Subgroups — #72227241

Order 2048.

398 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225126	C₂	1	1	yes	maximal, characteristic
2	#72216247	C₂	1	1	yes	maximal, characteristic
3	#72216243	C₂	1	1	yes	maximal, characteristic
4	#151981	—	2	1	no	—
5	#151981	—	2	1	no	—
6	C₂×#7459	C₄	1	2	yes	—
7	C₂×#7456	—	2	1	no	—
8	#135718	—	2	1	no	—
9	#135718	—	2	1	no	—
10	#135713	C₄	1	2	yes	—
11	C₈×#346	C₂²	1	1	yes	characteristic
12	#94019	—	2	1	no	—
13	#94018	C₄	1	2	yes	—
14	C₂×#1422	—	2	2	no	—
15	C₂×#1422	—	2	2	no	—
16	C₂×#1398	—	4	1	no	—
17	C₂×#1398	—	4	1	no	—
18	#12900	—	2	2	no	—
19	C₄×#346	D₄	1	1	yes	characteristic
20	C₄×#346	D₄	1	1	yes	characteristic
21	C₂×#1369	—	4	1	no	—
22	C₂×#1369	—	4	1	no	—
23	C₂×#1369	—	4	1	no	—
24	C₂×#1369	—	4	1	no	—
25	#11996	—	4	2	no	—
26	C₂×#1311	D₄	1	1	yes	characteristic
27	C₂×#1311	Q₈	1	1	yes	characteristic
28	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈²	C₂×C₄	1	1	yes	characteristic
31	#7468	—	4	1	no	—
32	#7468	—	4	1	no	—
33	#7459	—	2	2	no	—
34	#7459	C₈	1	4	yes	—
35	#7456	—	2	4	no	—
36	C₂×M₆₄	—	4	2	no	—
37	C₂×M₆₄	—	4	2	no	—
38	C₂×#364	—	4	2	no	—
39	C₂×#364	—	4	2	no	—
40	#2150	—	4	2	no	—
41	C₂×#346	D₈	1	1	yes	characteristic
42	C₂×#346	D₈	1	1	yes	characteristic
43	C₂×#346	SD₁₆	1	1	yes	characteristic
44	C₂×#346	SD₁₆	1	1	yes	characteristic
45	C₂×#346	SD₁₆	1	1	yes	characteristic
46	C₂×#346	SD₁₆	1	1	yes	characteristic
47	C₂×#346	Q₁₆	1	1	yes	characteristic
48	C₂×#346	Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈	—	2	1	no	—
54	C₄²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈²	—	2	1	no	—
56	C₂×C₈²	C₄²	1	1	yes	characteristic
57	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
58	#1422	—	2	4	no	—
59	#1422	—	2	4	no	—
60	#1422	—	4	2	no	—
61	#1422	—	4	2	no	—
62	#1398	—	8	2	no	—
63	#1398	—	8	2	no	—
64	#1369	—	8	2	no	—
65	#1369	—	8	2	no	—
66	#1369	—	8	2	no	—
67	#1369	—	8	2	no	—
68	#1367	—	8	4	no	—
69	#1311	—	2	2	no	—
70	#1311	—	2	2	no	—
71	C₈×C₁₆	—	4	1	no	—
72	C₈×C₁₆	C₂×C₈	1	2	yes	—
73	C₈×C₁₆	M₁₆	1	2	yes	—
74	C₂×M₃₂	—	8	2	no	—
75	C₂²×C₁₆	#102	1	1	yes	characteristic
76	C₂²×C₁₆	#102	1	1	yes	characteristic
77	C₂²×C₁₆	#103	1	1	yes	characteristic
78	C₂²×C₁₆	#103	1	1	yes	characteristic
79	C₂²×C₁₆	#106	1	1	yes	characteristic
80	C₂²×C₁₆	#106	1	1	yes	characteristic
81	C₂²×C₁₆	#107	1	1	yes	characteristic
82	C₂²×C₁₆	#107	1	1	yes	characteristic
83	C₂×C₄×C₈	—	2	1	no	—
84	C₂×C₄×C₈	—	2	1	no	—
85	C₂×C₄×C₈	—	4	2	no	—
86	C₂×C₄×C₈	#95	1	1	yes	characteristic
87	C₂×C₄×C₈	#100	1	1	yes	characteristic
88	C₂×C₄×C₈	#102	1	1	yes	characteristic
89	C₂×C₄×C₈	#102	1	1	yes	characteristic
90	C₂×C₄×C₈	#103	1	1	yes	characteristic
91	C₂×C₄×C₈	#103	1	1	yes	characteristic
92	C₄³	#101	1	1	yes	characteristic
93	M₆₄	—	4	4	no	—
94	M₆₄	—	4	4	no	—
95	M₆₄	—	8	2	no	—
96	M₆₄	—	8	2	no	—
97	C₂×C₃₂	—	8	2	no	—
98	C₂×C₃₂	—	8	2	no	—
99	#364	—	8	2	no	—
100	#364	—	8	2	no	—
101	#364	—	8	2	no	—
102	#364	—	8	2	no	—
103	C₄⋊C₁₆	—	8	4	no	—
104	#346	—	2	2	no	—
105	#346	—	2	2	no	—
106	#346	—	2	1	no	—
107	#346	—	2	1	no	—
108	#346	—	2	2	no	—
109	#346	—	2	2	no	—
110	#346	—	2	2	no	—
111	#346	—	4	1	no	—
112	#346	—	4	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₄×C₁₆	—	2	1	no	—
115	C₄×C₁₆	—	2	1	no	—
116	C₄×C₁₆	—	2	1	no	—
117	C₄×C₁₆	C₂²⋊C₈	1	2	yes	—
118	C₄×C₁₆	#104	1	2	yes	—
119	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
120	C₄×C₁₆	#108	1	2	yes	—
121	C₈²	—	2	4	no	—
122	C₈²	—	2	1	no	—
123	C₈²	—	2	1	no	—
124	C₈²	—	2	1	no	—
125	C₈²	C₄×C₈	1	1	yes	characteristic
126	C₈²	#97	1	1	yes	characteristic
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	4	1	no	—
129	C₂²×C₈	#331	1	1	yes	characteristic
130	C₂²×C₈	#333	1	1	yes	characteristic
131	C₂²×C₈	#340	1	1	yes	characteristic
132	C₂²×C₈	#340	1	1	yes	characteristic
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	#332	1	1	yes	characteristic
135	C₂×C₄²	#332	1	1	yes	characteristic
136	C₂×C₄²	#343	1	1	yes	characteristic
137	M₃₂	—	16	2	no	—
138	M₃₂	—	16	2	no	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	—	2	1	no	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	2	1	no	—
147	C₂×C₁₆	—	2	1	no	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	—	2	1	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	—	16	2	no	—
154	C₂×C₁₆	D₄⋊C₈	1	2	yes	—
155	C₂×C₁₆	Q₈⋊C₈	1	2	yes	—
156	C₂×C₁₆	#334	1	2	yes	—
157	C₂×C₁₆	#335	1	2	yes	—
158	C₂×C₁₆	#360	1	2	yes	—
159	C₂×C₁₆	#360	1	2	yes	—
160	C₂×C₁₆	#361	1	1	yes	characteristic
161	C₂×C₁₆	#362	1	1	yes	characteristic
162	C₂×C₁₆	#365	1	2	yes	—
163	C₂×C₁₆	#365	1	2	yes	—
164	C₂×C₁₆	#365	1	2	yes	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₄×C₈	—	8	4	no	—
180	C₄×C₈	#336	1	1	yes	characteristic
181	C₄×C₈	#339	1	1	yes	characteristic
182	C₄×C₈	#341	1	1	yes	characteristic
183	C₄×C₈	#344	1	1	yes	characteristic
184	C₃₂	—	8	4	no	—
185	C₃₂	—	8	4	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	#1296	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	8	2	no	—
200	C₂×C₈	#1277	1	1	yes	characteristic
201	C₂×C₈	#1279	1	1	yes	characteristic
202	C₂×C₈	#1301	1	1	yes	characteristic
203	C₂×C₈	#1303	1	1	yes	characteristic
204	C₂×C₈	#1359	1	2	yes	—
205	C₂×C₈	#1383	1	1	yes	characteristic
206	C₂×C₈	#1383	1	1	yes	characteristic
207	C₂×C₈	#1383	1	1	yes	characteristic
208	C₂×C₈	#1385	1	1	yes	characteristic
209	C₂×C₈	#1386	1	1	yes	characteristic
210	C₂×C₈	#1386	1	1	yes	characteristic
211	C₂×C₈	#1387	1	1	yes	characteristic
212	C₂×C₈	#1388	1	1	yes	characteristic
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	8	2	no	—
219	C₄²	#1283	1	1	yes	characteristic
220	C₄²	#1299	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	—	2	2	no	—
224	C₁₆	—	2	2	no	—
225	C₁₆	—	4	1	no	—
226	C₁₆	—	4	1	no	—
227	C₁₆	—	32	2	no	—
228	C₁₆	#1333	1	2	yes	—
229	C₁₆	#1334	1	2	yes	—
230	C₁₆	#1368	1	4	yes	—
231	C₂³	#7270	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	#7031	1	1	yes	characteristic
240	C₂×C₄	#7268	1	1	yes	characteristic
241	C₂×C₄	#7278	1	1	yes	characteristic
242	C₂×C₄	#7278	1	1	yes	characteristic
243	C₂×C₄	#7280	1	1	yes	characteristic
244	C₂×C₄	#7281	1	1	yes	characteristic
245	C₈	—	2	4	no	—
246	C₈	—	2	2	no	—
247	C₈	—	4	2	no	—
248	C₈	—	4	1	no	—
249	C₈	—	8	2	no	—
250	C₈	#7156	1	1	yes	characteristic
251	C₈	#7158	1	1	yes	characteristic
252	C₈	#7173	1	1	yes	characteristic
253	C₈	#7178	1	1	yes	characteristic
254	C₈	#7292	1	1	yes	characteristic
255	C₈	#7295	1	1	yes	characteristic
256	C₈	#7300	1	1	yes	characteristic
257	C₈	#7302	1	1	yes	characteristic
258	C₂²	—	4	1	no	—
259	C₂²	#93079	1	1	yes	characteristic
260	C₂²	#93081	1	1	yes	characteristic
261	C₂²	#93444	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	8	1	no	—
265	C₄	—	8	1	no	—
266	C₄	#93073	1	1	yes	characteristic
267	C₄	#93076	1	1	yes	characteristic
268	C₄	#93452	1	1	yes	characteristic
269	C₄	#93458	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	#72216557	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#93073
commutator	C₈²	C₄×C₈
Frattini, cosocle	C₈×#346	C₂²
socle	C₂	#72216557
prime radical	C₂³	#7270
LCS	G ▷ C₈² ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#72225126	2
#72216247	2
#72216243	2