Subgroups — #72225135

Order 4096.

1059 conjugacy classes of subgroups.

The subgroup lattice has 233 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225132	C₂	1	4	yes	maximal
2	#72225131	—	2	8	no	—
3	#72225130	C₂	1	1	yes	maximal, characteristic
4	#72225128	—	2	2	no	—
5	#72225126	C₂²	1	2	yes	—
6	#72216231	C₂	1	1	yes	maximal, characteristic
7	#72216226	C₄	1	2	yes	—
8	C₈×#1398	C₂²	1	2	yes	—
9	#72216220	C₄	1	1	yes	characteristic
10	C₁₆×M₆₄	C₄	1	1	yes	characteristic
11	#72216190	C₂	1	1	yes	maximal, characteristic
12	#72216182	C₂²	1	1	yes	characteristic
13	#72216168	—	2	2	no	—
14	#72216168	C₂²	1	2	yes	—
15	C₄×#1398	D₄	1	4	yes	—
16	C₂×#7457	—	4	16	no	—
17	#149495	Q₈	1	4	yes	—
18	C₈×M₆₄	C₂×C₄	1	2	yes	—
19	C₈×M₆₄	C₂×C₄	1	2	yes	—
20	#135718	—	2	4	no	—
21	#135715	—	2	4	no	—
22	#135715	—	4	4	no	—
23	#135713	—	2	2	no	—
24	C₈×#346	D₄	1	2	yes	—
25	C₈×#346	C₂³	1	1	yes	characteristic
26	#122921	—	4	4	no	—
27	C₂×#7052	C₂×C₄	1	1	yes	characteristic
28	C₂×#7052	Q₈	1	2	yes	—
29	C₂×C₁₆²	C₂×C₄	1	1	yes	characteristic
30	#99295	—	2	2	no	—
31	#93666	—	2	4	no	—
32	#93665	—	2	2	no	—
33	#93664	C₈	1	2	yes	complemented
34	C₁₆×C₃₂	C₈	1	2	yes	complemented
35	C₂×#1398	SD₁₆	1	16	yes	—
36	C₄×M₆₄	C₄⋊C₄	1	4	yes	—
37	C₄×M₆₄	C₄⋊C₄	1	4	yes	—
38	#12900	—	4	2	no	—
39	C₄×#346	SD₁₆	1	4	yes	—
40	C₄×#346	C₂×D₄	1	2	yes	—
41	C₂×#1369	—	4	8	no	—
42	C₂×#1368	—	4	8	no	—
43	C₂×#1368	—	8	8	no	—
44	#11996	—	4	4	no	—
45	C₂×#1311	SD₁₆	1	4	yes	—
46	C₂×#1311	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₁₆	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₆	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
51	#8805	—	4	2	no	—
52	#8805	—	4	4	no	—
53	C₄×C₈²	C₂×D₄	1	1	yes	characteristic
54	#7457	—	8	32	no	—
55	#7330	—	2	8	no	—
56	#7329	—	2	2	no	—
57	#7329	M₁₆	1	4	yes	—
58	C₈×C₃₂	—	2	2	no	—
59	C₈×C₃₂	C₂×C₈	1	4	yes	—
60	#7099	—	8	8	no	—
61	#7052	—	2	4	no	—
62	#7052	—	2	2	no	—
63	C₁₆²	C₂×C₈	1	1	yes	characteristic
64	C₁₆²	C₂×C₈	1	1	yes	characteristic
65	C₁₆²	M₁₆	1	2	yes	—
66	C₂×M₆₄	#106	1	8	yes	—
67	C₂×M₆₄	#106	1	8	yes	—
68	C₂×#364	—	8	4	no	—
69	#2150	—	8	2	no	—
70	C₂×#346	—	2	8	no	—
71	C₂×#346	C₂×SD₁₆	1	8	yes	—
72	C₂×C₄×C₁₆	#106	1	4	yes	—
73	C₂×C₄×C₁₆	#106	1	4	yes	—
74	C₂×C₄×C₁₆	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₆	C₄⋊Q₈	1	2	yes	—
76	#1842	—	8	2	no	—
77	C₄²×C₈	#127	1	1	yes	characteristic
78	C₄²×C₈	C₂×SD₁₆	1	2	yes	—
79	C₂×#334	—	8	8	no	—
80	C₂×#334	—	8	4	no	—
81	C₂×C₈²	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₈²	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₈²	C₂×SD₁₆	1	2	yes	—
84	#1398	—	2	32	no	—
85	#1397	—	2	4	no	—
86	#1397	#108	1	8	yes	—
87	C₄×C₃₂	—	2	4	no	—
88	C₄×C₃₂	C₄⋊C₈	1	8	yes	—
89	#1369	—	8	16	no	—
90	#1368	—	8	16	no	—
91	#1368	—	16	16	no	—
92	#1367	—	8	8	no	—
93	#1311	—	2	8	no	—
94	#1311	—	2	4	no	—
95	C₈×C₁₆	—	2	2	no	—
96	C₈×C₁₆	—	2	2	no	—
97	C₈×C₁₆	—	2	2	no	—
98	C₈×C₁₆	C₄⋊C₈	1	2	yes	—
99	C₈×C₁₆	C₄⋊C₈	1	2	yes	—
100	C₈×C₁₆	#108	1	4	yes	—
101	C₈×C₁₆	C₂²×C₈	1	1	yes	characteristic
102	C₈×C₁₆	C₂×M₁₆	1	1	yes	characteristic
103	C₈×C₁₆	C₂×M₁₆	1	2	yes	—
104	C₂×M₃₂	—	16	2	no	—
105	C₂²×C₁₆	—	2	4	no	—
106	C₂²×C₁₆	#365	1	8	yes	—
107	C₂²×C₁₆	C₂×#106	1	4	yes	—
108	C₂²×C₁₆	#498	1	4	yes	—
109	C₂×C₄⋊C₈	—	16	4	no	—
110	C₂×C₄⋊C₈	—	16	2	no	—
111	C₂×C₄×C₈	—	2	4	no	—
112	C₂×C₄×C₈	#384	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×#106	1	2	yes	—
114	C₂×C₄×C₈	#492	1	1	yes	characteristic
115	C₂×C₄×C₈	#492	1	4	yes	—
116	C₂×C₄×C₈	#498	1	2	yes	—
117	C₄³	#492	1	1	yes	characteristic
118	M₆₄	—	2	16	no	—
119	M₆₄	#365	1	32	yes	—
120	C₂×C₃₂	—	2	8	no	—
121	C₂×C₃₂	#334	1	16	yes	—
122	#364	—	16	8	no	—
123	C₄⋊C₁₆	—	16	4	no	—
124	#346	—	2	16	no	—
125	#346	—	4	16	no	—
126	C₄×C₁₆	—	2	4	no	—
127	C₄×C₁₆	—	2	2	no	—
128	C₄×C₁₆	—	2	2	no	—
129	C₄×C₁₆	—	2	4	no	—
130	C₄×C₁₆	#334	1	4	yes	—
131	C₄×C₁₆	#334	1	4	yes	—
132	C₄×C₁₆	#365	1	8	yes	—
133	C₄×C₁₆	C₂×C₄⋊C₈	1	2	yes	—
134	C₄×C₁₆	#423	1	2	yes	—
135	C₄×C₁₆	C₂×#108	1	4	yes	—
136	#334	—	16	8	no	—
137	#334	—	16	16	no	—
138	C₈²	—	2	2	no	—
139	C₈²	—	2	4	no	—
140	C₈²	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₈²	#423	1	1	yes	characteristic
142	C₈²	C₂×#108	1	2	yes	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	32	2	no	—
145	C₂²×C₈	#1705	1	2	yes	—
146	C₂²×C₈	#1844	1	2	yes	—
147	C₂²×C₈	C₂×#365	1	4	yes	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	#1705	1	2	yes	—
150	C₂×C₄²	#1844	1	1	yes	characteristic
151	M₃₂	—	32	4	no	—
152	C₂×C₁₆	—	2	16	no	—
153	C₂×C₁₆	—	2	8	no	—
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	—	4	8	no	—
156	C₂×C₁₆	—	4	4	no	—
157	C₂×C₁₆	—	32	2	no	—
158	C₂×C₁₆	#1368	1	8	yes	—
159	C₂×C₁₆	#1368	1	8	yes	—
160	C₂×C₁₆	C₂×#334	1	4	yes	—
161	C₂×C₁₆	#1566	1	4	yes	—
162	C₂×C₁₆	C₂×#365	1	16	yes	—
163	C₄⋊C₈	—	32	4	no	—
164	C₄⋊C₈	—	32	8	no	—
165	C₄×C₈	—	2	8	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	4	8	no	—
170	C₄×C₈	C₂×#334	1	2	yes	—
171	C₄×C₈	#1566	1	2	yes	—
172	C₄×C₈	#1842	1	1	yes	characteristic
173	C₄×C₈	#1848	1	1	yes	characteristic
174	C₄×C₈	#1848	1	2	yes	—
175	C₄×C₈	C₂×#365	1	4	yes	—
176	C₃₂	—	4	8	no	—
177	C₃₂	#1368	1	32	yes	complemented
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	#8813	1	1	yes	characteristic
180	C₂²×C₄	#12903	1	2	yes	—
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	64	4	no	—
187	C₂×C₈	—	64	2	no	—
188	C₂×C₈	#8805	1	2	yes	—
189	C₂×C₈	#8822	1	2	yes	—
190	C₂×C₈	C₂×#1368	1	4	yes	—
191	C₂×C₈	#12020	1	4	yes	—
192	C₂×C₈	#12903	1	8	yes	—
193	C₄²	—	2	4	no	—
194	C₄²	—	4	4	no	—
195	C₄²	#8805	1	1	yes	characteristic
196	C₄²	#8822	1	1	yes	characteristic
197	C₄²	#12903	1	2	yes	—
198	C₁₆	—	2	8	no	—
199	C₁₆	—	4	4	no	—
200	C₁₆	—	8	4	no	—
201	C₁₆	—	64	2	no	—
202	C₁₆	#7457	1	16	yes	complemented
203	C₁₆	C₂×#1368	1	8	yes	—
204	C₁₆	#12020	1	8	yes	—
205	C₂³	#135758	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	#99295	1	1	yes	characteristic
210	C₂×C₄	#135715	1	2	yes	—
211	C₂×C₄	#135758	1	4	yes	—
212	C₂×C₄	#135758	1	1	yes	characteristic
213	C₂×C₄	#135792	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	—	4	2	no	—
216	C₈	—	8	2	no	—
217	C₈	—	128	2	no	—
218	C₈	#135715	1	4	yes	—
219	C₈	#135792	1	4	yes	—
220	C₈	C₂×#7457	1	8	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	#72214006	1	2	yes	—
223	C₂²	#72216168	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#72214006	1	2	yes	—
227	C₄	#72216168	1	2	yes	—
228	C₄	#72225131	1	4	yes	—
229	C₂	—	2	2	no	—
230	C₂	#72216190	1	1	yes	characteristic
231	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	#135758
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
33. #93664	C₈	semidirect
34. C₁₆×C₃₂	C₈	semidirect
177. C₃₂	#1368	semidirect
202. C₁₆	#7457	semidirect

Maximal subgroups

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

subgroup	index
#72225132	2
#72225130	2
#72216231	2
#72216190	2