Subgroups — #134374

Order 512.

420 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
8	C₄²×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	—	2	1	no	—
11	C₂×C₄⋊C₁₆	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	2	yes	—
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×C₄⋊C₄	—	2	1	no	—
16	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
17	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₈	—	2	1	no	—
19	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
20	C₄²×C₈	—	2	1	no	—
21	C₄²×C₈	C₂²	1	1	yes	characteristic
22	#1374	—	2	1	no	—
23	Q₈⋊C₁₆	—	2	4	no	—
24	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
25	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
26	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	—	2	2	no	—
28	C₂²×C₁₆	—	2	2	no	—
29	C₈×Q₈	—	2	2	no	—
30	C₈×Q₈	—	2	2	no	—
31	C₈×Q₈	—	2	1	no	—
32	C₈×Q₈	—	2	1	no	—
33	C₈×Q₈	D₄	1	2	yes	—
34	C₈×Q₈	Q₈	1	2	yes	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₄⋊C₄	—	2	1	no	—
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
54	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
55	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄³	—	2	1	no	—
57	C₄³	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₁₆	—	2	4	no	—
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	—	2	2	no	—
61	C₄⋊C₁₆	—	4	2	no	—
62	C₄⋊C₁₆	—	4	2	no	—
63	C₄×C₁₆	—	2	2	no	—
64	C₄×C₁₆	—	2	2	no	—
65	#336	—	2	1	no	—
66	#336	—	2	1	no	—
67	#336	—	2	1	no	—
68	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	—	2	1	no	—
71	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	1	no	—
79	C₄×Q₈	—	2	1	no	—
80	C₄×Q₈	C₁₆	1	4	yes	complemented
81	C₄×Q₈	C₄⋊C₄	1	2	yes	—
82	C₄×Q₈	M₁₆	1	2	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄²	—	2	1	no	—
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	—	2	1	no	—
94	C₂×C₄²	—	2	1	no	—
95	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
99	C₂×C₄²	M₁₆	1	1	yes	characteristic
100	C₂×C₄²	M₁₆	1	1	yes	characteristic
101	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	4	2	no	—
104	C₂×C₁₆	—	4	2	no	—
105	C₂×C₁₆	—	4	2	no	—
106	C₂×C₁₆	—	4	2	no	—
107	C₂×C₁₆	—	4	2	no	—
108	C₄⋊C₈	—	2	4	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₈	M₁₆	1	2	yes	—
115	C₄×C₈	—	2	4	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	—	4	2	no	—
125	C₄×C₈	—	4	1	no	—
126	C₄×C₈	—	4	1	no	—
127	C₄×C₈	—	4	2	no	—
128	C₄×C₈	SD₁₆	1	2	yes	—
129	C₄×C₈	Q₁₆	1	2	yes	—
130	C₄×C₈	C₂×D₄	1	1	yes	characteristic
131	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
132	C₄×C₈	#41	1	1	yes	characteristic
133	C₄×C₈	#41	1	1	yes	characteristic
134	#95	—	2	1	no	—
135	#95	—	2	1	no	—
136	#95	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	C₄⋊C₈	1	2	yes	—
141	C₂×Q₈	C₂×C₁₆	1	2	yes	—
142	C₂×Q₈	M₃₂	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
148	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	#103	1	2	yes	—
168	C₂×C₈	#121	1	1	yes	characteristic
169	C₂×C₈	#121	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
173	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
174	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₈	#137	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
184	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
185	C₄⋊C₄	M₃₂	1	2	yes	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	#103	1	2	yes	—
201	C₄²	#104	1	2	yes	—
202	C₄²	C₂×C₁₆	1	2	yes	—
203	C₄²	M₃₂	1	2	yes	—
204	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₄²	#117	1	1	yes	characteristic
206	C₄²	C₂×M₁₆	1	1	yes	characteristic
207	C₄²	C₂×M₁₆	1	1	yes	characteristic
208	C₁₆	—	4	4	no	—
209	C₁₆	—	8	2	no	—
210	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
211	Q₈	—	2	2	no	—
212	Q₈	—	2	2	no	—
213	Q₈	—	2	2	no	—
214	Q₈	C₄⋊C₁₆	1	4	yes	complemented
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	Q₈⋊C₈	1	2	yes	—
237	C₂×C₄	Q₈⋊C₈	1	2	yes	—
238	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
239	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
240	C₂×C₄	#350	1	2	yes	—
241	C₂×C₄	#380	1	1	yes	characteristic
242	C₂×C₄	#407	1	1	yes	characteristic
243	C₂×C₄	C₂×#103	1	1	yes	characteristic
244	C₂×C₄	#417	1	1	yes	characteristic
245	C₂×C₄	C₂×#104	1	1	yes	characteristic
246	C₂×C₄	#421	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
248	C₂×C₄	#423	1	1	yes	characteristic
249	C₂×C₄	#431	1	1	yes	characteristic
250	C₂×C₄	#432	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
252	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
253	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
254	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	2	no	—
261	C₈	#460	1	1	yes	characteristic
262	C₈	#462	1	1	yes	characteristic
263	C₈	#475	1	1	yes	characteristic
264	C₈	#477	1	1	yes	characteristic
265	C₂²	C₂×Q₈⋊C₈	1	1	yes	characteristic
266	C₂²	#1477	1	1	yes	characteristic
267	C₂²	#1797	1	1	yes	characteristic
268	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
269	C₂²	#2112	1	1	yes	characteristic
270	C₂²	C₂×#350	1	1	yes	characteristic
271	C₂²	#2117	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	2	no	—
281	C₄	Q₈⋊C₁₆	1	4	yes	complemented
282	C₄	#1487	1	1	yes	characteristic
283	C₄	#1489	1	1	yes	characteristic
284	C₄	#1803	1	1	yes	characteristic
285	C₄	#1806	1	1	yes	characteristic
286	C₄	C₂×C₄⋊C₁₆	1	1	yes	characteristic
287	C₄	#2150	1	1	yes	characteristic
288	C₄	#2162	1	1	yes	characteristic
289	C₄	#2163	1	1	yes	characteristic
290	C₂	#8564	1	1	yes	characteristic
291	C₂	C₂×Q₈⋊C₁₆	1	1	yes	characteristic
292	C₂	#11607	1	1	yes	characteristic
293	C₂	#11621	1	1	yes	characteristic
294	C₂	#11623	1	1	yes	characteristic
295	C₂	#12783	1	1	yes	characteristic
296	C₂	#12787	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
80. C₄×Q₈	C₁₆	semidirect
214. Q₈	C₄⋊C₁₆	semidirect
281. C₄	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Q₈	2
#12898	2
C₄×C₄⋊C₁₆	2
C₂×Q₈⋊C₁₆	2
#7167	2