Subgroups — #134674

Order 512.

372 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1372	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
4	#8596	C₂	1	2	yes	maximal
5	#7154	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×M₃₂	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄×M₃₂	—	2	2	no	—
10	C₄×M₃₂	—	2	2	no	—
11	C₂×#346	C₄	1	1	yes	characteristic
12	C₂×#346	C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1797	—	2	2	no	—
16	C₂×#335	C₂²	1	2	yes	—
17	C₂×#335	C₂²	1	2	yes	—
18	#1372	—	2	2	no	—
19	#1371	—	2	2	no	—
20	#1277	—	2	2	no	—
21	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₈	Q₈	1	1	yes	characteristic
23	C₂²×C₄²	D₄	1	1	yes	characteristic
24	C₂×M₃₂	—	2	2	no	—
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	C₂×M₃₂	—	2	2	no	—
28	C₂×M₃₂	—	2	2	no	—
29	C₂×M₃₂	C₈	1	4	yes	complemented
30	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×C₄⋊C₈	—	2	2	no	—
36	C₂×C₂²⋊C₈	—	4	2	no	—
37	C₂×C₄×C₈	—	2	1	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	#346	—	2	2	no	—
46	#346	—	2	2	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	D₄	1	2	yes	—
50	C₄×C₁₆	D₄	1	2	yes	—
51	#336	—	4	2	no	—
52	#335	—	2	2	no	—
53	#335	—	2	2	no	—
54	#335	—	2	2	no	—
55	#335	—	2	2	no	—
56	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
57	C₂³×C₄	D₈	1	1	yes	characteristic
58	C₂³×C₄	Q₁₆	1	1	yes	characteristic
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	4	2	no	—
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×C₈	1	2	yes	—
72	C₂²×C₈	M₁₆	1	2	yes	—
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄²	—	2	1	no	—
76	C₂×C₄²	—	2	1	no	—
77	C₂×C₄²	—	2	1	no	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄²	D₈	1	1	yes	characteristic
81	C₂×C₄²	Q₁₆	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	M₃₂	—	2	8	no	—
84	M₃₂	—	2	8	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
92	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
93	C₂×C₁₆	C₂×C₈	1	2	yes	—
94	C₂×C₁₆	C₂×C₈	1	2	yes	—
95	C₂×C₁₆	M₁₆	1	2	yes	—
96	C₂×C₁₆	M₁₆	1	2	yes	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄⋊C₈	—	4	2	no	—
100	C₄⋊C₈	—	4	2	no	—
101	C₂²⋊C₈	—	8	4	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	C₂×D₄	1	1	yes	characteristic
109	C₄×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₄×C₈	#41	1	1	yes	characteristic
112	C₂⁴	#107	1	1	yes	characteristic
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₄⋊C₈	1	2	yes	—
125	C₂²×C₄	#107	1	1	yes	characteristic
126	C₂²×C₄	#107	1	1	yes	characteristic
127	C₂²×C₄	#107	1	1	yes	characteristic
128	C₂²×C₄	#108	1	2	yes	—
129	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
131	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	8	2	no	—
145	C₂×C₈	—	8	2	no	—
146	C₂×C₈	—	8	2	no	—
147	C₂×C₈	—	8	2	no	—
148	C₂×C₈	—	8	2	no	—
149	C₂×C₈	—	8	2	no	—
150	C₂×C₈	C₄⋊C₈	1	2	yes	—
151	C₂×C₈	#108	1	2	yes	—
152	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₈	#117	1	1	yes	characteristic
154	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
157	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
158	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
159	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
160	C₄²	—	2	1	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	#121	1	1	yes	characteristic
167	C₄²	#121	1	1	yes	characteristic
168	C₄²	#135	1	1	yes	characteristic
169	C₄²	#135	1	1	yes	characteristic
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	2	no	—
173	C₁₆	C₂²⋊C₈	1	4	yes	complemented
174	C₂³	—	2	1	no	—
175	C₂³	—	2	1	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	#335	1	2	yes	—
179	C₂³	C₂×#107	1	1	yes	characteristic
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#335	1	2	yes	—
197	C₂×C₄	#380	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
199	C₂×C₄	#423	1	1	yes	characteristic
200	C₂×C₄	#427	1	1	yes	characteristic
201	C₂×C₄	C₂×#108	1	1	yes	characteristic
202	C₂×C₄	C₂×#108	1	1	yes	characteristic
203	C₂×C₄	#466	1	1	yes	characteristic
204	C₂×C₄	#466	1	1	yes	characteristic
205	C₂×C₄	#467	1	1	yes	characteristic
206	C₂×C₄	#467	1	1	yes	characteristic
207	C₂×C₄	#509	1	2	yes	—
208	C₂×C₄	#510	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	16	2	no	—
214	C₈	—	16	2	no	—
215	C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
216	C₈	#407	1	1	yes	characteristic
217	C₈	#431	1	1	yes	characteristic
218	C₈	#432	1	1	yes	characteristic
219	C₂²	—	2	2	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂×#335	1	1	yes	characteristic
225	C₂²	#1567	1	1	yes	characteristic
226	C₂²	#1808	1	1	yes	characteristic
227	C₂²	#2159	1	2	yes	—
228	C₂²	#2161	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	#1564	1	1	yes	characteristic
235	C₄	#1568	1	1	yes	characteristic
236	C₄	#1797	1	1	yes	characteristic
237	C₄	#1810	1	1	yes	characteristic
238	C₄	#2218	1	1	yes	characteristic
239	C₄	#2219	1	1	yes	characteristic
240	C₄	#2220	1	2	yes	—
241	C₂	—	4	2	no	—
242	C₂	#8596	1	1	yes	characteristic
243	C₂	#12009	1	2	yes	—
244	C₂	#12025	1	2	yes	—
245	C₂	#12836	1	1	yes	characteristic
246	C₂	#12838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
2. C₂×#1372	C₂	semidirect
3. C₂×#1371	C₂	semidirect
5. #7154	C₂	semidirect
29. C₂×M₃₂	C₈	semidirect
173. C₁₆	C₂²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×#1372	2
C₂×#1371	2
#8596	2
#7154	2