Subgroups — #134656

Order 512.

349 conjugacy classes of subgroups.

The subgroup lattice has 271 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#131	C₂	1	1	yes	maximal, characteristic
2	#12787	C₂	1	2	yes	maximal
3	#11996	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	#7153	C₂	1	2	yes	maximal, complemented
6	C₈×#41	C₄	1	1	yes	characteristic
7	C₈×#41	C₄	1	1	yes	characteristic
8	C₄×#131	C₂²	1	1	yes	characteristic
9	#2150	C₂²	1	2	yes	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#350	—	2	2	no	—
12	#1450	—	2	1	no	—
13	#1450	C₂²	1	1	yes	characteristic
14	C₈×M₁₆	—	2	1	no	—
15	C₈×M₁₆	—	2	1	no	—
16	C₈×M₁₆	C₄	1	1	yes	characteristic
17	C₈×M₁₆	C₄	1	1	yes	characteristic
18	C₂×C₈²	—	2	1	no	—
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	#1379	—	2	2	no	—
21	#1367	—	2	1	no	—
22	#1367	—	2	1	no	—
23	#1367	—	2	1	no	—
24	#1367	—	2	1	no	—
25	C₂×#131	D₄	1	1	yes	characteristic
26	C₂×#131	Q₈	1	1	yes	characteristic
27	C₄×#41	C₂×C₄	1	1	yes	characteristic
28	C₂×M₃₂	—	2	2	no	—
29	C₂²×C₁₆	—	2	2	no	—
30	C₈×Q₈	—	2	1	no	—
31	C₈×Q₈	C₈	1	1	yes	characteristic
32	C₈×Q₈	C₈	1	1	yes	characteristic
33	C₈×D₄	—	2	1	no	—
34	C₈×D₄	—	2	1	no	—
35	C₈×D₄	—	2	1	no	—
36	C₈×D₄	—	2	2	no	—
37	C₈×D₄	C₈	1	1	yes	characteristic
38	C₈×D₄	C₈	1	1	yes	characteristic
39	#431	—	2	1	no	—
40	#431	—	2	1	no	—
41	#431	C₂×C₄	1	1	yes	characteristic
42	#431	C₂×C₄	1	1	yes	characteristic
43	#405	—	2	1	no	—
44	#405	—	2	1	no	—
45	#405	D₄	1	1	yes	characteristic
46	#405	Q₈	1	1	yes	characteristic
47	C₄×M₁₆	—	2	1	no	—
48	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	—	2	1	no	—
50	C₂×C₄×C₈	—	2	1	no	—
51	C₂×C₄×C₈	—	2	1	no	—
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₄⋊C₁₆	—	2	2	no	—
56	C₄⋊C₁₆	—	2	2	no	—
57	C₄⋊C₁₆	—	2	2	no	—
58	C₄⋊C₁₆	—	2	2	no	—
59	#350	—	4	4	no	—
60	#322	—	2	1	no	—
61	#322	—	2	1	no	—
62	#322	—	2	1	no	—
63	C₈²	—	2	1	no	—
64	C₈²	—	2	1	no	—
65	C₈²	—	2	1	no	—
66	C₈²	D₄	1	1	yes	characteristic
67	C₈²	D₄	1	1	yes	characteristic
68	C₈²	D₄	1	1	yes	characteristic
69	C₈²	D₄	1	1	yes	characteristic
70	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
71	#131	—	2	2	no	—
72	#131	—	2	2	no	—
73	#131	—	2	2	no	—
74	#131	SD₁₆	1	4	yes	—
75	C₂×M₁₆	—	2	1	no	—
76	C₂×M₁₆	—	2	1	no	—
77	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
78	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	M₁₆	1	1	yes	characteristic
86	C₂²×C₈	M₁₆	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
90	C₄×Q₈	M₁₆	1	1	yes	characteristic
91	C₄×D₄	—	2	1	no	—
92	C₄×D₄	—	2	1	no	—
93	C₄×D₄	C₂×C₈	1	1	yes	characteristic
94	C₄×D₄	M₁₆	1	1	yes	characteristic
95	#117	—	2	1	no	—
96	#117	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
99	M₃₂	—	4	2	no	—
100	M₃₂	—	4	2	no	—
101	C₂×C₁₆	—	4	2	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	—	4	2	no	—
104	C₂×C₁₆	—	4	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
110	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
111	C₂²⋊C₈	—	2	2	no	—
112	C₂²⋊C₈	—	2	1	no	—
113	C₂²⋊C₈	—	2	1	no	—
114	C₂²⋊C₈	—	2	1	no	—
115	C₂²⋊C₈	M₁₆	1	2	yes	—
116	#97	—	2	1	no	—
117	#97	—	2	1	no	—
118	#97	—	2	1	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₂×C₈	1	1	yes	characteristic
133	C₄×C₈	C₂×C₈	1	1	yes	characteristic
134	C₄×C₈	C₂×D₄	1	1	yes	characteristic
135	C₄×C₈	C₂×D₄	1	1	yes	characteristic
136	C₄×C₈	#41	1	1	yes	characteristic
137	C₄×C₈	#41	1	1	yes	characteristic
138	#41	—	2	1	no	—
139	#41	—	2	1	no	—
140	#41	#106	1	2	yes	—
141	#41	#108	1	2	yes	—
142	C₂×Q₈	C₄⋊C₈	1	1	yes	characteristic
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	C₄⋊C₈	1	1	yes	characteristic
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	2	4	no	—
155	M₁₆	#106	1	2	yes	—
156	M₁₆	#108	1	2	yes	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
177	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
178	C₂×C₈	#121	1	1	yes	characteristic
179	C₂×C₈	#121	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
182	C₂×C₈	C₂×SD₁₆	1	2	yes	—
183	C₂×C₈	#135	1	2	yes	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
187	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₄²	#117	1	1	yes	characteristic
196	C₄²	C₂²×C₈	1	1	yes	characteristic
197	C₄²	C₂×M₁₆	1	1	yes	characteristic
198	C₁₆	—	8	2	no	—
199	C₁₆	—	8	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	#423	1	1	yes	characteristic
202	Q₈	—	2	1	no	—
203	Q₈	#334	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	1	no	—
207	D₄	#334	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	#380	1	1	yes	characteristic
220	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
221	C₂×C₄	#407	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
223	C₂×C₄	C₂×#106	1	1	yes	characteristic
224	C₂×C₄	#427	1	1	yes	characteristic
225	C₂×C₄	C₂×#108	1	1	yes	characteristic
226	C₂×C₄	C₂×#108	1	1	yes	characteristic
227	C₂×C₄	#431	1	1	yes	characteristic
228	C₂×C₄	#432	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	—	4	1	no	—
239	C₈	—	4	1	no	—
240	C₈	—	4	1	no	—
241	C₈	#350	1	2	yes	complemented
242	C₈	#465	1	2	yes	—
243	C₈	#465	1	2	yes	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	#1566	1	1	yes	characteristic
250	C₂²	#1568	1	1	yes	characteristic
251	C₂²	#1797	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	C₂×#334	1	1	yes	characteristic
259	C₄	#1564	1	1	yes	characteristic
260	C₄	#1807	1	1	yes	characteristic
261	C₄	#1810	1	1	yes	characteristic
262	C₄	C₂×#350	1	1	yes	characteristic
263	C₄	#2117	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	4	2	no	—
267	C₂	#8595	1	1	yes	characteristic
268	C₂	#12787	1	1	yes	characteristic
269	C₂	#12792	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×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
3. #11996	C₂	semidirect
4. C₂×#1367	C₂	semidirect
5. #7153	C₂	semidirect
241. C₈	#350	semidirect

Maximal subgroups

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

subgroup	index
C₈×#131	2
#12787	2
#11996	2
C₂×#1367	2
#7153	2