Subgroups — #134588

Order 512.

362 conjugacy classes of subgroups.

The subgroup lattice has 251 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#500	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	C₂×#1364	C₂	1	2	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	#7274	C₂	1	2	yes	maximal
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₄×C₄⋊Q₈	—	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	#1990	—	2	1	no	—
12	#1942	—	2	1	no	—
13	#1942	C₄	1	2	yes	—
14	#1845	C₂²	1	1	yes	characteristic
15	#1842	—	2	1	no	—
16	C₄×#107	—	2	1	no	—
17	C₄×#107	C₄	1	2	yes	—
18	C₄²×C₈	C₂²	1	1	yes	characteristic
19	C₂×#335	—	2	1	no	—
20	C₂×#335	C₂²	1	2	yes	—
21	#1364	—	2	4	no	—
22	C₂×C₄⋊Q₈	—	2	1	no	—
23	C₂×C₄×Q₈	—	4	1	no	—
24	C₂²×C₁₆	—	2	2	no	—
25	#500	—	2	1	no	—
26	#500	—	2	1	no	—
27	#500	D₄	1	2	yes	—
28	#500	Q₈	1	2	yes	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	C₂×C₄	1	2	yes	—
32	C₂×#107	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	—	4	1	no	—
35	C₂×C₄⋊C₈	—	4	1	no	—
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#391	—	4	1	no	—
43	#389	—	2	2	no	—
44	#389	—	4	1	no	—
45	#384	—	2	1	no	—
46	C₄×C₄⋊C₄	—	2	2	no	—
47	C₄×C₄⋊C₄	—	4	1	no	—
48	C₄³	D₄	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	2	no	—
50	C₄⋊C₁₆	—	2	2	no	—
51	C₄⋊C₁₆	—	4	2	no	—
52	C₄⋊C₁₆	—	4	2	no	—
53	#340	—	2	2	no	—
54	#340	—	2	2	no	—
55	#335	—	2	2	no	—
56	#335	—	2	2	no	—
57	#335	—	4	2	no	—
58	C₂²×Q₈	—	4	1	no	—
59	C₂²×C₈	—	4	2	no	—
60	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₄⋊Q₈	—	4	1	no	—
68	C₄×Q₈	—	8	2	no	—
69	C₄×Q₈	—	8	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	4	1	no	—
73	C₂×C₄⋊C₄	—	4	2	no	—
74	C₂×C₄⋊C₄	—	4	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄²	—	4	2	no	—
77	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄²	D₈	1	1	yes	characteristic
80	C₂×C₄²	D₈	1	1	yes	characteristic
81	C₂×C₄²	SD₁₆	1	1	yes	characteristic
82	C₂×C₄²	SD₁₆	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₂×C₁₆	—	4	2	no	—
85	C₂×C₁₆	—	4	2	no	—
86	C₂×C₁₆	—	4	2	no	—
87	#107	—	2	2	no	—
88	#107	—	2	2	no	—
89	#107	—	2	2	no	—
90	#107	—	4	1	no	—
91	#107	—	4	1	no	—
92	#107	—	4	2	no	—
93	#107	C₄⋊C₄	1	4	yes	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄⋊C₈	—	4	2	no	—
96	C₄⋊C₈	—	8	2	no	—
97	C₄⋊C₈	—	8	2	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	SD₁₆	1	2	yes	—
105	C₄×C₈	Q₁₆	1	2	yes	—
106	C₄×C₈	C₂×D₄	1	1	yes	characteristic
107	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₄×C₈	#41	1	1	yes	characteristic
109	C₄×C₈	#41	1	1	yes	characteristic
110	#95	—	4	2	no	—
111	#95	—	4	2	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	8	1	no	—
115	C₂×Q₈	—	8	1	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂×Q₈	—	8	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	#102	1	1	yes	characteristic
121	C₂²×C₄	#102	1	1	yes	characteristic
122	C₂²×C₄	#102	1	1	yes	characteristic
123	C₂²×C₄	#102	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	8	2	no	—
135	C₂×C₈	—	8	2	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	#103	1	2	yes	—
138	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#117	1	1	yes	characteristic
140	C₂×C₈	#121	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₈	#137	1	1	yes	characteristic
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	8	2	no	—
153	C₄⋊C₄	—	8	4	no	—
154	C₄⋊C₄	—	8	1	no	—
155	C₄⋊C₄	—	8	1	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	8	4	no	—
163	C₄²	#101	1	2	yes	—
164	C₄²	#103	1	2	yes	—
165	C₄²	D₁₆	1	2	yes	—
166	C₄²	SD₃₂	1	2	yes	—
167	C₄²	#121	1	1	yes	characteristic
168	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
169	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₄²	#137	1	1	yes	characteristic
171	C₁₆	—	8	2	no	—
172	C₂³	C₂×#102	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	2	no	—
176	Q₈	—	8	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	#332	1	2	yes	—
193	C₂×C₄	#332	1	2	yes	—
194	C₂×C₄	#357	1	2	yes	—
195	C₂×C₄	#357	1	2	yes	—
196	C₂×C₄	#362	1	2	yes	—
197	C₂×C₄	#380	1	1	yes	characteristic
198	C₂×C₄	C₂×#101	1	1	yes	characteristic
199	C₂×C₄	#413	1	1	yes	characteristic
200	C₂×C₄	C₂×#103	1	1	yes	characteristic
201	C₂×C₄	#417	1	1	yes	characteristic
202	C₂×C₄	#417	1	1	yes	characteristic
203	C₂×C₄	#459	1	1	yes	characteristic
204	C₂×C₄	#461	1	1	yes	characteristic
205	C₂×C₄	#474	1	1	yes	characteristic
206	C₂×C₄	#476	1	1	yes	characteristic
207	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
208	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
209	C₂×C₄	#509	1	1	yes	characteristic
210	C₂×C₄	#510	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₈	—	4	1	no	—
215	C₈	—	16	2	no	—
216	C₈	#460	1	1	yes	characteristic
217	C₈	#462	1	1	yes	characteristic
218	C₈	#475	1	1	yes	characteristic
219	C₈	#477	1	1	yes	characteristic
220	C₂²	C₂×#332	1	1	yes	characteristic
221	C₂²	#1547	1	1	yes	characteristic
222	C₂²	#1802	1	1	yes	characteristic
223	C₂²	C₂×#357	1	1	yes	characteristic
224	C₂²	#2139	1	1	yes	characteristic
225	C₂²	C₂×#362	1	1	yes	characteristic
226	C₂²	#2148	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	8	4	no	—
233	C₄	—	16	2	no	—
234	C₄	#1364	1	4	yes	complemented
235	C₄	#1550	1	1	yes	characteristic
236	C₄	#1561	1	1	yes	characteristic
237	C₄	#1801	1	1	yes	characteristic
238	C₄	#1803	1	1	yes	characteristic
239	C₄	#2206	1	1	yes	characteristic
240	C₄	#2208	1	1	yes	characteristic
241	C₄	#2224	1	1	yes	characteristic
242	C₄	#2226	1	1	yes	characteristic
243	C₂	#8586	1	1	yes	characteristic
244	C₂	C₂×#1364	1	1	yes	characteristic
245	C₂	#11942	1	1	yes	characteristic
246	C₂	#11981	1	1	yes	characteristic
247	C₂	#11988	1	1	yes	characteristic
248	C₂	#12811	1	1	yes	characteristic
249	C₂	#12819	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
234. C₄	#1364	semidirect

Maximal subgroups

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

subgroup	index
C₄×#500	2
#12898	2
C₂×#1364	2
#8806	2
#7274	2