Subgroups — #149458

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×#1386	C₂	1	2	yes	maximal
3	C₂×#1383	C₂	1	2	yes	maximal
4	C₂×#1376	C₂	1	1	yes	maximal, characteristic
5	#10859	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Q₈	—	2	1	no	—
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂×#498	—	2	4	no	—
9	C₂×#498	—	2	4	no	—
10	C₂×#498	—	2	2	no	—
11	C₂×#498	C₄	1	4	yes	complemented
12	C₂²×#108	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂²×#106	—	2	1	no	—
15	C₂²×#106	C₂²	1	2	yes	—
16	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×#339	—	2	1	no	—
18	#1729	—	2	1	no	—
19	#1386	—	2	4	no	—
20	#1383	—	2	4	no	—
21	#1376	—	2	2	no	—
22	C₂³×Q₈	—	4	1	no	—
23	C₂²×M₁₆	—	2	1	no	—
24	C₂³×C₈	D₄	1	1	yes	characteristic
25	C₂³×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Q₈	—	2	4	no	—
27	C₂×C₄⋊Q₈	—	2	4	no	—
28	C₂×C₄⋊Q₈	—	2	4	no	—
29	C₂×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₄⋊Q₈	—	4	2	no	—
31	C₂×C₄⋊Q₈	—	4	2	no	—
32	C₂×#117	—	2	1	no	—
33	C₂²×C₄⋊C₄	—	2	2	no	—
34	C₂²×C₄⋊C₄	—	4	1	no	—
35	C₂²×C₄²	D₄	1	1	yes	characteristic
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	—	2	2	no	—
38	C₂×M₃₂	—	2	2	no	—
39	C₂×M₃₂	—	2	2	no	—
40	C₂×M₃₂	—	2	2	no	—
41	C₂×M₃₂	Q₈	1	4	yes	complemented
42	C₂²×C₁₆	—	2	2	no	—
43	#498	—	2	16	no	—
44	#498	—	2	16	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	—	2	1	no	—
48	C₂×#108	Q₈	1	2	yes	complemented
49	#427	—	2	2	no	—
50	#427	—	2	2	no	—
51	C₂×#107	Q₈	1	2	yes	complemented
52	C₂×#106	—	2	4	no	—
53	C₂×#106	—	2	4	no	—
54	C₂×#106	—	2	4	no	—
55	C₂×#106	—	2	2	no	—
56	C₂×#106	—	2	2	no	—
57	C₂×#106	—	2	2	no	—
58	C₂×#106	—	2	1	no	—
59	C₂×#106	C₂×C₄	1	4	yes	—
60	C₂×#106	D₄	1	4	yes	—
61	C₂×#106	D₄	1	4	yes	—
62	C₂×C₄×C₈	—	2	2	no	—
63	C₂×C₄×C₈	—	2	2	no	—
64	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
65	C₂×C₄×C₈	D₄	1	2	yes	—
66	#339	—	4	2	no	—
67	#336	—	4	2	no	—
68	C₂²×Q₈	—	4	4	no	—
69	C₂²×Q₈	—	4	4	no	—
70	C₂²×Q₈	—	4	4	no	—
71	C₂²×Q₈	—	8	8	no	—
72	C₂³×C₄	—	2	1	no	—
73	C₂³×C₄	—	4	2	no	—
74	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×M₁₆	—	2	2	no	—
76	C₂×M₁₆	—	4	1	no	—
77	C₂×M₁₆	—	4	1	no	—
78	C₂×M₁₆	—	4	1	no	—
79	C₂×M₁₆	—	4	1	no	—
80	C₂×M₁₆	—	4	1	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	—	4	1	no	—
86	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₈	SD₁₆	1	2	yes	—
88	C₂²×C₈	Q₁₆	1	2	yes	—
89	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	2	yes	—
91	C₂²×C₈	C₂×D₄	1	2	yes	—
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×Q₈	1	2	yes	—
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₈	#41	1	2	yes	—
97	C₂²×C₈	#41	1	1	yes	characteristic
98	C₂²×C₈	#41	1	1	yes	characteristic
99	C₄⋊Q₈	—	2	16	no	—
100	C₄⋊Q₈	—	4	8	no	—
101	C₄⋊Q₈	—	4	8	no	—
102	C₄⋊Q₈	—	4	8	no	—
103	#117	—	4	1	no	—
104	#117	—	4	1	no	—
105	#117	—	4	1	no	—
106	#117	—	4	1	no	—
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	4	4	no	—
112	C₂×C₄⋊C₄	—	4	4	no	—
113	C₂×C₄⋊C₄	—	4	4	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄⋊C₄	—	4	2	no	—
117	C₂×C₂²⋊C₄	—	4	1	no	—
118	C₂×C₄²	—	2	1	no	—
119	C₂×C₄²	—	2	1	no	—
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
123	C₂×C₄²	D₈	1	2	yes	—
124	C₂×C₄²	SD₁₆	1	2	yes	—
125	M₃₂	—	2	4	no	—
126	M₃₂	—	2	4	no	—
127	M₃₂	—	4	2	no	—
128	M₃₂	—	4	2	no	—
129	C₂×C₁₆	—	2	4	no	—
130	C₂×C₁₆	—	4	2	no	—
131	C₂×C₁₆	—	4	2	no	—
132	#108	—	2	4	no	—
133	#108	—	4	2	no	—
134	#107	—	2	4	no	—
135	#106	—	2	16	no	—
136	#106	—	2	4	no	—
137	#106	—	2	4	no	—
138	#106	—	2	4	no	—
139	#106	—	4	2	no	—
140	#106	C₂²⋊C₄	1	8	yes	—
141	C₄×C₈	—	2	4	no	—
142	C₄×C₈	—	2	4	no	—
143	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×Q₈	—	4	16	no	—
145	C₂×Q₈	—	8	16	no	—
146	C₂×Q₈	—	8	16	no	—
147	C₂×Q₈	—	8	16	no	—
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	4	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	8	8	no	—
157	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	#120	1	1	yes	characteristic
159	C₂²×C₄	#120	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	#124	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
165	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
167	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
168	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂²×C₄	#137	1	1	yes	characteristic
170	C₂²×C₄	#137	1	1	yes	characteristic
171	M₁₆	—	4	2	no	—
172	M₁₆	—	4	2	no	—
173	M₁₆	—	8	1	no	—
174	M₁₆	—	8	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	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	—	4	2	no	—
182	C₂×C₈	—	8	1	no	—
183	C₂×C₈	—	8	1	no	—
184	C₂×C₈	#103	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
187	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
188	C₂×C₈	C₄×Q₈	1	2	yes	—
189	C₂×C₈	#120	1	2	yes	—
190	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
191	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
192	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
193	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
194	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
195	C₂×C₈	#124	1	2	yes	—
196	C₂×C₈	#124	1	2	yes	—
197	C₂×C₈	#124	1	1	yes	characteristic
198	C₂×C₈	C₄⋊Q₈	1	2	yes	—
199	C₂×C₈	C₄⋊Q₈	1	2	yes	—
200	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	4	16	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	8	1	no	—
209	C₂²⋊C₄	—	8	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	8	1	no	—
214	C₄²	—	8	1	no	—
215	C₄²	#102	1	4	yes	—
216	C₁₆	—	4	4	no	—
217	C₂³	—	2	4	no	—
218	C₂³	#418	1	1	yes	characteristic
219	C₂³	#447	1	1	yes	characteristic
220	C₂³	#450	1	1	yes	characteristic
221	C₂³	#476	1	1	yes	characteristic
222	C₂³	#477	1	1	yes	characteristic
223	C₂³	#489	1	1	yes	characteristic
224	C₂³	#501	1	1	yes	characteristic
225	Q₈	—	8	32	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	8	8	no	—
232	C₂×C₄	—	8	8	no	—
233	C₂×C₄	—	8	8	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	#379	1	1	yes	characteristic
237	C₂×C₄	#391	1	1	yes	characteristic
238	C₂×C₄	#391	1	1	yes	characteristic
239	C₂×C₄	#393	1	1	yes	characteristic
240	C₂×C₄	#393	1	1	yes	characteristic
241	C₂×C₄	#395	1	1	yes	characteristic
242	C₂×C₄	#395	1	1	yes	characteristic
243	C₂×C₄	C₂×#102	1	1	yes	characteristic
244	C₂×C₄	C₂×#103	1	1	yes	characteristic
245	C₂×C₄	#419	1	1	yes	characteristic
246	C₂×C₄	#447	1	1	yes	characteristic
247	C₂×C₄	#448	1	1	yes	characteristic
248	C₂×C₄	#448	1	1	yes	characteristic
249	C₂×C₄	#450	1	1	yes	characteristic
250	C₂×C₄	#451	1	1	yes	characteristic
251	C₂×C₄	#451	1	1	yes	characteristic
252	C₂×C₄	#474	1	1	yes	characteristic
253	C₂×C₄	#474	1	1	yes	characteristic
254	C₂×C₄	#475	1	1	yes	characteristic
255	C₂×C₄	#475	1	1	yes	characteristic
256	C₂×C₄	#476	1	1	yes	characteristic
257	C₂×C₄	#477	1	1	yes	characteristic
258	C₂×C₄	#486	1	1	yes	characteristic
259	C₂×C₄	#487	1	1	yes	characteristic
260	C₂×C₄	#489	1	1	yes	characteristic
261	C₂×C₄	#498	1	1	yes	characteristic
262	C₂×C₄	#500	1	1	yes	characteristic
263	C₂×C₄	#501	1	1	yes	characteristic
264	C₈	—	2	4	no	—
265	C₈	—	8	2	no	—
266	C₈	#391	1	2	yes	—
267	C₈	#391	1	2	yes	—
268	C₈	#393	1	2	yes	—
269	C₈	#395	1	2	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#1786	1	1	yes	characteristic
277	C₂²	#1987	1	1	yes	characteristic
278	C₂²	#1993	1	1	yes	characteristic
279	C₂²	#2023	1	1	yes	characteristic
280	C₂²	#2024	1	1	yes	characteristic
281	C₂²	#2057	1	1	yes	characteristic
282	C₂²	#2058	1	1	yes	characteristic
283	C₂²	#2230	1	4	yes	—
284	C₄	—	2	8	no	—
285	C₄	—	8	8	no	—
286	C₄	—	16	2	no	—
287	C₄	#1446	1	1	yes	characteristic
288	C₄	#1787	1	1	yes	characteristic
289	C₄	#1988	1	1	yes	characteristic
290	C₄	#1990	1	1	yes	characteristic
291	C₄	#2023	1	1	yes	characteristic
292	C₄	#2024	1	1	yes	characteristic
293	C₄	#2057	1	1	yes	characteristic
294	C₄	#2058	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	#11384	1	1	yes	characteristic
300	C₂	#13272	1	2	yes	—
301	C₂	#13371	1	2	yes	—
302	C₂	#13457	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
11. C₂×#498	C₄	semidirect
41. C₂×M₃₂	Q₈	semidirect
48. C₂×#108	Q₈	semidirect
51. C₂×#107	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#498	2
C₂×#1386	2
C₂×#1383	2
C₂×#1376	2
#10859	2