Subgroups — #109169

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20366	C₂	1	1	yes	maximal, characteristic
2	C₂×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#7924	C₂	1	2	yes	maximal
5	#7882	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	2	yes	—
7	C₂×#431	C₂²	1	2	yes	—
8	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄×#117	C₂²	1	1	yes	characteristic
11	#1746	—	2	2	no	—
12	#1744	—	2	1	no	—
13	#1744	—	2	1	no	—
14	#1729	C₄	1	4	yes	—
15	#1564	—	2	2	no	—
16	#1564	—	2	1	no	—
17	#1564	—	2	1	no	—
18	#1564	—	2	1	no	—
19	#1564	—	2	1	no	—
20	#1564	C₄	1	4	yes	complemented
21	C₂×#335	—	2	1	no	—
22	C₂×#335	C₄	1	2	yes	complemented
23	C₂×#334	—	2	1	no	—
24	C₂×#334	C₄	1	2	yes	complemented
25	#1282	—	2	4	no	—
26	#1278	—	2	4	no	—
27	C₂²×M₁₆	C₂×C₄	1	2	yes	—
28	C₂³×C₈	C₂×C₄	1	2	yes	—
29	C₂×#117	C₂×C₄	1	2	yes	—
30	C₂²×C₄²	C₂³	1	1	yes	characteristic
31	#431	—	2	2	no	—
32	#431	—	2	2	no	—
33	#431	—	2	2	no	—
34	#431	—	2	2	no	—
35	#431	—	2	2	no	—
36	#431	—	2	2	no	—
37	#431	—	2	2	no	—
38	#431	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₈	—	2	2	no	—
40	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₈	—	2	2	no	—
42	C₂×C₂²⋊C₈	—	2	2	no	—
43	C₄×M₁₆	—	2	1	no	—
44	C₄×M₁₆	—	2	1	no	—
45	C₄×M₁₆	D₄	1	2	yes	—
46	C₄×M₁₆	Q₈	1	2	yes	—
47	C₂×#97	—	2	1	no	—
48	C₂×C₄×C₈	—	2	1	no	—
49	C₂×C₄×C₈	—	2	2	no	—
50	C₂×C₄×C₈	—	2	2	no	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	2	yes	—
59	C₂×C₄×C₈	Q₈	1	2	yes	—
60	C₄×C₄⋊C₄	—	2	1	no	—
61	C₄×C₄⋊C₄	—	2	1	no	—
62	C₄×C₂²⋊C₄	D₄	1	2	yes	—
63	C₄×C₂²⋊C₄	Q₈	1	2	yes	—
64	#376	—	2	1	no	—
65	C₄³	—	2	1	no	—
66	#336	—	2	4	no	—
67	#336	—	2	4	no	—
68	#335	—	2	4	no	—
69	#335	—	2	2	no	—
70	#335	—	2	2	no	—
71	#334	—	2	4	no	—
72	#334	—	2	2	no	—
73	#334	—	2	2	no	—
74	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	2	8	no	—
79	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
80	C₂²×C₈	—	2	8	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	2	4	no	—
83	C₂²×C₈	—	4	2	no	—
84	C₂²×C₈	C₄²	1	2	yes	—
85	C₂²×C₈	C₄²	1	2	yes	—
86	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂²×C₈	C₄⋊C₄	1	2	yes	—
89	C₂²×C₈	C₄⋊C₄	1	2	yes	—
90	#117	—	2	8	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
93	C₂×C₄²	—	2	2	no	—
94	C₂×C₄²	—	2	2	no	—
95	C₂×C₄²	—	2	2	no	—
96	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₄²	#41	1	1	yes	characteristic
105	C₂×C₄²	#41	1	1	yes	characteristic
106	C₂×C₄²	#41	1	1	yes	characteristic
107	C₂×C₄²	#41	1	1	yes	characteristic
108	C₄⋊C₈	—	2	4	no	—
109	C₄⋊C₈	—	2	4	no	—
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₂²⋊C₈	—	2	4	no	—
113	C₂²⋊C₈	—	4	2	no	—
114	C₂²⋊C₈	—	4	2	no	—
115	C₂²⋊C₈	—	4	2	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	C₄×C₈	—	2	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	4	2	no	—
122	C₄×C₈	—	4	2	no	—
123	C₄×C₈	—	4	2	no	—
124	C₄×C₈	C₂²⋊C₄	1	2	yes	—
125	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊C₄	1	2	yes	—
128	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	2	yes	—
131	C₄×C₈	C₄⋊C₄	1	2	yes	—
132	C₄×C₈	C₄⋊C₄	1	2	yes	—
133	C₄×C₈	C₄⋊C₄	1	2	yes	—
134	#95	—	2	2	no	—
135	#95	—	2	2	no	—
136	C₂⁴	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	#117	1	1	yes	characteristic
145	C₂²×C₄	#117	1	1	yes	characteristic
146	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
150	C₂²×C₄	#121	1	1	yes	characteristic
151	C₂²×C₄	#121	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂²×C₄	#127	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄	#131	1	8	yes	—
159	M₁₆	—	2	8	no	—
160	M₁₆	—	4	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	8	4	no	—
172	C₂×C₈	—	8	4	no	—
173	C₂×C₈	#95	1	2	yes	—
174	C₂×C₈	#95	1	2	yes	—
175	C₂×C₈	#95	1	2	yes	—
176	C₂×C₈	#95	1	2	yes	—
177	C₄⋊C₄	—	4	4	no	—
178	C₂²⋊C₄	—	2	8	no	—
179	C₄²	—	4	4	no	—
180	C₄²	—	4	4	no	—
181	C₄²	—	4	4	no	—
182	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₄²	C₄×D₄	1	1	yes	characteristic
187	C₄²	C₄×D₄	1	1	yes	characteristic
188	C₄²	C₄×Q₈	1	1	yes	characteristic
189	C₄²	C₄×Q₈	1	1	yes	characteristic
190	C₄²	#121	1	1	yes	characteristic
191	C₄²	#121	1	1	yes	characteristic
192	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
193	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
194	C₄²	#124	1	1	yes	characteristic
195	C₄²	#124	1	1	yes	characteristic
196	C₄²	#125	1	1	yes	characteristic
197	C₄²	#125	1	1	yes	characteristic
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
201	C₂³	#380	1	1	yes	characteristic
202	C₂³	#384	1	1	yes	characteristic
203	C₂³	#424	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	C₂×#95	1	1	yes	characteristic
220	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₄	#380	1	1	yes	characteristic
222	C₂×C₄	#382	1	1	yes	characteristic
223	C₂×C₄	#389	1	1	yes	characteristic
224	C₂×C₄	#389	1	1	yes	characteristic
225	C₂×C₄	#389	1	1	yes	characteristic
226	C₂×C₄	#389	1	1	yes	characteristic
227	C₂×C₄	#390	1	1	yes	characteristic
228	C₂×C₄	#390	1	1	yes	characteristic
229	C₂×C₄	#391	1	1	yes	characteristic
230	C₂×C₄	#391	1	1	yes	characteristic
231	C₂×C₄	#405	1	4	yes	—
232	C₂×C₄	#408	1	4	yes	—
233	C₂×C₄	#424	1	4	yes	—
234	C₂×C₄	#424	1	4	yes	—
235	C₂×C₄	#433	1	4	yes	—
236	C₂×C₄	#433	1	4	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	4	4	no	—
242	C₈	—	8	4	no	—
243	C₈	—	8	4	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	C₄⋊#95	1	1	yes	characteristic
252	C₂²	#1768	1	2	yes	—
253	C₂²	#1798	1	2	yes	—
254	C₂²	#1843	1	2	yes	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	4	no	—
262	C₄	#1729	1	2	yes	—
263	C₄	#1768	1	2	yes	—
264	C₄	#1798	1	2	yes	—
265	C₄	#1832	1	2	yes	—
266	C₄	#1953	1	2	yes	—
267	C₄	#1976	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	2	2	no	—
270	C₂	—	2	2	no	—
271	C₂	#10833	1	1	yes	characteristic
272	C₂	#10871	1	1	yes	characteristic
273	C₂	#10873	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#121
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
20. #1564	C₄	semidirect
22. C₂×#335	C₄	semidirect
24. C₂×#334	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20366	2
C₂×#1564	2
#7924	2
#7882	2