Subgroups — #483461

Order 512.

409 conjugacy classes of subgroups.

The subgroup lattice has 239 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33420	C₂	1	1	yes	maximal, characteristic
2	#17799	C₂	1	2	yes	maximal
3	#12036	C₂	1	2	yes	maximal, complemented
4	#12035	C₂	1	2	yes	maximal, complemented
5	#12034	C₂	1	2	yes	maximal, complemented
6	#12032	C₂	1	2	yes	maximal, complemented
7	#12025	C₂	1	2	yes	maximal, complemented
8	#12024	C₂	1	1	yes	maximal, characteristic, complemented
9	#12022	C₂	1	1	yes	maximal, characteristic, complemented
10	#2989	—	2	2	no	—
11	C₄×#131	C₂²	1	1	yes	characteristic
12	#2164	C₄	1	4	yes	—
13	#2162	C₂²	1	2	yes	—
14	#2162	C₂²	1	2	yes	—
15	#2151	C₄	1	2	yes	—
16	#2113	C₄	1	2	yes	—
17	C₄×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#1569	—	2	2	no	—
20	#1569	—	2	2	no	—
21	#1568	C₂²	1	2	yes	—
22	#1568	C₂²	1	2	yes	—
23	#1567	C₂²	1	2	yes	—
24	#1567	C₂²	1	2	yes	—
25	#1565	—	2	2	no	—
26	#1564	C₂²	1	2	yes	—
27	C₂×#335	C₂²	1	2	yes	—
28	#1372	C₂²	1	2	yes	—
29	#1372	C₂²	1	2	yes	—
30	#1372	C₂²	1	2	yes	—
31	#1371	C₂²	1	1	yes	characteristic
32	#1371	C₂²	1	2	yes	—
33	#1371	C₂²	1	2	yes	—
34	#1371	C₂²	1	2	yes	—
35	#1371	C₂²	1	1	yes	characteristic
36	#1371	C₂²	1	1	yes	characteristic
37	#1371	C₂²	1	1	yes	characteristic
38	C₂×#131	C₂×C₄	1	1	yes	characteristic
39	C₂×#131	Q₈	1	1	yes	characteristic
40	C₄×#41	D₄	1	1	yes	characteristic
41	C₂×M₃₂	C₂×C₄	1	2	yes	—
42	C₂²×C₁₆	C₂×C₄	1	2	yes	—
43	#446	—	4	2	no	—
44	#436	—	4	2	no	—
45	#435	—	4	2	no	—
46	#435	—	4	2	no	—
47	#432	—	2	2	no	—
48	#432	—	2	2	no	—
49	#431	—	2	2	no	—
50	#423	—	2	2	no	—
51	#423	—	2	2	no	—
52	C₂×C₄⋊C₈	—	2	2	no	—
53	#405	C₂×C₄	1	1	yes	characteristic
54	#405	C₂×C₄	1	2	yes	—
55	#405	Q₈	1	1	yes	characteristic
56	#405	Q₈	1	2	yes	—
57	C₄×M₁₆	D₄	1	2	yes	—
58	C₄×M₁₆	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	2	yes	—
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
62	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
63	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
64	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
65	#346	C₂×C₄	1	2	yes	—
66	C₄×C₁₆	C₂×C₄	1	2	yes	—
67	C₄×C₁₆	C₂³	1	1	yes	characteristic
68	C₄×C₁₆	C₂³	1	1	yes	characteristic
69	C₄×C₁₆	C₂³	1	2	yes	—
70	#335	—	2	2	no	—
71	#335	—	2	2	no	—
72	#335	—	2	2	no	—
73	#335	C₂³	1	2	yes	—
74	#335	C₂³	1	2	yes	—
75	#335	C₂³	1	2	yes	—
76	#335	C₂³	1	2	yes	—
77	#334	—	2	2	no	—
78	#334	—	2	2	no	—
79	#334	—	2	2	no	—
80	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
81	#131	—	4	2	no	—
82	#131	—	4	2	no	—
83	C₂×M₁₆	—	4	2	no	—
84	C₂×M₁₆	—	4	2	no	—
85	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
86	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
87	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
88	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	—	4	2	no	—
90	C₂²×C₈	C₂²×C₄	1	2	yes	—
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₈	C₂×Q₈	1	2	yes	—
94	C₄×Q₈	D₈	1	1	yes	characteristic
95	C₄×Q₈	Q₁₆	1	1	yes	characteristic
96	C₄×D₄	D₈	1	1	yes	characteristic
97	C₄×D₄	D₈	1	2	yes	—
98	C₄×D₄	Q₁₆	1	2	yes	—
99	C₄×D₄	Q₁₆	1	1	yes	characteristic
100	#117	C₄⋊C₄	1	1	yes	characteristic
101	#117	C₄⋊C₄	1	2	yes	—
102	C₂×C₄²	C₂×D₄	1	2	yes	—
103	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
104	M₃₂	—	4	2	no	—
105	C₂×C₁₆	—	4	2	no	—
106	C₂×C₁₆	C₂²×C₄	1	2	yes	—
107	C₂×C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×C₁₆	C₂²×C₄	1	4	yes	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₂²⋊C₈	—	4	2	no	—
117	C₂²⋊C₈	—	4	2	no	—
118	C₂²⋊C₈	—	4	2	no	—
119	#97	—	4	2	no	—
120	#97	—	4	2	no	—
121	#97	C₄⋊C₄	1	2	yes	—
122	#97	C₄⋊C₄	1	2	yes	—
123	#97	C₄⋊C₄	1	2	yes	—
124	C₄×C₈	—	4	2	no	—
125	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×C₈	C₂²×C₄	1	2	yes	—
127	C₄×C₈	C₂×D₄	1	2	yes	—
128	C₄×C₈	C₂×D₄	1	1	yes	characteristic
129	C₄×C₈	C₂×Q₈	1	2	yes	—
130	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
131	C₄×C₈	C₂⁴	1	1	yes	characteristic
132	#41	—	4	1	no	—
133	#41	—	4	1	no	—
134	C₂×Q₈	#107	1	1	yes	characteristic
135	C₂×D₄	#107	1	1	yes	characteristic
136	C₂×D₄	#107	1	2	yes	—
137	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
139	C₂²×C₄	C₂×D₈	1	2	yes	—
140	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
142	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
143	M₁₆	—	2	4	no	—
144	M₁₆	—	4	2	no	—
145	M₁₆	—	4	1	no	—
146	M₁₆	—	4	1	no	—
147	M₁₆	—	16	2	no	—
148	M₁₆	—	16	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	16	2	no	—
158	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
164	C₄⋊C₄	#107	1	1	yes	characteristic
165	C₄⋊C₄	#107	1	2	yes	—
166	C₄⋊C₄	#107	1	1	yes	characteristic
167	C₄⋊C₄	#107	1	2	yes	—
168	C₂²⋊C₄	#107	1	2	yes	—
169	C₂²⋊C₄	#107	1	2	yes	—
170	C₂²⋊C₄	#107	1	2	yes	—
171	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₄²	C₂×C₄⋊C₄	1	2	yes	—
173	C₄²	C₂×D₈	1	2	yes	—
174	C₄²	C₂×D₈	1	1	yes	characteristic
175	C₄²	C₂×Q₁₆	1	2	yes	—
176	C₄²	C₂×Q₁₆	1	1	yes	characteristic
177	C₄²	C₂²×D₄	1	1	yes	characteristic
178	C₁₆	—	2	2	no	—
179	C₁₆	—	2	2	no	—
180	C₁₆	—	2	4	no	—
181	C₂³	C₂×#107	1	2	yes	—
182	C₂³	C₂×#107	1	1	yes	characteristic
183	Q₈	—	4	1	no	—
184	D₄	—	4	1	no	—
185	D₄	—	4	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	C₂×#107	1	2	yes	—
197	C₂×C₄	C₂×#107	1	1	yes	characteristic
198	C₂×C₄	C₂×#107	1	2	yes	—
199	C₂×C₄	C₂×#107	1	1	yes	characteristic
200	C₂×C₄	C₂×#107	1	2	yes	—
201	C₂×C₄	C₂×#107	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
204	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	2	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	—	8	2	no	—
210	C₈	—	8	2	no	—
211	C₈	—	8	2	no	—
212	C₈	—	8	2	no	—
213	C₈	#430	1	2	yes	—
214	C₈	#568	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	2	no	—
221	C₂²	C₂²×#107	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	#2905	1	1	yes	characteristic
230	C₄	C₂×#430	1	1	yes	characteristic
231	C₄	#3415	1	1	yes	characteristic
232	C₄	#3417	1	1	yes	characteristic
233	C₂	—	4	2	no	—
234	C₂	—	4	1	no	—
235	C₂	#17814	1	1	yes	characteristic
236	C₂	#33410	1	1	yes	characteristic
237	C₂	#33414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #12036	C₂	semidirect
4. #12035	C₂	semidirect
5. #12034	C₂	semidirect
6. #12032	C₂	semidirect
7. #12025	C₂	semidirect
8. #12024	C₂	semidirect
9. #12022	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33420	2
#17799	2
#12036	2
#12035	2
#12034	2
#12032	2
#12025	2
#12024	2
#12022	2