Subgroups — #72225529

Order 2048.

522 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225527	C₂	1	2	yes	maximal
2	#72225526	C₂	1	2	yes	maximal
3	#72225138	C₂	1	1	yes	maximal, characteristic
4	#72216243	C₂	1	1	yes	maximal, characteristic
5	#72216215	C₂	1	1	yes	maximal, characteristic
6	#151981	C₂²	1	2	yes	—
7	C₄×#1398	C₂²	1	1	yes	characteristic
8	C₄×#1398	C₂²	1	1	yes	characteristic
9	C₂×#7459	C₄	1	2	yes	—
10	#149504	—	2	2	no	—
11	#149502	C₄	1	2	yes	—
12	#149499	C₄	1	2	yes	—
13	#129090	C₄	1	1	yes	characteristic
14	#129088	C₄	1	1	yes	characteristic
15	#129051	C₂²	1	2	yes	—
16	C₈×#346	C₂²	1	1	yes	characteristic
17	#94019	C₄	1	2	yes	—
18	#93996	—	2	2	no	—
19	#93264	C₄	1	2	yes	—
20	C₂×#1422	C₂×C₄	1	4	yes	—
21	C₂×#1398	—	2	2	no	—
22	C₂×#1398	—	2	1	no	—
23	C₂×#1398	—	2	1	no	—
24	C₂×#1398	D₄	1	2	yes	—
25	C₂×#1398	Q₈	1	2	yes	—
26	C₄×M₆₄	C₂×C₄	1	2	yes	—
27	C₄×M₆₄	C₂×C₄	1	2	yes	—
28	C₄×#364	C₂×C₄	1	2	yes	—
29	C₄×#346	C₂×C₄	1	1	yes	characteristic
30	C₄×#346	C₂³	1	1	yes	characteristic
31	C₂×#1369	—	2	1	no	—
32	#11999	—	2	1	no	—
33	#11405	—	2	2	no	—
34	C₂×#1311	C₂×C₄	1	1	yes	characteristic
35	C₂×#1311	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
38	#8266	C₂×C₄	1	2	yes	—
39	C₄×C₈²	C₂×C₄	1	1	yes	characteristic
40	#7468	—	2	2	no	—
41	#7459	C₈	1	4	yes	complemented
42	#7459	C₈	1	4	yes	complemented
43	#7333	—	4	4	no	—
44	#7332	—	2	4	no	—
45	#7331	—	2	4	no	—
46	#7161	—	2	1	no	—
47	#7157	—	2	1	no	—
48	#7140	—	2	2	no	—
49	C₂×M₆₄	—	2	2	no	—
50	C₂×M₆₄	—	2	4	no	—
51	C₂×M₆₄	C₄⋊C₄	1	4	yes	—
52	C₂×#364	—	2	2	no	—
53	C₄×M₃₂	—	2	2	no	—
54	C₂×#346	M₁₆	1	1	yes	characteristic
55	C₂×#346	M₁₆	1	1	yes	characteristic
56	C₂×#346	M₁₆	1	1	yes	characteristic
57	C₂×#346	M₁₆	1	1	yes	characteristic
58	C₂×#346	C₂×D₄	1	1	yes	characteristic
59	C₂×#346	C₂×Q₈	1	1	yes	characteristic
60	C₂×#346	#41	1	1	yes	characteristic
61	C₂×#346	#41	1	1	yes	characteristic
62	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
63	C₂×C₄×C₁₆	C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈	—	2	1	no	—
67	C₄²×C₈	C₂²×C₄	1	1	yes	characteristic
68	#1564	—	2	2	no	—
69	#1450	—	2	2	no	—
70	C₂×C₈²	—	2	1	no	—
71	C₂×C₈²	C₄²	1	1	yes	characteristic
72	C₂×C₈²	C₄²	1	1	yes	characteristic
73	#1422	—	2	4	no	—
74	#1422	C₂×C₈	1	8	yes	—
75	#1398	—	2	2	no	—
76	#1398	—	2	2	no	—
77	#1398	—	4	2	no	—
78	#1398	—	4	2	no	—
79	#1398	—	4	2	no	—
80	#1398	—	4	2	no	—
81	#1398	—	4	2	no	—
82	#1397	—	2	2	no	—
83	#1397	—	2	2	no	—
84	C₄×C₃₂	—	2	2	no	—
85	C₄×C₃₂	M₁₆	1	4	yes	—
86	#1376	—	2	2	no	—
87	#1369	—	4	2	no	—
88	#1368	—	4	2	no	—
89	#1313	—	4	2	no	—
90	#1313	—	4	2	no	—
91	#1311	—	2	2	no	—
92	#1311	M₁₆	1	4	yes	—
93	C₈×C₁₆	—	2	2	no	—
94	C₈×C₁₆	C₂×C₈	1	2	yes	—
95	C₈×C₁₆	C₂×C₈	1	2	yes	—
96	#1276	—	2	2	no	—
97	C₂×M₃₂	—	4	2	no	—
98	C₂²×C₁₆	#97	1	1	yes	characteristic
99	C₂²×C₁₆	#97	1	1	yes	characteristic
100	C₂²×C₁₆	#97	1	1	yes	characteristic
101	C₂²×C₁₆	#97	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₁₆	#117	1	1	yes	characteristic
104	C₂²×C₁₆	C₄×D₄	1	1	yes	characteristic
105	C₂²×C₁₆	C₄×Q₈	1	1	yes	characteristic
106	#431	—	4	2	no	—
107	C₂×C₄×C₈	—	2	1	no	—
108	C₂×C₄×C₈	—	2	1	no	—
109	C₂×C₄×C₈	—	4	2	no	—
110	C₂×C₄×C₈	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₈	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₈	#131	1	1	yes	characteristic
116	C₄³	#131	1	1	yes	characteristic
117	M₆₄	—	2	4	no	—
118	M₆₄	—	2	4	no	—
119	M₆₄	—	2	8	no	—
120	M₆₄	—	4	2	no	—
121	M₆₄	—	4	2	no	—
122	M₆₄	—	4	4	no	—
123	C₂×C₃₂	—	2	4	no	—
124	C₂×C₃₂	—	2	4	no	—
125	C₂×C₃₂	—	4	4	no	—
126	#364	—	4	2	no	—
127	#364	—	4	2	no	—
128	#346	—	2	2	no	—
129	#346	—	2	2	no	—
130	#346	—	2	2	no	—
131	#346	—	2	2	no	—
132	#346	—	2	2	no	—
133	#346	—	4	1	no	—
134	#346	—	4	1	no	—
135	#346	—	4	2	no	—
136	#346	AGL(1,ℤ/8)	1	2	yes	—
137	#346	#137	1	2	yes	—
138	C₄×C₁₆	—	2	1	no	—
139	C₄×C₁₆	—	2	1	no	—
140	C₄×C₁₆	—	4	2	no	—
141	C₄×C₁₆	C₄×C₈	1	2	yes	—
142	C₄×C₁₆	#97	1	2	yes	—
143	C₄×C₁₆	C₂²×C₈	1	2	yes	—
144	C₄×C₁₆	C₂×M₁₆	1	2	yes	—
145	C₄×C₁₆	C₂×M₁₆	1	2	yes	—
146	C₄×C₁₆	#131	1	2	yes	—
147	#335	—	4	2	no	—
148	#334	—	4	2	no	—
149	#322	—	4	2	no	—
150	C₈²	—	2	1	no	—
151	C₈²	—	2	1	no	—
152	C₈²	—	4	2	no	—
153	C₈²	—	4	2	no	—
154	C₈²	C₄×C₈	1	1	yes	characteristic
155	C₈²	C₄×C₈	1	1	yes	characteristic
156	C₈²	#97	1	2	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	4	1	no	—
159	C₂²×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₈	C₂×#97	1	1	yes	characteristic
161	C₂²×C₈	#436	1	1	yes	characteristic
162	C₂²×C₈	#446	1	1	yes	characteristic
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₄²	#405	1	1	yes	characteristic
165	C₂×C₄²	#436	1	1	yes	characteristic
166	C₂×C₄²	#446	1	1	yes	characteristic
167	M₃₂	—	8	2	no	—
168	M₃₂	—	8	2	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	2	1	no	—
171	C₂×C₁₆	—	2	1	no	—
172	C₂×C₁₆	—	2	1	no	—
173	C₂×C₁₆	—	2	1	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	8	2	no	—
182	C₂×C₁₆	#322	1	2	yes	—
183	C₂×C₁₆	#322	1	2	yes	—
184	C₂×C₁₆	#347	1	2	yes	—
185	C₂×C₁₆	#347	1	2	yes	—
186	C₂×C₁₆	#423	1	2	yes	—
187	C₂×C₁₆	#428	1	2	yes	—
188	C₂×C₁₆	#430	1	2	yes	—
189	C₂×C₁₆	#432	1	2	yes	—
190	C₂×C₁₆	C₈×D₄	1	2	yes	—
191	C₂×C₁₆	#441	1	1	yes	characteristic
192	C₂×C₁₆	#442	1	1	yes	characteristic
193	C₂×C₁₆	#444	1	2	yes	—
194	C₂×C₁₆	C₈×Q₈	1	2	yes	—
195	C₄⋊C₈	—	8	2	no	—
196	C₂²⋊C₈	—	8	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	—	4	1	no	—
208	C₄×C₈	—	4	1	no	—
209	C₄×C₈	—	8	4	no	—
210	C₄×C₈	—	8	2	no	—
211	C₄×C₈	C₂×C₄×C₈	1	1	yes	characteristic
212	C₄×C₈	C₂×#97	1	1	yes	characteristic
213	C₄×C₈	C₄×M₁₆	1	1	yes	characteristic
214	C₄×C₈	#405	1	1	yes	characteristic
215	C₄×C₈	#440	1	2	yes	—
216	C₄×C₈	#444	1	2	yes	—
217	C₃₂	—	4	4	no	—
218	C₃₂	—	4	4	no	—
219	C₃₂	—	4	8	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	#1770	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	8	2	no	—
234	C₂×C₈	—	16	2	no	—
235	C₂×C₈	C₂×#322	1	1	yes	characteristic
236	C₂×C₈	#1451	1	1	yes	characteristic
237	C₂×C₈	#1592	1	2	yes	—
238	C₂×C₈	#1767	1	1	yes	characteristic
239	C₂×C₈	C₈×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₈	#1773	1	1	yes	characteristic
241	C₂×C₈	#1773	1	1	yes	characteristic
242	C₂×C₈	#1776	1	1	yes	characteristic
243	C₂×C₈	#1778	1	1	yes	characteristic
244	C₂×C₈	C₂×#347	1	1	yes	characteristic
245	C₂×C₈	C₂×#347	1	1	yes	characteristic
246	C₂×C₈	#2110	1	1	yes	characteristic
247	C₂×C₈	#2110	1	1	yes	characteristic
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	8	2	no	—
253	C₄²	C₈×M₁₆	1	1	yes	characteristic
254	C₄²	#1452	1	1	yes	characteristic
255	C₄²	#1592	1	2	yes	—
256	C₁₆	—	2	2	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	4	1	no	—
260	C₁₆	—	4	1	no	—
261	C₁₆	—	16	2	no	—
262	C₁₆	#1313	1	4	yes	—
263	C₁₆	#1569	1	4	yes	—
264	C₁₆	#1579	1	2	yes	—
265	C₁₆	#1580	1	2	yes	—
266	C₂³	#12667	1	1	yes	characteristic
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	#8261	1	1	yes	characteristic
275	C₂×C₄	#8286	1	1	yes	characteristic
276	C₂×C₄	#8295	1	1	yes	characteristic
277	C₂×C₄	#12666	1	1	yes	characteristic
278	C₂×C₄	#12666	1	1	yes	characteristic
279	C₂×C₄	#12667	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	—	2	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	8	2	no	—
285	C₈	—	32	2	no	—
286	C₈	#8269	1	1	yes	characteristic
287	C₈	#8276	1	1	yes	characteristic
288	C₈	#8284	1	1	yes	characteristic
289	C₈	#8294	1	1	yes	characteristic
290	C₈	C₂×#1313	1	1	yes	characteristic
291	C₈	#11406	1	1	yes	characteristic
292	C₈	#11414	1	1	yes	characteristic
293	C₈	#11414	1	1	yes	characteristic
294	C₂²	—	4	1	no	—
295	C₂²	#129063	1	1	yes	characteristic
296	C₂²	#129172	1	1	yes	characteristic
297	C₂²	#129199	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	1	no	—
302	C₄	#96894	1	1	yes	characteristic
303	C₄	#129062	1	1	yes	characteristic
304	C₄	#129172	1	1	yes	characteristic
305	C₄	#129199	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	#72225528	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#8269
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72225528
prime radical	C₂³	#12667
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#346 ◁ C₄×#1398 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
41. #7459	C₈	semidirect
42. #7459	C₈	semidirect

Maximal subgroups

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

subgroup	index
#72225527	2
#72225526	2
#72225138	2
#72216243	2
#72216215	2