Subgroups — #109276

Order 512.

565 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#117	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	1	yes	maximal, characteristic
3	C₂×#1566	C₂	1	1	yes	maximal, characteristic
4	#8097	C₂	1	2	yes	maximal
5	#7934	C₂	1	2	yes	maximal
6	#2920	C₂²	1	2	yes	—
7	C₂×#431	C₂²	1	1	yes	characteristic
8	C₂×#423	C₂²	1	2	yes	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄×#117	C₂²	1	1	yes	characteristic
11	C₈×C₄⋊C₄	—	2	2	no	—
12	C₈×C₂²⋊C₄	C₄	1	4	yes	—
13	#1744	—	2	1	no	—
14	C₄²×C₈	—	2	1	no	—
15	#1568	—	2	2	no	—
16	#1567	C₄	1	2	yes	complemented
17	#1566	—	2	2	no	—
18	#1566	—	2	1	no	—
19	#1566	—	2	1	no	—
20	#1566	C₄	1	2	yes	complemented
21	#1564	—	2	2	no	—
22	C₂×#334	C₄	1	4	yes	complemented
23	#1303	—	2	4	no	—
24	#1283	—	2	4	no	—
25	C₂²×M₁₆	—	2	2	no	—
26	C₂³×C₈	C₂×C₄	1	2	yes	—
27	C₂×#117	—	2	1	no	—
28	C₂²×C₄²	C₂³	1	1	yes	characteristic
29	#432	—	2	2	no	—
30	#432	—	2	2	no	—
31	#431	—	2	1	no	—
32	#431	—	2	1	no	—
33	#431	—	2	2	no	—
34	#431	—	2	2	no	—
35	#431	D₄	1	2	yes	—
36	#431	Q₈	1	2	yes	—
37	#423	—	2	2	no	—
38	#423	—	2	2	no	—
39	#423	—	2	2	no	—
40	#423	—	2	2	no	—
41	#423	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₈	—	2	1	no	—
43	C₂×C₄⋊C₈	—	2	2	no	—
44	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	#407	—	2	2	no	—
46	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
47	C₄×M₁₆	—	2	2	no	—
48	C₂×C₄×C₈	—	2	1	no	—
49	C₂×C₄×C₈	—	2	4	no	—
50	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	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	2	yes	—
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
59	C₄×C₄⋊C₄	—	2	1	no	—
60	C₄×C₄⋊C₄	D₄	1	2	yes	—
61	C₄×C₂²⋊C₄	—	2	1	no	—
62	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
63	#376	Q₈	1	2	yes	—
64	C₄³	—	2	1	no	—
65	#336	—	2	4	no	—
66	#335	—	2	4	no	—
67	#334	—	2	4	no	—
68	#334	—	2	4	no	—
69	#334	—	2	4	no	—
70	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₄	#41	1	1	yes	characteristic
72	C₂³×C₄	#41	1	1	yes	characteristic
73	C₂×M₁₆	—	2	4	no	—
74	C₂×M₁₆	—	4	2	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂×M₁₆	—	8	4	no	—
77	C₂²×C₈	—	2	2	no	—
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	—	4	4	no	—
80	C₂²×C₈	C₄²	1	2	yes	—
81	C₂²×C₈	C₄²	1	2	yes	—
82	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₈	C₄⋊C₄	1	2	yes	—
85	C₂²×C₈	C₄⋊C₄	1	2	yes	—
86	#117	—	2	8	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	—	2	2	no	—
90	C₂×C₄²	—	2	2	no	—
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₄²	#41	1	1	yes	characteristic
103	C₂×C₄²	#41	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	2	4	no	—
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	2	4	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	4	2	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₂²⋊C₈	—	4	2	no	—
114	C₂²⋊C₈	—	4	2	no	—
115	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
116	#97	—	4	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	4	8	no	—
120	C₄×C₈	—	4	2	no	—
121	C₄×C₈	C₂²⋊C₄	1	2	yes	—
122	C₄×C₈	C₂²⋊C₄	1	2	yes	—
123	C₄×C₈	C₂²⋊C₄	1	2	yes	—
124	C₄×C₈	C₂²⋊C₄	1	2	yes	—
125	C₄×C₈	C₄⋊C₄	1	2	yes	—
126	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₄⋊C₄	1	2	yes	—
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	#95	—	2	2	no	—
132	#95	C₄⋊C₄	1	4	yes	—
133	C₂⁴	#117	1	1	yes	characteristic
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	#117	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
145	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
146	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
147	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
148	C₂²×C₄	#121	1	1	yes	characteristic
149	C₂²×C₄	#121	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂²×C₄	#124	1	1	yes	characteristic
153	C₂²×C₄	#124	1	1	yes	characteristic
154	C₂²×C₄	#125	1	1	yes	characteristic
155	C₂²×C₄	#125	1	1	yes	characteristic
156	M₁₆	—	4	8	no	—
157	M₁₆	—	8	4	no	—
158	M₁₆	—	8	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	8	4	no	—
170	C₂×C₈	#95	1	2	yes	—
171	C₂×C₈	#95	1	2	yes	—
172	C₂×C₈	#95	1	2	yes	—
173	C₂×C₈	#95	1	2	yes	—
174	C₄⋊C₄	—	2	8	no	—
175	C₂²⋊C₄	—	2	8	no	—
176	C₄²	—	2	8	no	—
177	C₄²	—	4	4	no	—
178	C₄²	—	4	4	no	—
179	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₄²	C₄×D₄	1	1	yes	characteristic
184	C₄²	C₄×D₄	1	1	yes	characteristic
185	C₄²	C₄×Q₈	1	1	yes	characteristic
186	C₄²	C₄×Q₈	1	1	yes	characteristic
187	C₄²	#121	1	1	yes	characteristic
188	C₄²	#121	1	1	yes	characteristic
189	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
190	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
191	C₄²	#127	1	1	yes	characteristic
192	C₄²	C₄⋊Q₈	1	1	yes	characteristic
193	C₄²	C₄⋊Q₈	1	1	yes	characteristic
194	C₄²	C₄⋊Q₈	1	1	yes	characteristic
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	4	1	no	—
198	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
199	C₂³	#380	1	1	yes	characteristic
200	C₂³	#382	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
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	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₂×#95	1	1	yes	characteristic
221	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₄	#380	1	1	yes	characteristic
223	C₂×C₄	#384	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	#389	1	1	yes	characteristic
228	C₂×C₄	#390	1	1	yes	characteristic
229	C₂×C₄	#390	1	1	yes	characteristic
230	C₂×C₄	#391	1	1	yes	characteristic
231	C₂×C₄	#391	1	1	yes	characteristic
232	C₂×C₄	#443	1	4	yes	—
233	C₂×C₄	#471	1	4	yes	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	2	2	no	—
237	C₈	—	4	4	no	—
238	C₈	—	8	4	no	—
239	C₈	—	8	4	no	—
240	C₈	#344	1	4	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	4	no	—
249	C₂²	C₄⋊#95	1	1	yes	characteristic
250	C₂²	#1772	1	2	yes	—
251	C₂²	#1815	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	8	no	—
260	C₄	C₂×#344	1	2	yes	—
261	C₄	#1945	1	2	yes	—
262	C₄	#1951	1	2	yes	—
263	C₄	#1974	1	2	yes	—
264	C₄	#1977	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	—	2	2	no	—
267	C₂	—	4	1	no	—
268	C₂	#10856	1	1	yes	characteristic
269	C₂	#10859	1	1	yes	characteristic
270	C₂	#10869	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
16. #1567	C₄	semidirect
20. #1566	C₄	semidirect
22. 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
C₈×#117	2
#17793	2
C₂×#1566	2
#8097	2
#7934	2