Subgroups — #72225215

Order 4096.

871 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225214	C₂	1	4	yes	maximal
2	#72225212	—	2	8	no	—
3	#72225130	C₂	1	2	yes	maximal
4	#72225128	—	2	2	no	—
5	#72225128	C₂²	1	4	yes	—
6	#72225126	—	2	2	no	—
7	#72216231	C₂	1	1	yes	maximal, characteristic
8	#72216226	C₄	1	2	yes	—
9	C₈×#1398	C₂²	1	2	yes	—
10	#72216220	C₄	1	1	yes	characteristic
11	C₁₆×M₆₄	C₄	1	1	yes	characteristic
12	#72216182	C₂²	1	1	yes	characteristic
13	C₄×#1398	Q₈	1	4	yes	—
14	#149495	D₄	1	4	yes	—
15	C₈×M₆₄	C₂×C₄	1	2	yes	—
16	C₈×M₆₄	C₂×C₄	1	2	yes	—
17	#135718	—	4	4	no	—
18	#135713	—	2	4	no	—
19	C₈×#346	D₄	1	2	yes	—
20	C₈×#346	C₂³	1	1	yes	characteristic
21	#122921	—	2	8	no	—
22	#122921	—	4	4	no	—
23	C₂×#7052	C₂×C₄	1	1	yes	characteristic
24	C₂×#7052	Q₈	1	2	yes	—
25	C₂×C₁₆²	C₂×C₄	1	1	yes	characteristic
26	#93877	—	4	16	no	—
27	#93666	—	2	4	no	—
28	#93665	—	2	2	no	—
29	#93664	C₈	1	2	yes	—
30	C₁₆×C₃₂	C₈	1	1	yes	characteristic
31	C₁₆×C₃₂	C₈	1	1	yes	characteristic
32	C₂×#1398	—	2	8	no	—
33	C₄×M₆₄	C₄⋊C₄	1	4	yes	—
34	C₄×M₆₄	C₄⋊C₄	1	4	yes	—
35	#12900	—	4	4	no	—
36	C₄×#346	SD₁₆	1	4	yes	—
37	C₄×#346	C₂×Q₈	1	2	yes	—
38	C₂×#1369	—	8	8	no	—
39	#11996	—	4	4	no	—
40	#11996	—	4	4	no	—
41	C₂×#1311	SD₁₆	1	4	yes	—
42	C₂×#1311	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₁₆	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₆	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
47	C₄×C₈²	C₂×D₄	1	1	yes	characteristic
48	#7330	SD₁₆	1	16	yes	—
49	#7329	—	2	2	no	—
50	#7329	M₁₆	1	4	yes	—
51	C₈×C₃₂	—	2	2	no	—
52	C₈×C₃₂	C₂×C₈	1	2	yes	—
53	C₈×C₃₂	C₂×C₈	1	2	yes	—
54	#7099	—	4	16	no	—
55	#7099	—	8	8	no	—
56	#7052	—	2	4	no	—
57	#7052	—	2	2	no	—
58	C₁₆²	C₂×C₈	1	1	yes	characteristic
59	C₁₆²	C₂×C₈	1	1	yes	characteristic
60	C₁₆²	M₁₆	1	2	yes	—
61	C₂×M₆₄	—	2	4	no	—
62	C₂×M₆₄	#108	1	8	yes	—
63	C₂×#364	—	8	8	no	—
64	#2150	—	8	4	no	—
65	C₂×#346	—	2	8	no	—
66	C₂×#346	—	2	4	no	—
67	C₂×C₄×C₁₆	#106	1	4	yes	—
68	C₂×C₄×C₁₆	#106	1	4	yes	—
69	C₂×C₄×C₁₆	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₆	C₄⋊Q₈	1	2	yes	—
71	C₄²×C₈	C₄⋊Q₈	1	1	yes	characteristic
72	C₄²×C₈	C₂×SD₁₆	1	2	yes	—
73	C₂×C₈²	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₈²	#127	1	1	yes	characteristic
75	C₂×C₈²	C₂×SD₁₆	1	2	yes	—
76	#1398	—	4	16	no	—
77	#1397	#106	1	8	yes	—
78	#1397	#106	1	8	yes	—
79	C₄×C₃₂	—	2	4	no	—
80	C₄×C₃₂	C₄⋊C₈	1	4	yes	—
81	C₄×C₃₂	C₄⋊C₈	1	4	yes	—
82	#1369	—	16	16	no	—
83	#1367	—	8	8	no	—
84	#1367	—	8	8	no	—
85	#1311	—	2	8	no	—
86	#1311	C₂×SD₁₆	1	8	yes	—
87	C₈×C₁₆	—	2	2	no	—
88	C₈×C₁₆	—	2	2	no	—
89	C₈×C₁₆	—	2	2	no	—
90	C₈×C₁₆	C₄⋊C₈	1	2	yes	—
91	C₈×C₁₆	C₄⋊C₈	1	2	yes	—
92	C₈×C₁₆	#108	1	4	yes	—
93	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
94	C₈×C₁₆	C₂×M₁₆	1	1	yes	characteristic
95	C₈×C₁₆	C₂×M₁₆	1	2	yes	—
96	C₂×M₃₂	—	16	4	no	—
97	C₂²×C₁₆	—	2	2	no	—
98	C₂²×C₁₆	—	2	4	no	—
99	C₂²×C₁₆	#365	1	8	yes	—
100	C₂²×C₁₆	C₂×#108	1	4	yes	—
101	C₂×C₄×C₈	—	2	2	no	—
102	C₂×C₄×C₈	—	2	4	no	—
103	C₂×C₄×C₈	#384	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×#106	1	2	yes	—
105	C₂×C₄×C₈	#492	1	2	yes	—
106	C₂×C₄×C₈	#498	1	1	yes	characteristic
107	C₄³	#498	1	1	yes	characteristic
108	M₆₄	—	2	16	no	—
109	M₆₄	—	4	8	no	—
110	C₂×C₃₂	—	2	8	no	—
111	C₂×C₃₂	#334	1	8	yes	—
112	C₂×C₃₂	#334	1	8	yes	—
113	#364	—	16	16	no	—
114	C₄⋊C₁₆	—	16	4	no	—
115	C₄⋊C₁₆	—	16	4	no	—
116	#346	—	4	16	no	—
117	#346	—	4	8	no	—
118	C₄×C₁₆	—	2	4	no	—
119	C₄×C₁₆	—	2	2	no	—
120	C₄×C₁₆	—	2	4	no	—
121	C₄×C₁₆	#334	1	4	yes	—
122	C₄×C₁₆	#334	1	4	yes	—
123	C₄×C₁₆	#365	1	8	yes	—
124	C₄×C₁₆	C₂×C₄⋊C₈	1	2	yes	—
125	C₄×C₁₆	#423	1	2	yes	—
126	C₄×C₁₆	C₂×#106	1	4	yes	—
127	C₄×C₁₆	#498	1	4	yes	—
128	C₈²	—	2	4	no	—
129	C₈²	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₈²	#423	1	1	yes	characteristic
131	C₈²	C₂×#108	1	2	yes	—
132	C₈²	#492	1	4	yes	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	#1848	1	2	yes	—
136	C₂²×C₈	C₂×#365	1	4	yes	—
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	—	2	1	no	—
139	C₂×C₄²	#1844	1	1	yes	characteristic
140	M₃₂	—	32	8	no	—
141	C₂×C₁₆	—	2	16	no	—
142	C₂×C₁₆	—	2	8	no	—
143	C₂×C₁₆	—	2	4	no	—
144	C₂×C₁₆	—	4	4	no	—
145	C₂×C₁₆	—	4	8	no	—
146	C₂×C₁₆	—	4	4	no	—
147	C₂×C₁₆	—	32	4	no	—
148	C₂×C₁₆	#1368	1	8	yes	—
149	C₂×C₁₆	#1368	1	8	yes	—
150	C₂×C₁₆	C₂×#334	1	4	yes	—
151	C₂×C₁₆	#1566	1	4	yes	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	4	8	no	—
155	C₄×C₈	—	4	4	no	—
156	C₄×C₈	C₂×#334	1	2	yes	—
157	C₄×C₈	#1566	1	2	yes	—
158	C₄×C₈	#1705	1	4	yes	—
159	C₄×C₈	#1842	1	1	yes	characteristic
160	C₄×C₈	#1844	1	2	yes	—
161	C₄×C₈	#1848	1	1	yes	characteristic
162	C₄×C₈	C₂×#365	1	4	yes	—
163	C₃₂	—	2	8	no	—
164	C₃₂	—	4	8	no	—
165	C₃₂	#1369	1	16	yes	complemented
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	#8822	1	1	yes	characteristic
168	C₂²×C₄	#12903	1	2	yes	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	#8805	1	2	yes	—
176	C₂×C₈	#8822	1	2	yes	—
177	C₂×C₈	C₂×#1368	1	4	yes	—
178	C₂×C₈	#12020	1	4	yes	—
179	C₄²	—	4	2	no	—
180	C₄²	—	4	4	no	—
181	C₄²	#8805	1	1	yes	characteristic
182	C₄²	#8813	1	1	yes	characteristic
183	C₄²	#12903	1	2	yes	—
184	C₁₆	—	2	4	no	—
185	C₁₆	—	2	8	no	—
186	C₁₆	—	4	4	no	—
187	C₁₆	—	8	4	no	—
188	C₁₆	—	64	4	no	—
189	C₁₆	#7457	1	16	yes	—
190	C₁₆	C₂×#1369	1	8	yes	—
191	C₂³	#135792	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	#99295	1	1	yes	characteristic
197	C₂×C₄	#135715	1	2	yes	—
198	C₂×C₄	#135758	1	2	yes	—
199	C₂×C₄	#135792	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	4	no	—
202	C₈	—	4	4	no	—
203	C₈	—	8	1	no	—
204	C₈	#135718	1	4	yes	—
205	C₈	C₂×#7457	1	8	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	2	no	—
208	C₂²	#72216168	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	16	1	no	—
212	C₄	#72225126	1	2	yes	—
213	C₄	#72225131	1	4	yes	—
214	C₂	—	4	1	no	—
215	C₂	#72225130	1	1	yes	characteristic
216	C₂	#72225132	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#72216168
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	C₂³	#135792
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72216182 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
165. C₃₂	#1369	semidirect

Maximal subgroups

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

subgroup	index
#72225214	2
#72225130	2
#72216231	2