Subgroups — #134581

Order 512.

404 conjugacy classes of subgroups.

The subgroup lattice has 259 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#494	C₂	1	1	yes	maximal, characteristic
2	#12898	C₂	1	1	yes	maximal, characteristic
3	C₂×#1362	C₂	1	2	yes	maximal
4	#8807	C₂	1	1	yes	maximal, characteristic
5	#7274	C₂	1	2	yes	maximal
6	C₂×#494	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₁₆	—	2	1	no	—
8	C₂×C₄×Q₁₆	C₄	1	2	yes	—
9	C₄×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₄⋊C₁₆	—	2	1	no	—
11	C₂×C₄⋊C₁₆	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
13	#1969	—	2	1	no	—
14	#1969	C₄	1	2	yes	—
15	#1956	—	2	1	no	—
16	#1845	C₂²	1	1	yes	characteristic
17	#1831	—	2	1	no	—
18	C₄²×C₈	C₂²	1	1	yes	characteristic
19	C₂×#335	C₂²	1	2	yes	—
20	C₂×#334	—	2	1	no	—
21	#1362	—	2	4	no	—
22	C₂²×Q₁₆	—	2	1	no	—
23	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Q₈	—	2	1	no	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	C₂²×C₁₆	—	2	2	no	—
27	#494	—	2	1	no	—
28	#494	—	2	1	no	—
29	#494	D₄	1	2	yes	—
30	#494	Q₈	1	2	yes	—
31	C₄×Q₁₆	—	4	4	no	—
32	C₄×Q₁₆	—	4	2	no	—
33	C₄×Q₁₆	—	4	2	no	—
34	C₂×#107	—	2	1	no	—
35	C₂×#107	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×#103	—	2	2	no	—
38	C₂×#103	—	2	2	no	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#391	—	2	2	no	—
46	#389	—	4	1	no	—
47	#389	—	4	1	no	—
48	#384	—	2	1	no	—
49	C₄×C₄⋊C₄	—	4	1	no	—
50	C₄×C₄⋊C₄	—	4	1	no	—
51	C₄³	D₄	1	1	yes	characteristic
52	C₄⋊C₁₆	—	2	2	no	—
53	C₄⋊C₁₆	—	2	2	no	—
54	C₄⋊C₁₆	—	4	2	no	—
55	C₄⋊C₁₆	—	4	2	no	—
56	#336	—	4	1	no	—
57	#336	—	4	1	no	—
58	#335	—	2	2	no	—
59	#335	—	2	2	no	—
60	#334	—	4	2	no	—
61	C₂²×Q₈	—	2	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₂×Q₁₆	—	2	2	no	—
64	C₂×Q₁₆	—	2	2	no	—
65	C₂×Q₁₆	—	4	2	no	—
66	C₂×Q₁₆	—	4	2	no	—
67	C₂×Q₁₆	—	4	1	no	—
68	C₂×Q₁₆	—	4	1	no	—
69	C₂×Q₁₆	—	4	1	no	—
70	C₂×Q₁₆	—	4	1	no	—
71	C₂×Q₁₆	C₄⋊C₄	1	4	yes	—
72	C₂²×C₈	—	4	2	no	—
73	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	4	1	no	—
80	C₄⋊Q₈	—	4	1	no	—
81	C₄×Q₈	—	4	4	no	—
82	C₄×Q₈	—	4	4	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	—	4	1	no	—
85	C₂×C₄⋊C₄	—	4	1	no	—
86	C₂×C₄⋊C₄	—	4	2	no	—
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄²	—	4	2	no	—
90	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄²	D₈	1	1	yes	characteristic
93	C₂×C₄²	D₈	1	1	yes	characteristic
94	C₂×C₄²	SD₁₆	1	1	yes	characteristic
95	C₂×C₄²	SD₁₆	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₁₆	—	4	2	no	—
98	C₂×C₁₆	—	4	2	no	—
99	C₂×C₁₆	—	4	2	no	—
100	#107	—	4	1	no	—
101	#107	—	4	1	no	—
102	#107	—	4	2	no	—
103	C₄⋊C₈	—	4	2	no	—
104	C₄⋊C₈	—	4	2	no	—
105	#103	—	4	4	no	—
106	#103	—	4	4	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	D₈	1	2	yes	—
114	C₄×C₈	SD₁₆	1	2	yes	—
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
117	C₄×C₈	#41	1	1	yes	characteristic
118	C₄×C₈	#41	1	1	yes	characteristic
119	#95	—	4	2	no	—
120	#95	—	4	2	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂×Q₈	—	4	2	no	—
123	C₂×Q₈	—	4	2	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	#102	1	1	yes	characteristic
130	C₂²×C₄	#102	1	1	yes	characteristic
131	C₂²×C₄	#102	1	1	yes	characteristic
132	C₂²×C₄	#102	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
136	Q₁₆	—	4	4	no	—
137	Q₁₆	—	4	4	no	—
138	Q₁₆	—	4	2	no	—
139	Q₁₆	—	4	2	no	—
140	Q₁₆	—	4	2	no	—
141	Q₁₆	—	4	2	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	8	2	no	—
150	C₂×C₈	—	8	2	no	—
151	C₂×C₈	—	8	2	no	—
152	C₂×C₈	#102	1	2	yes	—
153	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₈	#117	1	1	yes	characteristic
155	C₂×C₈	#121	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₈	C₂×D₈	1	1	yes	characteristic
158	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
159	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×C₈	#137	1	1	yes	characteristic
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	8	2	no	—
168	C₄⋊C₄	—	8	4	no	—
169	C₄⋊C₄	—	8	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	8	4	no	—
177	C₄²	#100	1	2	yes	—
178	C₄²	#102	1	2	yes	—
179	C₄²	#121	1	1	yes	characteristic
180	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
181	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄²	#137	1	1	yes	characteristic
183	C₁₆	—	8	2	no	—
184	C₂³	C₂×#102	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	2	no	—
188	Q₈	—	8	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	#331	1	2	yes	—
207	C₂×C₄	#331	1	2	yes	—
208	C₂×C₄	#362	1	2	yes	—
209	C₂×C₄	#380	1	1	yes	characteristic
210	C₂×C₄	C₂×#100	1	1	yes	characteristic
211	C₂×C₄	#413	1	1	yes	characteristic
212	C₂×C₄	C₂×#102	1	1	yes	characteristic
213	C₂×C₄	#417	1	1	yes	characteristic
214	C₂×C₄	#417	1	1	yes	characteristic
215	C₂×C₄	#459	1	1	yes	characteristic
216	C₂×C₄	#461	1	1	yes	characteristic
217	C₂×C₄	#474	1	1	yes	characteristic
218	C₂×C₄	#476	1	1	yes	characteristic
219	C₂×C₄	#508	1	2	yes	—
220	C₂×C₄	#510	1	2	yes	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	2	no	—
224	C₈	—	4	1	no	—
225	C₈	—	16	2	no	—
226	C₈	#459	1	1	yes	characteristic
227	C₈	#461	1	1	yes	characteristic
228	C₈	#474	1	1	yes	characteristic
229	C₈	#476	1	1	yes	characteristic
230	C₂²	C₂×#331	1	1	yes	characteristic
231	C₂²	#1547	1	1	yes	characteristic
232	C₂²	#1802	1	1	yes	characteristic
233	C₂²	#2138	1	1	yes	characteristic
234	C₂²	#2141	1	1	yes	characteristic
235	C₂²	C₂×#362	1	1	yes	characteristic
236	C₂²	#2148	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	8	4	no	—
244	C₄	—	16	2	no	—
245	C₄	#1547	1	1	yes	characteristic
246	C₄	#1560	1	1	yes	characteristic
247	C₄	#1800	1	1	yes	characteristic
248	C₄	#1802	1	1	yes	characteristic
249	C₄	#2211	1	2	yes	—
250	C₄	#2229	1	2	yes	—
251	C₂	#8585	1	1	yes	characteristic
252	C₂	#11929	1	1	yes	characteristic
253	C₂	#11932	1	1	yes	characteristic
254	C₂	#11984	1	1	yes	characteristic
255	C₂	#11987	1	1	yes	characteristic
256	C₂	#12814	1	1	yes	characteristic
257	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

Maximal subgroups

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

subgroup	index
C₄×#494	2
#12898	2
C₂×#1362	2
#8807	2
#7274	2