Subgroups — #483738

Order 512.

387 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33499	C₂	1	1	yes	maximal, characteristic
2	#33497	C₂	1	1	yes	maximal, characteristic
3	#17903	C₂	1	1	yes	maximal, characteristic
4	#12061	C₂	1	2	yes	maximal, complemented
5	#12060	C₂	1	2	yes	maximal, complemented
6	#12054	C₂	1	2	yes	maximal, complemented
7	#12053	C₂	1	2	yes	maximal, complemented
8	#12014	C₂	1	1	yes	maximal, characteristic
9	#11639	C₂	1	1	yes	maximal, characteristic
10	#11638	C₂	1	1	yes	maximal, characteristic
11	#11412	C₂	1	1	yes	maximal, characteristic
12	#3104	C₄	1	1	yes	characteristic
13	#2960	C₂²	1	1	yes	characteristic
14	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
15	#2183	—	2	2	no	—
16	#2183	—	2	2	no	—
17	#2183	C₂²	1	2	yes	—
18	#2183	C₂²	1	2	yes	—
19	#2183	C₂²	1	2	yes	—
20	Q₈×C₁₆	C₂²	1	2	yes	—
21	#2164	—	2	2	no	—
22	#2163	—	2	2	no	—
23	#2163	C₂²	1	1	yes	characteristic
24	#2162	C₂²	1	1	yes	characteristic
25	#2162	C₂²	1	1	yes	characteristic
26	#2162	C₂²	1	1	yes	characteristic
27	#1584	C₄	1	2	yes	—
28	#1584	C₄	1	2	yes	—
29	C₈×Q₁₆	C₂²	1	2	yes	—
30	C₈×Q₁₆	C₂²	1	2	yes	—
31	#1571	C₄	1	1	yes	characteristic
32	#1487	C₄	1	1	yes	characteristic
33	#1481	C₄	1	1	yes	characteristic
34	#1450	C₂²	1	1	yes	characteristic
35	#1370	C₂²	1	2	yes	—
36	#1367	C₂²	1	2	yes	—
37	Q₈⋊C₁₆	C₂²	1	2	yes	—
38	Q₈⋊C₁₆	C₂²	1	2	yes	—
39	Q₈⋊C₁₆	C₂²	1	2	yes	—
40	Q₈⋊C₁₆	C₂²	1	2	yes	—
41	#1312	C₂²	1	1	yes	characteristic
42	#1312	C₂²	1	1	yes	characteristic
43	#1311	C₂²	1	1	yes	characteristic
44	C₈×C₁₆	C₂²	1	1	yes	characteristic
45	#533	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
47	#484	C₈	1	2	yes	—
48	#477	C₈	1	2	yes	—
49	#462	C₈	1	2	yes	—
50	#451	C₈	1	2	yes	—
51	#446	—	2	1	no	—
52	#446	C₂×C₄	1	2	yes	—
53	C₈×Q₈	—	2	1	no	—
54	C₈×Q₈	—	2	1	no	—
55	C₈×Q₈	—	2	1	no	—
56	C₈×Q₈	C₂×C₄	1	2	yes	—
57	C₈×Q₈	C₂³	1	2	yes	—
58	C₈×Q₈	C₂³	1	2	yes	—
59	C₄×Q₁₆	C₂×C₄	1	2	yes	—
60	C₄×Q₁₆	C₂×C₄	1	2	yes	—
61	#431	C₂×C₄	1	1	yes	characteristic
62	#431	C₂×C₄	1	1	yes	characteristic
63	#424	—	2	1	no	—
64	#423	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₈	—	2	1	no	—
66	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
67	#405	—	2	1	no	—
68	C₂×C₄×C₈	—	2	1	no	—
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₄⋊C₁₆	—	2	2	no	—
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	—	2	1	no	—
75	C₄⋊C₁₆	D₄	1	2	yes	—
76	C₄⋊C₁₆	C₂³	1	2	yes	—
77	C₄⋊C₁₆	C₂³	1	2	yes	—
78	C₂²⋊C₁₆	—	2	1	no	—
79	C₂²⋊C₁₆	—	2	1	no	—
80	C₂²⋊C₁₆	—	2	1	no	—
81	C₂²⋊C₁₆	D₄	1	2	yes	—
82	#346	—	2	2	no	—
83	#346	—	2	2	no	—
84	#346	—	2	2	no	—
85	#346	—	2	1	no	—
86	C₄×C₁₆	—	2	2	no	—
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	C₂³	1	1	yes	characteristic
89	C₄×C₁₆	C₂³	1	1	yes	characteristic
90	C₄×C₁₆	C₂³	1	1	yes	characteristic
91	C₄×C₁₆	C₂³	1	1	yes	characteristic
92	#335	C₂×C₄	1	2	yes	—
93	#335	C₂×C₄	1	2	yes	—
94	Q₈⋊C₈	C₂×C₄	1	2	yes	—
95	Q₈⋊C₈	C₂×C₄	1	2	yes	—
96	Q₈⋊C₈	C₂×C₄	1	2	yes	—
97	Q₈⋊C₈	C₂×C₄	1	2	yes	—
98	#322	C₂×C₄	1	1	yes	characteristic
99	#322	C₂×C₄	1	1	yes	characteristic
100	C₈²	C₂³	1	1	yes	characteristic
101	C₈²	C₂³	1	1	yes	characteristic
102	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
103	C₂×Q₁₆	C₂×C₈	1	2	yes	—
104	C₂×Q₁₆	C₂×C₈	1	2	yes	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	#41	1	1	yes	characteristic
109	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
110	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
111	#125	—	2	1	no	—
112	C₂²⋊Q₈	—	2	1	no	—
113	C₂²⋊Q₈	C₂×C₈	1	2	yes	—
114	C₄×Q₈	—	2	1	no	—
115	C₄×Q₈	—	2	1	no	—
116	C₄×Q₈	—	2	1	no	—
117	C₄×Q₈	C₂×C₈	1	2	yes	—
118	C₄×Q₈	C₂²×C₄	1	2	yes	—
119	C₄×Q₈	C₂²×C₄	1	2	yes	—
120	#117	—	2	1	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
123	C₂×C₄²	—	2	1	no	—
124	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	C₂×D₄	1	2	yes	—
130	C₂×C₁₆	C₂×D₄	1	2	yes	—
131	#107	C₂×C₈	1	2	yes	—
132	#107	C₂×C₈	1	2	yes	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	1	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	C₂×C₈	1	2	yes	—
141	C₄⋊C₈	C₂²×C₄	1	2	yes	—
142	C₄⋊C₈	C₂²×C₄	1	2	yes	—
143	#103	C₂×C₈	1	2	yes	—
144	#103	C₂×C₈	1	2	yes	—
145	#103	C₂×C₈	1	2	yes	—
146	#103	C₂×C₈	1	2	yes	—
147	C₂²⋊C₈	—	2	1	no	—
148	C₂²⋊C₈	C₂×C₈	1	2	yes	—
149	#97	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₄×C₈	C₂×D₄	1	1	yes	characteristic
159	C₄×C₈	#41	1	1	yes	characteristic
160	C₄×C₈	C₂⁴	1	1	yes	characteristic
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	C₂²×C₈	1	2	yes	—
165	C₂×Q₈	C₂²×C₈	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
170	C₂²×C₄	#131	1	1	yes	characteristic
171	Q₁₆	—	4	2	no	—
172	Q₁₆	—	4	2	no	—
173	M₁₆	—	8	1	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	8	1	no	—
181	C₂×C₈	C₄×D₄	1	1	yes	characteristic
182	C₂×C₈	C₄×D₄	1	1	yes	characteristic
183	C₂×C₈	C₂²×C₈	1	2	yes	—
184	C₂×C₈	C₂²×C₈	1	2	yes	—
185	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₈	C₂×#41	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₂²×C₈	1	2	yes	—
196	C₄⋊C₄	C₂²×C₈	1	2	yes	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₄×D₄	1	1	yes	characteristic
202	C₄²	C₂²×C₈	1	1	yes	characteristic
203	C₄²	#131	1	1	yes	characteristic
204	C₄²	C₂³×C₄	1	1	yes	characteristic
205	C₁₆	—	2	2	no	—
206	C₁₆	—	2	2	no	—
207	C₁₆	—	4	1	no	—
208	C₁₆	—	4	1	no	—
209	C₁₆	—	4	2	no	—
210	C₁₆	—	4	2	no	—
211	C₂³	C₈×D₄	1	1	yes	characteristic
212	Q₈	—	2	2	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	8	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	C₈×D₄	1	1	yes	characteristic
229	C₂×C₄	C₈×D₄	1	1	yes	characteristic
230	C₂×C₄	C₈×D₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
233	C₂×C₄	C₂×#131	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	4	1	no	—
240	C₈	—	4	1	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	2	no	—
243	C₈	#578	1	2	yes	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	#2950	1	1	yes	characteristic
257	C₄	#2957	1	1	yes	characteristic
258	C₄	#3407	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#17856	1	1	yes	characteristic
261	C₂	#33433	1	1	yes	characteristic
262	C₂	#33442	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ 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
4. #12061	C₂	semidirect
5. #12060	C₂	semidirect
6. #12054	C₂	semidirect
7. #12053	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33499	2
#33497	2
#17903	2
#12061	2
#12060	2
#12054	2
#12053	2
#12014	2
#11639	2
#11638	2
#11412	2