Subgroups — #72225494

Order 2048.

420 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225489	C₂	1	2	yes	maximal
2	#72225486	C₂	1	2	yes	maximal
3	#72225400	C₂	1	1	yes	maximal, characteristic
4	#72225365	C₂	1	1	yes	maximal, characteristic
5	#72216239	C₂	1	1	yes	maximal, characteristic
6	#151068	—	2	2	no	—
7	#151042	C₂²	1	2	yes	—
8	#135796	—	2	1	no	—
9	#135796	C₂²	1	2	yes	—
10	#135781	—	2	1	no	—
11	C₈×#367	C₂²	1	1	yes	characteristic
12	#129140	—	2	1	no	—
13	C₂×#7101	C₂²	1	1	yes	characteristic
14	C₂×C₁₆²	C₂²	1	1	yes	characteristic
15	#94415	—	2	2	no	—
16	#93883	C₄	1	2	yes	complemented
17	#93879	C₄	1	2	yes	complemented
18	#92931	—	2	1	no	—
19	#13640	—	2	2	no	—
20	#12909	—	4	2	no	—
21	C₄×#367	D₄	1	1	yes	characteristic
22	#12662	—	4	1	no	—
23	#12023	—	2	2	no	—
24	#12023	—	2	2	no	—
25	#12023	—	4	1	no	—
26	#12023	—	4	1	no	—
27	C₂×#1372	D₄	1	1	yes	characteristic
28	C₂×#1370	—	2	1	no	—
29	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
32	#8823	—	2	2	no	—
33	C₈×#107	—	2	1	no	—
34	#7463	—	4	4	no	—
35	#7458	C₂×C₄	1	4	yes	—
36	#7455	—	2	2	no	—
37	#7171	—	2	1	no	—
38	#7167	—	4	1	no	—
39	#7101	—	2	1	no	—
40	#7101	Q₈	1	2	yes	—
41	C₁₆²	C₂×C₄	1	1	yes	characteristic
42	C₁₆²	C₂×C₄	1	1	yes	characteristic
43	C₁₆²	Q₈	1	1	yes	characteristic
44	C₁₆²	Q₈	1	1	yes	characteristic
45	C₂×#368	—	8	4	no	—
46	C₂×#367	—	2	1	no	—
47	C₂×C₄⋊C₁₆	—	4	1	no	—
48	C₂×#346	—	8	1	no	—
49	C₂×C₄×C₁₆	—	2	1	no	—
50	C₂×C₄×C₁₆	D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₆	D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₆	#41	1	1	yes	characteristic
54	#1848	—	4	2	no	—
55	C₄×#107	—	2	1	no	—
56	C₈×C₄⋊C₄	—	4	1	no	—
57	#1567	—	4	2	no	—
58	#1567	—	4	2	no	—
59	C₂×#335	—	2	1	no	—
60	C₂×C₈²	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₃₂	—	2	4	no	—
62	C₂²⋊C₃₂	—	4	2	no	—
63	#1397	—	4	2	no	—
64	#1380	Q₁₆	1	2	yes	—
65	#1374	—	8	1	no	—
66	#1372	—	2	1	no	—
67	#1372	—	4	2	no	—
68	#1372	—	4	2	no	—
69	#1372	—	8	2	no	—
70	#1372	—	8	1	no	—
71	#1372	—	8	1	no	—
72	#1372	—	8	2	no	—
73	#1372	D₈	1	2	yes	—
74	#1370	—	2	2	no	—
75	#1370	—	4	1	no	—
76	C₈×C₁₆	—	2	1	no	—
77	C₈×C₁₆	C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₁₆	C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₁₆	C₄⋊C₄	1	1	yes	characteristic
80	C₈×C₁₆	C₄⋊C₄	1	1	yes	characteristic
81	C₈×C₁₆	Q₁₆	1	1	yes	characteristic
82	C₈×C₁₆	Q₁₆	1	1	yes	characteristic
83	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
84	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
85	C₈×C₁₆	C₂×Q₈	1	1	yes	characteristic
86	C₈×C₁₆	#41	1	1	yes	characteristic
87	#1279	—	4	1	no	—
88	C₂²×C₁₆	—	2	2	no	—
89	C₂²×C₁₆	—	2	1	no	—
90	C₂²×C₁₆	—	2	1	no	—
91	C₂²×C₁₆	—	8	1	no	—
92	C₂²×C₁₆	SD₃₂	1	1	yes	characteristic
93	C₂²×C₁₆	SD₃₂	1	1	yes	characteristic
94	C₂²×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
95	C₂²×C₁₆	#135	1	1	yes	characteristic
96	C₂×#108	—	8	4	no	—
97	C₂×#107	—	2	2	no	—
98	#423	—	8	2	no	—
99	C₂×C₄⋊C₈	—	4	1	no	—
100	C₂×C₄×C₈	—	8	1	no	—
101	C₂×C₄×C₈	#121	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	4	1	no	—
105	C₂×C₃₂	—	4	4	no	—
106	#368	—	16	8	no	—
107	#367	—	4	2	no	—
108	C₄⋊C₁₆	—	4	2	no	—
109	C₄⋊C₁₆	—	8	1	no	—
110	#346	—	16	2	no	—
111	C₄×C₁₆	—	2	2	no	—
112	C₄×C₁₆	—	2	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₄×C₁₆	—	2	1	no	—
115	C₄×C₁₆	—	2	1	no	—
116	C₄×C₁₆	#107	1	1	yes	characteristic
117	C₄×C₁₆	#107	1	1	yes	characteristic
118	C₄×C₁₆	#107	1	1	yes	characteristic
119	C₄×C₁₆	#107	1	1	yes	characteristic
120	C₄×C₁₆	SD₃₂	1	1	yes	characteristic
121	C₄×C₁₆	SD₃₂	1	1	yes	characteristic
122	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
123	C₄×C₁₆	C₄×Q₈	1	1	yes	characteristic
124	C₄×C₁₆	#125	1	1	yes	characteristic
125	C₄×C₁₆	C₄⋊Q₈	1	1	yes	characteristic
126	C₄×C₁₆	C₂×D₈	1	1	yes	characteristic
127	C₄×C₁₆	#135	1	1	yes	characteristic
128	#340	—	4	1	no	—
129	#336	—	8	1	no	—
130	#335	—	2	2	no	—
131	#335	—	4	1	no	—
132	#335	—	8	2	no	—
133	#335	—	8	2	no	—
134	#335	—	8	2	no	—
135	#335	—	8	2	no	—
136	C₈²	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₈²	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈²	C₂×Q₁₆	1	1	yes	characteristic
139	C₈²	#137	1	1	yes	characteristic
140	C₂×M₁₆	—	16	2	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	8	1	no	—
143	C₂²×C₈	#467	1	1	yes	characteristic
144	C₂²×C₈	#473	1	1	yes	characteristic
145	C₂²×C₈	C₂×SD₃₂	1	1	yes	characteristic
146	C₂²×C₈	#510	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	4	2	no	—
148	C₂×C₄²	—	8	1	no	—
149	C₂×C₄²	#459	1	1	yes	characteristic
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	8	no	—
153	C₂×C₁₆	—	2	1	no	—
154	C₂×C₁₆	—	2	1	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	4	1	no	—
160	C₂×C₁₆	—	4	1	no	—
161	C₂×C₁₆	—	4	2	no	—
162	C₂×C₁₆	—	4	2	no	—
163	C₂×C₁₆	—	8	2	no	—
164	C₂×C₁₆	—	16	1	no	—
165	C₂×C₁₆	—	16	1	no	—
166	C₂×C₁₆	#367	1	1	yes	characteristic
167	C₂×C₁₆	#367	1	1	yes	characteristic
168	C₂×C₁₆	#367	1	1	yes	characteristic
169	C₂×C₁₆	#367	1	1	yes	characteristic
170	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
171	C₂×C₁₆	C₄×Q₁₆	1	1	yes	characteristic
172	C₂×C₁₆	#499	1	1	yes	characteristic
173	C₂×C₁₆	#500	1	1	yes	characteristic
174	#108	—	16	8	no	—
175	#107	—	4	2	no	—
176	#107	—	4	2	no	—
177	C₄⋊C₈	—	4	2	no	—
178	C₄⋊C₈	—	8	1	no	—
179	C₄⋊C₈	—	16	2	no	—
180	C₄⋊C₈	—	16	2	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	16	1	no	—
183	C₄×C₈	—	16	1	no	—
184	C₄×C₈	#389	1	1	yes	characteristic
185	C₄×C₈	C₂×#107	1	1	yes	characteristic
186	C₄×C₈	#428	1	1	yes	characteristic
187	C₄×C₈	#462	1	1	yes	characteristic
188	C₄×C₈	#466	1	1	yes	characteristic
189	C₄×C₈	#472	1	1	yes	characteristic
190	C₄×C₈	#474	1	1	yes	characteristic
191	C₄×C₈	#477	1	1	yes	characteristic
192	C₄×C₈	C₂×SD₃₂	1	1	yes	characteristic
193	C₄×C₈	#509	1	1	yes	characteristic
194	#95	—	8	1	no	—
195	C₃₂	—	8	4	no	—
196	C₂²×C₄	—	8	2	no	—
197	C₂²×C₄	#1669	1	1	yes	characteristic
198	C₂²×C₄	#2208	1	1	yes	characteristic
199	C₂²×C₄	#2213	1	1	yes	characteristic
200	M₁₆	—	32	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	8	2	no	—
211	C₂×C₈	—	16	1	no	—
212	C₂×C₈	—	16	1	no	—
213	C₂×C₈	—	32	2	no	—
214	C₂×C₈	#1674	1	1	yes	characteristic
215	C₂×C₈	#1676	1	1	yes	characteristic
216	C₂×C₈	#1942	1	1	yes	characteristic
217	C₂×C₈	#1952	1	1	yes	characteristic
218	C₂×C₈	C₂×#367	1	1	yes	characteristic
219	C₂×C₈	#2159	1	1	yes	characteristic
220	C₂×C₈	#2225	1	1	yes	characteristic
221	C₂×C₈	#2226	1	1	yes	characteristic
222	C₄⋊C₄	—	8	2	no	—
223	C₄⋊C₄	—	8	2	no	—
224	C₄²	—	16	1	no	—
225	C₄²	—	16	1	no	—
226	C₄²	#1667	1	1	yes	characteristic
227	C₄²	#1947	1	1	yes	characteristic
228	C₄²	#2207	1	1	yes	characteristic
229	C₄²	#2214	1	1	yes	characteristic
230	C₁₆	—	2	4	no	—
231	C₁₆	—	2	4	no	—
232	C₁₆	—	2	2	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	1	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	2	1	no	—
237	C₁₆	—	2	1	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	—	2	1	no	—
240	C₁₆	—	4	1	no	—
241	C₁₆	—	4	1	no	—
242	C₁₆	—	16	2	no	—
243	C₁₆	#1426	1	2	yes	—
244	C₁₆	#2179	1	2	yes	—
245	C₂³	#12447	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	16	2	no	—
252	C₂×C₄	—	16	2	no	—
253	C₂×C₄	—	16	2	no	—
254	C₂×C₄	#9614	1	1	yes	characteristic
255	C₂×C₄	#12425	1	1	yes	characteristic
256	C₂×C₄	#12426	1	1	yes	characteristic
257	C₂×C₄	#12446	1	1	yes	characteristic
258	C₂×C₄	#13177	1	1	yes	characteristic
259	C₂×C₄	#13181	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	16	2	no	—
269	C₈	—	64	2	no	—
270	C₈	#13170	1	1	yes	characteristic
271	C₈	#13173	1	1	yes	characteristic
272	C₈	C₂×#1426	1	1	yes	characteristic
273	C₈	#13638	1	1	yes	characteristic
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	#140433	1	1	yes	characteristic
277	C₂²	#151659	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	32	2	no	—
283	C₄	#140431	1	1	yes	characteristic
284	C₄	#140438	1	1	yes	characteristic
285	C₄	#152871	1	1	yes	characteristic
286	C₄	#152874	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	#72216273	1	1	yes	characteristic
289	C₂	#72225492	1	1	yes	characteristic
290	C₂	#72225493	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#140433
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂³	#12447
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #93883	C₄	semidirect
17. #93879	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225489	2
#72225486	2
#72225400	2
#72225365	2
#72216239	2