Subgroups — #72225376

Order 2048.

475 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225373	C₂	1	2	yes	maximal
2	#72225362	C₂	1	1	yes	maximal, characteristic
3	#72225352	C₂	1	1	yes	maximal, characteristic
4	#72216243	C₂	1	2	yes	maximal
5	C₈×#1398	C₂	1	1	yes	maximal, characteristic
6	#151981	—	2	2	no	—
7	C₄×#1398	C₄	1	1	yes	characteristic
8	C₄×#1398	C₄	1	1	yes	characteristic
9	C₂×#7459	C₂²	1	2	yes	—
10	C₂×#7459	C₂²	1	2	yes	—
11	#149495	C₄	1	1	yes	characteristic
12	#149495	C₄	1	1	yes	characteristic
13	C₈×M₆₄	C₂²	1	1	yes	characteristic
14	C₈×M₆₄	C₂²	1	1	yes	characteristic
15	C₈×#346	C₂²	1	1	yes	characteristic
16	#94019	—	2	2	no	—
17	#93885	C₄	1	4	yes	complemented
18	#93882	C₄	1	4	yes	complemented
19	C₂×#1422	—	2	2	no	—
20	C₂×#1422	—	2	2	no	—
21	C₂×#1398	—	2	2	no	—
22	C₂×#1398	—	4	2	no	—
23	C₂×#1398	C₈	1	2	yes	—
24	C₂×#1398	C₈	1	2	yes	—
25	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
26	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
27	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
28	C₄×M₆₄	C₂×C₄	1	1	yes	characteristic
29	C₄×#346	C₂×C₄	1	1	yes	characteristic
30	C₄×#346	D₄	1	1	yes	characteristic
31	C₂×#1311	C₂×C₄	1	1	yes	characteristic
32	C₂×#1311	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₆	Q₈	1	1	yes	characteristic
34	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
35	C₄×C₈²	Q₈	1	1	yes	characteristic
36	#7468	—	4	2	no	—
37	#7459	—	2	2	no	—
38	#7459	—	2	2	no	—
39	#7459	C₂×C₄	1	4	yes	—
40	#7459	C₂×C₄	1	4	yes	—
41	#7330	—	2	2	no	—
42	#7330	C₈	1	4	yes	—
43	#7329	—	2	1	no	—
44	#7329	C₂×C₄	1	2	yes	—
45	C₈×C₃₂	—	2	1	no	—
46	C₈×C₃₂	C₂×C₄	1	2	yes	—
47	C₂×M₆₄	—	2	1	no	—
48	C₂×M₆₄	—	2	1	no	—
49	C₂×M₆₄	—	4	2	no	—
50	C₂×M₆₄	—	4	2	no	—
51	C₂×M₆₄	C₂×C₈	1	2	yes	—
52	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
53	C₂×M₆₄	C₂×C₈	1	1	yes	characteristic
54	C₂×#346	C₂×C₈	1	2	yes	—
55	C₂×#346	M₁₆	1	2	yes	—
56	C₂×#346	D₈	1	2	yes	—
57	C₂×#346	Q₁₆	1	2	yes	—
58	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₈	—	2	1	no	—
63	C₄²×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈²	—	2	1	no	—
65	C₂×C₈²	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈²	C₂×Q₈	1	1	yes	characteristic
67	#1422	—	2	4	no	—
68	#1422	—	2	4	no	—
69	#1422	—	4	2	no	—
70	#1422	—	4	2	no	—
71	#1398	—	2	8	no	—
72	#1398	—	2	4	no	—
73	#1398	—	2	4	no	—
74	#1398	—	8	4	no	—
75	#1397	—	2	1	no	—
76	#1397	—	2	1	no	—
77	#1397	C₂×C₈	1	2	yes	—
78	#1397	C₂×C₈	1	2	yes	—
79	C₄×C₃₂	—	2	1	no	—
80	C₄×C₃₂	—	2	1	no	—
81	C₄×C₃₂	C₄²	1	2	yes	—
82	C₄×C₃₂	C₄²	1	2	yes	—
83	#1311	—	2	1	no	—
84	#1311	—	2	1	no	—
85	#1311	C₂×C₈	1	2	yes	—
86	#1311	M₁₆	1	2	yes	—
87	C₈×C₁₆	—	2	2	no	—
88	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
89	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
90	C₈×C₁₆	#41	1	1	yes	characteristic
91	C₈×C₁₆	#41	1	1	yes	characteristic
92	C₂²×C₁₆	C₄⋊C₈	1	2	yes	—
93	C₂²×C₁₆	#107	1	2	yes	—
94	C₂²×C₁₆	C₂²×C₈	1	1	yes	characteristic
95	C₂²×C₁₆	C₂×M₁₆	1	1	yes	characteristic
96	C₂²×C₁₆	C₂×D₈	1	1	yes	characteristic
97	C₂²×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₈	—	2	1	no	—
99	C₂×C₄×C₈	—	2	1	no	—
100	C₂×C₄×C₈	—	4	2	no	—
101	C₂×C₄×C₈	C₄⋊C₈	1	2	yes	—
102	C₂×C₄×C₈	#107	1	2	yes	—
103	C₂×C₄×C₈	#108	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₄³	#108	1	1	yes	characteristic
106	M₆₄	—	2	4	no	—
107	M₆₄	—	2	4	no	—
108	M₆₄	—	2	2	no	—
109	M₆₄	—	2	2	no	—
110	M₆₄	—	4	4	no	—
111	M₆₄	—	4	4	no	—
112	M₆₄	—	8	2	no	—
113	M₆₄	—	8	2	no	—
114	M₆₄	M₃₂	1	4	yes	—
115	M₆₄	M₃₂	1	4	yes	—
116	C₂×C₃₂	—	2	2	no	—
117	C₂×C₃₂	—	2	2	no	—
118	C₂×C₃₂	—	2	2	no	—
119	C₂×C₃₂	—	8	2	no	—
120	C₂×C₃₂	—	8	2	no	—
121	C₂×C₃₂	C₄×C₈	1	2	yes	—
122	C₂×C₃₂	C₄×C₈	1	2	yes	—
123	#346	—	2	4	no	—
124	#346	—	2	2	no	—
125	#346	—	2	2	no	—
126	#346	—	2	2	no	—
127	#346	—	2	2	no	—
128	#346	—	4	2	no	—
129	C₄×C₁₆	—	2	1	no	—
130	C₄×C₁₆	—	2	1	no	—
131	C₄×C₁₆	C₄⋊C₈	1	2	yes	—
132	C₄×C₁₆	#108	1	2	yes	—
133	C₄×C₁₆	C₂×C₄²	1	1	yes	characteristic
134	C₄×C₁₆	#117	1	1	yes	characteristic
135	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₆	C₄×D₄	1	1	yes	characteristic
137	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
138	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
139	C₄×C₁₆	#135	1	1	yes	characteristic
140	C₄×C₁₆	#135	1	1	yes	characteristic
141	C₈²	—	2	1	no	—
142	C₈²	—	2	1	no	—
143	C₈²	—	4	2	no	—
144	C₈²	C₄⋊C₈	1	2	yes	—
145	C₈²	C₄×Q₈	1	1	yes	characteristic
146	C₈²	C₄×Q₈	1	1	yes	characteristic
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	4	1	no	—
149	C₂²×C₈	#335	1	2	yes	—
150	C₂²×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₂²×C₈	C₂×#107	1	1	yes	characteristic
152	C₂×C₄²	—	4	1	no	—
153	C₂×C₄²	#335	1	2	yes	—
154	C₂×C₄²	C₂×#108	1	1	yes	characteristic
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	#335	1	4	yes	—
162	C₂×C₁₆	#364	1	4	yes	—
163	C₂×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
164	C₂×C₁₆	C₄×M₁₆	1	1	yes	characteristic
165	C₂×C₁₆	#431	1	1	yes	characteristic
166	C₂×C₁₆	#431	1	1	yes	characteristic
167	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
168	C₂×C₁₆	C₄×D₈	1	1	yes	characteristic
169	C₂×C₁₆	C₄×Q₁₆	1	1	yes	characteristic
170	C₂×C₁₆	C₄×Q₁₆	1	1	yes	characteristic
171	C₂×C₁₆	C₂×M₃₂	1	2	yes	—
172	C₂×C₁₆	C₂×M₃₂	1	2	yes	—
173	C₂×C₁₆	#509	1	2	yes	—
174	C₂×C₁₆	#510	1	2	yes	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	4	1	no	—
184	C₄×C₈	—	4	1	no	—
185	C₄×C₈	—	8	4	no	—
186	C₄×C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
188	C₄×C₈	#427	1	1	yes	characteristic
189	C₄×C₈	C₂×#108	1	1	yes	characteristic
190	C₃₂	—	2	4	no	—
191	C₃₂	—	2	4	no	—
192	C₃₂	—	2	2	no	—
193	C₃₂	—	2	4	no	—
194	C₃₂	—	8	4	no	—
195	C₃₂	—	8	4	no	—
196	C₃₂	#346	1	4	yes	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂×#335	1	1	yes	characteristic
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	8	2	no	—
208	C₂×C₈	C₂×#335	1	1	yes	characteristic
209	C₂×C₈	#1564	1	1	yes	characteristic
210	C₂×C₈	C₄×C₄⋊C₈	1	1	yes	characteristic
211	C₂×C₈	C₄×#107	1	1	yes	characteristic
212	C₂×C₈	#2150	1	2	yes	—
213	C₂×C₈	C₂×#364	1	2	yes	—
214	C₂×C₈	#2159	1	2	yes	—
215	C₂×C₈	#2161	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	8	2	no	—
220	C₄²	#1564	1	1	yes	characteristic
221	C₄²	C₄×#108	1	1	yes	characteristic
222	C₁₆	—	2	1	no	—
223	C₁₆	—	2	1	no	—
224	C₁₆	—	2	1	no	—
225	C₁₆	—	2	1	no	—
226	C₁₆	—	2	4	no	—
227	C₁₆	—	2	4	no	—
228	C₁₆	C₂×#346	1	1	yes	characteristic
229	C₁₆	C₄×M₃₂	1	1	yes	characteristic
230	C₁₆	#2163	1	2	yes	—
231	C₁₆	#2175	1	2	yes	—
232	C₁₆	#2176	1	1	yes	characteristic
233	C₁₆	#2177	1	1	yes	characteristic
234	C₂³	#12011	1	1	yes	characteristic
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₄×#335	1	1	yes	characteristic
241	C₂×C₄	#12004	1	2	yes	—
242	C₂×C₄	#12009	1	2	yes	—
243	C₂×C₄	#12011	1	1	yes	characteristic
244	C₈	—	2	4	no	—
245	C₈	—	4	2	no	—
246	C₈	—	4	2	no	—
247	C₈	—	8	2	no	—
248	C₈	#12009	1	2	yes	—
249	C₈	#12013	1	2	yes	—
250	C₈	#12653	1	1	yes	characteristic
251	C₈	C₄×#364	1	1	yes	characteristic
252	C₈	#12674	1	1	yes	characteristic
253	C₈	#12676	1	1	yes	characteristic
254	C₂²	—	4	1	no	—
255	C₂²	#122935	1	2	yes	—
256	C₂²	#129101	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	8	2	no	—
260	C₄	#122935	1	2	yes	—
261	C₄	#129093	1	1	yes	characteristic
262	C₄	#129099	1	1	yes	characteristic
263	C₂	—	2	1	no	—
264	C₂	—	2	2	no	—
265	C₂	#72225375	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#12674
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225375
prime radical	C₂³	#12011
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ 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
17. #93885	C₄	semidirect
18. #93882	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72225373	2
#72225362	2
#72225352	2
#72216243	2
C₈×#1398	2