Subgroups — #146986

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#1846	C₂	1	1	yes	maximal, characteristic
3	C₂×#1845	C₂	1	1	yes	maximal, characteristic
4	C₂×#1380	C₂	1	4	yes	maximal
5	C₂³×C₁₆	C₂²	1	2	yes	—
6	C₂²×#107	—	2	1	no	—
7	C₂²×#107	C₂²	1	2	yes	—
8	C₂²×#107	C₂²	1	2	yes	—
9	C₂²×#106	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#384	—	2	1	no	—
12	C₂×#382	—	2	1	no	—
13	C₂×#346	—	2	2	no	—
14	C₂×#346	—	2	2	no	—
15	C₂×#346	C₄	1	4	yes	complemented
16	#1846	—	2	2	no	—
17	#1846	C₄	1	4	yes	complemented
18	#1845	—	2	2	no	—
19	#1845	C₄	1	4	yes	complemented
20	#1380	—	2	8	no	—
21	C₂³×C₈	D₄	1	1	yes	characteristic
22	C₂³×C₈	C₂³	1	1	yes	characteristic
23	C₂²×C₄⋊C₄	—	2	2	no	—
24	C₂²×C₄⋊C₄	—	2	2	no	—
25	C₂²×C₄⋊C₄	—	4	2	no	—
26	C₂²×C₄²	D₄	1	1	yes	characteristic
27	C₂²×C₁₆	—	2	2	no	—
28	C₂²×C₁₆	—	2	2	no	—
29	C₂²×C₁₆	—	2	2	no	—
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₁₆	D₄	1	4	yes	—
33	C₂²×C₁₆	Q₈	1	4	yes	—
34	C₂×#107	—	2	4	no	—
35	C₂×#107	—	2	4	no	—
36	C₂×#107	—	2	4	no	—
37	C₂×#107	—	2	4	no	—
38	C₂×#107	—	2	4	no	—
39	C₂×#107	—	2	2	no	—
40	C₂×#107	—	2	2	no	—
41	C₂×#107	—	4	2	no	—
42	C₂×#107	—	4	2	no	—
43	C₂×#107	C₂×C₄	1	4	yes	—
44	C₂×#107	C₂×C₄	1	4	yes	—
45	C₂×#106	—	2	4	no	—
46	C₂×#106	—	4	2	no	—
47	C₂×#106	—	4	2	no	—
48	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
49	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	D₄	1	2	yes	—
52	C₂×C₄×C₈	D₄	1	2	yes	—
53	C₂×C₄×C₈	Q₈	1	2	yes	—
54	#384	—	2	4	no	—
55	#384	—	4	2	no	—
56	#382	—	2	4	no	—
57	#382	—	4	2	no	—
58	C₂×#95	—	4	2	no	—
59	#346	—	2	8	no	—
60	C₂³×C₄	—	4	2	no	—
61	C₂³×C₄	—	4	2	no	—
62	C₂³×C₄	D₈	1	2	yes	—
63	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₈	C₄⋊C₄	1	2	yes	—
67	C₂²×C₈	D₈	1	2	yes	—
68	C₂²×C₈	SD₁₆	1	2	yes	—
69	C₂²×C₈	SD₁₆	1	2	yes	—
70	C₂²×C₈	Q₁₆	1	2	yes	—
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₄	1	2	yes	—
77	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₈	C₂×Q₈	1	2	yes	—
79	C₂²×C₈	#41	1	2	yes	—
80	C₂²×C₈	#41	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	4	no	—
82	C₂×C₄⋊C₄	—	2	4	no	—
83	C₂×C₄⋊C₄	—	4	2	no	—
84	C₂×C₄⋊C₄	—	4	2	no	—
85	C₂×C₄⋊C₄	—	4	2	no	—
86	C₂×C₄⋊C₄	—	4	2	no	—
87	C₂×C₄⋊C₄	—	4	2	no	—
88	C₂×C₄⋊C₄	—	4	2	no	—
89	C₂×C₄⋊C₄	—	4	2	no	—
90	C₂×C₄⋊C₄	—	4	2	no	—
91	C₂×C₄⋊C₄	—	4	8	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄⋊C₄	—	4	2	no	—
94	C₂×C₄⋊C₄	—	8	8	no	—
95	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
96	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
97	C₂×C₄²	C₄⋊C₄	1	2	yes	—
98	C₂×C₄²	SD₁₆	1	2	yes	—
99	C₂×C₄²	SD₁₆	1	2	yes	—
100	C₂×C₄²	Q₁₆	1	2	yes	—
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	—	2	4	no	—
105	C₂×C₁₆	—	2	4	no	—
106	C₂×C₁₆	—	2	4	no	—
107	C₂×C₁₆	C₄⋊C₄	1	8	yes	complemented
108	#107	—	2	8	no	—
109	#107	—	2	8	no	—
110	#107	—	4	8	no	—
111	#107	—	4	4	no	—
112	#107	—	4	4	no	—
113	#106	—	4	8	no	—
114	C₄×C₈	C₄²	1	4	yes	—
115	C₄×C₈	C₂²⋊C₄	1	4	yes	—
116	C₄×C₈	C₄⋊C₄	1	4	yes	—
117	C₄×C₈	C₄⋊C₄	1	4	yes	—
118	#95	—	4	8	no	—
119	C₂⁴	C₂×D₈	1	1	yes	characteristic
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	4	no	—
122	C₂²×C₄	—	4	4	no	—
123	C₂²×C₄	—	4	4	no	—
124	C₂²×C₄	—	8	2	no	—
125	C₂²×C₄	—	8	2	no	—
126	C₂²×C₄	—	8	2	no	—
127	C₂²×C₄	—	8	2	no	—
128	C₂²×C₄	—	8	8	no	—
129	C₂²×C₄	—	8	4	no	—
130	C₂²×C₄	—	8	4	no	—
131	C₂²×C₄	#102	1	4	yes	—
132	C₂²×C₄	#102	1	4	yes	—
133	C₂²×C₄	#107	1	4	yes	—
134	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄	#124	1	1	yes	characteristic
138	C₂²×C₄	#124	1	1	yes	characteristic
139	C₂²×C₄	#127	1	1	yes	characteristic
140	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
143	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
146	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
147	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
148	C₂×C₈	#95	1	4	yes	—
149	C₂×C₈	#102	1	4	yes	—
150	C₂×C₈	#102	1	4	yes	—
151	C₂×C₈	#103	1	4	yes	—
152	C₂×C₈	#103	1	4	yes	—
153	C₂×C₈	#106	1	4	yes	—
154	C₂×C₈	#107	1	4	yes	—
155	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₈	#117	1	2	yes	—
161	C₂×C₈	C₄×D₄	1	2	yes	—
162	C₂×C₈	C₄×D₄	1	2	yes	—
163	C₂×C₈	C₄×Q₈	1	2	yes	—
164	C₂×C₈	C₄×Q₈	1	2	yes	—
165	C₂×C₈	#121	1	2	yes	—
166	C₂×C₈	#121	1	2	yes	—
167	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
168	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
169	C₂×C₈	#125	1	2	yes	—
170	C₂×C₈	C₄⋊Q₈	1	2	yes	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	8	16	no	—
176	C₄⋊C₄	—	8	2	no	—
177	C₄⋊C₄	—	8	2	no	—
178	C₄⋊C₄	—	8	2	no	—
179	C₄⋊C₄	—	8	2	no	—
180	C₄²	#95	1	4	yes	—
181	C₄²	#103	1	4	yes	—
182	C₄²	#103	1	4	yes	—
183	C₄²	#106	1	4	yes	—
184	C₁₆	—	2	8	no	—
185	C₂³	C₂×#102	1	1	yes	characteristic
186	C₂³	C₂×#102	1	1	yes	characteristic
187	C₂³	C₂×#107	1	1	yes	characteristic
188	C₂³	#486	1	1	yes	characteristic
189	C₂³	#486	1	1	yes	characteristic
190	C₂³	#493	1	1	yes	characteristic
191	C₂³	#500	1	1	yes	characteristic
192	C₂³	#509	1	4	yes	—
193	C₂³	#510	1	4	yes	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	8	8	no	—
199	C₂×C₄	—	8	8	no	—
200	C₂×C₄	—	8	4	no	—
201	C₂×C₄	—	8	4	no	—
202	C₂×C₄	—	16	4	no	—
203	C₂×C₄	—	16	4	no	—
204	C₂×C₄	—	16	2	no	—
205	C₂×C₄	—	16	4	no	—
206	C₂×C₄	—	16	2	no	—
207	C₂×C₄	#340	1	8	yes	—
208	C₂×C₄	C₂×#95	1	1	yes	characteristic
209	C₂×C₄	#382	1	1	yes	characteristic
210	C₂×C₄	#384	1	1	yes	characteristic
211	C₂×C₄	#390	1	1	yes	characteristic
212	C₂×C₄	#390	1	1	yes	characteristic
213	C₂×C₄	#391	1	1	yes	characteristic
214	C₂×C₄	#391	1	1	yes	characteristic
215	C₂×C₄	C₂×#102	1	1	yes	characteristic
216	C₂×C₄	C₂×#102	1	1	yes	characteristic
217	C₂×C₄	C₂×#103	1	1	yes	characteristic
218	C₂×C₄	C₂×#103	1	1	yes	characteristic
219	C₂×C₄	C₂×#103	1	1	yes	characteristic
220	C₂×C₄	C₂×#103	1	1	yes	characteristic
221	C₂×C₄	C₂×#106	1	1	yes	characteristic
222	C₂×C₄	C₂×#106	1	1	yes	characteristic
223	C₂×C₄	C₂×#107	1	1	yes	characteristic
224	C₂×C₄	#486	1	1	yes	characteristic
225	C₂×C₄	#486	1	1	yes	characteristic
226	C₂×C₄	#487	1	1	yes	characteristic
227	C₂×C₄	#487	1	1	yes	characteristic
228	C₂×C₄	#487	1	1	yes	characteristic
229	C₂×C₄	#487	1	1	yes	characteristic
230	C₂×C₄	#492	1	1	yes	characteristic
231	C₂×C₄	#492	1	1	yes	characteristic
232	C₂×C₄	#494	1	1	yes	characteristic
233	C₂×C₄	#498	1	1	yes	characteristic
234	C₂×C₄	#498	1	1	yes	characteristic
235	C₂×C₄	#500	1	1	yes	characteristic
236	C₈	#340	1	8	yes	—
237	C₈	C₄×C₄⋊C₄	1	2	yes	—
238	C₈	#380	1	2	yes	—
239	C₈	#389	1	2	yes	—
240	C₈	#389	1	2	yes	—
241	C₂²	C₂×#340	1	1	yes	characteristic
242	C₂²	#1834	1	1	yes	characteristic
243	C₂²	#1845	1	1	yes	characteristic
244	C₂²	#1957	1	1	yes	characteristic
245	C₂²	#1957	1	1	yes	characteristic
246	C₂²	#1990	1	1	yes	characteristic
247	C₂²	#1990	1	1	yes	characteristic
248	C₂²	#2139	1	4	yes	—
249	C₂²	#2140	1	2	yes	—
250	C₂²	#2141	1	2	yes	—
251	C₂²	#2159	1	4	yes	—
252	C₂²	#2243	1	4	yes	—
253	C₂²	#2244	1	2	yes	—
254	C₂²	#2245	1	2	yes	—
255	C₂²	#2250	1	2	yes	—
256	C₂²	#2251	1	2	yes	—
257	C₂²	#2255	1	4	yes	—
258	C₄	—	2	8	no	—
259	C₄	—	16	4	no	—
260	C₄	—	16	4	no	—
261	C₄	C₄⋊#95	1	1	yes	characteristic
262	C₄	C₂×#340	1	1	yes	characteristic
263	C₄	#1834	1	1	yes	characteristic
264	C₄	#1844	1	1	yes	characteristic
265	C₄	#1956	1	1	yes	characteristic
266	C₄	#1956	1	1	yes	characteristic
267	C₄	#1990	1	1	yes	characteristic
268	C₄	#1990	1	1	yes	characteristic
269	C₂	#10841	1	1	yes	characteristic
270	C₂	#12557	1	2	yes	—
271	C₂	#12890	1	2	yes	—
272	C₂	#12908	1	2	yes	—
273	C₂	#13205	1	2	yes	—
274	C₂	#13206	1	1	yes	characteristic
275	C₂	#13207	1	1	yes	characteristic
276	C₂	#13283	1	2	yes	—
277	C₂	#13284	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
15. C₂×#346	C₄	semidirect
17. #1846	C₄	semidirect
19. #1845	C₄	semidirect
107. C₂×C₁₆	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#346	2
C₂×#1846	2
C₂×#1845	2
C₂×#1380	2