Subgroups — #483571

Order 512.

424 conjugacy classes of subgroups.

The subgroup lattice has 251 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33474	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#17903	C₂	1	1	yes	maximal, characteristic
4	#12060	C₂	1	4	yes	maximal, complemented
5	#12059	C₂	1	4	yes	maximal, complemented
6	#11997	C₂	1	1	yes	maximal, characteristic
7	#11607	C₂	1	1	yes	maximal, characteristic
8	#11606	C₂	1	1	yes	maximal, characteristic
9	#11401	C₂	1	1	yes	maximal, characteristic
10	#3104	C₄	1	1	yes	characteristic
11	#2960	C₂²	1	1	yes	characteristic
12	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
13	#2183	—	2	1	no	—
14	#2183	C₂²	1	2	yes	—
15	Q₈×C₁₆	—	2	1	no	—
16	Q₈×C₁₆	—	2	1	no	—
17	Q₈×C₁₆	—	2	1	no	—
18	Q₈×C₁₆	C₂²	1	2	yes	—
19	Q₈×C₁₆	C₂²	1	2	yes	—
20	Q₈×C₁₆	C₂²	1	2	yes	—
21	#2151	—	2	1	no	—
22	#2150	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₁₆	—	2	1	no	—
24	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	#2108	—	2	1	no	—
26	C₄×M₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₁₆	—	2	1	no	—
28	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
29	#1584	C₄	1	4	yes	—
30	C₈×Q₁₆	C₂²	1	4	yes	—
31	#1571	C₄	1	1	yes	characteristic
32	#1487	C₄	1	1	yes	characteristic
33	#1481	C₄	1	1	yes	characteristic
34	#1450	C₂²	1	1	yes	characteristic
35	#1367	C₂²	1	4	yes	—
36	Q₈⋊C₁₆	C₂²	1	4	yes	—
37	Q₈⋊C₁₆	C₂²	1	4	yes	—
38	#1312	C₂²	1	2	yes	—
39	#1312	C₂²	1	2	yes	—
40	#533	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂²×C₁₆	—	2	1	no	—
45	C₂²×C₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₁₆	D₄	1	1	yes	characteristic
47	#484	C₈	1	2	yes	—
48	#477	C₈	1	2	yes	—
49	#462	C₈	1	2	yes	—
50	#451	C₈	1	2	yes	—
51	#446	—	2	1	no	—
52	#446	C₂×C₄	1	2	yes	—
53	C₈×Q₈	—	2	1	no	—
54	C₈×Q₈	—	2	1	no	—
55	C₈×Q₈	—	2	1	no	—
56	C₈×Q₈	C₂×C₄	1	2	yes	—
57	C₈×Q₈	C₂³	1	2	yes	—
58	C₈×Q₈	C₂³	1	2	yes	—
59	C₄×Q₁₆	C₂×C₄	1	4	yes	—
60	#431	C₂×C₄	1	1	yes	characteristic
61	#431	C₂×C₄	1	1	yes	characteristic
62	#424	—	2	1	no	—
63	#423	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₈	—	2	1	no	—
65	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
66	#405	—	2	1	no	—
67	C₂×C₄×C₈	—	2	1	no	—
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₄⋊C₁₆	—	2	2	no	—
70	C₄⋊C₁₆	—	2	2	no	—
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	—	2	1	no	—
75	C₄⋊C₁₆	C₂³	1	2	yes	—
76	C₄⋊C₁₆	C₂³	1	2	yes	—
77	#346	—	2	2	no	—
78	#346	—	2	1	no	—
79	C₄×C₁₆	—	2	2	no	—
80	C₄×C₁₆	—	2	2	no	—
81	C₄×C₁₆	—	2	2	no	—
82	C₄×C₁₆	D₄	1	1	yes	characteristic
83	C₄×C₁₆	D₄	1	1	yes	characteristic
84	C₄×C₁₆	C₂³	1	2	yes	—
85	C₄×C₁₆	C₂³	1	1	yes	characteristic
86	C₄×C₁₆	C₂³	1	1	yes	characteristic
87	#335	C₂×C₄	1	4	yes	—
88	Q₈⋊C₈	C₂×C₄	1	4	yes	—
89	Q₈⋊C₈	C₂×C₄	1	4	yes	—
90	#322	C₂×C₄	1	2	yes	—
91	C₈²	C₂³	1	2	yes	—
92	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
93	C₂×Q₁₆	C₂×C₈	1	4	yes	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	#41	1	1	yes	characteristic
98	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
99	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
100	#125	—	2	1	no	—
101	C₂²⋊Q₈	—	2	1	no	—
102	C₂²⋊Q₈	C₂×C₈	1	2	yes	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	—	2	1	no	—
105	C₄×Q₈	—	2	1	no	—
106	C₄×Q₈	C₂×C₈	1	2	yes	—
107	C₄×Q₈	C₂²×C₄	1	2	yes	—
108	C₄×Q₈	C₂²×C₄	1	2	yes	—
109	#117	—	2	1	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
112	C₂×C₄²	—	2	1	no	—
113	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
114	M₃₂	—	4	2	no	—
115	M₃₂	—	4	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	4	2	no	—
122	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
123	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
124	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
125	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
126	#107	C₂×C₈	1	4	yes	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	C₂×C₈	1	2	yes	—
135	C₄⋊C₈	C₂²×C₄	1	2	yes	—
136	C₄⋊C₈	C₂²×C₄	1	2	yes	—
137	#103	C₂×C₈	1	4	yes	—
138	#103	C₂×C₈	1	4	yes	—
139	C₂²⋊C₈	—	2	1	no	—
140	C₂²⋊C₈	C₂×C₈	1	2	yes	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	C₂²×C₄	1	2	yes	—
146	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₄	1	2	yes	—
148	C₄×C₈	C₂×D₄	1	1	yes	characteristic
149	C₄×C₈	#41	1	1	yes	characteristic
150	C₄×C₈	C₂⁴	1	1	yes	characteristic
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	1	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	C₂²×C₈	1	2	yes	—
155	C₂×Q₈	C₂²×C₈	1	2	yes	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
160	C₂²×C₄	#131	1	1	yes	characteristic
161	Q₁₆	—	2	8	no	—
162	M₁₆	—	4	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₄×D₄	1	1	yes	characteristic
169	C₂×C₈	C₄×D₄	1	1	yes	characteristic
170	C₂×C₈	C₂²×C₈	1	4	yes	—
171	C₂×C₈	#137	1	2	yes	—
172	C₂×C₈	#137	1	2	yes	—
173	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₈	C₂×#41	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	C₂²×C₈	1	2	yes	—
184	C₄⋊C₄	C₂²×C₈	1	2	yes	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	C₄×D₄	1	1	yes	characteristic
190	C₄²	C₂²×C₈	1	1	yes	characteristic
191	C₄²	#131	1	1	yes	characteristic
192	C₄²	C₂³×C₄	1	1	yes	characteristic
193	C₁₆	—	2	1	no	—
194	C₁₆	—	2	1	no	—
195	C₁₆	—	2	1	no	—
196	C₁₆	—	2	1	no	—
197	C₁₆	—	4	2	no	—
198	C₁₆	—	4	2	no	—
199	C₁₆	—	4	2	no	—
200	C₂³	C₈×D₄	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	#504	1	4	yes	—
206	C₂×C₄	—	2	4	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	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	C₈×D₄	1	1	yes	characteristic
215	C₂×C₄	C₈×D₄	1	1	yes	characteristic
216	C₂×C₄	C₈×D₄	1	1	yes	characteristic
217	C₂×C₄	#441	1	2	yes	—
218	C₂×C₄	#444	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
221	C₂×C₄	C₂×#131	1	1	yes	characteristic
222	C₈	—	2	4	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	4	2	no	—
228	C₈	#441	1	2	yes	—
229	C₈	C₂×#137	1	1	yes	characteristic
230	C₈	C₂×#137	1	1	yes	characteristic
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	#1579	1	2	yes	—
234	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	#1579	1	2	yes	—
243	C₄	C₂×#441	1	1	yes	characteristic
244	C₄	C₂×#444	1	1	yes	characteristic
245	C₄	C₂×#504	1	2	yes	—
246	C₂	—	2	2	no	—
247	C₂	C₂×#1579	1	1	yes	characteristic
248	C₂	#33432	1	1	yes	characteristic
249	C₂	#33441	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
4. #12060	C₂	semidirect
5. #12059	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33474	2
C₂×Q₈×C₁₆	2
#17903	2
#12060	2
#12059	2
#11997	2
#11607	2
#11606	2
#11401	2