Subgroups — #203944

Order 512.

645 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1837	C₂	1	1	yes	maximal, characteristic
2	C₈×#117	C₂	1	1	yes	maximal, characteristic
3	#17809	C₂	1	2	yes	maximal
4	#17793	C₂	1	1	yes	maximal, characteristic
5	#9629	C₂	1	8	yes	maximal, complemented
6	#8097	C₂	1	2	yes	maximal
7	#2920	C₂²	1	2	yes	—
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#423	C₂²	1	2	yes	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×#117	C₂²	1	1	yes	characteristic
12	#1837	—	2	1	no	—
13	#1837	—	2	1	no	—
14	#1837	C₂²	1	4	yes	—
15	C₈×C₄⋊C₄	C₂²	1	4	yes	—
16	C₈×C₂²⋊C₄	—	2	2	no	—
17	#1744	—	2	1	no	—
18	C₂×#341	—	2	2	no	—
19	C₄²×C₈	—	2	1	no	—
20	#1574	—	2	4	no	—
21	#1571	C₂²	1	4	yes	—
22	#1570	C₂²	1	4	yes	—
23	#1568	C₄	1	4	yes	complemented
24	#1567	C₄	1	2	yes	complemented
25	#1566	C₄	1	2	yes	complemented
26	#1565	—	2	4	no	—
27	#1564	C₂²	1	4	yes	—
28	#1303	C₂²	1	8	yes	—
29	C₂²×M₁₆	—	2	2	no	—
30	C₂³×C₈	D₄	1	2	yes	—
31	C₂×#117	—	2	1	no	—
32	C₂²×C₄²	C₂³	1	1	yes	characteristic
33	#432	—	2	2	no	—
34	#432	C₂×C₄	1	4	yes	—
35	#431	—	2	2	no	—
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	D₄	1	2	yes	—
39	#431	D₄	1	2	yes	—
40	#431	C₂³	1	4	yes	—
41	#423	—	2	2	no	—
42	#423	—	2	2	no	—
43	#423	—	2	2	no	—
44	#423	C₂×C₄	1	4	yes	—
45	#423	C₂³	1	4	yes	—
46	C₂×C₄⋊C₈	—	2	2	no	—
47	C₂×C₄⋊C₈	—	2	2	no	—
48	C₂×C₄⋊C₈	C₂³	1	2	yes	—
49	#407	—	2	2	no	—
50	C₂×C₂²⋊C₈	—	2	1	no	—
51	C₄×M₁₆	—	2	2	no	—
52	C₂×C₄×C₈	—	2	1	no	—
53	C₂×C₄×C₈	—	2	4	no	—
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈	Q₈	1	2	yes	—
62	C₂×C₄×C₈	C₂³	1	2	yes	—
63	C₄×C₄⋊C₄	—	2	1	no	—
64	C₄×C₄⋊C₄	C₂³	1	2	yes	—
65	C₄×C₂²⋊C₄	—	2	1	no	—
66	C₄×C₂²⋊C₄	Q₈	1	2	yes	—
67	#376	Q₈	1	2	yes	—
68	C₄³	—	2	1	no	—
69	#341	—	2	4	no	—
70	#341	—	2	4	no	—
71	#336	—	2	4	no	—
72	#335	—	2	8	no	—
73	#335	C₂×C₄	1	8	yes	—
74	#334	—	2	8	no	—
75	#334	C₂×C₄	1	8	yes	—
76	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂³×C₄	#41	1	1	yes	characteristic
79	C₂×M₁₆	—	2	4	no	—
80	C₂×M₁₆	—	2	4	no	—
81	C₂×M₁₆	—	4	2	no	—
82	C₂×M₁₆	—	4	8	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	4	4	no	—
85	C₂²×C₈	—	4	2	no	—
86	C₂²×C₈	C₄⋊C₄	1	2	yes	—
87	C₂²×C₈	C₄⋊C₄	1	2	yes	—
88	C₂²×C₈	C₄⋊C₄	1	2	yes	—
89	C₂²×C₈	C₄⋊C₄	1	2	yes	—
90	C₂²×C₈	C₂×D₄	1	2	yes	—
91	C₂²×C₈	C₂×D₄	1	2	yes	—
92	#117	—	2	8	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²	—	2	2	no	—
96	C₂×C₄²	—	2	2	no	—
97	C₂×C₄²	—	2	2	no	—
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₄	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×D₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₄²	#41	1	1	yes	characteristic
107	C₂×C₄²	#41	1	1	yes	characteristic
108	C₂×C₄²	#41	1	1	yes	characteristic
109	C₂×C₄²	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄⋊C₈	C₂²×C₄	1	8	yes	—
117	C₄⋊C₈	C₂×D₄	1	4	yes	—
118	C₂²⋊C₈	—	2	4	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	4	no	—
122	#97	—	2	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	4	no	—
126	C₄×C₈	—	4	8	no	—
127	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₄⋊C₄	1	2	yes	—
132	C₄×C₈	C₄⋊C₄	1	2	yes	—
133	C₄×C₈	C₂²×C₄	1	2	yes	—
134	C₄×C₈	C₂²×C₄	1	2	yes	—
135	C₄×C₈	C₂×Q₈	1	2	yes	—
136	C₄×C₈	C₂×Q₈	1	2	yes	—
137	#95	—	2	2	no	—
138	#95	C₂×Q₈	1	4	yes	—
139	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄⋊C₄	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	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄	#125	1	1	yes	characteristic
156	C₂²×C₄	#125	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
161	C₂²×C₄	C₂×#41	1	1	yes	characteristic
162	M₁₆	—	4	8	no	—
163	M₁₆	—	4	8	no	—
164	M₁₆	—	8	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	8	4	no	—
176	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
177	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
179	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
180	C₄⋊C₄	—	2	8	no	—
181	C₂²⋊C₄	—	2	8	no	—
182	C₄²	—	2	8	no	—
183	C₄²	—	4	4	no	—
184	C₄²	—	4	4	no	—
185	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₄²	C₄×D₄	1	1	yes	characteristic
188	C₄²	C₄×D₄	1	1	yes	characteristic
189	C₄²	C₄×Q₈	1	1	yes	characteristic
190	C₄²	C₄×Q₈	1	1	yes	characteristic
191	C₄²	#121	1	1	yes	characteristic
192	C₄²	#121	1	1	yes	characteristic
193	C₄²	#125	1	1	yes	characteristic
194	C₄²	#125	1	1	yes	characteristic
195	C₄²	C₄⋊Q₈	1	1	yes	characteristic
196	C₄²	C₄⋊Q₈	1	1	yes	characteristic
197	C₄²	C₂³×C₄	1	1	yes	characteristic
198	C₄²	C₂²×D₄	1	1	yes	characteristic
199	C₄²	C₂²×Q₈	1	1	yes	characteristic
200	C₄²	C₂×#41	1	1	yes	characteristic
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	4	1	no	—
204	C₂³	#389	1	1	yes	characteristic
205	C₂³	#389	1	1	yes	characteristic
206	C₂³	C₂×C₂²⋊Q₈	1	1	yes	characteristic
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	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	#389	1	1	yes	characteristic
227	C₂×C₄	#389	1	1	yes	characteristic
228	C₂×C₄	#389	1	1	yes	characteristic
229	C₂×C₄	#389	1	1	yes	characteristic
230	C₂×C₄	#389	1	1	yes	characteristic
231	C₂×C₄	#389	1	1	yes	characteristic
232	C₂×C₄	#443	1	4	yes	—
233	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
236	C₂×C₄	C₂×#121	1	1	yes	characteristic
237	C₂×C₄	C₂×#125	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
239	C₂×C₄	#575	1	4	yes	—
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	4	8	no	—
245	C₈	—	4	4	no	—
246	C₈	—	8	4	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	#1945	1	2	yes	—
256	C₂²	C₂×#389	1	1	yes	characteristic
257	C₂²	#3080	1	2	yes	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	4	8	no	—
266	C₄	#1945	1	2	yes	—
267	C₄	#1953	1	2	yes	—
268	C₄	#2911	1	2	yes	—
269	C₄	C₂×#443	1	2	yes	—
270	C₄	#3040	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	—	4	1	no	—
274	C₂	C₂×#1945	1	1	yes	characteristic
275	C₂	#21172	1	1	yes	characteristic
276	C₂	C₂×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #9629	C₂	semidirect
23. #1568	C₄	semidirect
24. #1567	C₄	semidirect
25. #1566	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1837	2
C₈×#117	2
#17809	2
#17793	2
#9629	2
#8097	2