Subgroups — #134596

Order 512.

308 conjugacy classes of subgroups.

The subgroup lattice has 218 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	#12900	C₂	1	1	yes	maximal, characteristic
3	#12900	C₂	1	1	yes	maximal, characteristic
4	C₂×#1366	C₂	1	2	yes	maximal
5	#7271	C₂	1	2	yes	maximal
6	C₂×#499	C₂²	1	1	yes	characteristic
7	C₄×#125	C₄	1	2	yes	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	—	2	1	no	—
10	C₂×#364	C₂²	1	2	yes	—
11	C₂×#364	C₂²	1	2	yes	—
12	#2150	—	2	1	no	—
13	#2150	—	2	1	no	—
14	#1991	C₄	1	2	yes	—
15	#1942	—	2	1	no	—
16	#1941	—	2	1	no	—
17	#1846	C₂²	1	1	yes	characteristic
18	C₄×#107	—	2	1	no	—
19	C₄×#106	—	2	1	no	—
20	C₄²×C₈	C₂²	1	1	yes	characteristic
21	#1366	—	2	4	no	—
22	C₂×#125	C₂×C₄	1	2	yes	—
23	C₂×M₃₂	—	2	2	no	—
24	C₂×M₃₂	—	2	2	no	—
25	#499	—	2	1	no	—
26	#499	—	2	1	no	—
27	#499	D₄	1	2	yes	—
28	#499	Q₈	1	2	yes	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#106	—	2	1	no	—
32	C₂×#106	—	2	1	no	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#389	—	2	2	no	—
40	#387	—	4	2	no	—
41	#382	C₂×C₄	1	2	yes	—
42	C₄×C₄⋊C₄	—	2	2	no	—
43	C₄×C₄⋊C₄	—	4	2	no	—
44	C₄³	C₂×C₄	1	1	yes	characteristic
45	#364	—	2	2	no	—
46	#364	—	2	2	no	—
47	#364	—	2	2	no	—
48	#364	—	2	2	no	—
49	#364	—	4	2	no	—
50	#364	—	4	2	no	—
51	C₄⋊C₁₆	—	4	1	no	—
52	C₄⋊C₁₆	—	4	1	no	—
53	C₄⋊C₁₆	—	4	1	no	—
54	C₄⋊C₁₆	—	4	1	no	—
55	#340	—	2	2	no	—
56	#340	—	2	2	no	—
57	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	#125	—	2	2	no	—
62	#125	—	4	2	no	—
63	#125	C₄⋊C₄	1	4	yes	—
64	C₂×C₄⋊C₄	—	2	2	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄²	—	2	1	no	—
69	C₂×C₄²	—	2	1	no	—
70	C₂×C₄²	—	4	2	no	—
71	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
74	M₃₂	—	4	2	no	—
75	M₃₂	—	4	2	no	—
76	M₃₂	—	4	2	no	—
77	M₃₂	—	4	2	no	—
78	C₂×C₁₆	—	4	2	no	—
79	C₂×C₁₆	—	4	2	no	—
80	#107	—	2	2	no	—
81	#107	—	2	2	no	—
82	#107	—	4	1	no	—
83	#107	—	4	1	no	—
84	#106	—	2	2	no	—
85	#106	—	2	2	no	—
86	#106	—	4	1	no	—
87	#106	—	4	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	4	1	no	—
93	C₄×C₈	SD₁₆	1	1	yes	characteristic
94	C₄×C₈	SD₁₆	1	1	yes	characteristic
95	C₄×C₈	SD₁₆	1	1	yes	characteristic
96	C₄×C₈	SD₁₆	1	1	yes	characteristic
97	C₄×C₈	Q₁₆	1	1	yes	characteristic
98	C₄×C₈	Q₁₆	1	1	yes	characteristic
99	C₄×C₈	Q₁₆	1	1	yes	characteristic
100	C₄×C₈	Q₁₆	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	C₄×C₈	#41	1	1	yes	characteristic
105	#95	—	4	2	no	—
106	#95	—	4	2	no	—
107	#95	—	4	2	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	#100	1	1	yes	characteristic
113	C₂²×C₄	#100	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	#103	1	1	yes	characteristic
120	C₂×C₈	#103	1	1	yes	characteristic
121	C₂×C₈	#103	1	1	yes	characteristic
122	C₂×C₈	#103	1	1	yes	characteristic
123	C₂×C₈	#121	1	1	yes	characteristic
124	C₂×C₈	#121	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₈	#137	1	1	yes	characteristic
134	C₂×C₈	#137	1	1	yes	characteristic
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	8	4	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	4	2	no	—
146	C₄²	—	4	2	no	—
147	C₄²	—	8	4	no	—
148	C₄²	#103	1	1	yes	characteristic
149	C₄²	#103	1	1	yes	characteristic
150	C₄²	#103	1	1	yes	characteristic
151	C₄²	#103	1	1	yes	characteristic
152	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₄²	#117	1	1	yes	characteristic
154	C₁₆	—	8	2	no	—
155	C₁₆	—	8	2	no	—
156	C₂³	C₂×#100	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	#333	1	1	yes	characteristic
173	C₂×C₄	#333	1	1	yes	characteristic
174	C₂×C₄	#333	1	1	yes	characteristic
175	C₂×C₄	#333	1	1	yes	characteristic
176	C₂×C₄	#380	1	1	yes	characteristic
177	C₂×C₄	#413	1	1	yes	characteristic
178	C₂×C₄	C₂×#103	1	1	yes	characteristic
179	C₂×C₄	C₂×#103	1	1	yes	characteristic
180	C₂×C₄	#417	1	1	yes	characteristic
181	C₂×C₄	#417	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	#460	1	1	yes	characteristic
187	C₈	#460	1	1	yes	characteristic
188	C₈	#462	1	1	yes	characteristic
189	C₈	#462	1	1	yes	characteristic
190	C₈	#475	1	1	yes	characteristic
191	C₈	#475	1	1	yes	characteristic
192	C₈	#477	1	1	yes	characteristic
193	C₈	#477	1	1	yes	characteristic
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	C₂×#333	1	1	yes	characteristic
197	C₂²	#1560	1	1	yes	characteristic
198	C₂²	#1800	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	1	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	16	2	no	—
207	C₄	#1366	1	2	yes	complemented
208	C₄	#1550	1	1	yes	characteristic
209	C₄	#1550	1	1	yes	characteristic
210	C₄	#1803	1	1	yes	characteristic
211	C₄	#1803	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#8587	1	1	yes	characteristic
215	C₂	C₂×#1366	1	1	yes	characteristic
216	C₂	#11956	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#333
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
207. C₄	#1366	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
#12900	2
C₂×#1366	2
#7271	2