Subgroups — #127761

Order 512.

488 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	1	yes	maximal, characteristic
2	C₂×#1846	C₂	1	1	yes	maximal, characteristic
3	#17793	C₂	1	1	yes	maximal, characteristic
4	#7279	C₂	1	4	yes	maximal, complemented
5	#2920	—	2	1	no	—
6	C₂²×#107	—	2	1	no	—
7	C₂²×#106	C₄	1	2	yes	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#382	—	2	1	no	—
10	#2163	—	2	1	no	—
11	#2163	C₄	1	2	yes	complemented
12	#2162	—	2	1	no	—
13	#2162	C₄	1	2	yes	complemented
14	#2150	C₂²	1	2	yes	—
15	#2112	—	2	1	no	—
16	C₄×M₃₂	—	2	1	no	—
17	#1846	—	2	1	no	—
18	#1846	—	2	1	no	—
19	#1846	C₄	1	2	yes	—
20	#1846	C₂²	1	2	yes	—
21	#1568	—	2	1	no	—
22	#1568	C₄	1	2	yes	complemented
23	#1567	—	2	1	no	—
24	#1566	C₄	1	2	yes	complemented
25	#1564	—	2	1	no	—
26	#1564	C₂²	1	2	yes	—
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₄⋊C₄	—	2	2	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	2	no	—
32	#432	—	2	2	no	—
33	#432	—	4	1	no	—
34	#431	—	2	2	no	—
35	#431	—	4	1	no	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#107	—	4	4	no	—
39	C₂×#106	—	2	2	no	—
40	C₂×#106	—	4	4	no	—
41	C₂×#106	C₂×C₄	1	4	yes	—
42	#423	—	2	2	no	—
43	#407	—	4	1	no	—
44	C₄×M₁₆	—	4	1	no	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
55	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	#382	—	2	2	no	—
58	#382	—	2	2	no	—
59	#382	—	4	1	no	—
60	#382	—	4	1	no	—
61	C₂×#95	—	4	1	no	—
62	C₂×#95	—	4	1	no	—
63	C₄⋊C₁₆	—	2	2	no	—
64	C₄⋊C₁₆	C₂×C₄	1	4	yes	complemented
65	C₂²⋊C₁₆	—	2	2	no	—
66	C₂²⋊C₁₆	—	2	2	no	—
67	#346	—	2	2	no	—
68	C₄×C₁₆	—	2	2	no	—
69	#335	—	2	2	no	—
70	#335	—	2	2	no	—
71	#334	—	2	2	no	—
72	#334	C₂×C₄	1	4	yes	complemented
73	C₂³×C₄	—	4	2	no	—
74	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×C₄	D₈	1	1	yes	characteristic
76	C₂³×C₄	SD₁₆	1	1	yes	characteristic
77	C₂×M₁₆	—	4	2	no	—
78	C₂²×C₈	—	2	2	no	—
79	C₂²×C₈	—	2	2	no	—
80	C₂²×C₈	—	2	2	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	C₄²	1	1	yes	characteristic
83	C₂²×C₈	C₄²	1	1	yes	characteristic
84	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	2	4	no	—
95	C₂×C₄⋊C₄	—	4	4	no	—
96	C₂×C₄⋊C₄	—	4	4	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	D₈	1	1	yes	characteristic
104	C₂×C₄²	SD₁₆	1	1	yes	characteristic
105	C₂×C₄²	SD₁₆	1	1	yes	characteristic
106	C₂×C₄²	SD₁₆	1	1	yes	characteristic
107	C₂×C₄²	Q₁₆	1	1	yes	characteristic
108	C₂×C₄²	Q₁₆	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
110	M₃₂	—	4	2	no	—
111	M₃₂	—	4	2	no	—
112	C₂×C₁₆	—	2	4	no	—
113	#107	—	4	4	no	—
114	#107	—	4	4	no	—
115	#106	—	4	4	no	—
116	#106	—	4	4	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	4	2	no	—
119	C₂²⋊C₈	—	4	2	no	—
120	C₂²⋊C₈	—	4	2	no	—
121	#97	—	4	2	no	—
122	C₄×C₈	—	4	2	no	—
123	C₄×C₈	C₄²	1	1	yes	characteristic
124	C₄×C₈	C₄²	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₄×C₈	C₂×D₄	1	1	yes	characteristic
138	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
139	#95	—	4	2	no	—
140	#95	—	4	2	no	—
141	#95	—	4	2	no	—
142	#95	—	4	2	no	—
143	C₂⁴	#102	1	1	yes	characteristic
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	—	8	4	no	—
152	C₂²×C₄	—	8	4	no	—
153	C₂²×C₄	—	8	4	no	—
154	C₂²×C₄	#95	1	1	yes	characteristic
155	C₂²×C₄	#95	1	1	yes	characteristic
156	C₂²×C₄	#102	1	1	yes	characteristic
157	C₂²×C₄	#102	1	1	yes	characteristic
158	C₂²×C₄	#102	1	1	yes	characteristic
159	C₂²×C₄	#102	1	1	yes	characteristic
160	C₂²×C₄	#102	1	1	yes	characteristic
161	C₂²×C₄	#103	1	1	yes	characteristic
162	C₂²×C₄	#103	1	1	yes	characteristic
163	C₂²×C₄	#103	1	1	yes	characteristic
164	C₂²×C₄	#103	1	1	yes	characteristic
165	C₂²×C₄	#106	1	1	yes	characteristic
166	C₂²×C₄	#106	1	1	yes	characteristic
167	C₂²×C₄	#107	1	1	yes	characteristic
168	C₂²×C₄	#107	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
171	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
172	M₁₆	—	8	4	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	#95	1	1	yes	characteristic
187	C₂×C₈	#95	1	1	yes	characteristic
188	C₂×C₈	#95	1	1	yes	characteristic
189	C₂×C₈	#95	1	1	yes	characteristic
190	C₂×C₈	#100	1	2	yes	—
191	C₂×C₈	#101	1	2	yes	—
192	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
193	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×C₈	#137	1	2	yes	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	8	4	no	—
201	C₄²	#95	1	1	yes	characteristic
202	C₄²	#95	1	1	yes	characteristic
203	C₄²	#102	1	1	yes	characteristic
204	C₄²	#102	1	1	yes	characteristic
205	C₄²	#103	1	1	yes	characteristic
206	C₄²	#103	1	1	yes	characteristic
207	C₄²	#103	1	1	yes	characteristic
208	C₄²	#103	1	1	yes	characteristic
209	C₄²	#106	1	1	yes	characteristic
210	C₄²	#106	1	1	yes	characteristic
211	C₄²	#107	1	1	yes	characteristic
212	C₄²	#107	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
215	C₄²	C₂×SD₁₆	1	1	yes	characteristic
216	C₄²	C₂×Q₁₆	1	1	yes	characteristic
217	C₁₆	—	4	4	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	#340	1	1	yes	characteristic
223	C₂³	#340	1	1	yes	characteristic
224	C₂³	C₂×#102	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	16	8	no	—
251	C₂×C₄	#340	1	1	yes	characteristic
252	C₂×C₄	#340	1	1	yes	characteristic
253	C₂×C₄	#340	1	1	yes	characteristic
254	C₂×C₄	#340	1	1	yes	characteristic
255	C₂×C₄	#340	1	1	yes	characteristic
256	C₂×C₄	#340	1	1	yes	characteristic
257	C₂×C₄	C₂×#95	1	1	yes	characteristic
258	C₂×C₄	C₂×#100	1	1	yes	characteristic
259	C₂×C₄	C₂×#101	1	1	yes	characteristic
260	C₂×C₄	C₂×#102	1	1	yes	characteristic
261	C₂×C₄	C₂×#103	1	1	yes	characteristic
262	C₂×C₄	C₂×#103	1	1	yes	characteristic
263	C₂×C₄	#418	1	1	yes	characteristic
264	C₂×C₄	#419	1	1	yes	characteristic
265	C₂×C₄	C₂×#106	1	1	yes	characteristic
266	C₂×C₄	C₂×#107	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	8	4	no	—
272	C₈	#343	1	2	yes	—
273	C₈	#343	1	2	yes	—
274	C₈	#417	1	2	yes	—
275	C₈	#428	1	2	yes	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	#1384	1	2	yes	—
285	C₂²	#1389	1	2	yes	—
286	C₂²	#1545	1	1	yes	characteristic
287	C₂²	#1546	1	1	yes	characteristic
288	C₂²	C₂×#340	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	16	4	no	—
300	C₄	#1548	1	1	yes	characteristic
301	C₄	#1549	1	1	yes	characteristic
302	C₄	#1736	1	1	yes	characteristic
303	C₄	C₂×#343	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	2	no	—
308	C₂	#8043	1	1	yes	characteristic
309	C₂	C₂×#1384	1	1	yes	characteristic
310	C₂	C₂×#1389	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7279	C₂	semidirect
11. #2163	C₄	semidirect
13. #2162	C₄	semidirect
22. #1568	C₄	semidirect
24. #1566	C₄	semidirect
64. C₄⋊C₁₆	C₂×C₄	semidirect
72. #334	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
C₂×#1846	2
#17793	2
#7279	2