Subgroups — #99046

Order 512.

372 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21712	C₂	1	1	yes	maximal, characteristic
2	#8806	C₂	1	1	yes	maximal, characteristic
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#8587	C₂	1	2	yes	maximal
5	#7050	C₂	1	2	yes	maximal
6	C₂×#446	C₄	1	2	yes	—
7	C₄×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	#1984	C₄	1	2	yes	—
9	#1939	—	2	1	no	—
10	#1939	—	2	1	no	—
11	#1842	C₂²	1	2	yes	—
12	#1842	C₂²	1	2	yes	—
13	#1842	C₂²	1	1	yes	characteristic
14	#1770	—	2	1	no	—
15	#1770	—	2	1	no	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#335	—	2	1	no	—
18	C₂×#335	—	2	1	no	—
19	C₂×#334	—	2	1	no	—
20	C₂×#334	—	2	1	no	—
21	C₂×#333	—	2	2	no	—
22	#1296	—	2	2	no	—
23	C₂×C₄⋊Q₈	D₄	1	1	yes	characteristic
24	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
25	#446	—	4	4	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄⋊C₈	—	2	2	no	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×#97	—	2	2	no	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	#391	Q₈	1	2	yes	—
44	#389	—	2	1	no	—
45	#389	—	2	1	no	—
46	#384	D₄	1	1	yes	characteristic
47	C₄×C₄⋊C₄	—	2	1	no	—
48	C₄×C₄⋊C₄	—	2	1	no	—
49	C₄³	C₂³	1	1	yes	characteristic
50	#336	—	2	2	no	—
51	#336	—	2	2	no	—
52	#335	—	4	2	no	—
53	#335	—	4	2	no	—
54	#334	—	4	2	no	—
55	#334	—	4	2	no	—
56	#333	—	4	4	no	—
57	C₂²×Q₈	C₄⋊C₄	1	2	yes	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	—	4	2	no	—
60	C₂²×C₈	—	4	2	no	—
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	4	2	no	—
68	C₄×Q₈	—	4	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	2	1	no	—
71	C₂×C₄⋊C₄	—	2	1	no	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₄⋊C₄	—	2	1	no	—
74	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₄²	—	2	2	no	—
76	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄²	#41	1	1	yes	characteristic
82	C₂×C₄²	#41	1	1	yes	characteristic
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	4	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	4	1	no	—
88	C₄⋊C₈	—	4	1	no	—
89	C₄⋊C₈	—	4	2	no	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	2	no	—
94	C₄⋊C₈	—	4	1	no	—
95	C₄⋊C₈	—	4	1	no	—
96	C₄⋊C₈	—	4	2	no	—
97	C₄⋊C₈	—	8	2	no	—
98	C₄⋊C₈	—	8	2	no	—
99	#97	—	4	4	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	SD₁₆	1	2	yes	—
107	C₄×C₈	SD₁₆	1	2	yes	—
108	C₄×C₈	Q₁₆	1	2	yes	—
109	C₄×C₈	Q₁₆	1	2	yes	—
110	#95	—	2	2	no	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	2	4	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	4	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄	#117	1	1	yes	characteristic
120	C₂²×C₄	#121	1	1	yes	characteristic
121	C₂²×C₄	#121	1	1	yes	characteristic
122	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
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	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	8	2	no	—
135	C₂×C₈	—	8	2	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	—	8	2	no	—
138	C₂×C₈	—	8	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	#100	1	2	yes	—
141	C₂×C₈	#103	1	2	yes	—
142	C₂×C₈	#103	1	2	yes	—
143	C₂×C₈	#103	1	2	yes	—
144	C₂×C₈	#103	1	2	yes	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	4	1	no	—
159	C₄²	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	C₂×SD₁₆	1	1	yes	characteristic
162	C₄²	C₂×SD₁₆	1	1	yes	characteristic
163	C₄²	C₂×Q₁₆	1	1	yes	characteristic
164	C₄²	C₂×Q₁₆	1	1	yes	characteristic
165	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄²	AGL(1,ℤ/8)	1	2	yes	—
168	C₄²	#137	1	1	yes	characteristic
169	C₄²	#137	1	1	yes	characteristic
170	C₄²	#137	1	2	yes	—
171	C₂³	#380	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	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	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×#100	1	1	yes	characteristic
189	C₂×C₄	#413	1	1	yes	characteristic
190	C₂×C₄	C₂×#103	1	1	yes	characteristic
191	C₂×C₄	C₂×#103	1	1	yes	characteristic
192	C₂×C₄	#417	1	1	yes	characteristic
193	C₂×C₄	#417	1	1	yes	characteristic
194	C₂×C₄	#428	1	2	yes	—
195	C₂×C₄	#460	1	1	yes	characteristic
196	C₂×C₄	#460	1	1	yes	characteristic
197	C₂×C₄	#462	1	1	yes	characteristic
198	C₂×C₄	#462	1	1	yes	characteristic
199	C₂×C₄	#468	1	2	yes	—
200	C₂×C₄	#469	1	1	yes	characteristic
201	C₂×C₄	#470	1	1	yes	characteristic
202	C₂×C₄	#471	1	2	yes	—
203	C₂×C₄	#472	1	1	yes	characteristic
204	C₂×C₄	#473	1	1	yes	characteristic
205	C₂×C₄	#475	1	1	yes	characteristic
206	C₂×C₄	#475	1	1	yes	characteristic
207	C₂×C₄	#477	1	1	yes	characteristic
208	C₂×C₄	#477	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	1	no	—
212	C₈	—	8	2	no	—
213	C₈	—	16	2	no	—
214	C₈	—	16	2	no	—
215	C₈	#333	1	4	yes	complemented
216	C₂²	#1800	1	1	yes	characteristic
217	C₂²	#1803	1	1	yes	characteristic
218	C₂²	#1803	1	1	yes	characteristic
219	C₂²	#1809	1	1	yes	characteristic
220	C₂²	#1813	1	1	yes	characteristic
221	C₂²	#1815	1	1	yes	characteristic
222	C₂²	#1816	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	8	2	no	—
230	C₄	—	8	2	no	—
231	C₄	C₂×#333	1	1	yes	characteristic
232	C₄	#1550	1	1	yes	characteristic
233	C₄	#1550	1	1	yes	characteristic
234	C₄	#1560	1	1	yes	characteristic
235	C₄	#1686	1	2	yes	—
236	C₄	#1688	1	1	yes	characteristic
237	C₄	#1690	1	1	yes	characteristic
238	C₄	#1692	1	2	yes	—
239	C₄	#1694	1	1	yes	characteristic
240	C₄	#1696	1	1	yes	characteristic
241	C₂	#8587	1	1	yes	characteristic
242	C₂	#8618	1	1	yes	characteristic
243	C₂	#8640	1	1	yes	characteristic
244	C₂	#8651	1	1	yes	characteristic
245	C₂	#8653	1	1	yes	characteristic
246	C₂	#8683	1	1	yes	characteristic
247	C₂	#8685	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
215. C₈	#333	semidirect

Maximal subgroups

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

subgroup	index
#21712	2
#8806	2
#8805	2
#8587	2
#7050	2