Subgroups — #483564

Order 512.

457 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	#33476	C₂	1	1	yes	maximal, characteristic, complemented
2	#33438	C₂	1	1	yes	maximal, characteristic
3	#17957	C₂	1	1	yes	maximal, characteristic
4	#12058	C₂	1	2	yes	maximal, complemented
5	#12057	C₂	1	2	yes	maximal, complemented
6	#12056	C₂	1	2	yes	maximal, complemented
7	SD₁₆⋊C₁₆	C₂	1	2	yes	maximal, complemented
8	C₂×#1370	C₂	1	1	yes	maximal, characteristic, complemented
9	#11606	C₂	1	1	yes	maximal, characteristic, complemented
10	#11582	C₂	1	1	yes	maximal, characteristic
11	#11402	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#504	—	2	1	no	—
13	#3167	C₄	1	1	yes	characteristic
14	#2960	C₂²	1	1	yes	characteristic
15	#2931	C₂²	1	1	yes	characteristic
16	#2183	—	2	2	no	—
17	#2183	C₂²	1	2	yes	—
18	Q₈×C₁₆	C₂²	1	2	yes	—
19	#2169	C₂²	1	2	yes	—
20	#2168	—	2	2	no	—
21	D₄×C₁₆	C₂²	1	2	yes	—
22	#2151	—	2	1	no	—
23	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	#2113	—	2	1	no	—
26	#2108	—	2	1	no	—
27	C₄×M₃₂	C₂²	1	1	yes	characteristic
28	C₄×M₃₂	C₂²	1	1	yes	characteristic
29	#1590	C₄	1	2	yes	—
30	#1590	C₄	1	2	yes	—
31	C₈×SD₁₆	C₂²	1	2	yes	—
32	C₈×SD₁₆	C₂²	1	2	yes	—
33	#1566	C₄	1	1	yes	characteristic
34	#1493	C₄	1	1	yes	characteristic
35	#1492	C₄	1	1	yes	characteristic
36	C₂×C₈²	C₂²	1	1	yes	characteristic
37	#1370	C₂²	1	1	yes	characteristic
38	#1370	C₂²	1	1	yes	characteristic
39	#1370	C₂²	1	1	yes	characteristic
40	#1370	C₂²	1	1	yes	characteristic
41	Q₈⋊C₁₆	C₂²	1	2	yes	—
42	Q₈⋊C₁₆	C₂²	1	2	yes	—
43	D₄⋊C₁₆	C₂²	1	2	yes	—
44	D₄⋊C₁₆	C₂²	1	2	yes	—
45	#1312	C₂²	1	2	yes	—
46	#1312	C₂²	1	2	yes	—
47	C₂×#131	—	2	1	no	—
48	#533	C₂×C₄	1	1	yes	characteristic
49	#532	C₂×C₄	1	1	yes	characteristic
50	#504	—	4	4	no	—
51	C₂×M₃₂	—	2	1	no	—
52	C₂×M₃₂	—	2	2	no	—
53	C₂×M₃₂	D₄	1	1	yes	characteristic
54	C₂×M₃₂	D₄	1	1	yes	characteristic
55	C₂²×C₁₆	—	2	1	no	—
56	C₂²×C₁₆	—	2	1	no	—
57	C₂²×C₁₆	—	2	2	no	—
58	#498	C₈	1	1	yes	characteristic
59	#492	C₈	1	1	yes	characteristic
60	#487	C₈	1	1	yes	characteristic
61	#486	C₈	1	1	yes	characteristic
62	#465	C₈	1	2	yes	—
63	#465	C₈	1	2	yes	—
64	#446	—	2	1	no	—
65	#446	C₂×C₄	1	2	yes	—
66	C₈×Q₈	—	2	1	no	—
67	C₈×Q₈	C₂³	1	2	yes	—
68	C₄×SD₁₆	C₂×C₄	1	2	yes	—
69	C₄×SD₁₆	C₂×C₄	1	2	yes	—
70	#436	—	2	1	no	—
71	#436	C₂×C₄	1	2	yes	—
72	C₈×D₄	—	2	1	no	—
73	C₈×D₄	C₂³	1	2	yes	—
74	#424	—	2	1	no	—
75	#423	C₂×C₄	1	1	yes	characteristic
76	#423	C₂×C₄	1	1	yes	characteristic
77	#408	—	2	1	no	—
78	#405	—	2	1	no	—
79	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
82	C₄⋊C₁₆	—	2	2	no	—
83	C₄⋊C₁₆	—	2	2	no	—
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	—	2	1	no	—
86	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
87	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
88	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
89	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
90	C₂²⋊C₁₆	—	2	2	no	—
91	C₂²⋊C₁₆	—	2	2	no	—
92	#346	—	2	2	no	—
93	#346	—	2	1	no	—
94	#346	D₄	1	2	yes	—
95	C₄×C₁₆	—	2	2	no	—
96	C₄×C₁₆	C₂³	1	2	yes	—
97	C₄×C₁₆	C₂³	1	2	yes	—
98	#334	C₂×C₄	1	2	yes	—
99	#334	C₂×C₄	1	2	yes	—
100	Q₈⋊C₈	C₂×C₄	1	2	yes	—
101	Q₈⋊C₈	C₂×C₄	1	2	yes	—
102	D₄⋊C₈	C₂×C₄	1	2	yes	—
103	D₄⋊C₈	C₂×C₄	1	2	yes	—
104	C₈²	C₂×C₄	1	1	yes	characteristic
105	C₈²	C₂×C₄	1	1	yes	characteristic
106	C₈²	C₂³	1	1	yes	characteristic
107	C₈²	C₂³	1	1	yes	characteristic
108	C₂×#41	—	2	1	no	—
109	C₂×SD₁₆	C₂×C₈	1	2	yes	—
110	C₂×SD₁₆	C₂×C₈	1	2	yes	—
111	#131	—	4	2	no	—
112	#131	—	4	2	no	—
113	C₂×M₁₆	—	2	1	no	—
114	C₂×M₁₆	—	2	1	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₈	#41	1	1	yes	characteristic
122	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
123	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
125	#127	C₂×C₈	1	1	yes	characteristic
126	#125	—	2	1	no	—
127	#124	—	2	1	no	—
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	C₂×C₈	1	2	yes	—
130	#121	—	2	1	no	—
131	#121	C₂×C₈	1	2	yes	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	C₂²×C₄	1	2	yes	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	C₂²×C₄	1	2	yes	—
136	#117	—	2	1	no	—
137	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
138	M₃₂	—	2	2	no	—
139	M₃₂	—	2	2	no	—
140	M₃₂	—	4	4	no	—
141	M₃₂	—	4	2	no	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	4	4	no	—
148	C₂×C₁₆	—	4	2	no	—
149	C₂×C₁₆	—	4	2	no	—
150	C₂×C₁₆	C₂×D₄	1	2	yes	—
151	C₂×C₁₆	C₂×D₄	1	2	yes	—
152	#106	C₂×C₈	1	2	yes	—
153	#106	C₂×C₈	1	2	yes	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	C₂²×C₄	1	2	yes	—
159	C₄⋊C₈	C₂²×C₄	1	2	yes	—
160	#103	C₂×C₈	1	2	yes	—
161	#103	C₂×C₈	1	2	yes	—
162	#102	C₂×C₈	1	2	yes	—
163	#102	C₂×C₈	1	2	yes	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	#97	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	C₂×C₈	1	1	yes	characteristic
170	C₄×C₈	C₂×C₈	1	1	yes	characteristic
171	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
172	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
173	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
174	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
175	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
176	C₄×C₈	C₂×D₄	1	1	yes	characteristic
177	C₄×C₈	#41	1	1	yes	characteristic
178	C₄×C₈	C₂⁴	1	1	yes	characteristic
179	#41	—	4	2	no	—
180	#41	—	4	2	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	C₂²×C₈	1	2	yes	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	C₂²×C₈	1	2	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
191	C₂²×C₄	#131	1	1	yes	characteristic
192	SD₁₆	—	2	4	no	—
193	SD₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	4	4	no	—
197	C₂×C₈	—	2	1	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	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	C₄×D₄	1	1	yes	characteristic
208	C₂×C₈	C₄×D₄	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
213	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
214	C₂×C₈	#137	1	2	yes	—
215	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₈	C₂×#41	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	2	1	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	C₂²×C₈	1	2	yes	—
224	C₄⋊C₄	C₂²×C₈	1	2	yes	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	C₄×D₄	1	1	yes	characteristic
230	C₄²	C₂²×C₈	1	1	yes	characteristic
231	C₄²	#131	1	1	yes	characteristic
232	C₄²	C₂³×C₄	1	1	yes	characteristic
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	4	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	4	1	no	—
238	C₁₆	—	4	1	no	—
239	C₁₆	—	4	1	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₈×D₄	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₈×D₄	1	1	yes	characteristic
260	C₂×C₄	C₈×D₄	1	1	yes	characteristic
261	C₂×C₄	C₈×D₄	1	1	yes	characteristic
262	C₂×C₄	#440	1	2	yes	—
263	C₂×C₄	#444	1	2	yes	—
264	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
265	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×#131	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	#440	1	2	yes	—
277	C₈	#504	1	2	yes	complemented
278	C₈	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
279	C₈	C₂×#137	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	SD₁₆⋊C₈	1	2	yes	—
286	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	SD₁₆⋊C₈	1	2	yes	—
294	C₄	C₂×#440	1	1	yes	characteristic
295	C₄	C₂×#444	1	1	yes	characteristic
296	C₄	C₂×#504	1	1	yes	characteristic
297	C₄	#3407	1	1	yes	characteristic
298	C₂	—	2	2	no	—
299	C₂	—	8	2	no	—
300	C₂	C₂×SD₁₆⋊C₈	1	1	yes	characteristic
301	C₂	#33438	1	1	yes	characteristic
302	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33476	C₂	semidirect
4. #12058	C₂	semidirect
5. #12057	C₂	semidirect
6. #12056	C₂	semidirect
7. SD₁₆⋊C₁₆	C₂	semidirect
8. C₂×#1370	C₂	semidirect
9. #11606	C₂	semidirect
11. #11402	C₂	semidirect
277. C₈	#504	semidirect

Maximal subgroups

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

subgroup	index
#33476	2
#33438	2
#17957	2
#12058	2
#12057	2
#12056	2
SD₁₆⋊C₁₆	2
C₂×#1370	2
#11606	2
#11582	2
#11402	2