Subgroups — #496746

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 328 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	#20826	C₂	1	1	yes	maximal, characteristic
3	C₂×#1848	C₂	1	1	yes	maximal, characteristic
4	#13179	C₂	1	4	yes	maximal, complemented
5	#12922	C₂	1	4	yes	maximal
6	#12910	C₂	1	2	yes	maximal, complemented
7	#12892	C₂	1	2	yes	maximal, complemented
8	#2920	—	2	1	no	—
9	C₂²×#108	—	2	1	no	—
10	C₂²×#108	C₂²	1	2	yes	—
11	C₂×#423	—	2	1	no	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#2163	C₄	1	2	yes	—
14	#2163	C₄	1	2	yes	—
15	#2163	C₄	1	2	yes	—
16	#2163	C₄	1	2	yes	—
17	C₂×#368	C₂²	1	8	yes	—
18	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₁₆	C₂²	1	2	yes	—
20	C₂×#346	C₂²	1	2	yes	—
21	#1848	—	2	1	no	—
22	#1848	—	2	1	no	—
23	#1848	C₂²	1	2	yes	—
24	#1848	C₂²	1	2	yes	—
25	#1848	C₂²	1	1	yes	characteristic
26	#1848	C₂²	1	1	yes	characteristic
27	#1837	—	2	1	no	—
28	#1810	—	2	1	no	—
29	#1810	C₂²	1	2	yes	—
30	C₄×#108	—	2	1	no	—
31	C₄×#108	C₂²	1	2	yes	—
32	#1381	C₂²	1	8	yes	—
33	C₂²×M₁₆	—	2	2	no	—
34	C₂³×C₈	D₄	1	1	yes	characteristic
35	C₂³×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄²	D₄	1	1	yes	characteristic
37	C₂²×C₁₆	C₂³	1	4	yes	—
38	#432	—	4	1	no	—
39	#432	—	4	1	no	—
40	#431	—	4	1	no	—
41	#431	—	4	1	no	—
42	C₂×#108	—	2	1	no	—
43	C₂×#108	—	2	1	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	—	4	4	no	—
48	C₂×#108	—	4	4	no	—
49	C₂×#108	C₂³	1	2	yes	—
50	C₂×#108	C₂³	1	2	yes	—
51	C₂×#108	C₂³	1	4	yes	—
52	#423	—	2	1	no	—
53	#423	—	2	1	no	—
54	#423	—	2	2	no	—
55	#423	—	2	2	no	—
56	#423	—	4	1	no	—
57	#423	—	4	1	no	—
58	C₂×C₄⋊C₈	—	4	1	no	—
59	C₂×C₄⋊C₈	—	4	1	no	—
60	#407	—	2	2	no	—
61	C₄×M₁₆	—	2	2	no	—
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	#368	—	4	8	no	—
75	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
76	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
77	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
78	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
79	#346	C₂×C₄	1	4	yes	—
80	#346	C₂×C₄	1	4	yes	—
81	#341	—	2	2	no	—
82	#341	—	2	2	no	—
83	C₂³×C₄	Q₁₆	1	1	yes	characteristic
84	C₂³×C₄	Q₁₆	1	1	yes	characteristic
85	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×M₁₆	—	2	4	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂×M₁₆	—	4	2	no	—
90	C₂×M₁₆	—	4	4	no	—
91	C₂×M₁₆	—	4	4	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	—	2	2	no	—
96	C₂²×C₈	—	4	2	no	—
97	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
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	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₈	C₂⁴	1	1	yes	characteristic
109	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄²	D₈	1	1	yes	characteristic
114	C₂×C₄²	D₈	1	1	yes	characteristic
115	C₂×C₄²	D₈	1	1	yes	characteristic
116	C₂×C₄²	D₈	1	1	yes	characteristic
117	C₂×C₄²	Q₁₆	1	1	yes	characteristic
118	C₂×C₄²	Q₁₆	1	1	yes	characteristic
119	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
121	C₂×C₁₆	—	2	4	no	—
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	C₂²×C₄	1	8	yes	—
124	#108	—	2	4	no	—
125	#108	—	2	8	no	—
126	#108	—	2	4	no	—
127	#108	—	4	4	no	—
128	#108	—	4	4	no	—
129	#108	—	4	2	no	—
130	#108	—	4	4	no	—
131	#108	—	4	2	no	—
132	C₄⋊C₈	—	4	2	no	—
133	C₄⋊C₈	—	4	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄⋊C₈	—	4	1	no	—
137	C₄⋊C₈	—	4	1	no	—
138	C₄⋊C₈	—	4	1	no	—
139	C₄⋊C₈	—	4	1	no	—
140	C₂²⋊C₈	—	4	2	no	—
141	C₂²⋊C₈	—	4	2	no	—
142	#97	—	4	2	no	—
143	C₄×C₈	—	4	2	no	—
144	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
157	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
158	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
159	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
160	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	#107	1	1	yes	characteristic
168	C₂²×C₄	#107	1	1	yes	characteristic
169	C₂²×C₄	#107	1	1	yes	characteristic
170	C₂²×C₄	#107	1	1	yes	characteristic
171	C₂²×C₄	#107	1	1	yes	characteristic
172	C₂²×C₄	#107	1	1	yes	characteristic
173	C₂²×C₄	#107	1	1	yes	characteristic
174	C₂²×C₄	#107	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
185	M₁₆	—	4	4	no	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	4	2	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	8	4	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	8	4	no	—
208	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
215	C₂×C₈	#137	1	2	yes	—
216	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
217	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
218	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
219	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
220	C₄²	#107	1	1	yes	characteristic
221	C₄²	#107	1	1	yes	characteristic
222	C₄²	#107	1	1	yes	characteristic
223	C₄²	#107	1	1	yes	characteristic
224	C₄²	#107	1	1	yes	characteristic
225	C₄²	#107	1	1	yes	characteristic
226	C₄²	#107	1	1	yes	characteristic
227	C₄²	#107	1	1	yes	characteristic
228	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
230	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
231	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
232	C₄²	C₂×D₈	1	1	yes	characteristic
233	C₄²	C₂×D₈	1	1	yes	characteristic
234	C₄²	C₂×D₈	1	1	yes	characteristic
235	C₄²	C₂×D₈	1	1	yes	characteristic
236	C₁₆	—	2	8	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	4	1	no	—
241	C₂³	C₂×#107	1	1	yes	characteristic
242	C₂³	C₂×#107	1	1	yes	characteristic
243	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	C₂×#107	1	1	yes	characteristic
267	C₂×C₄	C₂×#107	1	1	yes	characteristic
268	C₂×C₄	C₂×#107	1	1	yes	characteristic
269	C₂×C₄	C₂×#107	1	1	yes	characteristic
270	C₂×C₄	C₂×#107	1	1	yes	characteristic
271	C₂×C₄	C₂×#107	1	1	yes	characteristic
272	C₂×C₄	C₂×#107	1	1	yes	characteristic
273	C₂×C₄	C₂×#107	1	1	yes	characteristic
274	C₂×C₄	C₂×#107	1	1	yes	characteristic
275	C₂×C₄	C₂×#107	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₄	#535	1	1	yes	characteristic
278	C₂×C₄	#536	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
280	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₂×C₄	C₂×#137	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	8	4	no	—
287	C₈	—	8	2	no	—
288	C₈	—	8	2	no	—
289	C₈	#428	1	2	yes	—
290	C₈	#428	1	2	yes	—
291	C₈	#520	1	2	yes	—
292	C₈	#544	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	#2161	1	2	yes	—
303	C₂²	C₂²×#107	1	1	yes	characteristic
304	C₂²	#3209	1	1	yes	characteristic
305	C₂²	#3230	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	#2161	1	2	yes	—
316	C₄	#2317	1	1	yes	characteristic
317	C₄	C₂×#428	1	1	yes	characteristic
318	C₄	#3205	1	1	yes	characteristic
319	C₄	#3226	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	4	1	no	—
324	C₂	#21982	1	1	yes	characteristic
325	C₂	C₂×#2161	1	1	yes	characteristic
326	C₂	C₂×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13179	C₂	semidirect
6. #12910	C₂	semidirect
7. #12892	C₂	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
#20826	2
C₂×#1848	2
#13179	2
#12922	2
#12910	2
#12892	2