Subgroups — #203727

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20826	C₂	1	1	yes	maximal, characteristic
2	#17808	C₂	1	2	yes	maximal
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#17323	C₂	1	1	yes	maximal, characteristic
5	#9609	C₂	1	4	yes	maximal, complemented
6	#9608	C₂	1	4	yes	maximal, complemented
7	#7936	C₂	1	2	yes	maximal
8	#2920	C₂²	1	1	yes	characteristic
9	#2920	C₂²	1	1	yes	characteristic
10	#2920	C₂²	1	2	yes	—
11	C₂×#431	C₂²	1	2	yes	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#1848	C₄	1	2	yes	—
14	#1837	C₄	1	2	yes	—
15	#1810	C₄	1	2	yes	—
16	#1810	C₄	1	2	yes	—
17	C₄×#108	C₂²	1	2	yes	—
18	C₄×#108	C₂²	1	2	yes	—
19	#1574	—	2	4	no	—
20	#1572	—	2	4	no	—
21	#1565	C₂²	1	8	yes	—
22	#1564	—	2	1	no	—
23	#1564	—	2	1	no	—
24	#1564	—	2	1	no	—
25	#1564	—	2	1	no	—
26	C₂×#335	C₂²	1	2	yes	—
27	C₂×#334	C₂²	1	2	yes	—
28	#1452	C₂²	1	2	yes	—
29	#1451	—	2	1	no	—
30	#1451	—	2	1	no	—
31	#1450	C₂²	1	2	yes	—
32	C₈×M₁₆	—	2	1	no	—
33	C₈×M₁₆	—	2	1	no	—
34	#1283	C₂²	1	8	yes	—
35	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄²	C₂³	1	1	yes	characteristic
38	#432	—	2	1	no	—
39	#432	—	2	1	no	—
40	#432	—	2	2	no	—
41	#432	—	2	2	no	—
42	#432	D₄	1	2	yes	—
43	#432	Q₈	1	2	yes	—
44	#431	—	2	2	no	—
45	#431	—	2	2	no	—
46	#431	—	2	1	no	—
47	#431	—	2	1	no	—
48	#431	—	2	2	no	—
49	#431	—	2	2	no	—
50	#431	—	2	2	no	—
51	#431	—	2	2	no	—
52	#431	C₂³	1	2	yes	—
53	#431	C₂³	1	2	yes	—
54	C₂×#108	C₂×C₄	1	4	yes	—
55	C₂×#108	C₂×C₄	1	4	yes	—
56	#423	—	2	1	no	—
57	#423	C₂×C₄	1	4	yes	—
58	#423	D₄	1	2	yes	—
59	C₂×C₄⋊C₈	C₂³	1	4	yes	—
60	#407	—	2	1	no	—
61	#407	C₂×C₄	1	4	yes	—
62	#407	Q₈	1	2	yes	—
63	C₂×C₂²⋊C₈	—	2	2	no	—
64	C₄×M₁₆	—	2	1	no	—
65	C₄×M₁₆	—	2	1	no	—
66	C₄×M₁₆	C₂³	1	4	yes	—
67	C₂×C₄×C₈	—	2	2	no	—
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	D₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
80	#341	C₂×C₄	1	4	yes	—
81	#341	C₂×C₄	1	4	yes	—
82	#335	—	2	8	no	—
83	#335	—	2	2	no	—
84	#335	—	2	2	no	—
85	#334	—	2	8	no	—
86	#334	—	2	2	no	—
87	#334	—	2	2	no	—
88	#322	—	2	2	no	—
89	#322	—	2	2	no	—
90	#322	—	2	2	no	—
91	C₈²	—	2	2	no	—
92	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
94	C₂³×C₄	#41	1	1	yes	characteristic
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	C₂²×C₄	1	8	yes	—
98	C₂²×C₈	—	2	4	no	—
99	C₂²×C₈	—	2	4	no	—
100	C₂²×C₈	—	4	2	no	—
101	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
120	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₄²	#41	1	1	yes	characteristic
122	C₂×C₄²	#41	1	1	yes	characteristic
123	C₂×C₄²	#41	1	1	yes	characteristic
124	C₂×C₄²	C₂⁴	1	1	yes	characteristic
125	#108	—	2	8	no	—
126	#108	—	4	4	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	C₂×D₄	1	4	yes	—
135	C₂²⋊C₈	—	2	4	no	—
136	C₂²⋊C₈	—	2	4	no	—
137	C₂²⋊C₈	—	2	2	no	—
138	C₂²⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	—	2	2	no	—
140	C₂²⋊C₈	—	4	2	no	—
141	C₂²⋊C₈	—	4	2	no	—
142	C₂²⋊C₈	C₂×Q₈	1	4	yes	—
143	#97	—	2	2	no	—
144	#97	—	2	2	no	—
145	#97	—	2	4	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	4	no	—
149	C₄×C₈	—	4	2	no	—
150	C₄×C₈	—	4	2	no	—
151	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₈	C₂×D₄	1	1	yes	characteristic
160	C₄×C₈	C₂×D₄	1	1	yes	characteristic
161	C₄×C₈	C₂×D₄	1	1	yes	characteristic
162	C₄×C₈	C₂×D₄	1	1	yes	characteristic
163	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
164	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
165	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
166	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
167	C₂⁴	C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
176	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
178	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
179	C₂²×C₄	#121	1	1	yes	characteristic
180	C₂²×C₄	#121	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
183	C₂²×C₄	#125	1	1	yes	characteristic
184	C₂²×C₄	#125	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×#41	1	1	yes	characteristic
190	M₁₆	—	2	8	no	—
191	M₁₆	—	2	8	no	—
192	M₁₆	—	4	4	no	—
193	M₁₆	—	4	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	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	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	—	8	4	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₈	#131	1	4	yes	—
219	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₄²	C₄×D₄	1	1	yes	characteristic
222	C₄²	C₄×D₄	1	1	yes	characteristic
223	C₄²	#121	1	1	yes	characteristic
224	C₄²	#121	1	1	yes	characteristic
225	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
226	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
227	C₄²	#125	1	1	yes	characteristic
228	C₄²	#125	1	1	yes	characteristic
229	C₄²	C₄⋊Q₈	1	1	yes	characteristic
230	C₄²	C₄⋊Q₈	1	1	yes	characteristic
231	C₄²	C₂²×D₄	1	1	yes	characteristic
232	C₄²	C₂²×D₄	1	1	yes	characteristic
233	C₄²	C₂²×Q₈	1	1	yes	characteristic
234	C₄²	C₂×#41	1	1	yes	characteristic
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	#389	1	1	yes	characteristic
239	C₂³	#389	1	1	yes	characteristic
240	C₂³	C₂×C₄×Q₈	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	#389	1	1	yes	characteristic
261	C₂×C₄	#389	1	1	yes	characteristic
262	C₂×C₄	#389	1	1	yes	characteristic
263	C₂×C₄	#389	1	1	yes	characteristic
264	C₂×C₄	#389	1	1	yes	characteristic
265	C₂×C₄	#389	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
268	C₂×C₄	C₂×#121	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂×C₄	C₂×#125	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
272	C₂×C₄	C₂×#131	1	2	yes	—
273	C₂×C₄	#568	1	2	yes	—
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	4	4	no	—
281	C₈	—	4	8	no	—
282	C₈	—	4	4	no	—
283	C₈	—	8	4	no	—
284	C₈	#424	1	2	yes	—
285	C₈	#424	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	C₂×#389	1	1	yes	characteristic
296	C₂²	#2962	1	2	yes	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	#1953	1	2	yes	—
306	C₄	C₂×#424	1	1	yes	characteristic
307	C₄	#2905	1	1	yes	characteristic
308	C₄	#2935	1	2	yes	—
309	C₄	#3067	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	4	1	no	—
312	C₂	—	4	1	no	—
313	C₂	#21145	1	1	yes	characteristic
314	C₂	#21181	1	1	yes	characteristic
315	C₂	C₂×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #9609	C₂	semidirect
6. #9608	C₂	semidirect

Maximal subgroups

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

subgroup	index
#20826	2
#17808	2
C₂×#1564	2
#17323	2
#9609	2
#9608	2
#7936	2