Subgroups — #109442

Order 512.

661 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	#20826	C₂	1	1	yes	maximal, characteristic
2	C₄×#405	C₂	1	1	yes	maximal, characteristic
3	C₂×#1566	C₂	1	1	yes	maximal, characteristic
4	#8097	C₂	1	2	yes	maximal
5	#8063	C₂	1	2	yes	maximal
6	#2920	C₂²	1	2	yes	—
7	C₂×#423	C₂²	1	2	yes	—
8	C₂×#405	—	2	2	no	—
9	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×#117	C₂²	1	1	yes	characteristic
12	#1848	C₄	1	2	yes	complemented
13	#1837	C₄	1	2	yes	complemented
14	#1810	—	2	2	no	—
15	C₄×#108	—	2	2	no	—
16	#1726	C₄	1	4	yes	—
17	C₄×#97	—	2	1	no	—
18	C₄²×C₈	—	2	1	no	—
19	#1566	—	2	1	no	—
20	#1566	—	2	1	no	—
21	#1566	C₄	1	4	yes	complemented
22	C₂×#334	—	2	2	no	—
23	#1303	—	2	4	no	—
24	#1299	—	2	4	no	—
25	C₂²×M₁₆	—	2	1	no	—
26	C₂²×M₁₆	—	2	2	no	—
27	C₂³×C₈	D₄	1	2	yes	—
28	C₂×#117	—	2	1	no	—
29	C₂²×C₄²	C₂³	1	1	yes	characteristic
30	#432	—	2	2	no	—
31	#432	—	2	2	no	—
32	#431	—	2	2	no	—
33	#431	—	2	2	no	—
34	C₂×#108	—	2	4	no	—
35	#423	—	2	2	no	—
36	#423	—	2	2	no	—
37	#423	—	2	2	no	—
38	#423	C₂×C₄	1	4	yes	—
39	#423	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	#407	—	2	2	no	—
43	#405	—	2	16	no	—
44	C₄×M₁₆	—	2	1	no	—
45	C₄×M₁₆	—	2	1	no	—
46	C₄×M₁₆	—	2	2	no	—
47	C₄×M₁₆	D₄	1	2	yes	—
48	C₄×M₁₆	Q₈	1	2	yes	—
49	C₂×#97	—	2	4	no	—
50	C₂×#97	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	—	2	4	no	—
52	C₂×C₄×C₈	—	2	1	no	—
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
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	2	yes	—
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
62	C₄×C₄⋊C₄	—	2	1	no	—
63	C₄×C₄⋊C₄	D₄	1	2	yes	—
64	C₄×C₂²⋊C₄	—	2	1	no	—
65	C₄×C₂²⋊C₄	Q₈	1	2	yes	—
66	#376	C₂×C₄	1	2	yes	—
67	C₄³	—	2	1	no	—
68	#341	—	2	4	no	—
69	#334	—	2	4	no	—
70	#334	—	2	4	no	—
71	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₄	#41	1	1	yes	characteristic
74	C₂×M₁₆	—	2	8	no	—
75	C₂×M₁₆	—	2	2	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂×M₁₆	—	2	4	no	—
78	C₂×M₁₆	—	2	4	no	—
79	C₂×M₁₆	—	4	2	no	—
80	C₂×M₁₆	—	8	4	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	2	8	no	—
83	C₂²×C₈	—	4	2	no	—
84	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₈	C₂²⋊C₄	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	—	2	2	no	—
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	C₂²×C₄	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×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	#41	1	1	yes	characteristic
106	C₂×C₄²	#41	1	1	yes	characteristic
107	C₂×C₄²	#41	1	1	yes	characteristic
108	#108	—	4	8	no	—
109	C₄⋊C₈	—	2	4	no	—
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₂²⋊C₈	—	4	2	no	—
116	C₂²⋊C₈	—	4	2	no	—
117	#97	—	2	16	no	—
118	#97	—	2	2	no	—
119	#97	—	4	2	no	—
120	#97	C₄⋊C₄	1	4	yes	—
121	C₄×C₈	—	2	16	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	4	2	no	—
125	C₄×C₈	C₄²	1	2	yes	—
126	C₄×C₈	C₄²	1	2	yes	—
127	C₄×C₈	C₂²⋊C₄	1	2	yes	—
128	C₄×C₈	C₂²⋊C₄	1	2	yes	—
129	C₄×C₈	C₄⋊C₄	1	2	yes	—
130	C₄×C₈	C₄⋊C₄	1	2	yes	—
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	#95	—	2	2	no	—
136	#95	C₄⋊C₄	1	4	yes	—
137	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	#117	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	C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₄	#121	1	1	yes	characteristic
152	C₂²×C₄	#121	1	1	yes	characteristic
153	C₂²×C₄	#121	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄	#124	1	1	yes	characteristic
157	C₂²×C₄	#125	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	M₁₆	—	2	8	no	—
162	M₁₆	—	4	8	no	—
163	M₁₆	—	8	4	no	—
164	M₁₆	—	8	4	no	—
165	C₂×C₈	—	2	8	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	8	4	no	—
175	C₂×C₈	#95	1	2	yes	—
176	C₂×C₈	#95	1	2	yes	—
177	C₂×C₈	#95	1	2	yes	—
178	C₂×C₈	#95	1	2	yes	—
179	C₂×C₈	#104	1	8	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×C₄²	1	1	yes	characteristic
186	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₄²	#117	1	1	yes	characteristic
191	C₄²	C₄×D₄	1	1	yes	characteristic
192	C₄²	C₄×D₄	1	1	yes	characteristic
193	C₄²	C₄×D₄	1	1	yes	characteristic
194	C₄²	C₄×Q₈	1	1	yes	characteristic
195	C₄²	#121	1	1	yes	characteristic
196	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
197	C₄²	#124	1	1	yes	characteristic
198	C₄²	#125	1	1	yes	characteristic
199	C₄²	#127	1	1	yes	characteristic
200	C₄²	C₄⋊Q₈	1	1	yes	characteristic
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	4	1	no	—
204	C₂³	#380	1	1	yes	characteristic
205	C₂³	#389	1	1	yes	characteristic
206	C₂³	#391	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	C₂×#95	1	1	yes	characteristic
227	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₄	#380	1	1	yes	characteristic
230	C₂×C₄	#382	1	1	yes	characteristic
231	C₂×C₄	#384	1	1	yes	characteristic
232	C₂×C₄	#389	1	1	yes	characteristic
233	C₂×C₄	#389	1	1	yes	characteristic
234	C₂×C₄	#389	1	1	yes	characteristic
235	C₂×C₄	#390	1	1	yes	characteristic
236	C₂×C₄	#390	1	1	yes	characteristic
237	C₂×C₄	#391	1	1	yes	characteristic
238	C₂×C₄	C₂×#104	1	4	yes	—
239	C₂×C₄	#471	1	4	yes	—
240	C₈	—	2	8	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	8	4	no	—
244	C₈	—	8	4	no	—
245	C₈	#339	1	4	yes	—
246	C₈	#339	1	4	yes	—
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	C₄⋊#95	1	1	yes	characteristic
256	C₂²	#1815	1	2	yes	—
257	C₂²	#1994	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	C₂×#339	1	2	yes	—
267	C₄	#1951	1	2	yes	—
268	C₄	#1953	1	2	yes	—
269	C₄	#1962	1	2	yes	—
270	C₄	#1977	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	—	4	1	no	—
274	C₂	#10839	1	1	yes	characteristic
275	C₂	#10869	1	1	yes	characteristic
276	C₂	#10873	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ 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
12. #1848	C₄	semidirect
13. #1837	C₄	semidirect
21. #1566	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20826	2
C₄×#405	2
C₂×#1566	2
#8097	2
#8063	2