Subgroups — #134586

Order 512.

336 conjugacy classes of subgroups.

The subgroup lattice has 265 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#499	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	C₂×#1363	C₂	1	2	yes	maximal
4	#8805	C₂	1	1	yes	maximal, characteristic
5	#7275	C₂	1	2	yes	maximal
6	C₂×#499	C₂²	1	1	yes	characteristic
7	C₄×#125	—	2	1	no	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	#1991	—	2	1	no	—
12	#1942	C₄	1	1	yes	characteristic
13	#1942	C₄	1	1	yes	characteristic
14	#1941	—	2	1	no	—
15	#1846	C₂²	1	1	yes	characteristic
16	#1842	—	2	1	no	—
17	C₄×#107	C₄	1	1	yes	characteristic
18	C₄×#107	C₄	1	1	yes	characteristic
19	C₄×#106	—	2	1	no	—
20	C₄²×C₈	C₂²	1	1	yes	characteristic
21	C₂×#334	—	2	1	no	—
22	C₂×#334	C₂²	1	2	yes	—
23	#1363	—	2	4	no	—
24	C₂×#125	—	2	1	no	—
25	C₂²×C₁₆	—	2	2	no	—
26	#499	—	2	1	no	—
27	#499	—	2	1	no	—
28	#499	D₄	1	2	yes	—
29	#499	Q₈	1	2	yes	—
30	C₂×#107	C₂×C₄	1	1	yes	characteristic
31	C₂×#107	C₂×C₄	1	1	yes	characteristic
32	C₂×#107	C₂×C₄	1	1	yes	characteristic
33	C₂×#107	C₂×C₄	1	1	yes	characteristic
34	C₂×#106	—	2	1	no	—
35	C₂×#106	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×C₄⋊C₈	—	4	1	no	—
38	C₂×C₄⋊C₈	—	4	1	no	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#389	—	2	1	no	—
46	#389	—	2	1	no	—
47	#387	—	4	1	no	—
48	#387	—	4	1	no	—
49	#382	—	2	1	no	—
50	C₄×C₄⋊C₄	—	2	1	no	—
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄×C₄⋊C₄	—	4	1	no	—
53	C₄×C₄⋊C₄	—	4	1	no	—
54	C₄³	D₄	1	1	yes	characteristic
55	C₄⋊C₁₆	—	2	2	no	—
56	C₄⋊C₁₆	—	2	2	no	—
57	C₄⋊C₁₆	—	4	2	no	—
58	C₄⋊C₁₆	—	4	2	no	—
59	#340	—	2	1	no	—
60	#340	—	2	1	no	—
61	#340	—	2	1	no	—
62	#340	—	2	1	no	—
63	#334	—	2	2	no	—
64	#334	—	2	2	no	—
65	#334	—	4	2	no	—
66	C₂²×C₈	—	4	2	no	—
67	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	#125	—	2	2	no	—
72	#125	—	2	2	no	—
73	#125	—	4	1	no	—
74	#125	—	4	1	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	—	4	1	no	—
80	C₂×C₄⋊C₄	—	4	1	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄²	—	4	1	no	—
84	C₂×C₄²	—	4	1	no	—
85	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄²	D₈	1	1	yes	characteristic
88	C₂×C₄²	D₈	1	1	yes	characteristic
89	C₂×C₄²	SD₁₆	1	1	yes	characteristic
90	C₂×C₄²	SD₁₆	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₁₆	—	4	2	no	—
93	C₂×C₁₆	—	4	2	no	—
94	C₂×C₁₆	—	4	2	no	—
95	#107	—	2	1	no	—
96	#107	—	2	1	no	—
97	#107	—	4	1	no	—
98	#107	—	4	1	no	—
99	#107	C₄⋊C₄	1	2	yes	—
100	#107	C₄⋊C₄	1	2	yes	—
101	#106	—	2	2	no	—
102	#106	—	2	2	no	—
103	#106	—	4	1	no	—
104	#106	—	4	1	no	—
105	C₄⋊C₈	—	4	2	no	—
106	C₄⋊C₈	—	4	2	no	—
107	C₄⋊C₈	—	8	2	no	—
108	C₄⋊C₈	—	8	2	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	4	1	no	—
115	C₄×C₈	SD₁₆	1	2	yes	—
116	C₄×C₈	Q₁₆	1	2	yes	—
117	C₄×C₈	C₂×D₄	1	1	yes	characteristic
118	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
119	C₄×C₈	#41	1	1	yes	characteristic
120	C₄×C₈	#41	1	1	yes	characteristic
121	#95	—	4	1	no	—
122	#95	—	4	1	no	—
123	#95	—	4	1	no	—
124	#95	—	4	1	no	—
125	#95	—	4	1	no	—
126	#95	—	4	1	no	—
127	C₂²×C₄	—	4	1	no	—
128	C₂²×C₄	—	4	1	no	—
129	C₂²×C₄	—	4	1	no	—
130	C₂²×C₄	—	4	1	no	—
131	C₂²×C₄	#102	1	1	yes	characteristic
132	C₂²×C₄	#102	1	1	yes	characteristic
133	C₂²×C₄	#102	1	1	yes	characteristic
134	C₂²×C₄	#102	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	8	2	no	—
146	C₂×C₈	—	8	2	no	—
147	C₂×C₈	—	8	2	no	—
148	C₂×C₈	#103	1	2	yes	—
149	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₈	#117	1	1	yes	characteristic
151	C₂×C₈	#121	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
155	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₈	#137	1	1	yes	characteristic
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	1	no	—
166	C₄⋊C₄	—	4	1	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	8	2	no	—
170	C₄⋊C₄	—	8	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	4	1	no	—
177	C₄²	—	8	2	no	—
178	C₄²	—	8	2	no	—
179	C₄²	#101	1	2	yes	—
180	C₄²	#103	1	2	yes	—
181	C₄²	#121	1	1	yes	characteristic
182	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
183	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄²	#137	1	1	yes	characteristic
185	C₁₆	—	8	2	no	—
186	C₂³	C₂×#102	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	1	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	—	8	1	no	—
206	C₂×C₄	—	8	1	no	—
207	C₂×C₄	—	8	1	no	—
208	C₂×C₄	—	8	1	no	—
209	C₂×C₄	—	8	1	no	—
210	C₂×C₄	#332	1	2	yes	—
211	C₂×C₄	#332	1	2	yes	—
212	C₂×C₄	#359	1	2	yes	—
213	C₂×C₄	#360	1	2	yes	—
214	C₂×C₄	#360	1	2	yes	—
215	C₂×C₄	#380	1	1	yes	characteristic
216	C₂×C₄	C₂×#101	1	1	yes	characteristic
217	C₂×C₄	#413	1	1	yes	characteristic
218	C₂×C₄	C₂×#103	1	1	yes	characteristic
219	C₂×C₄	#417	1	1	yes	characteristic
220	C₂×C₄	#417	1	1	yes	characteristic
221	C₂×C₄	#459	1	1	yes	characteristic
222	C₂×C₄	#461	1	1	yes	characteristic
223	C₂×C₄	#474	1	1	yes	characteristic
224	C₂×C₄	#476	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	4	1	no	—
229	C₈	—	16	2	no	—
230	C₈	#460	1	1	yes	characteristic
231	C₈	#462	1	1	yes	characteristic
232	C₈	#475	1	1	yes	characteristic
233	C₈	#477	1	1	yes	characteristic
234	C₂²	C₂×#332	1	1	yes	characteristic
235	C₂²	#1547	1	1	yes	characteristic
236	C₂²	#1802	1	1	yes	characteristic
237	C₂²	C₂×#359	1	1	yes	characteristic
238	C₂²	#2143	1	1	yes	characteristic
239	C₂²	C₂×#360	1	1	yes	characteristic
240	C₂²	C₂×#360	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	8	2	no	—
250	C₄	—	16	1	no	—
251	C₄	—	16	1	no	—
252	C₄	#1363	1	4	yes	complemented
253	C₄	#1550	1	1	yes	characteristic
254	C₄	#1561	1	1	yes	characteristic
255	C₄	#1801	1	1	yes	characteristic
256	C₄	#1803	1	1	yes	characteristic
257	C₂	#8586	1	1	yes	characteristic
258	C₂	C₂×#1363	1	1	yes	characteristic
259	C₂	#11936	1	1	yes	characteristic
260	C₂	#11977	1	1	yes	characteristic
261	C₂	#11980	1	1	yes	characteristic
262	C₂	#12815	1	1	yes	characteristic
263	C₂	#12816	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ 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
252. C₄	#1363	semidirect

Maximal subgroups

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

subgroup	index
C₄×#499	2
#12898	2
C₂×#1363	2
#8805	2
#7275	2