Subgroups — #72231878

Order 2048.

1190 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72231877	C₂	1	3	yes	maximal, complemented
2	#72231876	C₂	1	1	yes	maximal, characteristic
3	#72231875	C₂	1	1	yes	maximal, characteristic
4	#72231870	C₂	1	3	yes	maximal, complemented
5	C₂×#93668	C₂	1	3	yes	maximal, complemented
6	#72231854	C₂	1	1	yes	maximal, characteristic
7	#72225141	C₂	1	3	yes	maximal, complemented
8	#512545	C₄	1	2	yes	—
9	#512545	C₄	1	1	yes	characteristic
10	#512545	C₄	1	1	yes	characteristic
11	#479948	C₄	1	2	yes	—
12	#479899	C₂²	1	1	yes	characteristic
13	#149502	C₂²	1	3	yes	—
14	#149501	C₄	1	6	yes	—
15	C₂×#7332	C₂²	1	3	yes	—
16	#149496	C₄	1	3	yes	—
17	#149496	C₄	1	3	yes	—
18	#149495	C₂²	1	3	yes	—
19	C₂×#7330	C₂²	1	3	yes	—
20	#122214	C₂²	1	3	yes	—
21	#122213	C₄	1	6	yes	—
22	C₂×#7053	C₂²	1	3	yes	—
23	#93668	C₂²	1	3	yes	—
24	#93668	C₂²	1	6	yes	—
25	#93668	C₂²	1	1	yes	characteristic
26	#93668	C₂²	1	3	yes	—
27	#93668	C₂²	1	3	yes	—
28	C₂×#2258	—	2	2	no	—
29	C₂×#2258	C₈	1	2	yes	—
30	C₂×#2258	C₈	1	2	yes	—
31	C₁₆×#41	—	2	2	no	—
32	C₄×#504	C₂×C₄	1	1	yes	characteristic
33	C₄×#504	C₂×C₄	1	1	yes	characteristic
34	C₈×#131	C₂×C₄	1	1	yes	characteristic
35	#13549	—	2	6	no	—
36	#13549	C₈	1	6	yes	—
37	#13549	C₈	1	6	yes	—
38	C₄×M₆₄	C₂×C₄	1	6	yes	—
39	C₄×M₆₄	C₂×C₄	1	3	yes	—
40	C₄×M₆₄	C₂×C₄	1	3	yes	—
41	C₂×#1397	C₂×C₄	1	6	yes	—
42	C₂×#1397	C₂×C₄	1	3	yes	—
43	C₂×#1397	C₂×C₄	1	3	yes	—
44	#11410	C₂×C₄	1	6	yes	—
45	C₁₆×M₁₆	—	2	6	no	—
46	C₂×#1312	C₂×C₄	1	6	yes	—
47	#11399	C₂×C₄	1	3	yes	—
48	#11399	C₂×C₄	1	3	yes	—
49	C₈×M₃₂	C₂×C₄	1	3	yes	—
50	C₂×#1311	C₂³	1	3	yes	—
51	#7332	C₂×C₄	1	6	yes	—
52	#7332	C₂×C₄	1	6	yes	—
53	#7332	C₂³	1	1	yes	characteristic
54	#7332	C₂³	1	3	yes	—
55	#7330	C₂×C₄	1	3	yes	—
56	#7330	C₂×C₄	1	3	yes	—
57	#7330	C₂×C₄	1	6	yes	—
58	#7330	C₂³	1	1	yes	characteristic
59	#7330	C₂³	1	3	yes	—
60	#7053	C₂×C₄	1	6	yes	—
61	#7053	C₂×C₄	1	6	yes	—
62	#7053	C₂³	1	1	yes	characteristic
63	#7053	C₂³	1	3	yes	—
64	C₂×#504	C₂×C₈	1	1	yes	characteristic
65	C₂×#504	C₂×C₈	1	1	yes	characteristic
66	C₂×#504	M₁₆	1	2	yes	—
67	C₈×#41	M₁₆	1	2	yes	—
68	C₄×#131	C₄²	1	1	yes	characteristic
69	#2258	—	2	8	no	—
70	#2258	—	4	8	no	—
71	#2258	C₁₆	1	16	yes	complemented
72	C₂×M₆₄	—	2	6	no	—
73	C₂×M₆₄	C₂×C₈	1	6	yes	—
74	C₂×M₆₄	C₂×C₈	1	6	yes	—
75	C₂²×C₃₂	—	2	6	no	—
76	C₂²×C₃₂	C₂×C₈	1	6	yes	—
77	C₂²×C₃₂	C₂×C₈	1	6	yes	—
78	Q₈×C₁₆	—	4	2	no	—
79	D₄×C₁₆	—	4	6	no	—
80	#2162	—	2	6	no	—
81	#2108	C₂×C₈	1	3	yes	—
82	#2108	C₂×C₈	1	3	yes	—
83	#2108	M₁₆	1	6	yes	—
84	C₄×M₃₂	C₄²	1	3	yes	—
85	C₄×M₃₂	C₄²	1	3	yes	—
86	C₂×C₄×C₁₆	—	2	6	no	—
87	C₂×C₄×C₁₆	C₂²×C₄	1	3	yes	—
88	C₂×C₄×C₁₆	C₂²×C₄	1	3	yes	—
89	#1450	C₄²	1	3	yes	—
90	C₈×M₁₆	M₁₆	1	6	yes	—
91	C₂×C₈²	C₂²×C₄	1	3	yes	—
92	#1397	—	2	6	no	—
93	#1397	C₂×C₈	1	6	yes	—
94	#1397	C₂×C₈	1	6	yes	—
95	#1397	C₂²×C₄	1	2	yes	—
96	#1397	C₂²×C₄	1	6	yes	—
97	#1397	C₂²×C₄	1	1	yes	characteristic
98	#1397	C₂²×C₄	1	1	yes	characteristic
99	#1397	C₂²×C₄	1	3	yes	—
100	#1397	C₂²×C₄	1	3	yes	—
101	C₄×C₃₂	—	2	6	no	—
102	C₄×C₃₂	C₂×C₈	1	6	yes	—
103	C₄×C₃₂	C₂×C₈	1	6	yes	—
104	#1312	—	2	6	no	—
105	#1312	C₂²×C₄	1	2	yes	—
106	#1312	C₂²×C₄	1	6	yes	—
107	#1311	C₂²×C₄	1	3	yes	—
108	#1311	C₂²×C₄	1	3	yes	—
109	#1311	C₂²×C₄	1	3	yes	—
110	#1311	C₂⁴	1	1	yes	characteristic
111	C₈×C₁₆	—	2	6	no	—
112	C₈×C₁₆	C₄²	1	6	yes	—
113	C₂×#131	C₄×C₈	1	1	yes	characteristic
114	C₂×#131	#97	1	1	yes	characteristic
115	C₄×#41	#97	1	1	yes	characteristic
116	#504	—	2	8	no	—
117	#504	C₂×C₁₆	1	8	yes	—
118	#504	M₃₂	1	8	yes	—
119	C₂×M₃₂	C₄×C₈	1	3	yes	—
120	C₂×M₃₂	C₄×C₈	1	3	yes	—
121	C₂×M₃₂	#97	1	6	yes	—
122	C₂²×C₁₆	—	4	6	no	—
123	C₂²×C₁₆	C₂²×C₈	1	3	yes	—
124	C₂²×C₁₆	C₂²×C₈	1	3	yes	—
125	C₂²×C₁₆	C₂×M₁₆	1	6	yes	—
126	C₈×Q₈	—	2	2	no	—
127	C₈×D₄	—	2	6	no	—
128	#431	#97	1	6	yes	—
129	#405	C₄×C₈	1	3	yes	—
130	#405	#97	1	3	yes	—
131	C₄×M₁₆	#97	1	3	yes	—
132	C₂×C₄×C₈	C₂×C₄²	1	3	yes	—
133	C₂×C₄×C₈	C₂×M₁₆	1	6	yes	—
134	M₆₄	—	2	12	no	—
135	M₆₄	—	4	12	no	—
136	M₆₄	C₂×C₁₆	1	24	yes	complemented
137	C₂×C₃₂	—	2	12	no	—
138	C₂×C₃₂	—	2	6	no	—
139	C₂×C₃₂	—	2	2	no	—
140	C₂×C₃₂	—	4	12	no	—
141	C₂×C₃₂	C₂×C₁₆	1	24	yes	complemented
142	C₂×C₃₂	C₂²×C₈	1	6	yes	—
143	C₂×C₃₂	C₂²×C₈	1	2	yes	—
144	C₂×C₃₂	C₂²×C₈	1	6	yes	—
145	C₂×C₃₂	C₂²×C₈	1	2	yes	—
146	C₄⋊C₁₆	—	4	6	no	—
147	C₂²⋊C₁₆	—	4	6	no	—
148	#346	C₄×C₈	1	6	yes	—
149	#346	#97	1	6	yes	—
150	C₄×C₁₆	—	2	6	no	—
151	C₄×C₁₆	—	2	2	no	—
152	C₄×C₁₆	—	4	6	no	—
153	C₄×C₁₆	C₂×C₄²	1	3	yes	—
154	C₄×C₁₆	C₂×C₄²	1	3	yes	—
155	C₄×C₁₆	C₂²×C₈	1	3	yes	—
156	C₄×C₁₆	C₂²×C₈	1	3	yes	—
157	C₄×C₁₆	C₂×M₁₆	1	6	yes	—
158	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
159	C₄×C₁₆	C₂³×C₄	1	1	yes	characteristic
160	#322	#97	1	6	yes	—
161	C₈²	C₂×C₄²	1	3	yes	—
162	C₈²	C₂×M₁₆	1	6	yes	—
163	C₈²	C₂³×C₄	1	1	yes	characteristic
164	C₂×#41	#322	1	1	yes	characteristic
165	#131	C₄×C₁₆	1	4	yes	—
166	#131	#346	1	4	yes	—
167	#131	#347	1	8	yes	—
168	C₂×M₁₆	#322	1	3	yes	—
169	C₂×M₁₆	#322	1	3	yes	—
170	C₂²×C₈	—	2	6	no	—
171	C₂²×C₈	C₂×C₄×C₈	1	3	yes	—
172	C₂²×C₈	C₂×#97	1	3	yes	—
173	C₄×Q₈	#347	1	2	yes	—
174	C₄×D₄	#347	1	6	yes	—
175	#117	#322	1	3	yes	—
176	C₂×C₄²	C₂×#97	1	3	yes	—
177	M₃₂	—	2	12	no	—
178	M₃₂	C₄×C₁₆	1	12	yes	—
179	M₃₂	#346	1	12	yes	—
180	C₂×C₁₆	—	2	12	no	—
181	C₂×C₁₆	—	4	2	no	—
182	C₂×C₁₆	—	4	6	no	—
183	C₂×C₁₆	—	16	6	no	—
184	C₂×C₁₆	C₂×C₄×C₈	1	3	yes	—
185	C₂×C₁₆	C₂×C₄×C₈	1	3	yes	—
186	C₂×C₁₆	C₂×#97	1	6	yes	—
187	C₂×C₁₆	C₂²×C₁₆	1	12	yes	—
188	C₂×C₁₆	C₂×M₃₂	1	12	yes	—
189	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
190	C₂×C₁₆	C₂³×C₈	1	1	yes	characteristic
191	C₂×C₁₆	C₂²×M₁₆	1	2	yes	—
192	C₄⋊C₈	—	2	6	no	—
193	C₂²⋊C₈	—	2	6	no	—
194	#97	#322	1	6	yes	—
195	C₄×C₈	—	2	6	no	—
196	C₄×C₈	C₂×C₄×C₈	1	3	yes	—
197	C₄×C₈	C₂×#97	1	6	yes	—
198	C₄×C₈	C₂×#97	1	3	yes	—
199	C₄×C₈	C₂×#97	1	3	yes	—
200	C₄×C₈	C₂²×C₄²	1	1	yes	characteristic
201	C₄×C₈	C₂²×M₁₆	1	2	yes	—
202	C₃₂	—	2	2	no	—
203	C₃₂	—	2	6	no	—
204	C₃₂	—	4	2	no	—
205	C₃₂	—	4	6	no	—
206	C₃₂	C₂²×C₁₆	1	4	yes	—
207	C₃₂	C₂²×C₁₆	1	12	yes	—
208	#41	#1312	1	4	yes	—
209	#41	#1313	1	4	yes	—
210	C₂×Q₈	#1313	1	1	yes	characteristic
211	C₂×D₄	#1313	1	3	yes	—
212	C₂²×C₄	C₂×#322	1	3	yes	—
213	C₂²×C₄	C₂×#347	1	6	yes	—
214	M₁₆	#1312	1	12	yes	—
215	M₁₆	#1313	1	12	yes	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	6	no	—
218	C₂×C₈	—	8	6	no	—
219	C₂×C₈	C₂×#322	1	3	yes	—
220	C₂×C₈	C₂×#322	1	3	yes	—
221	C₂×C₈	C₂×C₄×C₁₆	1	6	yes	—
222	C₂×C₈	C₂×#346	1	6	yes	—
223	C₂×C₈	C₂×#347	1	12	yes	—
224	C₂×C₈	C₂²×C₄×C₈	1	1	yes	characteristic
225	C₂×C₈	C₂²×#97	1	1	yes	characteristic
226	C₄⋊C₄	#1313	1	6	yes	—
227	C₂²⋊C₄	#1313	1	6	yes	—
228	C₄²	C₂×#322	1	3	yes	—
229	C₄²	C₂×#347	1	6	yes	—
230	C₄²	C₂²×#97	1	1	yes	characteristic
231	C₁₆	—	2	2	no	—
232	C₁₆	—	2	6	no	—
233	C₁₆	—	8	6	no	—
234	C₁₆	—	8	2	no	—
235	C₁₆	C₂×C₄×C₁₆	1	6	yes	—
236	C₁₆	C₂×#346	1	6	yes	—
237	C₁₆	C₂³×C₁₆	1	2	yes	—
238	C₁₆	C₂²×M₃₂	1	2	yes	—
239	C₂³	C₂×#1313	1	3	yes	—
240	Q₈	#7053	1	4	yes	—
241	D₄	#7053	1	12	yes	—
242	C₂×C₄	—	4	6	no	—
243	C₂×C₄	C₂×#1312	1	6	yes	—
244	C₂×C₄	C₂×#1313	1	6	yes	—
245	C₂×C₄	C₂×#1313	1	3	yes	—
246	C₂×C₄	C₂×#1313	1	6	yes	—
247	C₂×C₄	C₂²×#322	1	1	yes	characteristic
248	C₂×C₄	C₂²×#347	1	2	yes	—
249	C₈	—	4	2	no	—
250	C₈	—	4	6	no	—
251	C₈	C₂×#1312	1	6	yes	—
252	C₈	C₂×#1313	1	6	yes	—
253	C₈	C₂²×C₄×C₁₆	1	1	yes	characteristic
254	C₈	C₂²×#346	1	1	yes	characteristic
255	C₈	C₂²×#347	1	2	yes	—
256	C₂²	—	2	6	no	—
257	C₂²	C₂×#7053	1	6	yes	—
258	C₂²	C₂²×#1313	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	6	no	—
261	C₄	C₂×#7053	1	6	yes	—
262	C₄	C₂²×#1312	1	1	yes	characteristic
263	C₄	C₂²×#1313	1	1	yes	characteristic
264	C₂	—	2	6	no	—
265	C₂	#72217199	1	2	yes	—
266	C₂	C₂²×#7053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#322
commutator	C₈	C₂²×C₄×C₁₆
Frattini, cosocle	#1311	C₂⁴
socle	C₂²	C₂²×#1313
prime radical	C₂×#41	#322
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #479899 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72231877	C₂	semidirect
4. #72231870	C₂	semidirect
5. C₂×#93668	C₂	semidirect
7. #72225141	C₂	semidirect
71. #2258	C₁₆	semidirect
136. M₆₄	C₂×C₁₆	semidirect
141. C₂×C₃₂	C₂×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72231877	2
#72231876	2
#72231875	2
#72231870	2
C₂×#93668	2
#72231854	2
#72225141	2