Subgroups — #127359

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2163	C₂	1	1	yes	maximal, characteristic
2	#20818	C₂	1	1	yes	maximal, characteristic
3	#20817	C₂	1	1	yes	maximal, characteristic
4	#7230	C₂	1	4	yes	maximal
5	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
6	#2285	—	2	1	no	—
7	#2283	—	2	1	no	—
8	#2163	—	2	1	no	—
9	#2163	—	2	1	no	—
10	#2163	—	2	1	no	—
11	#2163	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₄	1	2	yes	complemented
13	C₂×C₂²⋊C₁₆	C₂²	1	2	yes	—
14	C₂×#346	C₄	1	2	yes	complemented
15	#1846	—	2	1	no	—
16	#1845	—	2	1	no	—
17	#1835	C₄	1	2	yes	complemented
18	#1834	C₄	1	2	yes	complemented
19	#1808	—	2	1	no	—
20	#1808	C₂²	1	2	yes	—
21	#1808	C₂²	1	2	yes	—
22	#1807	—	2	1	no	—
23	C₄×#107	—	2	1	no	—
24	C₄×#107	—	2	1	no	—
25	C₄×#107	C₄	1	2	yes	complemented
26	C₄×#106	C₄	1	2	yes	complemented
27	C₂³×C₈	D₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	—	4	1	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂×C₄	1	4	yes	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#106	—	2	2	no	—
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
47	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
48	#384	—	2	2	no	—
49	#383	—	4	1	no	—
50	#382	—	2	2	no	—
51	#382	—	4	1	no	—
52	#381	—	4	1	no	—
53	#380	—	2	2	no	—
54	#380	—	2	2	no	—
55	#380	—	4	1	no	—
56	C₄×C₄⋊C₄	—	2	2	no	—
57	C₄×C₄⋊C₄	—	2	2	no	—
58	C₄×C₄⋊C₄	—	4	1	no	—
59	C₄×C₂²⋊C₄	—	4	1	no	—
60	#376	—	4	1	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	C₄⋊C₁₆	—	2	2	no	—
63	C₂²⋊C₁₆	—	2	2	no	—
64	C₂²⋊C₁₆	—	2	2	no	—
65	#346	—	2	2	no	—
66	#346	—	2	2	no	—
67	#340	—	2	2	no	—
68	#340	—	2	2	no	—
69	#340	C₂×C₄	1	4	yes	—
70	#340	C₂×C₄	1	4	yes	—
71	C₂³×C₄	D₈	1	1	yes	characteristic
72	C₂³×C₄	D₈	1	1	yes	characteristic
73	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	—	4	2	no	—
75	C₂²×C₈	—	4	2	no	—
76	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
88	#117	—	8	4	no	—
89	C₂×C₄⋊C₄	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	4	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₂²⋊C₄	—	4	2	no	—
94	C₂×C₂²⋊C₄	—	4	2	no	—
95	C₂×C₄²	—	4	2	no	—
96	C₂×C₄²	—	4	2	no	—
97	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	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×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	4	4	no	—
111	#107	—	4	4	no	—
112	#107	—	4	4	no	—
113	#107	—	4	4	no	—
114	#106	—	4	4	no	—
115	C₄×C₈	C₄²	1	1	yes	characteristic
116	C₄×C₈	C₄²	1	1	yes	characteristic
117	C₄×C₈	C₄²	1	1	yes	characteristic
118	C₄×C₈	C₄²	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	#95	—	4	2	no	—
132	#95	—	4	2	no	—
133	#95	—	4	2	no	—
134	#95	—	4	2	no	—
135	#95	—	4	2	no	—
136	#95	—	4	2	no	—
137	C₂⁴	C₂×D₈	1	1	yes	characteristic
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	4	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	#102	1	1	yes	characteristic
146	C₂²×C₄	#102	1	1	yes	characteristic
147	C₂²×C₄	#102	1	1	yes	characteristic
148	C₂²×C₄	#102	1	1	yes	characteristic
149	C₂²×C₄	#102	1	1	yes	characteristic
150	C₂²×C₄	#102	1	1	yes	characteristic
151	C₂²×C₄	#102	1	1	yes	characteristic
152	C₂²×C₄	#102	1	1	yes	characteristic
153	C₂²×C₄	#107	1	1	yes	characteristic
154	C₂²×C₄	#107	1	1	yes	characteristic
155	C₂²×C₄	#107	1	1	yes	characteristic
156	C₂²×C₄	#107	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
162	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	#95	1	1	yes	characteristic
176	C₂×C₈	#95	1	1	yes	characteristic
177	C₂×C₈	#95	1	1	yes	characteristic
178	C₂×C₈	#95	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	8	4	no	—
188	C₄⋊C₄	—	8	4	no	—
189	C₂²⋊C₄	—	8	2	no	—
190	C₂²⋊C₄	—	8	4	no	—
191	C₂²⋊C₄	—	8	2	no	—
192	C₄²	—	8	4	no	—
193	C₄²	—	8	4	no	—
194	C₄²	#95	1	1	yes	characteristic
195	C₄²	#95	1	1	yes	characteristic
196	C₄²	#95	1	1	yes	characteristic
197	C₄²	#95	1	1	yes	characteristic
198	C₄²	#103	1	1	yes	characteristic
199	C₄²	#103	1	1	yes	characteristic
200	C₄²	#103	1	1	yes	characteristic
201	C₄²	#103	1	1	yes	characteristic
202	C₄²	#103	1	1	yes	characteristic
203	C₄²	#103	1	1	yes	characteristic
204	C₄²	#103	1	1	yes	characteristic
205	C₄²	#103	1	1	yes	characteristic
206	C₄²	#106	1	1	yes	characteristic
207	C₄²	#106	1	1	yes	characteristic
208	C₄²	#106	1	1	yes	characteristic
209	C₄²	#106	1	1	yes	characteristic
210	C₁₆	—	4	4	no	—
211	C₂³	—	4	1	no	—
212	C₂³	—	4	1	no	—
213	C₂³	C₂×#102	1	1	yes	characteristic
214	C₂³	C₂×#102	1	1	yes	characteristic
215	C₂³	C₂×#107	1	1	yes	characteristic
216	C₂³	#509	1	2	yes	—
217	C₂³	#510	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	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	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	#340	1	1	yes	characteristic
248	C₂×C₄	#340	1	1	yes	characteristic
249	C₂×C₄	#340	1	1	yes	characteristic
250	C₂×C₄	#340	1	1	yes	characteristic
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	C₂×#95	1	1	yes	characteristic
256	C₂×C₄	C₂×#103	1	1	yes	characteristic
257	C₂×C₄	C₂×#103	1	1	yes	characteristic
258	C₂×C₄	C₂×#106	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	H(3,ℤ/4)	1	2	yes	—
266	C₈	#410	1	2	yes	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	C₂×#340	1	1	yes	characteristic
276	C₂²	#2139	1	2	yes	—
277	C₂²	#2140	1	1	yes	characteristic
278	C₂²	#2141	1	1	yes	characteristic
279	C₂²	#2159	1	2	yes	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	16	4	no	—
291	C₄	—	16	4	no	—
292	C₄	#1384	1	2	yes	—
293	C₄	C₂×H(3,ℤ/4)	1	1	yes	characteristic
294	C₄	#1740	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#7981	1	1	yes	characteristic
300	C₂	#12557	1	1	yes	characteristic
301	C₂	C₂×#1384	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. C₂×C₄⋊C₁₆	C₄	semidirect
14. C₂×#346	C₄	semidirect
17. #1835	C₄	semidirect
18. #1834	C₄	semidirect
25. C₄×#107	C₄	semidirect
26. C₄×#106	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2163	2
#20818	2
#20817	2
#7230	2