Subgroups — #134686

Order 512.

322 conjugacy classes of subgroups.

The subgroup lattice has 259 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17335	C₂	1	1	yes	maximal, characteristic
2	#12787	C₂	1	2	yes	maximal
3	#12004	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	#2965	C₄	1	1	yes	characteristic
7	#2965	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	#1455	—	2	1	no	—
13	#1453	—	2	1	no	—
14	#1451	C₄	1	1	yes	characteristic
15	#1451	C₄	1	1	yes	characteristic
16	#1450	C₂²	1	1	yes	characteristic
17	C₈×M₁₆	—	2	1	no	—
18	C₈×M₁₆	—	2	1	no	—
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	#1379	—	2	2	no	—
21	#1370	—	2	1	no	—
22	#1370	—	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	#446	—	2	1	no	—
31	#446	C₈	1	1	yes	characteristic
32	#446	C₈	1	1	yes	characteristic
33	#436	—	2	1	no	—
34	#436	C₈	1	1	yes	characteristic
35	#436	C₈	1	1	yes	characteristic
36	#435	—	2	1	no	—
37	#435	—	2	1	no	—
38	#435	—	2	2	no	—
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	—	2	1	no	—
49	C₄×M₁₆	—	2	1	no	—
50	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
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	#322	—	2	1	no	—
64	#322	—	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	—	4	2	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	2	1	no	—
76	C₂×M₁₆	—	2	1	no	—
77	C₂×M₁₆	—	2	1	no	—
78	C₂×M₁₆	—	2	1	no	—
79	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
80	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₂²×C₈	M₁₆	1	1	yes	characteristic
85	C₂²×C₈	M₁₆	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
89	C₄×Q₈	M₁₆	1	1	yes	characteristic
90	C₄×D₄	—	2	1	no	—
91	C₄×D₄	—	2	1	no	—
92	C₄×D₄	C₂×C₈	1	1	yes	characteristic
93	C₄×D₄	M₁₆	1	1	yes	characteristic
94	#117	—	2	1	no	—
95	#117	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄²	—	2	1	no	—
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	M₃₂	—	4	2	no	—
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	4	2	no	—
101	C₂×C₁₆	—	4	2	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	—	4	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
109	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
110	C₂²⋊C₈	—	2	2	no	—
111	C₂²⋊C₈	—	2	1	no	—
112	C₂²⋊C₈	—	2	1	no	—
113	C₂²⋊C₈	—	2	1	no	—
114	C₂²⋊C₈	M₁₆	1	2	yes	—
115	#97	—	2	2	no	—
116	#97	—	2	1	no	—
117	#97	—	2	1	no	—
118	#97	—	2	1	no	—
119	#97	—	2	1	no	—
120	#97	—	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	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×C₈	1	1	yes	characteristic
132	C₄×C₈	C₂×C₈	1	1	yes	characteristic
133	C₄×C₈	C₂×D₄	1	1	yes	characteristic
134	C₄×C₈	C₂×D₄	1	1	yes	characteristic
135	C₄×C₈	#41	1	1	yes	characteristic
136	C₄×C₈	#41	1	1	yes	characteristic
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	C₄⋊C₈	1	1	yes	characteristic
140	C₂×D₄	—	2	1	no	—
141	C₂×D₄	C₄⋊C₈	1	1	yes	characteristic
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
147	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	—	4	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	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	—	4	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
173	C₂×C₈	#121	1	1	yes	characteristic
174	C₂×C₈	#121	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
177	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₈	#137	1	2	yes	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
182	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₄²	#117	1	1	yes	characteristic
191	C₄²	C₂²×C₈	1	1	yes	characteristic
192	C₄²	C₂×M₁₆	1	1	yes	characteristic
193	C₁₆	—	8	2	no	—
194	C₁₆	—	8	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	#423	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	4	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	#380	1	1	yes	characteristic
212	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
213	C₂×C₄	#407	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
215	C₂×C₄	#428	1	2	yes	—
216	C₂×C₄	#430	1	2	yes	—
217	C₂×C₄	#431	1	1	yes	characteristic
218	C₂×C₄	#432	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	1	no	—
226	C₈	—	4	2	no	—
227	C₈	—	4	1	no	—
228	C₈	—	4	1	no	—
229	C₈	—	4	1	no	—
230	C₈	—	4	1	no	—
231	C₈	#350	1	2	yes	complemented
232	C₈	#468	1	2	yes	—
233	C₈	#469	1	1	yes	characteristic
234	C₈	#470	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	#1569	1	2	yes	—
241	C₂²	#1797	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	1	no	—
248	C₄	#1565	1	2	yes	—
249	C₄	#1809	1	1	yes	characteristic
250	C₄	#1811	1	1	yes	characteristic
251	C₄	C₂×#350	1	1	yes	characteristic
252	C₄	#2117	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	8	1	no	—
255	C₂	#8597	1	1	yes	characteristic
256	C₂	#12787	1	1	yes	characteristic
257	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. #12004	C₂	semidirect
4. C₂×#1367	C₂	semidirect
5. #7153	C₂	semidirect
231. C₈	#350	semidirect

Maximal subgroups

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

subgroup	index
#17335	2
#12787	2
#12004	2
C₂×#1367	2
#7153	2