Subgroups — #72225214

Order 2048.

509 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225212	C₂	1	4	yes	maximal
2	#72225128	C₂	1	2	yes	maximal
3	C₈×#1398	C₂	1	1	yes	maximal, characteristic
4	C₄×#1398	C₄	1	2	yes	—
5	#149495	C₂²	1	2	yes	—
6	C₈×M₆₄	C₄	1	1	yes	characteristic
7	C₈×M₆₄	C₄	1	1	yes	characteristic
8	#135713	—	2	2	no	—
9	C₈×#346	C₂²	1	1	yes	characteristic
10	#122921	—	2	2	no	—
11	#122921	C₂²	1	4	yes	—
12	#93877	—	2	8	no	—
13	C₂×#1398	—	2	4	no	—
14	C₄×M₆₄	C₂×C₄	1	2	yes	—
15	C₄×M₆₄	C₂×C₄	1	2	yes	—
16	#12900	—	4	2	no	—
17	C₄×#346	C₂×C₄	1	1	yes	characteristic
18	C₄×#346	Q₈	1	1	yes	characteristic
19	#11996	—	2	4	no	—
20	#11996	—	4	2	no	—
21	C₂×#1311	D₄	1	1	yes	characteristic
22	C₂×#1311	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₆	Q₈	1	1	yes	characteristic
25	C₄×C₈²	D₄	1	1	yes	characteristic
26	#7330	D₄	1	4	yes	—
27	#7330	Q₈	1	4	yes	—
28	#7329	—	2	1	no	—
29	#7329	C₈	1	2	yes	—
30	C₈×C₃₂	—	2	1	no	—
31	C₈×C₃₂	C₈	1	1	yes	characteristic
32	C₈×C₃₂	C₈	1	1	yes	characteristic
33	#7099	—	2	8	no	—
34	#7099	—	4	4	no	—
35	C₂×M₆₄	—	2	2	no	—
36	C₂×M₆₄	M₁₆	1	4	yes	—
37	C₂×#364	—	8	4	no	—
38	#2150	—	4	4	no	—
39	C₂×#346	—	2	2	no	—
40	C₂×#346	—	2	2	no	—
41	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₈	SD₁₆	1	2	yes	—
47	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈²	SD₁₆	1	2	yes	—
49	C₂×C₈²	C₂×D₄	1	1	yes	characteristic
50	#1398	—	2	16	no	—
51	#1397	C₄⋊C₄	1	4	yes	—
52	#1397	C₄⋊C₄	1	4	yes	—
53	C₄×C₃₂	—	2	2	no	—
54	C₄×C₃₂	C₂×C₈	1	2	yes	—
55	C₄×C₃₂	C₂×C₈	1	2	yes	—
56	#1367	—	4	4	no	—
57	#1367	—	4	4	no	—
58	#1367	—	8	4	no	—
59	#1311	SD₁₆	1	4	yes	—
60	#1311	C₂×D₄	1	2	yes	—
61	#1311	C₂×Q₈	1	2	yes	—
62	C₈×C₁₆	—	2	2	no	—
63	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
64	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
65	C₈×C₁₆	M₁₆	1	1	yes	characteristic
66	C₈×C₁₆	M₁₆	1	1	yes	characteristic
67	C₂×M₃₂	—	8	4	no	—
68	C₂²×C₁₆	—	2	1	no	—
69	C₂²×C₁₆	—	2	1	no	—
70	C₂²×C₁₆	#108	1	2	yes	—
71	C₂²×C₁₆	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₈	—	2	1	no	—
73	C₂×C₄×C₈	—	2	4	no	—
74	C₂×C₄×C₈	—	2	1	no	—
75	C₂×C₄×C₈	#106	1	1	yes	characteristic
76	C₂×C₄×C₈	#106	1	2	yes	—
77	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×SD₁₆	1	2	yes	—
79	C₄³	#106	1	1	yes	characteristic
80	M₆₄	—	2	8	no	—
81	M₆₄	—	2	8	no	—
82	C₂×C₃₂	—	2	4	no	—
83	C₂×C₃₂	C₄⋊C₈	1	4	yes	—
84	C₂×C₃₂	C₄⋊C₈	1	4	yes	—
85	#364	—	16	8	no	—
86	C₄⋊C₁₆	—	8	4	no	—
87	C₄⋊C₁₆	—	8	4	no	—
88	#346	—	2	8	no	—
89	#346	—	4	4	no	—
90	C₄×C₁₆	—	2	1	no	—
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
93	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₁₆	#106	1	2	yes	—
95	C₄×C₁₆	#106	1	2	yes	—
96	C₄×C₁₆	C₂×C₄⋊C₄	1	2	yes	—
97	C₄×C₁₆	C₄⋊Q₈	1	2	yes	—
98	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
99	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
100	C₈²	—	2	4	no	—
101	C₈²	C₄⋊C₈	1	1	yes	characteristic
102	C₈²	C₄⋊C₈	1	1	yes	characteristic
103	C₈²	#108	1	2	yes	—
104	C₈²	#127	1	1	yes	characteristic
105	C₈²	C₄⋊Q₈	1	1	yes	characteristic
106	C₈²	C₂×SD₁₆	1	2	yes	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	#365	1	4	yes	—
110	C₂²×C₈	#423	1	1	yes	characteristic
111	C₂²×C₈	C₂×#108	1	1	yes	characteristic
112	C₂×C₄²	—	2	1	no	—
113	C₂×C₄²	—	2	2	no	—
114	C₂×C₄²	C₂×#106	1	1	yes	characteristic
115	M₃₂	—	16	8	no	—
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	4	no	—
119	C₂×C₁₆	—	2	4	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	4	2	no	—
122	C₂×C₁₆	—	16	4	no	—
123	C₂×C₁₆	#334	1	2	yes	—
124	C₂×C₁₆	#334	1	2	yes	—
125	C₂×C₁₆	C₂×C₄⋊C₈	1	2	yes	—
126	C₂×C₁₆	#423	1	2	yes	—
127	C₄×C₈	—	2	4	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	4	2	no	—
131	C₄×C₈	—	4	8	no	—
132	C₄×C₈	#334	1	2	yes	—
133	C₄×C₈	#334	1	2	yes	—
134	C₄×C₈	#365	1	4	yes	—
135	C₄×C₈	#384	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₈	C₂×#106	1	1	yes	characteristic
138	C₄×C₈	C₂×#108	1	1	yes	characteristic
139	C₄×C₈	#492	1	2	yes	—
140	C₄×C₈	#498	1	2	yes	—
141	C₃₂	—	2	8	no	—
142	C₃₂	—	2	4	no	—
143	C₃₂	#364	1	8	yes	complemented
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	#1566	1	1	yes	characteristic
146	C₂²×C₄	C₂×#365	1	2	yes	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	#1368	1	4	yes	—
157	C₂×C₈	#1368	1	4	yes	—
158	C₂×C₈	C₂×#334	1	1	yes	characteristic
159	C₂×C₈	#1566	1	1	yes	characteristic
160	C₂×C₈	#1842	1	1	yes	characteristic
161	C₂×C₈	#1848	1	1	yes	characteristic
162	C₄²	—	2	2	no	—
163	C₄²	—	4	1	no	—
164	C₄²	—	4	4	no	—
165	C₄²	C₂×#334	1	1	yes	characteristic
166	C₄²	#1844	1	1	yes	characteristic
167	C₄²	C₂×#365	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₁₆	—	2	2	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	4	2	no	—
172	C₁₆	—	32	4	no	—
173	C₁₆	#1369	1	4	yes	—
174	C₁₆	C₂×#364	1	4	yes	—
175	C₂³	#12020	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	#8805	1	1	yes	characteristic
182	C₂×C₄	C₂×#1368	1	2	yes	—
183	C₂×C₄	#12020	1	1	yes	characteristic
184	C₂×C₄	#12903	1	2	yes	—
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	2	4	no	—
188	C₈	—	4	2	no	—
189	C₈	—	4	2	no	—
190	C₈	—	4	2	no	—
191	C₈	#7457	1	8	yes	—
192	C₈	C₂×#1369	1	2	yes	—
193	C₈	#12900	1	2	yes	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#135715	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	8	2	no	—
200	C₄	#135713	1	1	yes	characteristic
201	C₄	#135718	1	1	yes	characteristic
202	C₄	C₂×#7457	1	4	yes	—
203	C₂	—	4	1	no	—
204	C₂	#72225128	1	1	yes	characteristic
205	C₂	#72225131	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#135715
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₂×#1311	C₂³
prime radical	C₂³	#12020
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
143. C₃₂	#364	semidirect

Maximal subgroups

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

subgroup	index
#72225212	2
#72225128	2
C₈×#1398	2