Subgroups — #72225408

Order 2048.

404 conjugacy classes of subgroups.

The subgroup lattice has 253 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225405	C₂	1	2	yes	maximal
2	#72225404	C₂	1	2	yes	maximal
3	#72225395	C₂	1	2	yes	maximal
4	C₈×#1398	C₂	1	1	yes	maximal, characteristic
5	#153582	C₂²	1	4	yes	—
6	C₄×#1398	C₄	1	1	yes	characteristic
7	C₄×#1398	C₂²	1	1	yes	characteristic
8	#149495	C₄	1	1	yes	characteristic
9	#149495	C₂²	1	1	yes	characteristic
10	C₈×M₆₄	C₄	1	1	yes	characteristic
11	C₈×M₆₄	C₂²	1	1	yes	characteristic
12	C₈×#346	C₄	1	1	yes	characteristic
13	#94748	C₄	1	4	yes	complemented
14	#94746	C₄	1	4	yes	complemented
15	#94509	—	2	2	no	—
16	#94506	—	2	2	no	—
17	C₂×#1398	—	2	2	no	—
18	C₂×#1398	—	2	2	no	—
19	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
20	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
21	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
22	C₄×M₆₄	C₂³	1	1	yes	characteristic
23	C₄×#346	C₈	1	1	yes	characteristic
24	C₄×#346	C₂×C₄	1	1	yes	characteristic
25	C₂×#1311	C₈	1	1	yes	characteristic
26	C₂×#1311	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₆	C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈²	C₈	1	1	yes	characteristic
30	#7544	C₂×C₄	1	8	yes	—
31	#7513	—	2	4	no	—
32	#7511	—	2	4	no	—
33	#7330	—	2	1	no	—
34	#7330	—	2	1	no	—
35	#7330	D₄	1	1	yes	characteristic
36	#7330	D₄	1	1	yes	characteristic
37	#7330	Q₈	1	1	yes	characteristic
38	#7330	Q₈	1	1	yes	characteristic
39	#7329	C₈	1	2	yes	—
40	#7329	C₂×C₄	1	2	yes	—
41	C₈×C₃₂	C₈	1	2	yes	—
42	C₈×C₃₂	C₂×C₄	1	2	yes	—
43	C₂×M₆₄	—	2	1	no	—
44	C₂×M₆₄	—	2	1	no	—
45	C₂×M₆₄	M₁₆	1	2	yes	—
46	C₂×M₆₄	#41	1	2	yes	—
47	C₂×#346	—	2	1	no	—
48	C₂×#346	—	2	1	no	—
49	C₂×#346	—	2	1	no	—
50	C₂×#346	—	2	1	no	—
51	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₈	—	2	1	no	—
56	C₄²×C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈²	—	2	1	no	—
58	C₂×C₈²	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈²	C₂×C₈	1	1	yes	characteristic
60	M₁₂₈	—	2	8	no	—
61	#1398	—	2	8	no	—
62	#1398	—	2	8	no	—
63	#1397	—	2	1	no	—
64	#1397	—	2	1	no	—
65	#1397	C₄⋊C₄	1	1	yes	characteristic
66	#1397	C₄⋊C₄	1	1	yes	characteristic
67	#1397	C₂×D₄	1	1	yes	characteristic
68	#1397	C₂×Q₈	1	1	yes	characteristic
69	C₄×C₃₂	C₄²	1	2	yes	—
70	C₄×C₃₂	C₂×C₈	1	2	yes	—
71	C₄×C₃₂	C₂×C₈	1	2	yes	—
72	C₄×C₃₂	C₂²×C₄	1	2	yes	—
73	#1311	—	2	1	no	—
74	#1311	—	2	1	no	—
75	#1311	C₄⋊C₄	1	1	yes	characteristic
76	#1311	C₄⋊C₄	1	1	yes	characteristic
77	#1311	M₁₆	1	1	yes	characteristic
78	#1311	M₁₆	1	1	yes	characteristic
79	C₈×C₁₆	—	2	2	no	—
80	C₈×C₁₆	C₄²	1	1	yes	characteristic
81	C₈×C₁₆	C₄²	1	1	yes	characteristic
82	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
83	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₁₆	—	2	1	no	—
85	C₂²×C₁₆	—	2	1	no	—
86	C₂²×C₁₆	M₃₂	1	1	yes	characteristic
87	C₂²×C₁₆	M₃₂	1	1	yes	characteristic
88	C₂²×C₁₆	#117	1	1	yes	characteristic
89	C₂²×C₁₆	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₈	—	2	1	no	—
91	C₂×C₄×C₈	—	2	1	no	—
92	C₂×C₄×C₈	—	2	1	no	—
93	C₂×C₄×C₈	—	2	1	no	—
94	C₂×C₄×C₈	—	4	2	no	—
95	C₂×C₄×C₈	M₃₂	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₈	1	1	yes	characteristic
97	C₄³	M₃₂	1	1	yes	characteristic
98	M₆₄	—	2	4	no	—
99	M₆₄	—	2	4	no	—
100	M₆₄	—	2	4	no	—
101	M₆₄	—	2	4	no	—
102	C₂×C₃₂	—	2	2	no	—
103	C₂×C₃₂	—	2	2	no	—
104	C₂×C₃₂	C₄⋊C₈	1	2	yes	—
105	C₂×C₃₂	C₄⋊C₈	1	2	yes	—
106	C₂×C₃₂	C₄×D₄	1	2	yes	—
107	C₂×C₃₂	C₄×Q₈	1	2	yes	—
108	#346	—	2	4	no	—
109	#346	—	2	4	no	—
110	#346	—	4	2	no	—
111	#346	—	4	2	no	—
112	C₄×C₁₆	—	2	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
115	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
116	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
117	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₁₆	M₃₂	1	2	yes	—
119	C₄×C₁₆	M₃₂	1	1	yes	characteristic
120	C₄×C₁₆	M₃₂	1	1	yes	characteristic
121	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
122	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
124	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
125	C₈²	—	2	1	no	—
126	C₈²	—	2	1	no	—
127	C₈²	—	4	2	no	—
128	C₈²	C₄×C₈	1	1	yes	characteristic
129	C₈²	C₄×C₈	1	1	yes	characteristic
130	C₈²	C₄⋊C₈	1	1	yes	characteristic
131	C₈²	C₄⋊C₈	1	1	yes	characteristic
132	C₆₄	—	4	8	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	—	4	1	no	—
136	C₂²×C₈	#431	1	1	yes	characteristic
137	C₂²×C₈	C₂×M₃₂	1	1	yes	characteristic
138	C₂×C₄²	—	2	1	no	—
139	C₂×C₄²	—	4	1	no	—
140	C₂×C₄²	C₂×M₃₂	1	1	yes	characteristic
141	C₂×C₁₆	—	2	2	no	—
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	—	2	1	no	—
149	C₂×C₁₆	—	2	1	no	—
150	C₂×C₁₆	—	4	1	no	—
151	C₂×C₁₆	—	4	1	no	—
152	C₂×C₁₆	#346	1	1	yes	characteristic
153	C₂×C₁₆	#346	1	1	yes	characteristic
154	C₂×C₁₆	#346	1	1	yes	characteristic
155	C₂×C₁₆	#346	1	1	yes	characteristic
156	C₂×C₁₆	C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₁₆	C₄×M₁₆	1	1	yes	characteristic
158	C₂×C₁₆	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₂×C₁₆	#431	1	1	yes	characteristic
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	4	1	no	—
168	C₄×C₈	—	4	1	no	—
169	C₄×C₈	—	4	1	no	—
170	C₄×C₈	—	4	1	no	—
171	C₄×C₈	—	8	4	no	—
172	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
173	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
174	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
175	C₄×C₈	C₂×M₃₂	1	1	yes	characteristic
176	C₃₂	—	2	4	no	—
177	C₃₂	—	2	4	no	—
178	C₃₂	—	2	2	no	—
179	C₃₂	—	2	2	no	—
180	C₃₂	#364	1	4	yes	—
181	C₃₂	#443	1	4	yes	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	#2163	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	8	2	no	—
198	C₂×C₈	C₄×C₄⋊C₈	1	1	yes	characteristic
199	C₂×C₈	C₂×#346	1	1	yes	characteristic
200	C₂×C₈	#2150	1	1	yes	characteristic
201	C₂×C₈	#2163	1	1	yes	characteristic
202	C₄²	—	2	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	8	2	no	—
206	C₄²	C₄×M₃₂	1	1	yes	characteristic
207	C₄²	#2150	1	1	yes	characteristic
208	C₁₆	—	2	4	no	—
209	C₁₆	—	2	1	no	—
210	C₁₆	—	2	1	no	—
211	C₁₆	—	2	1	no	—
212	C₁₆	—	2	1	no	—
213	C₁₆	—	4	1	no	—
214	C₁₆	—	4	1	no	—
215	C₁₆	#1398	1	2	yes	—
216	C₁₆	#1398	1	2	yes	—
217	C₁₆	#1772	1	2	yes	—
218	C₁₆	C₂×#364	1	2	yes	—
219	C₂³	#13642	1	1	yes	characteristic
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	#12653	1	1	yes	characteristic
228	C₂×C₄	#13642	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	4	2	no	—
233	C₈	—	4	2	no	—
234	C₈	—	8	1	no	—
235	C₈	—	8	1	no	—
236	C₈	#8266	1	1	yes	characteristic
237	C₈	C₄×#364	1	1	yes	characteristic
238	C₈	C₂×#1398	1	1	yes	characteristic
239	C₈	C₂×#1398	1	1	yes	characteristic
240	C₂²	—	2	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	#151981	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	8	2	no	—
247	C₄	#129076	1	1	yes	characteristic
248	C₄	#151979	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	1	no	—
251	C₂	#72225407	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#8266
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×M₆₄	C₂³
socle	C₂	#72225407
prime radical	C₂³	#13642
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #7330 ◁ #149495 ◁ G
LECS	G ▷ C₄×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
13. #94748	C₄	semidirect
14. #94746	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225405	2
#72225404	2
#72225395	2
C₈×#1398	2