Subgroups — #149457

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×#1389	C₂	1	2	yes	maximal
3	C₂×#1384	C₂	1	2	yes	maximal
4	C₂×#1376	C₂	1	1	yes	maximal, characteristic
5	#10859	C₂	1	1	yes	maximal, characteristic
6	C₂×#533	—	2	1	no	—
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	#3160	—	2	4	no	—
9	#3160	—	2	4	no	—
10	C₂×#500	—	2	1	no	—
11	C₂×#499	—	2	1	no	—
12	C₂×#499	C₄	1	2	yes	complemented
13	C₂×#498	C₄	1	2	yes	complemented
14	C₂²×#108	C₂²	1	1	yes	characteristic
15	C₂×#427	—	2	1	no	—
16	C₂×#427	C₂²	1	2	yes	—
17	C₂×#427	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×#339	—	2	1	no	—
20	#1729	—	2	1	no	—
21	#1389	—	2	4	no	—
22	#1384	—	2	4	no	—
23	#1376	—	2	1	no	—
24	#1376	—	2	1	no	—
25	C₂²×M₁₆	—	2	1	no	—
26	C₂³×C₈	D₄	1	1	yes	characteristic
27	C₂³×C₈	C₂³	1	1	yes	characteristic
28	#533	—	2	4	no	—
29	#533	—	2	4	no	—
30	#533	—	4	2	no	—
31	#533	—	4	2	no	—
32	C₂×C₄⋊Q₈	—	2	2	no	—
33	C₂×#125	—	2	2	no	—
34	C₂×C₂²⋊Q₈	—	4	2	no	—
35	C₂×C₄×Q₈	—	4	2	no	—
36	C₂×#117	—	2	2	no	—
37	C₂×#117	—	2	1	no	—
38	C₂²×C₄²	D₄	1	1	yes	characteristic
39	C₂×M₃₂	—	2	2	no	—
40	C₂×M₃₂	—	2	2	no	—
41	C₂×M₃₂	—	2	2	no	—
42	C₂×M₃₂	—	2	2	no	—
43	C₂×M₃₂	—	2	2	no	—
44	C₂×M₃₂	Q₈	1	4	yes	complemented
45	C₂²×C₁₆	—	2	2	no	—
46	#500	—	2	8	no	—
47	#499	—	2	8	no	—
48	#499	—	2	8	no	—
49	#498	—	2	8	no	—
50	C₂×#108	—	2	2	no	—
51	C₂×#108	—	2	2	no	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	Q₈	1	2	yes	complemented
54	#427	—	2	4	no	—
55	#427	—	2	4	no	—
56	#427	—	2	2	no	—
57	#427	—	2	2	no	—
58	#427	—	2	2	no	—
59	#427	—	2	2	no	—
60	#427	D₄	1	4	yes	—
61	#427	D₄	1	4	yes	—
62	C₂×#107	—	2	2	no	—
63	C₂×#107	—	2	2	no	—
64	C₂×#107	—	2	1	no	—
65	C₂×#106	—	2	2	no	—
66	C₂×#106	C₂×C₄	1	4	yes	—
67	C₂×#106	Q₈	1	2	yes	complemented
68	C₂×C₄×C₈	—	2	2	no	—
69	C₂×C₄×C₈	—	2	2	no	—
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×C₈	D₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	#339	—	4	2	no	—
75	#336	—	4	2	no	—
76	C₂²×Q₈	—	4	2	no	—
77	C₂³×C₄	—	2	1	no	—
78	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×M₁₆	—	2	2	no	—
80	C₂×M₁₆	—	4	1	no	—
81	C₂×M₁₆	—	4	1	no	—
82	C₂×M₁₆	—	4	1	no	—
83	C₂×M₁₆	—	4	1	no	—
84	C₂×M₁₆	—	4	1	no	—
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	—	4	1	no	—
90	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	SD₁₆	1	2	yes	—
93	C₂²×C₈	Q₁₆	1	2	yes	—
94	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₄	1	2	yes	—
96	C₂²×C₈	C₂×D₄	1	2	yes	—
97	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×Q₈	1	2	yes	—
99	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₈	#41	1	2	yes	—
102	C₂²×C₈	#41	1	1	yes	characteristic
103	C₂²×C₈	#41	1	1	yes	characteristic
104	C₄⋊Q₈	—	2	8	no	—
105	C₄⋊Q₈	—	4	4	no	—
106	#125	—	2	8	no	—
107	#125	—	4	4	no	—
108	C₂²⋊Q₈	—	4	8	no	—
109	C₂²⋊Q₈	—	4	8	no	—
110	C₄×Q₈	—	4	8	no	—
111	C₄×Q₈	—	4	8	no	—
112	#117	—	2	4	no	—
113	#117	—	2	4	no	—
114	#117	—	4	2	no	—
115	#117	—	4	2	no	—
116	#117	—	4	1	no	—
117	#117	—	4	1	no	—
118	#117	—	4	1	no	—
119	#117	—	4	1	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	4	4	no	—
123	C₂×C₄⋊C₄	—	4	2	no	—
124	C₂×C₂²⋊C₄	—	4	2	no	—
125	C₂×C₂²⋊C₄	—	4	1	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	—	4	2	no	—
130	C₂×C₄²	—	4	1	no	—
131	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄²	D₈	1	2	yes	—
134	C₂×C₄²	SD₁₆	1	2	yes	—
135	M₃₂	—	2	4	no	—
136	M₃₂	—	2	4	no	—
137	M₃₂	—	4	2	no	—
138	M₃₂	—	4	2	no	—
139	C₂×C₁₆	—	2	4	no	—
140	C₂×C₁₆	—	4	2	no	—
141	C₂×C₁₆	—	4	2	no	—
142	#108	—	2	4	no	—
143	#108	—	4	2	no	—
144	#107	—	2	8	no	—
145	#107	—	2	4	no	—
146	#107	—	2	4	no	—
147	#107	—	4	2	no	—
148	#106	—	2	8	no	—
149	#106	—	2	4	no	—
150	#106	—	2	4	no	—
151	#106	C₂²⋊C₄	1	8	yes	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×Q₈	—	4	8	no	—
158	C₂×Q₈	—	8	8	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	#120	1	1	yes	characteristic
168	C₂²×C₄	#120	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
171	C₂²×C₄	#124	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
174	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
177	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂²×C₄	#137	1	1	yes	characteristic
179	C₂²×C₄	#137	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	M₁₆	—	8	1	no	—
183	M₁₆	—	8	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	8	1	no	—
194	C₂×C₈	—	8	1	no	—
195	C₂×C₈	#103	1	2	yes	—
196	C₂×C₈	#103	1	2	yes	—
197	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
198	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
199	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
200	C₂×C₈	C₄×Q₈	1	2	yes	—
201	C₂×C₈	#120	1	2	yes	—
202	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
203	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
204	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
205	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
206	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
207	C₂×C₈	#124	1	2	yes	—
208	C₂×C₈	#124	1	2	yes	—
209	C₂×C₈	#124	1	1	yes	characteristic
210	C₂×C₈	C₄⋊Q₈	1	2	yes	—
211	C₂×C₈	C₄⋊Q₈	1	2	yes	—
212	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	4	16	no	—
215	C₄⋊C₄	—	4	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	8	1	no	—
222	C₂²⋊C₄	—	8	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	4	4	no	—
228	C₄²	—	4	4	no	—
229	C₄²	—	8	1	no	—
230	C₄²	—	8	1	no	—
231	C₄²	#102	1	2	yes	—
232	C₄²	#102	1	2	yes	—
233	C₁₆	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	#418	1	1	yes	characteristic
236	C₂³	#447	1	1	yes	characteristic
237	C₂³	#450	1	1	yes	characteristic
238	C₂³	#476	1	1	yes	characteristic
239	C₂³	#477	1	1	yes	characteristic
240	C₂³	#489	1	1	yes	characteristic
241	C₂³	#501	1	1	yes	characteristic
242	Q₈	—	8	16	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	—	8	8	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	#379	1	1	yes	characteristic
254	C₂×C₄	#391	1	1	yes	characteristic
255	C₂×C₄	#391	1	1	yes	characteristic
256	C₂×C₄	#393	1	1	yes	characteristic
257	C₂×C₄	#393	1	1	yes	characteristic
258	C₂×C₄	#395	1	1	yes	characteristic
259	C₂×C₄	#395	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	#419	1	1	yes	characteristic
263	C₂×C₄	#447	1	1	yes	characteristic
264	C₂×C₄	#448	1	1	yes	characteristic
265	C₂×C₄	#448	1	1	yes	characteristic
266	C₂×C₄	#450	1	1	yes	characteristic
267	C₂×C₄	#451	1	1	yes	characteristic
268	C₂×C₄	#451	1	1	yes	characteristic
269	C₂×C₄	#474	1	1	yes	characteristic
270	C₂×C₄	#474	1	1	yes	characteristic
271	C₂×C₄	#475	1	1	yes	characteristic
272	C₂×C₄	#475	1	1	yes	characteristic
273	C₂×C₄	#476	1	1	yes	characteristic
274	C₂×C₄	#477	1	1	yes	characteristic
275	C₂×C₄	#486	1	1	yes	characteristic
276	C₂×C₄	#487	1	1	yes	characteristic
277	C₂×C₄	#489	1	1	yes	characteristic
278	C₂×C₄	#498	1	1	yes	characteristic
279	C₂×C₄	#500	1	1	yes	characteristic
280	C₂×C₄	#501	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₈	—	8	2	no	—
284	C₈	#391	1	2	yes	—
285	C₈	#391	1	2	yes	—
286	C₈	#393	1	2	yes	—
287	C₈	#395	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#1786	1	1	yes	characteristic
296	C₂²	#1987	1	1	yes	characteristic
297	C₂²	#1993	1	1	yes	characteristic
298	C₂²	#2023	1	1	yes	characteristic
299	C₂²	#2024	1	1	yes	characteristic
300	C₂²	#2057	1	1	yes	characteristic
301	C₂²	#2058	1	1	yes	characteristic
302	C₄	—	2	4	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	8	no	—
305	C₄	—	16	2	no	—
306	C₄	#1446	1	1	yes	characteristic
307	C₄	#1787	1	1	yes	characteristic
308	C₄	#1988	1	1	yes	characteristic
309	C₄	#1990	1	1	yes	characteristic
310	C₄	#2023	1	1	yes	characteristic
311	C₄	#2024	1	1	yes	characteristic
312	C₄	#2057	1	1	yes	characteristic
313	C₄	#2058	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	#11384	1	1	yes	characteristic
321	C₂	#13271	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1987
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ 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
12. C₂×#499	C₄	semidirect
13. C₂×#498	C₄	semidirect
44. C₂×M₃₂	Q₈	semidirect
53. C₂×#108	Q₈	semidirect
67. C₂×#106	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3160	2
C₂×#1389	2
C₂×#1384	2
C₂×#1376	2
#10859	2