Subgroups — #72227183

Order 2048.

475 conjugacy classes of subgroups.

The subgroup lattice has 251 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72227181	C₂	1	2	yes	maximal
2	#72227180	C₂	1	2	yes	maximal
3	#72216247	C₂	1	2	yes	maximal
4	C₈×#1398	C₂	1	1	yes	maximal, characteristic
5	#151981	—	2	2	no	—
6	C₄×#1398	C₄	1	1	yes	characteristic
7	C₄×#1398	C₄	1	1	yes	characteristic
8	C₂×#7456	C₂²	1	4	yes	—
9	#149495	C₂²	1	1	yes	characteristic
10	#149495	C₂²	1	1	yes	characteristic
11	C₈×M₆₄	C₄	1	1	yes	characteristic
12	C₈×M₆₄	C₄	1	1	yes	characteristic
13	C₈×#346	C₂²	1	1	yes	characteristic
14	#94018	—	2	2	no	—
15	#93878	C₄	1	8	yes	complemented
16	C₂×#1422	—	2	4	no	—
17	C₂×#1398	—	2	2	no	—
18	C₂×#1398	—	4	2	no	—
19	C₂×#1398	C₈	1	2	yes	—
20	C₂×#1398	C₈	1	2	yes	—
21	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
22	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
23	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
24	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
25	C₄×#346	C₂×C₄	1	1	yes	characteristic
26	C₄×#346	D₄	1	1	yes	characteristic
27	C₂×#1311	Q₈	1	1	yes	characteristic
28	C₂×#1311	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
31	C₄×C₈²	Q₈	1	1	yes	characteristic
32	#7468	—	4	2	no	—
33	#7456	—	2	4	no	—
34	#7456	C₂×C₄	1	8	yes	—
35	#7330	—	2	2	no	—
36	#7330	C₂×C₄	1	4	yes	—
37	#7329	—	2	1	no	—
38	#7329	C₈	1	2	yes	—
39	C₈×C₃₂	—	2	1	no	—
40	C₈×C₃₂	C₈	1	2	yes	—
41	C₂×M₆₄	—	2	1	no	—
42	C₂×M₆₄	—	2	1	no	—
43	C₂×M₆₄	—	4	4	no	—
44	C₂×M₆₄	C₂×C₈	1	2	yes	—
45	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
46	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
47	C₂×#346	C₂×C₈	1	2	yes	—
48	C₂×#346	M₁₆	1	2	yes	—
49	C₂×#346	SD₁₆	1	2	yes	—
50	C₂×#346	SD₁₆	1	2	yes	—
51	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₈	—	2	1	no	—
56	C₄²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈²	—	2	1	no	—
58	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈²	C₂×Q₈	1	1	yes	characteristic
60	#1422	—	2	8	no	—
61	#1422	—	4	4	no	—
62	#1398	—	2	8	no	—
63	#1398	—	2	4	no	—
64	#1398	—	2	4	no	—
65	#1398	—	8	4	no	—
66	#1397	—	2	1	no	—
67	#1397	—	2	1	no	—
68	#1397	C₄²	1	2	yes	—
69	#1397	C₄²	1	2	yes	—
70	C₄×C₃₂	—	2	1	no	—
71	C₄×C₃₂	—	2	1	no	—
72	C₄×C₃₂	C₂×C₈	1	2	yes	—
73	C₄×C₃₂	C₂×C₈	1	2	yes	—
74	#1311	—	2	1	no	—
75	#1311	—	2	1	no	—
76	#1311	C₂²×C₄	1	2	yes	—
77	#1311	#41	1	2	yes	—
78	C₈×C₁₆	—	2	2	no	—
79	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
80	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
81	C₈×C₁₆	M₁₆	1	1	yes	characteristic
82	C₈×C₁₆	M₁₆	1	1	yes	characteristic
83	C₂²×C₁₆	C₄⋊C₈	1	2	yes	—
84	C₂²×C₁₆	#106	1	1	yes	characteristic
85	C₂²×C₁₆	#106	1	1	yes	characteristic
86	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
87	C₂²×C₁₆	C₂×M₁₆	1	1	yes	characteristic
88	C₂²×C₁₆	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₈	—	2	1	no	—
90	C₂×C₄×C₈	—	2	1	no	—
91	C₂×C₄×C₈	—	4	2	no	—
92	C₂×C₄×C₈	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₈	#106	1	2	yes	—
94	C₂×C₄×C₈	#108	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄³	#108	1	1	yes	characteristic
97	M₆₄	—	2	4	no	—
98	M₆₄	—	2	4	no	—
99	M₆₄	—	2	2	no	—
100	M₆₄	—	2	2	no	—
101	M₆₄	—	4	8	no	—
102	M₆₄	—	8	4	no	—
103	M₆₄	M₃₂	1	4	yes	—
104	M₆₄	M₃₂	1	4	yes	—
105	C₂×C₃₂	—	2	2	no	—
106	C₂×C₃₂	—	2	2	no	—
107	C₂×C₃₂	—	2	2	no	—
108	C₂×C₃₂	—	8	4	no	—
109	C₂×C₃₂	C₄×C₈	1	2	yes	—
110	C₂×C₃₂	C₄×C₈	1	2	yes	—
111	#346	—	2	4	no	—
112	#346	—	2	4	no	—
113	#346	—	2	2	no	—
114	#346	—	2	2	no	—
115	#346	—	4	2	no	—
116	C₄×C₁₆	—	2	1	no	—
117	C₄×C₁₆	—	2	1	no	—
118	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
119	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
120	C₄×C₁₆	#108	1	1	yes	characteristic
121	C₄×C₁₆	#108	1	1	yes	characteristic
122	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
123	C₄×C₁₆	#117	1	1	yes	characteristic
124	C₄×C₁₆	C₄×D₄	1	2	yes	—
125	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
126	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
127	C₄×C₁₆	#135	1	2	yes	—
128	C₈²	—	2	1	no	—
129	C₈²	—	2	1	no	—
130	C₈²	—	4	2	no	—
131	C₈²	C₄⋊C₈	1	1	yes	characteristic
132	C₈²	C₄⋊C₈	1	1	yes	characteristic
133	C₈²	C₄×Q₈	1	2	yes	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	4	1	no	—
136	C₂²×C₈	#334	1	2	yes	—
137	C₂²×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₈	C₂×#106	1	1	yes	characteristic
139	C₂×C₄²	—	4	1	no	—
140	C₂×C₄²	#334	1	2	yes	—
141	C₂×C₄²	C₂×#108	1	1	yes	characteristic
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	#334	1	1	yes	characteristic
149	C₂×C₁₆	#334	1	1	yes	characteristic
150	C₂×C₁₆	#334	1	1	yes	characteristic
151	C₂×C₁₆	#334	1	1	yes	characteristic
152	C₂×C₁₆	#364	1	4	yes	—
153	C₂×C₁₆	#365	1	4	yes	—
154	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
155	C₂×C₁₆	C₄×M₁₆	1	1	yes	characteristic
156	C₂×C₁₆	#431	1	1	yes	characteristic
157	C₂×C₁₆	#431	1	1	yes	characteristic
158	C₂×C₁₆	C₄×SD₁₆	1	4	yes	—
159	C₂×C₁₆	C₂×M₃₂	1	2	yes	—
160	C₂×C₁₆	C₂×M₃₂	1	2	yes	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	4	1	no	—
170	C₄×C₈	—	4	1	no	—
171	C₄×C₈	—	8	4	no	—
172	C₄×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
174	C₄×C₈	#427	1	1	yes	characteristic
175	C₄×C₈	C₂×#108	1	1	yes	characteristic
176	C₃₂	—	2	4	no	—
177	C₃₂	—	2	4	no	—
178	C₃₂	—	2	2	no	—
179	C₃₂	—	2	4	no	—
180	C₃₂	—	8	8	no	—
181	C₃₂	#346	1	4	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂×#334	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	8	2	no	—
193	C₂×C₈	C₂×#334	1	1	yes	characteristic
194	C₂×C₈	#1564	1	1	yes	characteristic
195	C₂×C₈	C₄×C₄⋊C₈	1	1	yes	characteristic
196	C₂×C₈	C₄×#106	1	1	yes	characteristic
197	C₂×C₈	#2150	1	2	yes	—
198	C₂×C₈	C₂×#364	1	2	yes	—
199	C₂×C₈	C₂×#365	1	2	yes	—
200	C₂×C₈	#2155	1	2	yes	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	8	2	no	—
205	C₄²	#1564	1	1	yes	characteristic
206	C₄²	C₄×#108	1	1	yes	characteristic
207	C₁₆	—	2	1	no	—
208	C₁₆	—	2	1	no	—
209	C₁₆	—	2	4	no	—
210	C₁₆	—	2	1	no	—
211	C₁₆	—	2	1	no	—
212	C₁₆	—	2	4	no	—
213	C₁₆	#1368	1	2	yes	—
214	C₁₆	#1368	1	2	yes	—
215	C₁₆	C₂×#346	1	1	yes	characteristic
216	C₁₆	C₄×M₃₂	1	1	yes	characteristic
217	C₁₆	#2163	1	2	yes	—
218	C₂³	#12002	1	1	yes	characteristic
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	C₄×#334	1	1	yes	characteristic
225	C₂×C₄	#11996	1	2	yes	—
226	C₂×C₄	#12000	1	2	yes	—
227	C₂×C₄	#12002	1	1	yes	characteristic
228	C₈	—	2	4	no	—
229	C₈	—	4	2	no	—
230	C₈	—	4	2	no	—
231	C₈	—	8	2	no	—
232	C₈	C₂×#1368	1	1	yes	characteristic
233	C₈	#11999	1	1	yes	characteristic
234	C₈	#12013	1	2	yes	—
235	C₈	#12653	1	1	yes	characteristic
236	C₈	C₄×#364	1	1	yes	characteristic
237	C₈	C₄×#365	1	2	yes	—
238	C₂²	—	4	1	no	—
239	C₂²	#122923	1	2	yes	—
240	C₂²	#129090	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	8	2	no	—
244	C₄	#122923	1	2	yes	—
245	C₄	#129086	1	1	yes	characteristic
246	C₄	C₄×#1368	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	#72227182	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#11999
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×#1311	C₂³
socle	C₂	#72227182
prime radical	C₂³	#12002
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ 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
15. #93878	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72227181	2
#72227180	2
#72216247	2
C₈×#1398	2