Subgroups — #483409

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 243 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#504	C₂	1	1	yes	maximal, characteristic
2	#17799	C₂	1	2	yes	maximal
3	#12025	C₂	1	4	yes	maximal, complemented
4	#12024	C₂	1	2	yes	maximal, complemented
5	#12023	C₂	1	2	yes	maximal, complemented
6	#12009	C₂	1	2	yes	maximal, complemented
7	#12008	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#1372	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#504	C₄	1	1	yes	characteristic
10	C₂×#504	C₄	1	1	yes	characteristic
11	#2989	—	2	2	no	—
12	C₄×#131	C₂²	1	1	yes	characteristic
13	#2108	C₄	1	1	yes	characteristic
14	#2108	C₄	1	2	yes	—
15	#2108	C₄	1	2	yes	—
16	#2108	C₄	1	1	yes	characteristic
17	C₄×M₃₂	C₂²	1	1	yes	characteristic
18	C₄×M₃₂	C₂²	1	2	yes	—
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	2	yes	—
21	#1569	—	2	4	no	—
22	#1568	C₂²	1	4	yes	—
23	#1567	C₂²	1	4	yes	—
24	#1565	—	2	2	no	—
25	#1564	C₂²	1	2	yes	—
26	C₂×#335	C₂²	1	2	yes	—
27	#1372	C₂²	1	2	yes	—
28	#1372	C₂²	1	2	yes	—
29	#1372	C₂²	1	1	yes	characteristic
30	#1372	C₂²	1	1	yes	characteristic
31	#1372	C₂²	1	4	yes	—
32	#1371	C₂²	1	2	yes	—
33	#1371	C₂²	1	4	yes	—
34	C₂×#131	C₂×C₄	1	1	yes	characteristic
35	C₂×#131	Q₈	1	1	yes	characteristic
36	C₄×#41	D₄	1	1	yes	characteristic
37	#504	—	2	4	no	—
38	#504	—	2	4	no	—
39	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
40	C₂×M₃₂	C₂×C₄	1	2	yes	—
41	C₂×M₃₂	C₂×C₄	1	2	yes	—
42	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₁₆	C₂×C₄	1	2	yes	—
45	C₂²×C₁₆	C₂×C₄	1	2	yes	—
46	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
47	#446	—	4	2	no	—
48	#436	—	4	2	no	—
49	#435	—	4	4	no	—
50	#432	—	2	4	no	—
51	#431	—	2	2	no	—
52	#423	—	2	4	no	—
53	C₂×C₄⋊C₈	—	2	2	no	—
54	#405	C₂×C₄	1	2	yes	—
55	#405	C₂×C₄	1	1	yes	characteristic
56	#405	Q₈	1	1	yes	characteristic
57	#405	Q₈	1	2	yes	—
58	C₄×M₁₆	D₄	1	1	yes	characteristic
59	C₄×M₁₆	D₄	1	2	yes	—
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂³	1	2	yes	—
62	#346	C₂×C₄	1	2	yes	—
63	#346	C₂×C₄	1	2	yes	—
64	#346	C₂×C₄	1	2	yes	—
65	C₄×C₁₆	C₂×C₄	1	2	yes	—
66	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
67	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
68	C₄×C₁₆	C₂×C₄	1	2	yes	—
69	C₄×C₁₆	C₂³	1	1	yes	characteristic
70	C₄×C₁₆	C₂³	1	1	yes	characteristic
71	C₄×C₁₆	C₂³	1	2	yes	—
72	#335	—	2	2	no	—
73	#335	—	2	4	no	—
74	#335	C₂³	1	4	yes	—
75	#335	C₂³	1	2	yes	—
76	#335	C₂³	1	2	yes	—
77	#334	—	2	2	no	—
78	#334	—	2	4	no	—
79	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
80	#131	—	2	4	no	—
81	#131	—	2	2	no	—
82	#131	—	2	2	no	—
83	C₂×M₁₆	—	4	4	no	—
84	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
85	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
86	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
87	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	—	4	2	no	—
89	C₂²×C₈	C₂²×C₄	1	2	yes	—
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₈	C₂×Q₈	1	2	yes	—
93	C₄×Q₈	D₈	1	1	yes	characteristic
94	C₄×Q₈	Q₁₆	1	1	yes	characteristic
95	C₄×D₄	D₈	1	2	yes	—
96	C₄×D₄	D₈	1	1	yes	characteristic
97	C₄×D₄	Q₁₆	1	2	yes	—
98	C₄×D₄	Q₁₆	1	1	yes	characteristic
99	#117	C₄⋊C₄	1	2	yes	—
100	#117	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₄	1	2	yes	—
103	M₃₂	—	2	2	no	—
104	M₃₂	—	2	4	no	—
105	M₃₂	—	2	4	no	—
106	M₃₂	M₁₆	1	4	yes	complemented
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	4	no	—
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	M₁₆	1	4	yes	complemented
111	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₁₆	C₂²×C₄	1	2	yes	—
114	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₁₆	C₂²×C₄	1	2	yes	—
116	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	4	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₂²⋊C₈	—	4	2	no	—
123	C₂²⋊C₈	—	4	4	no	—
124	#97	—	4	4	no	—
125	#97	C₄⋊C₄	1	2	yes	—
126	#97	C₄⋊C₄	1	2	yes	—
127	#97	C₄⋊C₄	1	2	yes	—
128	C₄×C₈	—	4	2	no	—
129	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₄×C₈	C₂²×C₄	1	2	yes	—
131	C₄×C₈	C₂×D₄	1	1	yes	characteristic
132	C₄×C₈	C₂×D₄	1	2	yes	—
133	C₄×C₈	C₂×Q₈	1	2	yes	—
134	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
135	C₄×C₈	C₂⁴	1	1	yes	characteristic
136	#41	—	2	2	no	—
137	#41	—	2	2	no	—
138	C₂×Q₈	#107	1	1	yes	characteristic
139	C₂×D₄	#107	1	2	yes	—
140	C₂×D₄	#107	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
142	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₈	1	2	yes	—
145	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
146	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	2	4	no	—
150	M₁₆	—	16	4	no	—
151	M₁₆	#108	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	4	no	—
159	C₂×C₈	—	16	2	no	—
160	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₈	C₂×M₁₆	1	2	yes	—
165	C₂×C₈	C₂×M₁₆	1	2	yes	—
166	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
168	C₄⋊C₄	#107	1	2	yes	—
169	C₄⋊C₄	#107	1	1	yes	characteristic
170	C₄⋊C₄	#107	1	2	yes	—
171	C₄⋊C₄	#107	1	1	yes	characteristic
172	C₂²⋊C₄	#107	1	2	yes	—
173	C₂²⋊C₄	#107	1	2	yes	—
174	C₂²⋊C₄	#107	1	2	yes	—
175	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₄²	C₂×C₄⋊C₄	1	2	yes	—
177	C₄²	C₂×D₈	1	1	yes	characteristic
178	C₄²	C₂×D₈	1	2	yes	—
179	C₄²	C₂×Q₁₆	1	2	yes	—
180	C₄²	C₂×Q₁₆	1	1	yes	characteristic
181	C₄²	C₂²×D₄	1	1	yes	characteristic
182	C₁₆	—	2	1	no	—
183	C₁₆	—	2	1	no	—
184	C₁₆	—	2	2	no	—
185	C₁₆	C₂×M₁₆	1	2	yes	—
186	C₁₆	C₂×M₁₆	1	4	yes	—
187	C₁₆	C₂×M₁₆	1	2	yes	—
188	C₂³	C₂×#107	1	2	yes	—
189	C₂³	C₂×#107	1	1	yes	characteristic
190	Q₈	—	2	2	no	—
191	D₄	—	2	4	no	—
192	D₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#423	1	2	yes	—
200	C₂×C₄	C₂×#107	1	1	yes	characteristic
201	C₂×C₄	C₂×#107	1	2	yes	—
202	C₂×C₄	C₂×#107	1	1	yes	characteristic
203	C₂×C₄	C₂×#107	1	2	yes	—
204	C₂×C₄	C₂×#107	1	2	yes	—
205	C₂×C₄	C₂×#107	1	1	yes	characteristic
206	C₂×C₄	C₂×#108	1	2	yes	—
207	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
209	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	8	2	no	—
214	C₈	—	8	4	no	—
215	C₈	—	8	2	no	—
216	C₈	#423	1	2	yes	—
217	C₈	C₂×#108	1	4	yes	—
218	C₈	C₂²×M₁₆	1	1	yes	characteristic
219	C₈	C₂²×M₁₆	1	1	yes	characteristic
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	#1567	1	2	yes	—
226	C₂²	C₂²×#107	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	#1567	1	2	yes	—
234	C₄	C₂×#423	1	1	yes	characteristic
235	C₄	C₂²×#108	1	1	yes	characteristic
236	C₄	#3416	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	—	4	2	no	—
239	C₂	C₂×#1567	1	1	yes	characteristic
240	C₂	#33411	1	1	yes	characteristic
241	C₂	#33414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12025	C₂	semidirect
4. #12024	C₂	semidirect
5. #12023	C₂	semidirect
6. #12009	C₂	semidirect
7. #12008	C₂	semidirect
8. C₂×#1372	C₂	semidirect
106. M₃₂	M₁₆	semidirect
110. C₂×C₁₆	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×#504	2
#17799	2
#12025	2
#12024	2
#12023	2
#12009	2
#12008	2
C₂×#1372	2