₂	1	3	yes	—
154	C₄×C₈	C₂³×C₈	1	1	yes	characteristic
155	C₃₂	—	2	2	no	—
156	C₃₂	—	2	6	no	—
157	C₃₂	C₂×C₄×C₈	1	12	yes	—
158	C₃₂	C₂³×C₈	1	4	yes	—
159	#41	#1311	1	4	yes	—
160	#41	#1398	1	4	yes	—
161	C₂×Q₈	#1398	1	1	yes	characteristic
162	C₂×D₄	#1398	1	3	yes	—
163	C₂²×C₄	—	2	3	no	—
164	C₂²×C₄	C₂×#346	1	3	yes	—
165	M₁₆	—	2	6	no	—
166	M₁₆	#1311	1	12	yes	—
167	C₂×C₈	—	2	6	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	3	no	—
170	C₂×C₈	—	16	3	no	—
171	C₂×C₈	C₂×C₈²	1	6	yes	—
172	C₂×C₈	C₂×#346	1	3	yes	—
173	C₂×C₈	C₂×#346	1	6	yes	—
174	C₂×C₈	C₂×#346	1	3	yes	—
175	C₂×C₈	C₂²×C₄×C₈	1	1	yes	characteristic
176	C₂×C₈	C₂²×M₃₂	1	1	yes	characteristic
177	C₄⋊C₄	—	2	3	no	—
178	C₂²⋊C₄	—	2	3	no	—
179	C₄²	—	2	3	no	—
180	C₄²	C₂×#346	1	3	yes	—
181	C₄²	C₂²×M₃₂	1	1	yes	characteristic
182	C₁₆	—	4	1	no	—
183	C₁₆	—	4	3	no	—
184	C₁₆	C₂×C₈²	1	6	yes	—
185	C₁₆	C₂×#346	1	6	yes	—
186	C₁₆	C₂²×C₄×C₈	1	2	yes	—
187	C₁₆	C₂²×M₃₂	1	2	yes	—
188	C₂³	C₂×#1398	1	3	yes	—
189	Q₈	#7330	1	4	yes	—
190	D₄	#7330	1	12	yes	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	8	3	no	—
194	C₂×C₄	C₂×#1311	1	6	yes	—
195	C₂×C₄	C₂×#1398	1	6	yes	—
196	C₂×C₄	C₂×#1398	1	3	yes	—
197	C₂×C₄	C₂²×#346	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	3	no	—
200	C₈	—	8	3	no	—
201	C₈	—	8	1	no	—
202	C₈	C₂×#1311	1	6	yes	—
203	C₈	C₂²×C₈²	1	1	yes	characteristic
204	C₈	C₂²×#346	1	1	yes	characteristic
205	C₂²	—	4	3	no	—
206	C₂²	C₂×#7330	1	6	yes	—
207	C₂²	C₂²×#1398	1	1	yes	characteristic
208	C₄	—	4	1	no	—
209	C₄	—	4	3	no	—
210	C₄	C₂×#7330	1	6	yes	—
211	C₄	C₂²×#1311	1	1	yes	characteristic
212	C₄	C₂²×#1398	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	6	no	—
215	C₂	C₂²×#7330	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₈	C₂²×C₈²
Frattini, cosocle	#1397	C₂⁴
socle	C₂	C₂²×#7330
prime radical	C₂×#41	#346
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₈ ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#94506	C₂	semidirect
3. #72234379	C₂	semidirect
22. #13737	C₈	semidirect
62. M₁₂₈	C₂×C₈	semidirect
64. C₂×C₆₄	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
#72321329	2
C₂×#94506	2
#72234379	2
#72231854	2