Subgroups — #127308

Order 512.

395 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	1	yes	maximal, characteristic
2	#20826	C₂	1	1	yes	maximal, characteristic
3	#17793	C₂	1	1	yes	maximal, characteristic
4	#7224	C₂	1	4	yes	maximal, complemented
5	#2920	—	2	1	no	—
6	#2920	—	2	1	no	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#2163	C₄	1	2	yes	—
9	#2163	C₂²	1	2	yes	—
10	#2162	—	2	1	no	—
11	#2162	—	2	1	no	—
12	#2150	—	2	1	no	—
13	#2112	—	2	1	no	—
14	C₄×M₃₂	C₄	1	2	yes	—
15	#1848	—	2	1	no	—
16	#1837	C₂²	1	2	yes	—
17	#1810	—	2	1	no	—
18	#1810	C₄	1	2	yes	—
19	C₄×#108	—	2	1	no	—
20	C₄×#108	C₄	1	2	yes	—
21	#1568	—	2	1	no	—
22	#1568	C₂²	1	2	yes	—
23	#1567	—	2	1	no	—
24	#1566	C₄	1	2	yes	—
25	#1564	—	2	1	no	—
26	#1564	C₄	1	2	yes	—
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₄²	D₄	1	1	yes	characteristic
30	C₂×M₃₂	—	2	2	no	—
31	#432	—	2	2	no	—
32	#432	—	4	1	no	—
33	#432	—	4	1	no	—
34	#432	—	4	1	no	—
35	#431	—	2	2	no	—
36	#431	—	4	1	no	—
37	#431	—	4	1	no	—
38	#431	—	4	1	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	2	no	—
41	#423	—	2	2	no	—
42	#423	—	2	2	no	—
43	#407	—	2	2	no	—
44	#407	—	4	1	no	—
45	C₄×M₁₆	—	2	2	no	—
46	C₄×M₁₆	—	4	1	no	—
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	—	2	2	no	—
61	C₂²⋊C₁₆	—	2	2	no	—
62	C₂²⋊C₁₆	—	2	2	no	—
63	#346	C₂×C₄	1	4	yes	—
64	C₄×C₁₆	—	2	2	no	—
65	#341	—	2	2	no	—
66	#341	C₂×C₄	1	4	yes	—
67	#335	—	2	2	no	—
68	#335	—	2	2	no	—
69	#334	—	2	2	no	—
70	#334	C₂×C₄	1	4	yes	—
71	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₄	SD₁₆	1	1	yes	characteristic
73	C₂³×C₄	Q₁₆	1	1	yes	characteristic
74	C₂×M₁₆	—	2	4	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	—	4	2	no	—
78	C₂²×C₈	C₄²	1	1	yes	characteristic
79	C₂²×C₈	C₄²	1	1	yes	characteristic
80	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄²	D₈	1	1	yes	characteristic
96	C₂×C₄²	D₈	1	1	yes	characteristic
97	C₂×C₄²	SD₁₆	1	1	yes	characteristic
98	C₂×C₄²	SD₁₆	1	1	yes	characteristic
99	C₂×C₄²	SD₁₆	1	1	yes	characteristic
100	C₂×C₄²	Q₁₆	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	M₃₂	—	4	4	no	—
103	C₂×C₁₆	—	2	4	no	—
104	#108	—	4	4	no	—
105	#108	—	4	4	no	—
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	4	2	no	—
108	C₄⋊C₈	—	4	2	no	—
109	C₄⋊C₈	—	4	2	no	—
110	C₂²⋊C₈	—	4	2	no	—
111	C₂²⋊C₈	—	4	2	no	—
112	C₂²⋊C₈	—	4	2	no	—
113	C₂²⋊C₈	—	4	2	no	—
114	#97	—	4	2	no	—
115	#97	—	4	2	no	—
116	C₄×C₈	—	4	2	no	—
117	C₄×C₈	—	4	2	no	—
118	C₄×C₈	C₄²	1	1	yes	characteristic
119	C₄×C₈	C₄²	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
130	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₄×C₈	C₂×D₄	1	1	yes	characteristic
133	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
134	C₂⁴	#103	1	1	yes	characteristic
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	#95	1	1	yes	characteristic
140	C₂²×C₄	#95	1	1	yes	characteristic
141	C₂²×C₄	#102	1	1	yes	characteristic
142	C₂²×C₄	#102	1	1	yes	characteristic
143	C₂²×C₄	#102	1	1	yes	characteristic
144	C₂²×C₄	#102	1	1	yes	characteristic
145	C₂²×C₄	#103	1	1	yes	characteristic
146	C₂²×C₄	#103	1	1	yes	characteristic
147	C₂²×C₄	#103	1	1	yes	characteristic
148	C₂²×C₄	#103	1	1	yes	characteristic
149	C₂²×C₄	#103	1	1	yes	characteristic
150	C₂²×C₄	#106	1	1	yes	characteristic
151	C₂²×C₄	#106	1	1	yes	characteristic
152	C₂²×C₄	#107	1	1	yes	characteristic
153	C₂²×C₄	#107	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
157	M₁₆	—	4	4	no	—
158	M₁₆	—	4	4	no	—
159	M₁₆	—	8	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	#95	1	1	yes	characteristic
175	C₂×C₈	#95	1	1	yes	characteristic
176	C₂×C₈	#95	1	1	yes	characteristic
177	C₂×C₈	#95	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₄²	#95	1	1	yes	characteristic
183	C₄²	#95	1	1	yes	characteristic
184	C₄²	#102	1	1	yes	characteristic
185	C₄²	#102	1	1	yes	characteristic
186	C₄²	#102	1	1	yes	characteristic
187	C₄²	#102	1	1	yes	characteristic
188	C₄²	#103	1	1	yes	characteristic
189	C₄²	#103	1	1	yes	characteristic
190	C₄²	#106	1	1	yes	characteristic
191	C₄²	#106	1	1	yes	characteristic
192	C₄²	#107	1	1	yes	characteristic
193	C₄²	#107	1	1	yes	characteristic
194	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₄²	C₂×D₈	1	1	yes	characteristic
197	C₄²	C₂×SD₁₆	1	1	yes	characteristic
198	C₁₆	—	4	4	no	—
199	C₂³	—	4	1	no	—
200	C₂³	—	4	1	no	—
201	C₂³	—	4	1	no	—
202	C₂³	#340	1	1	yes	characteristic
203	C₂³	#340	1	1	yes	characteristic
204	C₂³	C₂×#103	1	1	yes	characteristic
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	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	#340	1	1	yes	characteristic
227	C₂×C₄	#340	1	1	yes	characteristic
228	C₂×C₄	#340	1	1	yes	characteristic
229	C₂×C₄	#340	1	1	yes	characteristic
230	C₂×C₄	#340	1	1	yes	characteristic
231	C₂×C₄	#340	1	1	yes	characteristic
232	C₂×C₄	C₂×#95	1	1	yes	characteristic
233	C₂×C₄	C₂×#102	1	1	yes	characteristic
234	C₂×C₄	C₂×#102	1	1	yes	characteristic
235	C₂×C₄	C₂×#103	1	1	yes	characteristic
236	C₂×C₄	C₂×#106	1	1	yes	characteristic
237	C₂×C₄	C₂×#107	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	8	4	no	—
245	C₈	—	8	4	no	—
246	C₈	H(3,ℤ/4)	1	2	yes	—
247	C₈	#344	1	2	yes	—
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	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₂×#340	1	1	yes	characteristic
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	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	#1388	1	2	yes	—
269	C₄	#1389	1	2	yes	—
270	C₄	C₂×H(3,ℤ/4)	1	1	yes	characteristic
271	C₄	C₂×#344	1	1	yes	characteristic
272	C₂	—	4	1	no	—
273	C₂	—	4	1	no	—
274	C₂	—	4	1	no	—
275	C₂	#7957	1	1	yes	characteristic
276	C₂	C₂×#1388	1	1	yes	characteristic
277	C₂	C₂×#1389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7224	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
#20826	2
#17793	2
#7224	2