Subgroups — #99054

Order 512.

362 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#8805	C₂	1	1	yes	maximal, characteristic
3	#8784	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×#334	—	2	1	no	—
18	C₂×#334	—	2	1	no	—
19	C₂×#333	—	2	2	no	—
20	#1296	—	2	2	no	—
21	#1277	—	2	1	no	—
22	#1277	—	2	1	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	#334	—	4	2	no	—
53	#334	—	4	2	no	—
54	#333	—	4	4	no	—
55	C₂²×Q₈	C₄⋊C₄	1	2	yes	—
56	C₂²×C₈	—	2	2	no	—
57	C₂²×C₈	—	4	2	no	—
58	C₂²×C₈	—	4	2	no	—
59	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄×Q₈	—	4	2	no	—
66	C₄×Q₈	—	4	2	no	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	—	2	1	no	—
71	C₂×C₄⋊C₄	—	2	1	no	—
72	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
73	C₂×C₄²	—	2	2	no	—
74	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₄²	#41	1	1	yes	characteristic
80	C₂×C₄²	#41	1	1	yes	characteristic
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	4	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄⋊C₈	—	4	1	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	2	no	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	2	no	—
94	C₄⋊C₈	—	8	2	no	—
95	C₄⋊C₈	—	8	2	no	—
96	#97	—	4	4	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	SD₁₆	1	2	yes	—
104	C₄×C₈	Q₁₆	1	2	yes	—
105	#95	—	2	2	no	—
106	#95	—	2	2	no	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	—	4	2	no	—
109	C₂×Q₈	—	4	2	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄	#117	1	1	yes	characteristic
115	C₂²×C₄	#121	1	1	yes	characteristic
116	C₂²×C₄	#121	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
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	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	8	2	no	—
132	C₂×C₈	—	8	2	no	—
133	C₂×C₈	—	8	2	no	—
134	C₂×C₈	—	8	2	no	—
135	C₂×C₈	—	8	2	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	#100	1	2	yes	—
138	C₂×C₈	#103	1	2	yes	—
139	C₂×C₈	#103	1	2	yes	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	4	1	no	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	C₂×SD₁₆	1	1	yes	characteristic
155	C₄²	C₂×Q₁₆	1	1	yes	characteristic
156	C₄²	#135	1	2	yes	—
157	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₄²	AGL(1,ℤ/8)	1	2	yes	—
159	C₄²	#137	1	1	yes	characteristic
160	C₄²	#137	1	2	yes	—
161	C₄²	#137	1	2	yes	—
162	C₂³	#380	1	1	yes	characteristic
163	Q₈	—	8	4	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×#100	1	1	yes	characteristic
180	C₂×C₄	#413	1	1	yes	characteristic
181	C₂×C₄	C₂×#103	1	1	yes	characteristic
182	C₂×C₄	#416	1	1	yes	characteristic
183	C₂×C₄	#417	1	1	yes	characteristic
184	C₂×C₄	#419	1	1	yes	characteristic
185	C₂×C₄	#428	1	2	yes	—
186	C₂×C₄	#460	1	1	yes	characteristic
187	C₂×C₄	#462	1	1	yes	characteristic
188	C₂×C₄	#464	1	2	yes	—
189	C₂×C₄	#468	1	2	yes	—
190	C₂×C₄	#469	1	1	yes	characteristic
191	C₂×C₄	#470	1	1	yes	characteristic
192	C₂×C₄	#471	1	2	yes	—
193	C₂×C₄	#471	1	2	yes	—
194	C₂×C₄	#475	1	1	yes	characteristic
195	C₂×C₄	#477	1	1	yes	characteristic
196	C₂×C₄	#479	1	2	yes	—
197	C₈	—	2	2	no	—
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	4	1	no	—
201	C₈	—	8	2	no	—
202	C₈	—	16	2	no	—
203	C₈	—	16	2	no	—
204	C₂²	#1800	1	1	yes	characteristic
205	C₂²	#1803	1	1	yes	characteristic
206	C₂²	#1805	1	1	yes	characteristic
207	C₂²	#1809	1	1	yes	characteristic
208	C₂²	#1813	1	1	yes	characteristic
209	C₂²	#1815	1	1	yes	characteristic
210	C₂²	#1815	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	1	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	#1542	1	1	yes	characteristic
219	C₄	#1548	1	1	yes	characteristic
220	C₄	#1551	1	1	yes	characteristic
221	C₄	#1559	1	1	yes	characteristic
222	C₄	#1686	1	2	yes	—
223	C₄	#1692	1	2	yes	—
224	C₄	#1702	1	2	yes	—
225	C₄	#1704	1	2	yes	—
226	C₂	#8590	1	1	yes	characteristic
227	C₂	#8618	1	1	yes	characteristic
228	C₂	#8636	1	1	yes	characteristic
229	C₂	#8651	1	1	yes	characteristic
230	C₂	#8657	1	1	yes	characteristic
231	C₂	#8683	1	1	yes	characteristic
232	C₂	#8687	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

Maximal subgroups

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

subgroup	index
#21712	2
#8805	2
#8784	2
#8587	2
#7050	2