Subgroups — #146817

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20331	C₂	1	1	yes	maximal, characteristic
2	#12554	C₂	1	2	yes	maximal
3	C₂×#1378	C₂	1	2	yes	maximal
4	C₂×#1375	C₂	1	1	yes	maximal, characteristic
5	C₂×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂×#405	C₄	1	1	yes	characteristic
9	C₂×#405	C₄	1	1	yes	characteristic
10	C₂×#405	C₂²	1	1	yes	characteristic
11	C₂×#405	C₂²	1	1	yes	characteristic
12	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
13	C₂²×#97	C₂²	1	1	yes	characteristic
14	C₄×#117	C₄	1	1	yes	characteristic
15	#1726	—	2	1	no	—
16	#1726	—	2	1	no	—
17	C₄×#97	—	2	1	no	—
18	C₄×#97	—	2	1	no	—
19	#1379	C₄	1	8	yes	complemented
20	#1378	—	2	4	no	—
21	#1375	—	2	1	no	—
22	#1375	—	2	1	no	—
23	#1375	—	2	1	no	—
24	#1375	—	2	1	no	—
25	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	C₂×C₄	1	1	yes	characteristic
30	C₂×#117	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
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	—	2	2	no	—
42	C₂×M₃₂	—	2	2	no	—
43	C₂×M₃₂	C₂×C₄	1	4	yes	—
44	C₂²×C₁₆	—	2	2	no	—
45	C₂²×C₁₆	C₂×C₄	1	4	yes	—
46	#405	—	2	2	no	—
47	#405	—	2	2	no	—
48	#405	—	2	2	no	—
49	#405	—	2	1	no	—
50	#405	—	2	1	no	—
51	#405	—	2	2	no	—
52	#405	C₈	1	4	yes	—
53	#405	C₂×C₄	1	4	yes	—
54	#405	D₄	1	2	yes	—
55	#405	Q₈	1	2	yes	—
56	C₄×M₁₆	—	2	2	no	—
57	C₄×M₁₆	C₈	1	4	yes	—
58	C₂×#97	—	2	1	no	—
59	C₂×#97	—	2	1	no	—
60	C₂×#97	—	2	1	no	—
61	C₂×#97	—	2	1	no	—
62	C₂×#97	—	2	2	no	—
63	C₂×#97	—	2	2	no	—
64	C₂×#97	—	2	1	no	—
65	C₂×#97	D₄	1	1	yes	characteristic
66	C₂×#97	D₄	1	1	yes	characteristic
67	C₂×#97	D₄	1	1	yes	characteristic
68	C₂×#97	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	—	2	1	no	—
70	C₂×C₄×C₈	—	2	1	no	—
71	C₂×C₄×C₈	—	2	1	no	—
72	C₂×C₄×C₈	—	2	1	no	—
73	C₂×C₄×C₈	—	2	1	no	—
74	C₄×C₄⋊C₄	—	2	2	no	—
75	C₄×C₂²⋊C₄	—	2	2	no	—
76	#376	—	2	1	no	—
77	C₄³	—	2	1	no	—
78	C₂³×C₄	—	2	1	no	—
79	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×M₁₆	—	2	1	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂×M₁₆	—	2	1	no	—
83	C₂×M₁₆	—	2	1	no	—
84	C₂×M₁₆	—	2	1	no	—
85	C₂×M₁₆	—	2	1	no	—
86	C₂×M₁₆	C₄²	1	2	yes	—
87	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
88	C₂×M₁₆	C₂×C₈	1	2	yes	—
89	C₂×M₁₆	C₂×C₈	1	2	yes	—
90	C₂×M₁₆	C₂×C₈	1	2	yes	—
91	C₂×M₁₆	M₁₆	1	2	yes	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	—	2	1	no	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₈	1	2	yes	—
101	C₂²×C₈	M₁₆	1	2	yes	—
102	C₂²×C₈	C₂²×C₄	1	2	yes	—
103	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₈	#41	1	1	yes	characteristic
108	C₂²×C₈	#41	1	1	yes	characteristic
109	C₂²×C₈	#41	1	2	yes	—
110	C₂²×C₈	#41	1	2	yes	—
111	C₂²×C₈	#41	1	2	yes	—
112	#117	—	2	1	no	—
113	#117	—	2	1	no	—
114	#117	—	2	1	no	—
115	#117	—	2	1	no	—
116	#117	C₄²	1	2	yes	—
117	#117	C₄⋊C₄	1	2	yes	—
118	#117	C₂×C₈	1	2	yes	—
119	#117	M₁₆	1	2	yes	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	M₁₆	1	2	yes	—
122	C₂×C₂²⋊C₄	—	2	1	no	—
123	C₂×C₂²⋊C₄	M₁₆	1	2	yes	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	—	2	4	no	—
130	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₈	1	2	yes	—
133	C₂×C₄²	M₁₆	1	2	yes	—
134	M₃₂	—	2	4	no	—
135	M₃₂	—	2	4	no	—
136	M₃₂	—	4	2	no	—
137	M₃₂	—	4	2	no	—
138	M₃₂	—	4	2	no	—
139	M₃₂	—	4	2	no	—
140	M₃₂	—	4	2	no	—
141	M₃₂	—	4	2	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	2	4	no	—
144	C₂×C₁₆	—	2	4	no	—
145	C₂×C₁₆	—	2	4	no	—
146	C₂×C₁₆	—	4	2	no	—
147	C₂×C₁₆	—	4	2	no	—
148	#97	—	2	4	no	—
149	#97	—	2	4	no	—
150	#97	—	2	4	no	—
151	#97	—	2	2	no	—
152	#97	—	2	2	no	—
153	#97	—	2	2	no	—
154	#97	—	2	2	no	—
155	#97	—	2	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	4	no	—
158	C₄×C₈	—	2	4	no	—
159	C₄×C₈	—	2	4	no	—
160	C₄×C₈	—	2	4	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	4	no	—
164	#95	—	2	4	no	—
165	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	8	2	no	—
173	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	#117	1	1	yes	characteristic
177	C₂²×C₄	#117	1	1	yes	characteristic
178	C₂²×C₄	#117	1	1	yes	characteristic
179	C₂²×C₄	#117	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
185	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
186	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
187	M₁₆	—	2	2	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	2	2	no	—
193	M₁₆	C₄×C₈	1	4	yes	—
194	M₁₆	C₄⋊C₈	1	4	yes	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂²⋊C₈	1	2	yes	—
210	C₂×C₈	C₂²⋊C₈	1	2	yes	—
211	C₂×C₈	#117	1	2	yes	—
212	C₂×C₈	#117	1	2	yes	—
213	C₂×C₈	C₄×D₄	1	2	yes	—
214	C₂×C₈	C₄×D₄	1	2	yes	—
215	C₂×C₈	C₄×D₄	1	2	yes	—
216	C₂×C₈	C₄×Q₈	1	2	yes	—
217	C₂×C₈	#121	1	1	yes	characteristic
218	C₂×C₈	#121	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
221	C₂×C₈	#123	1	2	yes	—
222	C₂×C₈	#123	1	2	yes	—
223	C₂×C₈	#124	1	2	yes	—
224	C₂×C₈	#125	1	2	yes	—
225	C₂×C₈	#126	1	2	yes	—
226	C₂×C₈	#126	1	2	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	#97	1	4	yes	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	C₄⋊C₈	1	4	yes	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	8	4	no	—
244	C₄²	C₂²⋊C₈	1	2	yes	—
245	C₄²	C₂²⋊C₈	1	2	yes	—
246	C₁₆	—	4	4	no	—
247	C₁₆	—	4	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	C₄×M₁₆	1	1	yes	characteristic
250	C₂³	#407	1	1	yes	characteristic
251	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
252	C₂³	#431	1	1	yes	characteristic
253	C₂³	#432	1	1	yes	characteristic
254	C₂³	#432	1	1	yes	characteristic
255	C₂³	#432	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	#376	1	1	yes	characteristic
267	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₄	#380	1	1	yes	characteristic
270	C₂×C₄	#381	1	1	yes	characteristic
271	C₂×C₄	#382	1	1	yes	characteristic
272	C₂×C₄	#383	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
274	C₂×C₄	C₂×#97	1	1	yes	characteristic
275	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
278	C₂×C₄	#407	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
280	C₂×C₄	#423	1	1	yes	characteristic
281	C₂×C₄	#423	1	1	yes	characteristic
282	C₂×C₄	#431	1	1	yes	characteristic
283	C₂×C₄	#431	1	1	yes	characteristic
284	C₂×C₄	#431	1	1	yes	characteristic
285	C₂×C₄	#431	1	1	yes	characteristic
286	C₂×C₄	#431	1	1	yes	characteristic
287	C₂×C₄	#431	1	1	yes	characteristic
288	C₂×C₄	#431	1	1	yes	characteristic
289	C₂×C₄	#432	1	1	yes	characteristic
290	C₂×C₄	#432	1	1	yes	characteristic
291	C₂×C₄	#432	1	1	yes	characteristic
292	C₂×C₄	#432	1	1	yes	characteristic
293	C₂×C₄	#432	1	1	yes	characteristic
294	C₈	—	2	4	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	#387	1	2	yes	—
299	C₈	#387	1	2	yes	—
300	C₈	#388	1	2	yes	—
301	C₈	#388	1	2	yes	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	#1745	1	1	yes	characteristic
310	C₂²	#1749	1	1	yes	characteristic
311	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
312	C₂²	#1797	1	1	yes	characteristic
313	C₂²	#1817	1	1	yes	characteristic
314	C₂²	#1831	1	1	yes	characteristic
315	C₂²	#1839	1	1	yes	characteristic
316	C₄	—	4	4	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	#1434	1	1	yes	characteristic
321	C₄	#1744	1	1	yes	characteristic
322	C₄	C₄×C₂²⋊C₈	1	1	yes	characteristic
323	C₄	#1767	1	1	yes	characteristic
324	C₄	#1797	1	1	yes	characteristic
325	C₄	#1817	1	1	yes	characteristic
326	C₄	#1831	1	1	yes	characteristic
327	C₄	#1839	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	#10795	1	1	yes	characteristic
335	C₂	#12609	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
19. #1379	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20331	2
#12554	2
C₂×#1378	2
C₂×#1375	2